Subgroups — C₂₃×#22873 #19170499

Order 5888.

1102 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#452	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2341	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1968	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1934	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1925	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1892	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1880	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
25	C₂₃×#452	C₄	1	8	yes	complemented
26	C₄×Q₁₆×C₂₃	—	2	4	no	—
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂₃×#416	—	2	4	no	—
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#391	C₂²	1	1	yes	characteristic
42	C₂₃×#390	—	2	1	no	—
43	C₂₃×#390	—	2	1	no	—
44	C₂₃×#390	C₂²	1	1	yes	characteristic
45	C₂₃×#390	C₂²	1	1	yes	characteristic
46	C₂₃×#389	C₂²	1	1	yes	characteristic
47	C₂₃×#388	—	2	1	no	—
48	C₂₃×#388	—	2	1	no	—
49	C₂₃×#387	—	2	1	no	—
50	C₂₃×#380	—	2	1	no	—
51	C₂₃×#380	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
55	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂×#41×C₂₃	—	2	2	no	—
63	C₂×#41×C₂₃	—	2	2	no	—
64	C₂×#41×C₂₃	—	2	2	no	—
65	C₂×#41×C₂₃	—	4	1	no	—
66	C₂×#41×C₂₃	—	4	1	no	—
67	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
68	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₂₃	—	2	1	no	—
71	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₂₃	—	2	1	no	—
73	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×Q₁₆×C₂₃	—	2	2	no	—
75	C₂×Q₁₆×C₂₃	—	2	2	no	—
76	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
77	C₂×SD₁₆×C₂₃	—	4	1	no	—
78	C₂×SD₁₆×C₂₃	—	4	1	no	—
79	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
80	C₂²×C₈×C₂₃	—	2	2	no	—
81	C₂²×C₈×C₂₃	—	2	2	no	—
82	C₂²×C₈×C₂₃	—	4	1	no	—
83	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₂₃×C₂²⋊Q₈	—	2	2	no	—
88	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	—	2	2	no	—
92	C₄×Q₈×C₂₃	—	4	1	no	—
93	C₄×Q₈×C₂₃	—	4	1	no	—
94	C₄×Q₈×C₂₃	—	4	1	no	—
95	C₂₃×#117	—	2	2	no	—
96	C₂₃×#117	—	4	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	—	2	1	no	—
118	C₂×C₄²×C₂₃	—	2	1	no	—
119	C₂×C₄²×C₂₃	—	2	1	no	—
120	C₂×C₄²×C₂₃	—	2	1	no	—
121	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂₃×#107	—	2	2	no	—
124	C₂₃×#106	—	4	1	no	—
125	C₂₃×#103	—	2	2	no	—
126	C₂₃×#103	—	2	2	no	—
127	C₂₃×#103	—	4	1	no	—
128	C₂₃×#103	C₂×C₄	1	4	yes	—
129	C₂₃×#103	D₄	1	4	yes	—
130	C₂₃×#102	—	2	2	no	—
131	C₂₃×#102	—	2	2	no	—
132	C₂₃×#102	C₂×C₄	1	4	yes	—
133	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
134	C₄×C₈×C₂₃	—	2	2	no	—
135	C₄×C₈×C₂₃	—	4	1	no	—
136	C₂₃×#95	—	2	1	no	—
137	C₂₃×#95	—	2	1	no	—
138	C₂₃×#95	—	2	1	no	—
139	C₂₃×#95	—	2	1	no	—
140	C₂₃×#95	—	2	1	no	—
141	C₂₃×#95	D₄	1	1	yes	characteristic
142	C₂₃×#95	D₄	1	1	yes	characteristic
143	C₂⁴×C₂₃	—	2	1	no	—
144	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
145	#41×C₂₃	—	2	8	no	—
146	#41×C₂₃	—	4	4	no	—
147	#41×C₂₃	—	4	4	no	—
148	#41×C₂₃	—	4	4	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	2	1	no	—
153	C₂×Q₈×C₂₃	—	2	1	no	—
154	C₂×Q₈×C₂₃	—	2	1	no	—
155	C₂×Q₈×C₂₃	—	2	1	no	—
156	C₂×Q₈×C₂₃	—	2	1	no	—
157	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
158	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
159	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
160	C₂×Q₈×C₂₃	#41	1	2	yes	—
161	C₂×D₄×C₂₃	—	2	4	no	—
162	C₂×D₄×C₂₃	—	2	1	no	—
163	C₂×D₄×C₂₃	—	2	1	no	—
164	C₂×D₄×C₂₃	—	2	1	no	—
165	C₂×D₄×C₂₃	—	2	1	no	—
166	C₂×D₄×C₂₃	—	2	1	no	—
167	C₂×D₄×C₂₃	—	4	2	no	—
168	C₂×D₄×C₂₃	—	4	2	no	—
169	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	2	1	no	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	2	4	no	—
179	C₂²×C₄×C₂₃	—	2	1	no	—
180	C₂²×C₄×C₂₃	—	2	1	no	—
181	C₂²×C₄×C₂₃	—	2	1	no	—
182	C₂²×C₄×C₂₃	—	4	2	no	—
183	C₂²×C₄×C₂₃	—	4	2	no	—
184	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
185	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	#41	1	2	yes	—
191	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
192	Q₁₆×C₂₃	—	2	8	no	—
193	SD₁₆×C₂₃	—	4	4	no	—
194	C₂×C₈×C₂₃	—	2	1	no	—
195	C₂×C₈×C₂₃	—	2	1	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	4	4	no	—
204	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
205	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
206	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₂₃	—	2	4	no	—
208	C₄⋊C₄×C₂₃	—	2	1	no	—
209	C₄⋊C₄×C₂₃	—	2	1	no	—
210	C₄⋊C₄×C₂₃	—	2	1	no	—
211	C₄⋊C₄×C₂₃	—	2	1	no	—
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	4	2	no	—
217	C₄⋊C₄×C₂₃	—	4	2	no	—
218	C₄⋊C₄×C₂₃	—	4	2	no	—
219	C₄⋊C₄×C₂₃	—	4	2	no	—
220	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
221	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
222	C₂²⋊C₄×C₂₃	—	2	4	no	—
223	C₂²⋊C₄×C₂₃	—	2	2	no	—
224	C₂²⋊C₄×C₂₃	—	2	2	no	—
225	C₂²⋊C₄×C₂₃	—	4	2	no	—
226	C₂²⋊C₄×C₂₃	—	4	2	no	—
227	C₂²⋊C₄×C₂₃	—	4	2	no	—
228	C₂²⋊C₄×C₂₃	—	8	1	no	—
229	C₂²⋊C₄×C₂₃	—	8	1	no	—
230	C₂²⋊C₄×C₂₃	—	8	1	no	—
231	C₄²×C₂₃	—	2	1	no	—
232	C₄²×C₂₃	—	2	1	no	—
233	C₄²×C₂₃	—	2	1	no	—
234	C₄²×C₂₃	—	2	1	no	—
235	C₄²×C₂₃	—	4	2	no	—
236	C₄²×C₂₃	—	4	2	no	—
237	C₄²×C₂₃	—	4	2	no	—
238	C₄²×C₂₃	—	4	2	no	—
239	C₄²×C₂₃	—	4	2	no	—
240	#22873	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂³×C₂₃	—	2	1	no	—
242	C₂³×C₂₃	—	2	1	no	—
243	C₂³×C₂₃	—	2	1	no	—
244	C₂³×C₂₃	—	2	1	no	—
245	C₂³×C₂₃	—	2	1	no	—
246	C₂³×C₂₃	—	2	1	no	—
247	C₂³×C₂₃	—	2	2	no	—
248	C₂³×C₂₃	—	4	1	no	—
249	C₂³×C₂₃	—	4	1	no	—
250	C₂³×C₂₃	—	4	1	no	—
251	C₂³×C₂₃	—	4	1	no	—
252	C₂³×C₂₃	—	4	1	no	—
253	C₂³×C₂₃	—	4	1	no	—
254	C₂³×C₂₃	C₄×D₄	1	2	yes	—
255	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
256	Q₈×C₂₃	—	2	2	no	—
257	Q₈×C₂₃	—	2	2	no	—
258	Q₈×C₂₃	—	2	2	no	—
259	Q₈×C₂₃	—	2	2	no	—
260	Q₈×C₂₃	—	2	2	no	—
261	Q₈×C₂₃	—	4	1	no	—
262	Q₈×C₂₃	—	4	1	no	—
263	Q₈×C₂₃	C₄×D₄	1	4	yes	—
264	D₄×C₂₃	—	2	2	no	—
265	D₄×C₂₃	—	2	2	no	—
266	D₄×C₂₃	—	4	8	no	—
267	D₄×C₂₃	—	4	1	no	—
268	D₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	2	1	no	—
271	C₂×C₄×C₂₃	—	2	1	no	—
272	C₂×C₄×C₂₃	—	2	2	no	—
273	C₂×C₄×C₂₃	—	2	1	no	—
274	C₂×C₄×C₂₃	—	2	2	no	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	1	no	—
277	C₂×C₄×C₂₃	—	2	1	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	2	2	no	—
281	C₂×C₄×C₂₃	—	2	2	no	—
282	C₂×C₄×C₂₃	—	2	2	no	—
283	C₂×C₄×C₂₃	—	2	2	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	2	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	4	8	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	—	4	1	no	—
302	C₂×C₄×C₂₃	—	4	1	no	—
303	C₂×C₄×C₂₃	—	4	1	no	—
304	C₂×C₄×C₂₃	—	4	1	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
308	C₂×C₄×C₂₃	#135	1	2	yes	—
309	C₂×C₄×C₂₃	#135	1	2	yes	—
310	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
311	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
313	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
314	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
315	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
316	C₈×C₂₃	—	2	2	no	—
317	C₈×C₂₃	—	2	2	no	—
318	C₈×C₂₃	—	2	2	no	—
319	C₈×C₂₃	—	4	1	no	—
320	C₂×#452	C₂×C₂₃	1	1	yes	characteristic
321	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
322	C₂×#416	C₂×C₂₃	1	1	yes	characteristic
323	#2362	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#2341	C₂×C₂₃	1	1	yes	characteristic
325	#1968	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1934	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1925	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1892	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1880	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1854	C₂×C₂₃	1	1	yes	characteristic, complemented
333	C₄×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
334	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
335	C₂²×C₂₃	—	2	2	no	—
336	C₂²×C₂₃	—	2	2	no	—
337	C₂²×C₂₃	—	2	2	no	—
338	C₂²×C₂₃	—	4	1	no	—
339	C₂²×C₂₃	—	4	1	no	—
340	C₂²×C₂₃	—	4	1	no	—
341	C₂²×C₂₃	—	4	1	no	—
342	C₂²×C₂₃	—	4	2	no	—
343	C₂²×C₂₃	—	4	2	no	—
344	C₂²×C₂₃	—	4	2	no	—
345	C₂²×C₂₃	—	8	1	no	—
346	C₂²×C₂₃	—	8	1	no	—
347	C₂²×C₂₃	—	8	1	no	—
348	C₂²×C₂₃	—	8	1	no	—
349	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
350	C₂²×C₂₃	D₄²	1	1	yes	characteristic
351	C₂²×C₂₃	#546	1	1	yes	characteristic
352	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
353	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
354	C₂²×C₂₃	#577	1	1	yes	characteristic
355	C₂²×C₂₃	#578	1	1	yes	characteristic
356	C₄×C₂₃	—	2	2	no	—
357	C₄×C₂₃	—	2	4	no	—
358	C₄×C₂₃	—	2	2	no	—
359	C₄×C₂₃	—	2	2	no	—
360	C₄×C₂₃	—	4	2	no	—
361	C₄×C₂₃	—	4	2	no	—
362	C₄×C₂₃	—	4	1	no	—
363	C₄×C₂₃	—	4	2	no	—
364	C₄×C₂₃	—	4	1	no	—
365	C₄×C₂₃	—	8	1	no	—
366	C₄×C₂₃	—	8	1	no	—
367	C₄×C₂₃	—	8	1	no	—
368	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
369	C₄×C₂₃	#518	1	1	yes	characteristic
370	C₄×C₂₃	#546	1	1	yes	characteristic
371	C₄×C₂₃	#551	1	1	yes	characteristic
372	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
373	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
374	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
375	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
381	#452	C₄×C₂₃	1	8	yes	complemented
382	C₄×Q₁₆	—	2	4	no	—
383	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
385	#416	—	2	4	no	—
386	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
389	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
392	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
394	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
397	#391	C₂²×C₂₃	1	1	yes	characteristic
398	#390	—	2	1	no	—
399	#390	—	2	1	no	—
400	#390	C₂²×C₂₃	1	1	yes	characteristic
401	#390	C₂²×C₂₃	1	1	yes	characteristic
402	#389	C₂²×C₂₃	1	1	yes	characteristic
403	#388	—	2	1	no	—
404	#388	—	2	1	no	—
405	#387	—	2	1	no	—
406	#380	—	2	1	no	—
407	#380	C₂²×C₂₃	1	1	yes	characteristic
408	C₄×C₄⋊C₄	—	2	1	no	—
409	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
410	C₄×C₂²⋊C₄	—	2	1	no	—
411	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
412	#340	C₂²×C₂₃	1	1	yes	characteristic
413	#340	C₂²×C₂₃	1	1	yes	characteristic
414	#340	C₂²×C₂₃	1	1	yes	characteristic
415	#340	C₂²×C₂₃	1	1	yes	characteristic
416	#336	C₂²×C₂₃	1	1	yes	characteristic
417	#336	C₂²×C₂₃	1	1	yes	characteristic
418	C₂×C₂₃	—	4	2	no	—
419	C₂×C₂₃	—	8	2	no	—
420	C₂×C₂₃	#2358	1	1	yes	characteristic
421	C₂×C₂₃	C₄×#135	1	1	yes	characteristic
422	C₂×C₂₃	#2947	1	1	yes	characteristic
423	C₂×C₂₃	#3284	1	1	yes	characteristic
424	C₂×C₂₃	#3285	1	1	yes	characteristic
425	C₂×C₂₃	#3300	1	1	yes	characteristic
426	C₂×C₂₃	#3304	1	1	yes	characteristic
427	C₂×#41	—	2	2	no	—
428	C₂×#41	—	2	2	no	—
429	C₂×#41	—	2	2	no	—
430	C₂×#41	—	4	1	no	—
431	C₂×#41	—	4	1	no	—
432	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
433	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₂²×D₄	—	2	1	no	—
436	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
437	C₂³×C₄	—	2	1	no	—
438	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×Q₁₆	—	2	2	no	—
440	C₂×Q₁₆	—	2	2	no	—
441	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
442	C₂×SD₁₆	—	4	1	no	—
443	C₂×SD₁₆	—	4	1	no	—
444	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
445	C₂²×C₈	—	2	2	no	—
446	C₂²×C₈	—	2	2	no	—
447	C₂²×C₈	—	4	1	no	—
448	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
454	C₄×Q₈	—	2	2	no	—
455	C₄×Q₈	—	2	2	no	—
456	C₄×Q₈	—	2	2	no	—
457	C₄×Q₈	—	4	1	no	—
458	C₄×Q₈	—	4	1	no	—
459	C₄×Q₈	—	4	1	no	—
460	#117	—	2	2	no	—
461	#117	—	4	1	no	—
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	—	2	1	no	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
470	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
471	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	—	2	1	no	—
474	C₂×C₂²⋊C₄	—	2	1	no	—
475	C₂×C₂²⋊C₄	—	2	1	no	—
476	C₂×C₂²⋊C₄	—	2	1	no	—
477	C₂×C₂²⋊C₄	—	2	1	no	—
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
481	C₂×C₄²	—	2	1	no	—
482	C₂×C₄²	—	2	1	no	—
483	C₂×C₄²	—	2	1	no	—
484	C₂×C₄²	—	2	1	no	—
485	C₂×C₄²	—	2	1	no	—
486	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
487	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
488	#107	—	2	2	no	—
489	#106	—	4	1	no	—
490	#103	—	2	2	no	—
491	#103	—	2	2	no	—
492	#103	—	4	1	no	—
493	#103	C₂×C₄×C₂₃	1	4	yes	—
494	#103	D₄×C₂₃	1	4	yes	—
495	#102	—	2	2	no	—
496	#102	—	2	2	no	—
497	#102	C₂×C₄×C₂₃	1	4	yes	—
498	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
499	C₄×C₈	—	2	2	no	—
500	C₄×C₈	—	4	1	no	—
501	#95	—	2	1	no	—
502	#95	—	2	1	no	—
503	#95	—	2	1	no	—
504	#95	—	2	1	no	—
505	#95	—	2	1	no	—
506	#95	D₄×C₂₃	1	1	yes	characteristic
507	#95	D₄×C₂₃	1	1	yes	characteristic
508	C₂₃	#22873	1	1	yes	characteristic, direct product factor, Sylow
509	C₂⁴	—	2	1	no	—
510	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
511	#41	—	2	8	no	—
512	#41	—	4	4	no	—
513	#41	—	4	4	no	—
514	#41	—	4	4	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
524	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
525	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
526	C₂×Q₈	#41×C₂₃	1	2	yes	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	4	2	no	—
534	C₂×D₄	—	4	2	no	—
535	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	4	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	4	2	no	—
549	C₂²×C₄	—	4	2	no	—
550	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
551	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
552	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
553	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
554	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	#41×C₂₃	1	2	yes	—
557	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
558	Q₁₆	—	2	8	no	—
559	SD₁₆	—	4	4	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	4	4	no	—
570	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
571	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
572	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
573	C₄⋊C₄	—	2	4	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	4	2	no	—
583	C₄⋊C₄	—	4	2	no	—
584	C₄⋊C₄	—	4	2	no	—
585	C₄⋊C₄	—	4	2	no	—
586	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
587	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
588	C₂²⋊C₄	—	2	4	no	—
589	C₂²⋊C₄	—	2	2	no	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	—	4	2	no	—
592	C₂²⋊C₄	—	4	2	no	—
593	C₂²⋊C₄	—	4	2	no	—
594	C₂²⋊C₄	—	8	1	no	—
595	C₂²⋊C₄	—	8	1	no	—
596	C₂²⋊C₄	—	8	1	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	1	no	—
599	C₄²	—	2	1	no	—
600	C₄²	—	2	1	no	—
601	C₄²	—	4	2	no	—
602	C₄²	—	4	2	no	—
603	C₄²	—	4	2	no	—
604	C₄²	—	4	2	no	—
605	C₄²	—	4	2	no	—
606	C₂³	—	2	1	no	—
607	C₂³	—	2	1	no	—
608	C₂³	—	2	1	no	—
609	C₂³	—	2	1	no	—
610	C₂³	—	2	1	no	—
611	C₂³	—	2	1	no	—
612	C₂³	—	2	2	no	—
613	C₂³	—	4	1	no	—
614	C₂³	—	4	1	no	—
615	C₂³	—	4	1	no	—
616	C₂³	—	4	1	no	—
617	C₂³	—	4	1	no	—
618	C₂³	—	4	1	no	—
619	C₂³	C₄×D₄×C₂₃	1	2	yes	—
620	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
621	Q₈	—	2	2	no	—
622	Q₈	—	2	2	no	—
623	Q₈	—	2	2	no	—
624	Q₈	—	2	2	no	—
625	Q₈	—	2	2	no	—
626	Q₈	—	4	1	no	—
627	Q₈	—	4	1	no	—
628	Q₈	C₄×D₄×C₂₃	1	4	yes	—
629	D₄	—	2	2	no	—
630	D₄	—	2	2	no	—
631	D₄	—	4	8	no	—
632	D₄	—	4	1	no	—
633	D₄	—	4	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	4	8	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
673	C₂×C₄	C₂₃×#135	1	2	yes	—
674	C₂×C₄	C₂₃×#135	1	2	yes	—
675	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
676	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
677	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
678	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
679	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
680	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
681	C₈	—	2	2	no	—
682	C₈	—	2	2	no	—
683	C₈	—	2	2	no	—
684	C₈	—	4	1	no	—
685	C₂²	—	2	2	no	—
686	C₂²	—	2	2	no	—
687	C₂²	—	2	2	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	1	no	—
690	C₂²	—	4	1	no	—
691	C₂²	—	4	1	no	—
692	C₂²	—	4	2	no	—
693	C₂²	—	4	2	no	—
694	C₂²	—	4	2	no	—
695	C₂²	—	8	1	no	—
696	C₂²	—	8	1	no	—
697	C₂²	—	8	1	no	—
698	C₂²	—	8	1	no	—
699	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
700	C₂²	D₄²×C₂₃	1	1	yes	characteristic
701	C₂²	C₂₃×#546	1	1	yes	characteristic
702	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
703	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
704	C₂²	C₂₃×#577	1	1	yes	characteristic
705	C₂²	C₂₃×#578	1	1	yes	characteristic
706	C₄	—	2	2	no	—
707	C₄	—	2	2	no	—
708	C₄	—	2	2	no	—
709	C₄	—	2	4	no	—
710	C₄	—	4	2	no	—
711	C₄	—	4	1	no	—
712	C₄	—	4	2	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	2	no	—
715	C₄	—	8	1	no	—
716	C₄	—	8	1	no	—
717	C₄	—	8	1	no	—
718	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
719	C₄	C₂₃×#518	1	1	yes	characteristic
720	C₄	C₂₃×#546	1	1	yes	characteristic
721	C₄	C₂₃×#551	1	1	yes	characteristic
722	C₂	—	4	2	no	—
723	C₂	—	8	2	no	—
724	C₂	C₂₃×#2358	1	1	yes	characteristic
725	C₂	C₄×C₂₃×#135	1	1	yes	characteristic
726	C₂	C₂₃×#2947	1	1	yes	characteristic
727	C₂	C₂₃×#3284	1	1	yes	characteristic
728	C₂	C₂₃×#3285	1	1	yes	characteristic
729	C₂	C₂₃×#3300	1	1	yes	characteristic
730	C₂	C₂₃×#3304	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₂₃	C₂	semidirect
4. C₂₃×#2362	C₂	semidirect
6. C₂₃×#1968	C₂	semidirect
7. C₂₃×#1934	C₂	semidirect
8. C₂₃×#1925	C₂	semidirect
9. C₂₃×#1892	C₂	semidirect
10. C₂₃×#1880	C₂	semidirect
11. C₂₃×#1878	C₂	semidirect
12. C₂₃×#1865	C₂	semidirect
13. C₂₃×#1854	C₂	semidirect
14. C₄×C₂₃×#103	C₂	semidirect
15. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
25. C₂₃×#452	C₄	semidirect
240. #22873	C₂₃	direct (N × H)
321. C₂×C₄×Q₁₆	C₂×C₂₃	semidirect
323. #2362	C₂×C₂₃	semidirect
325. #1968	C₂×C₂₃	semidirect
326. #1934	C₂×C₂₃	semidirect
327. #1925	C₂×C₂₃	semidirect
328. #1892	C₂×C₂₃	semidirect
329. #1880	C₂×C₂₃	semidirect
330. #1878	C₂×C₂₃	semidirect
331. #1865	C₂×C₂₃	semidirect
332. #1854	C₂×C₂₃	semidirect
333. C₄×#103	C₂×C₂₃	semidirect
334. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
381. #452	C₄×C₂₃	semidirect
508. C₂₃	#22873	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#452	2
C₂×C₄×Q₁₆×C₂₃	2
C₂×C₂₃×#416	2
C₂₃×#2362	2
C₂₃×#2341	2
C₂₃×#1968	2
C₂₃×#1934	2
C₂₃×#1925	2
C₂₃×#1892	2
C₂₃×#1880	2
C₂₃×#1878	2
C₂₃×#1865	2
C₂₃×#1854	2
C₄×C₂₃×#103	2
C₈×C₂²⋊C₄×C₂₃	2
#22873	23

Subgroups — C23×#22873 #19170499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22873 #19170499