Subgroups — C₂₃×#22054 #19169680

Order 5888.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#451	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2323	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2310	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1935	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1928	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1927	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1873	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1787	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1750	C₂	1	1	yes	maximal, characteristic
16	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
23	C₂₃×#451	C₄	1	8	yes	complemented
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#391	—	2	1	no	—
39	C₂₃×#391	—	2	1	no	—
40	C₂₃×#391	C₂²	1	1	yes	characteristic
41	C₂₃×#391	C₂²	1	1	yes	characteristic
42	C₂₃×#391	C₂²	1	1	yes	characteristic
43	C₂₃×#391	C₂²	1	1	yes	characteristic
44	C₂₃×#391	C₂²	1	1	yes	characteristic
45	C₂₃×#389	—	2	1	no	—
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×#381	—	2	1	no	—
51	C₂₃×#381	—	2	1	no	—
52	C₂₃×#380	C₂²	1	1	yes	characteristic
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₂₃	—	2	1	no	—
61	C₂²×Q₈×C₂₃	—	2	2	no	—
62	C₂²×Q₈×C₂₃	—	4	1	no	—
63	C₂²×Q₈×C₂₃	—	4	1	no	—
64	C₂²×Q₈×C₂₃	—	4	1	no	—
65	C₂²×Q₈×C₂₃	—	4	1	no	—
66	C₂²×Q₈×C₂₃	C₂×C₄	1	4	yes	—
67	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₂₃	—	2	1	no	—
71	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×Q₁₆×C₂₃	—	4	1	no	—
73	C₂×Q₁₆×C₂₃	—	4	1	no	—
74	C₂×Q₁₆×C₂₃	—	4	1	no	—
75	C₂×Q₁₆×C₂₃	—	4	1	no	—
76	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
77	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
84	C₄×Q₈×C₂₃	—	4	1	no	—
85	C₄×Q₈×C₂₃	—	4	1	no	—
86	C₂₃×#117	—	4	1	no	—
87	C₂₃×#117	—	4	1	no	—
88	C₂₃×#117	—	4	1	no	—
89	C₂₃×#117	—	4	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×#107	—	4	1	no	—
111	C₂₃×#106	—	4	1	no	—
112	C₂₃×#103	—	4	1	no	—
113	C₂₃×#103	—	4	1	no	—
114	C₂₃×#103	—	4	1	no	—
115	C₂₃×#103	—	4	1	no	—
116	C₂₃×#103	—	4	1	no	—
117	C₂₃×#103	—	4	1	no	—
118	C₂₃×#103	C₂×C₄	1	4	yes	—
119	C₂₃×#103	C₂×C₄	1	4	yes	—
120	C₂₃×C₂²⋊C₈	—	2	2	no	—
121	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
122	C₂₃×#97	—	4	1	no	—
123	C₄×C₈×C₂₃	—	4	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂₃×#95	—	2	1	no	—
134	C₂₃×#95	—	2	1	no	—
135	C₂₃×#95	—	2	1	no	—
136	C₂₃×#95	—	2	1	no	—
137	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₂₃	—	2	4	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	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	—	4	4	no	—
155	C₂×Q₈×C₂₃	—	4	4	no	—
156	C₂×Q₈×C₂₃	—	4	2	no	—
157	C₂×Q₈×C₂₃	—	4	2	no	—
158	C₂×Q₈×C₂₃	—	4	4	no	—
159	C₂×Q₈×C₂₃	—	4	4	no	—
160	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
161	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
162	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	4	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
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	—	4	2	no	—
177	C₂²×C₄×C₂₃	—	4	2	no	—
178	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₂₃	#41	1	2	yes	—
185	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
186	Q₁₆×C₂₃	—	4	4	no	—
187	Q₁₆×C₂₃	—	4	4	no	—
188	C₂×C₈×C₂₃	—	2	1	no	—
189	C₂×C₈×C₂₃	—	2	1	no	—
190	C₂×C₈×C₂₃	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	1	no	—
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	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	C₂²×C₄	1	2	yes	—
199	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
200	C₄⋊C₄×C₂₃	—	2	4	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	2	1	no	—
207	C₄⋊C₄×C₂₃	—	2	1	no	—
208	C₄⋊C₄×C₂₃	—	2	1	no	—
209	C₄⋊C₄×C₂₃	—	2	1	no	—
210	C₄⋊C₄×C₂₃	—	4	1	no	—
211	C₄⋊C₄×C₂₃	—	4	1	no	—
212	C₄⋊C₄×C₂₃	—	4	2	no	—
213	C₄⋊C₄×C₂₃	—	8	1	no	—
214	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
215	C₂²⋊C₄×C₂₃	—	2	4	no	—
216	C₂²⋊C₄×C₂₃	—	4	1	no	—
217	C₂²⋊C₄×C₂₃	—	4	1	no	—
218	C₂²⋊C₄×C₂₃	—	4	2	no	—
219	C₂²⋊C₄×C₂₃	—	8	1	no	—
220	C₄²×C₂₃	—	2	1	no	—
221	C₄²×C₂₃	—	2	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	—	4	2	no	—
225	C₄²×C₂₃	—	4	2	no	—
226	C₄²×C₂₃	—	8	1	no	—
227	#22054	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	—	2	1	no	—
231	C₂³×C₂₃	—	2	1	no	—
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	C₄×D₄	1	2	yes	—
235	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₂₃	—	2	2	no	—
237	Q₈×C₂₃	—	2	2	no	—
238	Q₈×C₂₃	—	2	2	no	—
239	Q₈×C₂₃	—	2	2	no	—
240	Q₈×C₂₃	—	2	2	no	—
241	Q₈×C₂₃	—	2	2	no	—
242	Q₈×C₂₃	—	4	8	no	—
243	Q₈×C₂₃	—	4	1	no	—
244	Q₈×C₂₃	—	4	1	no	—
245	Q₈×C₂₃	—	4	1	no	—
246	Q₈×C₂₃	—	4	1	no	—
247	Q₈×C₂₃	—	4	1	no	—
248	Q₈×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	1	no	—
270	C₂×C₄×C₂₃	—	2	1	no	—
271	C₂×C₄×C₂₃	—	4	8	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	4	1	no	—
281	C₂×C₄×C₂₃	—	4	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	4	1	no	—
285	C₂×C₄×C₂₃	—	4	1	no	—
286	C₂×C₄×C₂₃	—	4	1	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
293	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
294	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
295	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
296	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
299	C₈×C₂₃	—	2	2	no	—
300	C₈×C₂₃	—	2	2	no	—
301	C₈×C₂₃	—	4	1	no	—
302	C₈×C₂₃	—	4	1	no	—
303	C₂×#451	C₂×C₂₃	1	1	yes	characteristic
304	#2323	C₂×C₂₃	1	1	yes	characteristic
305	#2310	C₂×C₂₃	1	1	yes	characteristic
306	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1969	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#1935	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#1928	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#1927	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1873	C₂×C₂₃	1	1	yes	characteristic
315	#1854	C₂×C₂₃	1	1	yes	characteristic
316	#1787	C₂×C₂₃	1	1	yes	characteristic
317	#1750	C₂×C₂₃	1	1	yes	characteristic
318	C₂²×C₂₃	—	2	2	no	—
319	C₂²×C₂₃	—	2	2	no	—
320	C₂²×C₂₃	—	2	2	no	—
321	C₂²×C₂₃	—	4	1	no	—
322	C₂²×C₂₃	—	4	1	no	—
323	C₂²×C₂₃	—	4	1	no	—
324	C₂²×C₂₃	—	4	1	no	—
325	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
326	C₂²×C₂₃	#534	1	1	yes	characteristic
327	C₂²×C₂₃	#535	1	1	yes	characteristic
328	C₂²×C₂₃	#577	1	1	yes	characteristic
329	C₂²×C₂₃	#578	1	1	yes	characteristic
330	C₂²×C₂₃	#578	1	1	yes	characteristic
331	C₂²×C₂₃	#578	1	1	yes	characteristic
332	C₄×C₂₃	—	2	2	no	—
333	C₄×C₂₃	—	4	2	no	—
334	C₄×C₂₃	—	4	2	no	—
335	C₄×C₂₃	—	4	2	no	—
336	C₄×C₂₃	—	4	1	no	—
337	C₄×C₂₃	—	4	1	no	—
338	C₄×C₂₃	—	4	1	no	—
339	C₄×C₂₃	—	4	1	no	—
340	C₄×C₂₃	—	4	2	no	—
341	C₄×C₂₃	—	8	1	no	—
342	C₄×C₂₃	—	8	1	no	—
343	C₄×C₂₃	—	8	1	no	—
344	C₄×C₂₃	—	8	1	no	—
345	C₄×C₂₃	#518	1	1	yes	characteristic
346	C₄×C₂₃	#518	1	1	yes	characteristic
347	C₄×C₂₃	#538	1	1	yes	characteristic
348	C₄×C₂₃	#538	1	1	yes	characteristic
349	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
350	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
351	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
356	#451	C₄×C₂₃	1	8	yes	complemented
357	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
371	#391	—	2	1	no	—
372	#391	—	2	1	no	—
373	#391	C₂²×C₂₃	1	1	yes	characteristic
374	#391	C₂²×C₂₃	1	1	yes	characteristic
375	#391	C₂²×C₂₃	1	1	yes	characteristic
376	#391	C₂²×C₂₃	1	1	yes	characteristic
377	#391	C₂²×C₂₃	1	1	yes	characteristic
378	#389	—	2	1	no	—
379	#389	C₂²×C₂₃	1	1	yes	characteristic
380	#388	—	2	1	no	—
381	#388	—	2	1	no	—
382	#387	—	2	1	no	—
383	#381	—	2	1	no	—
384	#381	—	2	1	no	—
385	#380	C₂²×C₂₃	1	1	yes	characteristic
386	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
387	#340	C₂²×C₂₃	1	1	yes	characteristic
388	#340	C₂²×C₂₃	1	1	yes	characteristic
389	#340	C₂²×C₂₃	1	1	yes	characteristic
390	#340	C₂²×C₂₃	1	1	yes	characteristic
391	#336	C₂²×C₂₃	1	1	yes	characteristic
392	#336	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×C₂₃	—	4	2	no	—
394	C₂×C₂₃	#2321	1	1	yes	characteristic
395	C₂×C₂₃	#2947	1	1	yes	characteristic
396	C₂×C₂₃	#2950	1	1	yes	characteristic
397	C₂×C₂₃	#3201	1	1	yes	characteristic
398	C₂×C₂₃	#3202	1	1	yes	characteristic
399	C₂×C₂₃	#3223	1	1	yes	characteristic
400	C₂×C₂₃	#3225	1	1	yes	characteristic
401	C₂²×Q₈	—	2	1	no	—
402	C₂²×Q₈	—	2	2	no	—
403	C₂²×Q₈	—	4	1	no	—
404	C₂²×Q₈	—	4	1	no	—
405	C₂²×Q₈	—	4	1	no	—
406	C₂²×Q₈	—	4	1	no	—
407	C₂²×Q₈	C₂×C₄×C₂₃	1	4	yes	—
408	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₂³×C₄	—	2	1	no	—
412	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
413	C₂×Q₁₆	—	4	1	no	—
414	C₂×Q₁₆	—	4	1	no	—
415	C₂×Q₁₆	—	4	1	no	—
416	C₂×Q₁₆	—	4	1	no	—
417	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
418	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
419	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
423	C₂²⋊Q₈	—	2	2	no	—
424	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
425	C₄×Q₈	—	4	1	no	—
426	C₄×Q₈	—	4	1	no	—
427	#117	—	4	1	no	—
428	#117	—	4	1	no	—
429	#117	—	4	1	no	—
430	#117	—	4	1	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	1	no	—
443	C₂×C₂²⋊C₄	—	2	1	no	—
444	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
445	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	—	2	1	no	—
448	C₂×C₄²	—	2	1	no	—
449	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
451	#107	—	4	1	no	—
452	#106	—	4	1	no	—
453	#103	—	4	1	no	—
454	#103	—	4	1	no	—
455	#103	—	4	1	no	—
456	#103	—	4	1	no	—
457	#103	—	4	1	no	—
458	#103	—	4	1	no	—
459	#103	C₂×C₄×C₂₃	1	4	yes	—
460	#103	C₂×C₄×C₂₃	1	4	yes	—
461	C₂²⋊C₈	—	2	2	no	—
462	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
463	#97	—	4	1	no	—
464	C₄×C₈	—	4	1	no	—
465	#95	—	2	1	no	—
466	#95	—	2	1	no	—
467	#95	—	2	1	no	—
468	#95	—	2	1	no	—
469	#95	—	2	1	no	—
470	#95	—	2	1	no	—
471	#95	—	2	1	no	—
472	#95	—	2	1	no	—
473	#95	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	1	no	—
476	#95	—	2	1	no	—
477	#95	—	2	1	no	—
478	C₂₃	#22054	1	1	yes	characteristic, direct product factor, Sylow
479	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	4	4	no	—
497	C₂×Q₈	—	4	4	no	—
498	C₂×Q₈	—	4	2	no	—
499	C₂×Q₈	—	4	2	no	—
500	C₂×Q₈	—	4	4	no	—
501	C₂×Q₈	—	4	4	no	—
502	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
503	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
504	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	4	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
521	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
522	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
526	C₂²×C₄	#41×C₂₃	1	2	yes	—
527	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
528	Q₁₆	—	4	4	no	—
529	Q₁₆	—	4	4	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
541	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	4	1	no	—
553	C₄⋊C₄	—	4	1	no	—
554	C₄⋊C₄	—	4	2	no	—
555	C₄⋊C₄	—	8	1	no	—
556	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	4	1	no	—
559	C₂²⋊C₄	—	4	1	no	—
560	C₂²⋊C₄	—	4	2	no	—
561	C₂²⋊C₄	—	8	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	—	2	1	no	—
565	C₄²	—	2	1	no	—
566	C₄²	—	4	2	no	—
567	C₄²	—	4	2	no	—
568	C₄²	—	8	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	C₄×D₄×C₂₃	1	2	yes	—
576	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
577	Q₈	—	2	2	no	—
578	Q₈	—	2	2	no	—
579	Q₈	—	2	2	no	—
580	Q₈	—	2	2	no	—
581	Q₈	—	2	2	no	—
582	Q₈	—	2	2	no	—
583	Q₈	—	4	8	no	—
584	Q₈	—	4	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	Q₈	—	4	1	no	—
589	Q₈	—	4	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
634	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
635	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
636	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
637	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
638	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
639	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
640	C₈	—	2	2	no	—
641	C₈	—	2	2	no	—
642	C₈	—	4	1	no	—
643	C₈	—	4	1	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	2	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
652	C₂²	C₂₃×#534	1	1	yes	characteristic
653	C₂²	C₂₃×#535	1	1	yes	characteristic
654	C₂²	C₂₃×#577	1	1	yes	characteristic
655	C₂²	C₂₃×#578	1	1	yes	characteristic
656	C₂²	C₂₃×#578	1	1	yes	characteristic
657	C₂²	C₂₃×#578	1	1	yes	characteristic
658	C₄	—	2	2	no	—
659	C₄	—	4	1	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	2	no	—
667	C₄	—	8	1	no	—
668	C₄	—	8	1	no	—
669	C₄	—	8	1	no	—
670	C₄	—	8	1	no	—
671	C₄	C₂₃×#518	1	1	yes	characteristic
672	C₄	C₂₃×#518	1	1	yes	characteristic
673	C₄	C₂₃×#538	1	1	yes	characteristic
674	C₄	C₂₃×#538	1	1	yes	characteristic
675	C₂	—	4	2	no	—
676	C₂	C₂₃×#2321	1	1	yes	characteristic
677	C₂	C₂₃×#2947	1	1	yes	characteristic
678	C₂	C₂₃×#2950	1	1	yes	characteristic
679	C₂	C₂₃×#3201	1	1	yes	characteristic
680	C₂	C₂₃×#3202	1	1	yes	characteristic
681	C₂	C₂₃×#3223	1	1	yes	characteristic
682	C₂	C₂₃×#3225	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	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1971	C₂	semidirect
5. C₂₃×#1969	C₂	semidirect
6. C₂₃×#1935	C₂	semidirect
7. C₂₃×#1933	C₂	semidirect
8. C₂₃×#1928	C₂	semidirect
9. C₂₃×#1927	C₂	semidirect
10. C₂₃×#1878	C₂	semidirect
11. C₂₃×#1878	C₂	semidirect
23. C₂₃×#451	C₄	semidirect
227. #22054	C₂₃	direct (N × H)
306. #1971	C₂×C₂₃	semidirect
307. #1969	C₂×C₂₃	semidirect
308. #1935	C₂×C₂₃	semidirect
309. #1933	C₂×C₂₃	semidirect
310. #1928	C₂×C₂₃	semidirect
311. #1927	C₂×C₂₃	semidirect
312. #1878	C₂×C₂₃	semidirect
313. #1878	C₂×C₂₃	semidirect
356. #451	C₄×C₂₃	semidirect
478. C₂₃	#22054	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×#451	2
C₂₃×#2323	2
C₂₃×#2310	2
C₂₃×#1971	2
C₂₃×#1969	2
C₂₃×#1935	2
C₂₃×#1933	2
C₂₃×#1928	2
C₂₃×#1927	2
C₂₃×#1878	2
C₂₃×#1873	2
C₂₃×#1854	2
C₂₃×#1787	2
C₂₃×#1750	2
#22054	23

Subgroups — C23×#22054 #19169680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22054 #19169680