Subgroups — C₂₃×#26820 #19174446

Order 5888.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2452	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2446	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2055	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2013	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1968	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1934	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1877	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
21	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#487	—	2	4	no	—
24	C₄×SD₁₆×C₂₃	—	2	4	no	—
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊C₈	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×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#400	—	2	1	no	—
40	C₂₃×#398	—	2	1	no	—
41	C₂₃×#396	—	2	1	no	—
42	C₂₃×#393	—	2	1	no	—
43	C₂₃×#391	C₂²	1	1	yes	characteristic
44	C₂₃×#391	C₂²	1	1	yes	characteristic
45	C₂₃×#391	C₂²	1	1	yes	characteristic
46	C₂₃×#390	C₂²	1	1	yes	characteristic
47	C₂₃×#390	C₂²	1	1	yes	characteristic
48	C₂₃×#389	C₂²	1	1	yes	characteristic
49	C₂₃×#389	C₂²	1	1	yes	characteristic
50	C₂₃×#389	C₂²	1	1	yes	characteristic
51	C₂₃×#387	—	2	1	no	—
52	C₂₃×#387	—	2	1	no	—
53	C₂₃×#385	—	2	1	no	—
54	C₄×C₄⋊C₄×C₂₃	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×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂×SD₁₆×C₂₃	—	2	2	no	—
67	C₂×SD₁₆×C₂₃	—	2	2	no	—
68	C₂×SD₁₆×C₂₃	—	2	2	no	—
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	4	1	no	—
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	—	2	2	no	—
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	—	4	1	no	—
79	C₂₃×C₄⋊Q₈	—	4	1	no	—
80	C₂₃×C₂²⋊Q₈	—	4	4	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	4	1	no	—
83	C₄×D₄×C₂₃	—	2	2	no	—
84	C₄×D₄×C₂₃	—	4	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	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×C₂²⋊C₄×C₂₃	—	2	1	no	—
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	—	2	1	no	—
104	C₂×C₄²×C₂₃	—	2	1	no	—
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂₃×#106	—	2	2	no	—
109	C₂₃×#106	—	4	1	no	—
110	C₂₃×C₄⋊C₈	—	2	2	no	—
111	C₂₃×#103	—	2	2	no	—
112	C₂₃×#103	—	2	2	no	—
113	C₂₃×#103	—	2	2	no	—
114	C₂₃×#103	—	2	2	no	—
115	C₂₃×#103	—	2	2	no	—
116	C₂₃×#103	D₄	1	4	yes	—
117	C₂₃×#102	—	2	2	no	—
118	C₂₃×#102	—	2	2	no	—
119	C₄×C₈×C₂₃	—	2	2	no	—
120	C₄×C₈×C₂₃	—	2	2	no	—
121	C₄×C₈×C₂₃	—	2	2	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	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	D₄	1	1	yes	characteristic
132	C₂₃×#95	D₄	1	1	yes	characteristic
133	C₂⁴×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	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	—	4	2	no	—
146	C₂×Q₈×C₂₃	—	4	2	no	—
147	C₂×Q₈×C₂₃	—	8	1	no	—
148	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	#41	1	2	yes	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	4	2	no	—
163	C₂²×C₄×C₂₃	—	4	2	no	—
164	C₂²×C₄×C₂₃	—	8	1	no	—
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₂₃	—	2	8	no	—
173	SD₁₆×C₂₃	—	4	4	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
184	C₂×C₈×C₂₃	#41	1	2	yes	—
185	C₂×C₈×C₂₃	#41	1	2	yes	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	4	2	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	4	2	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	4	2	no	—
201	C₄⋊C₄×C₂₃	—	4	2	no	—
202	C₄⋊C₄×C₂₃	—	4	2	no	—
203	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	complemented
204	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
205	C₂²⋊C₄×C₂₃	—	4	2	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₂²⋊C₄×C₂₃	—	4	2	no	—
208	C₂²⋊C₄×C₂₃	—	4	2	no	—
209	C₂²⋊C₄×C₂₃	—	8	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	4	2	no	—
217	#26820	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	4	1	no	—
220	C₂³×C₂₃	—	4	1	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	C₂³×C₂₃	—	4	1	no	—
223	C₂³×C₂₃	—	4	1	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₂₃	—	2	2	no	—
227	Q₈×C₂₃	—	2	2	no	—
228	Q₈×C₂₃	—	4	1	no	—
229	Q₈×C₂₃	—	4	1	no	—
230	Q₈×C₂₃	—	4	1	no	—
231	Q₈×C₂₃	—	4	1	no	—
232	Q₈×C₂₃	—	4	1	no	—
233	Q₈×C₂₃	—	4	1	no	—
234	Q₈×C₂₃	—	8	4	no	—
235	D₄×C₂₃	—	2	2	no	—
236	D₄×C₂₃	—	2	2	no	—
237	D₄×C₂₃	—	4	1	no	—
238	D₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	8	4	no	—
275	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
276	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
277	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
278	C₂×C₄×C₂₃	#135	1	2	yes	—
279	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
282	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
285	C₈×C₂₃	—	2	2	no	—
286	C₈×C₂₃	—	2	2	no	—
287	C₈×C₂₃	—	2	2	no	—
288	C₈×C₂₃	—	4	1	no	—
289	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
290	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
291	#2452	C₂×C₂₃	1	1	yes	characteristic
292	#2446	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#2055	C₂×C₂₃	1	1	yes	characteristic
296	#2013	C₂×C₂₃	1	1	yes	characteristic
297	#1968	C₂×C₂₃	1	1	yes	characteristic
298	#1934	C₂×C₂₃	1	1	yes	characteristic
299	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1877	C₂×C₂₃	1	1	yes	characteristic
302	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
303	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
304	C₂²×C₂₃	—	4	2	no	—
305	C₂²×C₂₃	—	4	2	no	—
306	C₂²×C₂₃	—	4	2	no	—
307	C₂²×C₂₃	—	8	1	no	—
308	C₂²×C₂₃	—	8	1	no	—
309	C₂²×C₂₃	—	8	1	no	—
310	C₂²×C₂₃	—	8	1	no	—
311	C₂²×C₂₃	#532	1	1	yes	characteristic
312	C₂²×C₂₃	D₄²	1	1	yes	characteristic
313	C₂²×C₂₃	#548	1	1	yes	characteristic
314	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
315	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
316	C₂²×C₂₃	#577	1	1	yes	characteristic
317	C₂²×C₂₃	#578	1	1	yes	characteristic
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	2	2	no	—
320	C₄×C₂₃	—	2	2	no	—
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	4	1	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	—	4	2	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	8	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	—	8	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	#525	1	1	yes	characteristic
332	C₄×C₂₃	#542	1	1	yes	characteristic
333	C₄×C₂₃	#547	1	1	yes	characteristic
334	C₄×C₂₃	#551	1	1	yes	characteristic
335	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
336	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×C₂²⋊Q₈	—	2	1	no	—
340	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
342	#487	—	2	4	no	—
343	C₄×SD₁₆	—	2	4	no	—
344	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
358	#400	—	2	1	no	—
359	#398	—	2	1	no	—
360	#396	—	2	1	no	—
361	#393	—	2	1	no	—
362	#391	C₂²×C₂₃	1	1	yes	characteristic
363	#391	C₂²×C₂₃	1	1	yes	characteristic
364	#391	C₂²×C₂₃	1	1	yes	characteristic
365	#390	C₂²×C₂₃	1	1	yes	characteristic
366	#390	C₂²×C₂₃	1	1	yes	characteristic
367	#389	C₂²×C₂₃	1	1	yes	characteristic
368	#389	C₂²×C₂₃	1	1	yes	characteristic
369	#389	C₂²×C₂₃	1	1	yes	characteristic
370	#387	—	2	1	no	—
371	#387	—	2	1	no	—
372	#385	—	2	1	no	—
373	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#340	C₂²×C₂₃	1	1	yes	characteristic
378	#336	C₂²×C₂₃	1	1	yes	characteristic
379	#336	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×C₂₃	—	8	2	no	—
381	C₂×C₂₃	#2461	1	1	yes	characteristic
382	C₂×C₂₃	#3167	1	1	yes	characteristic
383	C₂×C₂₃	#3178	1	1	yes	characteristic
384	C₂×C₂₃	D₄×SD₁₆	1	1	yes	characteristic
385	C₂×C₂₃	#3285	1	1	yes	characteristic
386	C₂×C₂₃	#3359	1	1	yes	characteristic
387	C₂×C₂₃	#3360	1	1	yes	characteristic
388	C₂²×Q₈	—	2	1	no	—
389	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
390	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
391	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂³×C₄	—	2	1	no	—
393	C₂×SD₁₆	—	2	2	no	—
394	C₂×SD₁₆	—	2	2	no	—
395	C₂×SD₁₆	—	2	2	no	—
396	C₂×SD₁₆	—	2	2	no	—
397	C₂×SD₁₆	—	2	2	no	—
398	C₂×SD₁₆	—	4	1	no	—
399	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₄⋊Q₈	—	2	2	no	—
404	C₄⋊Q₈	—	2	2	no	—
405	C₄⋊Q₈	—	4	1	no	—
406	C₄⋊Q₈	—	4	1	no	—
407	C₂²⋊Q₈	—	4	4	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	—	4	1	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	4	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
435	#106	—	2	2	no	—
436	#106	—	4	1	no	—
437	C₄⋊C₈	—	2	2	no	—
438	#103	—	2	2	no	—
439	#103	—	2	2	no	—
440	#103	—	2	2	no	—
441	#103	—	2	2	no	—
442	#103	—	2	2	no	—
443	#103	D₄×C₂₃	1	4	yes	—
444	#102	—	2	2	no	—
445	#102	—	2	2	no	—
446	C₄×C₈	—	2	2	no	—
447	C₄×C₈	—	2	2	no	—
448	C₄×C₈	—	2	2	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	D₄×C₂₃	1	1	yes	characteristic
459	#95	D₄×C₂₃	1	1	yes	characteristic
460	C₂₃	#26820	1	1	yes	characteristic, direct product factor, Sylow
461	C₂⁴	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	4	2	no	—
474	C₂×Q₈	—	4	2	no	—
475	C₂×Q₈	—	8	1	no	—
476	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	#41×C₂₃	1	2	yes	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	—	8	1	no	—
493	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
500	SD₁₆	—	2	8	no	—
501	SD₁₆	—	4	4	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
512	C₂×C₈	#41×C₂₃	1	2	yes	—
513	C₂×C₈	#41×C₂₃	1	2	yes	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	4	2	no	—
525	C₄⋊C₄	—	4	2	no	—
526	C₄⋊C₄	—	4	2	no	—
527	C₄⋊C₄	—	4	2	no	—
528	C₄⋊C₄	—	4	2	no	—
529	C₄⋊C₄	—	4	2	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	complemented
532	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
533	C₂²⋊C₄	—	4	2	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₂²⋊C₄	—	4	2	no	—
536	C₂²⋊C₄	—	4	2	no	—
537	C₂²⋊C₄	—	8	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	4	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	4	1	no	—
547	C₂³	—	4	1	no	—
548	C₂³	—	4	1	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
553	Q₈	—	2	2	no	—
554	Q₈	—	2	2	no	—
555	Q₈	—	4	1	no	—
556	Q₈	—	4	1	no	—
557	Q₈	—	4	1	no	—
558	Q₈	—	4	1	no	—
559	Q₈	—	4	1	no	—
560	Q₈	—	4	1	no	—
561	Q₈	—	8	4	no	—
562	D₄	—	2	2	no	—
563	D₄	—	2	2	no	—
564	D₄	—	4	1	no	—
565	D₄	—	4	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	8	4	no	—
602	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
603	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
604	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
605	C₂×C₄	C₂₃×#135	1	2	yes	—
606	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
608	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
609	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
610	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
611	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
612	C₈	—	2	2	no	—
613	C₈	—	2	2	no	—
614	C₈	—	2	2	no	—
615	C₈	—	4	1	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	8	1	no	—
620	C₂²	—	8	1	no	—
621	C₂²	—	8	1	no	—
622	C₂²	—	8	1	no	—
623	C₂²	C₂₃×#532	1	1	yes	characteristic
624	C₂²	D₄²×C₂₃	1	1	yes	characteristic
625	C₂²	C₂₃×#548	1	1	yes	characteristic
626	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
627	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
628	C₂²	C₂₃×#577	1	1	yes	characteristic
629	C₂²	C₂₃×#578	1	1	yes	characteristic
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	4	1	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	1	no	—
638	C₄	—	8	1	no	—
639	C₄	—	8	1	no	—
640	C₄	—	8	1	no	—
641	C₄	—	8	1	no	—
642	C₄	—	8	1	no	—
643	C₄	C₂₃×#525	1	1	yes	characteristic
644	C₄	C₂₃×#542	1	1	yes	characteristic
645	C₄	C₂₃×#547	1	1	yes	characteristic
646	C₄	C₂₃×#551	1	1	yes	characteristic
647	C₂	—	8	2	no	—
648	C₂	C₂₃×#2461	1	1	yes	characteristic
649	C₂	C₂₃×#3167	1	1	yes	characteristic
650	C₂	C₂₃×#3178	1	1	yes	characteristic
651	C₂	D₄×SD₁₆×C₂₃	1	1	yes	characteristic
652	C₂	C₂₃×#3285	1	1	yes	characteristic
653	C₂	C₂₃×#3359	1	1	yes	characteristic
654	C₂	C₂₃×#3360	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#487	C₂	semidirect
4. C₂₃×#2446	C₂	semidirect
5. C₂₃×#2066	C₂	semidirect
6. C₂₃×#2058	C₂	semidirect
11. C₂₃×#1933	C₂	semidirect
12. C₂₃×#1933	C₂	semidirect
14. C₂₃×#1865	C₂	semidirect
15. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
203. C₄⋊C₄×C₂₃	SD₁₆	semidirect
217. #26820	C₂₃	direct (N × H)
289. C₂×#487	C₂×C₂₃	semidirect
292. #2446	C₂×C₂₃	semidirect
293. #2066	C₂×C₂₃	semidirect
294. #2058	C₂×C₂₃	semidirect
299. #1933	C₂×C₂₃	semidirect
300. #1933	C₂×C₂₃	semidirect
302. #1865	C₂×C₂₃	semidirect
303. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
460. C₂₃	#26820	direct (N × H)
531. C₄⋊C₄	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#487	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#2452	2
C₂₃×#2446	2
C₂₃×#2066	2
C₂₃×#2058	2
C₂₃×#2055	2
C₂₃×#2013	2
C₂₃×#1968	2
C₂₃×#1934	2
C₂₃×#1933	2
C₂₃×#1877	2
C₂₃×#1865	2
C₈×C₄⋊C₄×C₂₃	2
#26820	23

Subgroups — C23×#26820 #19174446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26820 #19174446