Subgroups — C₂₃×#3349 #15670968

Order 2944.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#556	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#552	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#435	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×#126	—	2	1	no	—
21	C₂₃×#126	—	2	1	no	—
22	C₂₃×#126	—	2	1	no	—
23	C₂₃×#125	—	2	1	no	—
24	C₂₃×#125	C₂²	1	1	yes	characteristic
25	C₂₃×#124	—	2	1	no	—
26	C₂₃×#123	—	2	1	no	—
27	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₂₃	—	2	1	no	—
32	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#117	—	2	1	no	—
36	C₂₃×#117	C₂²	1	1	yes	characteristic
37	C₂₃×#117	C₂²	1	1	yes	characteristic
38	C₂₃×#117	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#107	C₂²	1	1	yes	characteristic
41	C₂₃×#107	C₂²	1	1	yes	characteristic
42	C₂₃×#107	C₂²	1	1	yes	characteristic
43	C₂₃×#107	C₂²	1	1	yes	characteristic
44	C₂₃×#107	C₂²	1	1	yes	characteristic
45	C₂₃×#106	C₂²	1	1	yes	characteristic
46	C₂₃×#106	C₂²	1	1	yes	characteristic
47	C₂₃×#106	C₂²	1	1	yes	characteristic
48	C₂₃×#106	C₂²	1	1	yes	characteristic
49	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂₃×#103	C₂²	1	1	yes	characteristic
51	C₂₃×#103	C₂²	1	1	yes	characteristic
52	C₂₃×#103	C₂²	1	1	yes	characteristic
53	C₂₃×#103	C₂²	1	1	yes	characteristic
54	C₂₃×#103	C₂²	1	1	yes	characteristic
55	C₂₃×#103	C₂²	1	1	yes	characteristic
56	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×#97	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	—	2	1	no	—
60	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	Q₁₆×C₂₃	—	2	2	no	—
67	M₁₆×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
99	C₄²×C₂₃	—	2	1	no	—
100	C₄²×C₂₃	—	2	1	no	—
101	C₄²×C₂₃	—	2	1	no	—
102	C₄²×C₂₃	—	2	1	no	—
103	C₄²×C₂₃	—	2	1	no	—
104	C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₂₃	—	2	1	no	—
108	Q₈×C₂₃	—	2	1	no	—
109	Q₈×C₂₃	—	2	1	no	—
110	Q₈×C₂₃	—	2	1	no	—
111	Q₈×C₂₃	—	4	2	no	—
112	D₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
131	C₈×C₂₃	—	2	1	no	—
132	C₈×C₂₃	—	2	1	no	—
133	C₈×C₂₃	—	2	1	no	—
134	C₈×C₂₃	#41	1	2	yes	—
135	#3349	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₄×C₂₃	—	4	1	no	—
151	C₄×C₂₃	—	4	1	no	—
152	C₄×C₂₃	—	4	1	no	—
153	C₄×C₂₃	—	4	1	no	—
154	C₄×C₂₃	—	4	1	no	—
155	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
157	#556	C₂×C₂₃	1	1	yes	characteristic
158	#552	C₂×C₂₃	1	1	yes	characteristic
159	#501	C₂×C₂₃	1	1	yes	characteristic, complemented
160	#485	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#485	C₂×C₂₃	1	1	yes	characteristic, complemented
162	#485	C₂×C₂₃	1	1	yes	characteristic, complemented
163	#484	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#479	C₂×C₂₃	1	1	yes	characteristic, complemented
165	#477	C₂×C₂₃	1	1	yes	characteristic, complemented
166	#470	C₂×C₂₃	1	1	yes	characteristic, complemented
167	#467	C₂×C₂₃	1	1	yes	characteristic, complemented
168	#441	C₂×C₂₃	1	1	yes	characteristic, complemented
169	#435	C₂×C₂₃	1	1	yes	characteristic
170	#428	C₂×C₂₃	1	1	yes	characteristic, complemented
171	#427	C₂×C₂₃	1	1	yes	characteristic, complemented
172	C₂×C₂₃	—	4	1	no	—
173	C₂×C₂₃	—	4	1	no	—
174	C₂×C₂₃	#547	1	1	yes	characteristic
175	C₂×C₂₃	#575	1	1	yes	characteristic
176	C₂×C₂₃	#578	1	1	yes	characteristic
177	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
180	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
181	#126	—	2	1	no	—
182	#126	—	2	1	no	—
183	#126	—	2	1	no	—
184	#125	—	2	1	no	—
185	#125	C₂²×C₂₃	1	1	yes	characteristic
186	#124	—	2	1	no	—
187	#123	—	2	1	no	—
188	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
189	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
190	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
191	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
194	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
195	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
196	#117	—	2	1	no	—
197	#117	C₂²×C₂₃	1	1	yes	characteristic
198	#117	C₂²×C₂₃	1	1	yes	characteristic
199	#117	C₂²×C₂₃	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
201	#107	C₂²×C₂₃	1	1	yes	characteristic
202	#107	C₂²×C₂₃	1	1	yes	characteristic
203	#107	C₂²×C₂₃	1	1	yes	characteristic
204	#107	C₂²×C₂₃	1	1	yes	characteristic
205	#107	C₂²×C₂₃	1	1	yes	characteristic
206	#106	C₂²×C₂₃	1	1	yes	characteristic
207	#106	C₂²×C₂₃	1	1	yes	characteristic
208	#106	C₂²×C₂₃	1	1	yes	characteristic
209	#106	C₂²×C₂₃	1	1	yes	characteristic
210	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
211	#103	C₂²×C₂₃	1	1	yes	characteristic
212	#103	C₂²×C₂₃	1	1	yes	characteristic
213	#103	C₂²×C₂₃	1	1	yes	characteristic
214	#103	C₂²×C₂₃	1	1	yes	characteristic
215	#103	C₂²×C₂₃	1	1	yes	characteristic
216	#103	C₂²×C₂₃	1	1	yes	characteristic
217	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
218	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
219	#97	C₂²×C₂₃	1	1	yes	characteristic
220	C₂₃	#3349	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
223	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
224	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
232	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
233	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
246	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
247	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
248	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
251	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	2	1	no	—
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
260	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	C₂³×C₂₃	1	1	yes	characteristic
267	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	4	2	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	#41×C₂₃	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
317	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	C₂₃×#547	1	1	yes	characteristic
321	C₂	C₂₃×#575	1	1	yes	characteristic
322	C₂	C₂₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#501	C₂	semidirect
4. C₂₃×#485	C₂	semidirect
5. C₂₃×#485	C₂	semidirect
6. C₂₃×#485	C₂	semidirect
7. C₂₃×#484	C₂	semidirect
8. C₂₃×#479	C₂	semidirect
9. C₂₃×#477	C₂	semidirect
10. C₂₃×#470	C₂	semidirect
11. C₂₃×#467	C₂	semidirect
12. C₂₃×#441	C₂	semidirect
14. C₂₃×#428	C₂	semidirect
15. C₂₃×#427	C₂	semidirect
135. #3349	C₂₃	direct (N × H)
159. #501	C₂×C₂₃	semidirect
160. #485	C₂×C₂₃	semidirect
161. #485	C₂×C₂₃	semidirect
162. #485	C₂×C₂₃	semidirect
163. #484	C₂×C₂₃	semidirect
164. #479	C₂×C₂₃	semidirect
165. #477	C₂×C₂₃	semidirect
166. #470	C₂×C₂₃	semidirect
167. #467	C₂×C₂₃	semidirect
168. #441	C₂×C₂₃	semidirect
170. #428	C₂×C₂₃	semidirect
171. #427	C₂×C₂₃	semidirect
220. C₂₃	#3349	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#556	2
C₂₃×#552	2
C₂₃×#501	2
C₂₃×#485	2
C₂₃×#484	2
C₂₃×#479	2
C₂₃×#477	2
C₂₃×#470	2
C₂₃×#467	2
C₂₃×#441	2
C₂₃×#435	2
C₂₃×#428	2
C₂₃×#427	2
#3349	23

Subgroups — C23×#3349 #15670968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3349 #15670968