Subgroups — C₂₃×#3388 #15671007

Order 2944.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#554	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#478	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#478	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#476	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#440	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#440	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
16	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊Q₈	—	2	1	no	—
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×#125	—	2	1	no	—
21	C₂₃×#125	—	2	1	no	—
22	C₂₃×#125	C₂²	1	1	yes	characteristic
23	C₂₃×#125	C₂²	1	1	yes	characteristic
24	C₂₃×#125	C₂²	1	1	yes	characteristic
25	C₂₃×#125	C₂²	1	1	yes	characteristic
26	C₂₃×C₂²⋊Q₈	—	2	1	no	—
27	C₂₃×C₂²⋊Q₈	—	2	1	no	—
28	C₂₃×C₂²⋊Q₈	—	2	1	no	—
29	C₄×Q₈×C₂₃	—	2	1	no	—
30	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
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	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
38	C₂₃×#107	C₂²	1	1	yes	characteristic
39	C₂₃×#107	C₂²	1	1	yes	characteristic
40	C₂₃×#107	C₂²	1	1	yes	characteristic
41	C₂₃×#107	C₂²	1	1	yes	characteristic
42	C₂₃×#106	C₂²	1	1	yes	characteristic
43	C₂₃×#106	C₂²	1	1	yes	characteristic
44	C₂₃×#106	C₂²	1	1	yes	characteristic
45	C₂₃×#106	C₂²	1	1	yes	characteristic
46	C₂₃×#106	C₂²	1	1	yes	characteristic
47	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	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×#102	C₂²	1	1	yes	characteristic
54	C₂₃×#102	C₂²	1	1	yes	characteristic
55	C₂₃×#102	C₂²	1	1	yes	characteristic
56	C₂₃×#97	C₂²	1	1	yes	characteristic
57	C₂₃×#97	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	—	2	1	no	—
60	C₂×Q₈×C₂₃	—	2	1	no	—
61	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	SD₁₆×C₂₃	Q₈	1	4	yes	complemented
68	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	2	1	no	—
84	C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₄²×C₂₃	—	2	1	no	—
98	C₄²×C₂₃	—	2	1	no	—
99	C₄²×C₂₃	—	2	1	no	—
100	C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂³×C₂₃	—	2	1	no	—
104	Q₈×C₂₃	—	2	1	no	—
105	Q₈×C₂₃	—	4	1	no	—
106	Q₈×C₂₃	—	4	2	no	—
107	Q₈×C₂₃	—	8	1	no	—
108	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
109	D₄×C₂₃	—	2	1	no	—
110	D₄×C₂₃	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	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	—	4	2	no	—
119	C₂×C₄×C₂₃	—	8	1	no	—
120	C₂×C₄×C₂₃	—	8	1	no	—
121	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
128	C₈×C₂₃	—	2	1	no	—
129	C₈×C₂₃	—	2	1	no	—
130	C₈×C₂₃	—	2	1	no	—
131	C₈×C₂₃	C₂×Q₈	1	2	yes	—
132	#3388	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	—	4	1	no	—
135	C₂²×C₂₃	—	4	1	no	—
136	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	1	no	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	—	4	1	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
151	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
152	#557	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#554	C₂×C₂₃	1	1	yes	characteristic
154	#501	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#501	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#499	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#479	C₂×C₂₃	1	1	yes	characteristic, complemented
158	#479	C₂×C₂₃	1	1	yes	characteristic, complemented
159	#478	C₂×C₂₃	1	1	yes	characteristic
160	#478	C₂×C₂₃	1	1	yes	characteristic
161	#476	C₂×C₂₃	1	1	yes	characteristic
162	#475	C₂×C₂₃	1	1	yes	characteristic, complemented
163	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#440	C₂×C₂₃	1	1	yes	characteristic
165	#440	C₂×C₂₃	1	1	yes	characteristic
166	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
167	C₂×C₂₃	—	4	2	no	—
168	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
169	C₂×C₂₃	#575	1	1	yes	characteristic
170	C₂×C₂₃	#577	1	1	yes	characteristic
171	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	1	no	—
173	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
174	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
175	#125	—	2	1	no	—
176	#125	—	2	1	no	—
177	#125	C₂²×C₂₃	1	1	yes	characteristic
178	#125	C₂²×C₂₃	1	1	yes	characteristic
179	#125	C₂²×C₂₃	1	1	yes	characteristic
180	#125	C₂²×C₂₃	1	1	yes	characteristic
181	C₂²⋊Q₈	—	2	1	no	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
186	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
187	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
188	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
189	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
190	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
191	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	—	2	1	no	—
193	#107	C₂²×C₂₃	1	1	yes	characteristic
194	#107	C₂²×C₂₃	1	1	yes	characteristic
195	#107	C₂²×C₂₃	1	1	yes	characteristic
196	#107	C₂²×C₂₃	1	1	yes	characteristic
197	#106	C₂²×C₂₃	1	1	yes	characteristic
198	#106	C₂²×C₂₃	1	1	yes	characteristic
199	#106	C₂²×C₂₃	1	1	yes	characteristic
200	#106	C₂²×C₂₃	1	1	yes	characteristic
201	#106	C₂²×C₂₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
203	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
204	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
205	#103	C₂²×C₂₃	1	1	yes	characteristic
206	#103	C₂²×C₂₃	1	1	yes	characteristic
207	#103	C₂²×C₂₃	1	1	yes	characteristic
208	#102	C₂²×C₂₃	1	1	yes	characteristic
209	#102	C₂²×C₂₃	1	1	yes	characteristic
210	#102	C₂²×C₂₃	1	1	yes	characteristic
211	#97	C₂²×C₂₃	1	1	yes	characteristic
212	#97	C₂²×C₂₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
214	C₂₃	#3388	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	2	1	no	—
217	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
218	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
219	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	SD₁₆	Q₈×C₂₃	1	4	yes	complemented
224	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
225	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
226	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
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	D₄×C₂₃	1	1	yes	characteristic
242	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
243	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
244	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
245	C₄⋊C₄	C₂³×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	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂³×C₂₃	1	1	yes	characteristic
257	C₄²	C₂³×C₂₃	1	1	yes	characteristic
258	C₄²	C₂³×C₂₃	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	4	1	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	8	1	no	—
264	Q₈	C₂×Q₈×C₂₃	1	2	yes	—
265	D₄	—	2	1	no	—
266	D₄	C₂×Q₈×C₂₃	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	C₂×Q₈×C₂₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
306	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
309	C₂	C₂₃×#575	1	1	yes	characteristic
310	C₂	C₂₃×#577	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	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#557	C₂	semidirect
3. C₂₃×#501	C₂	semidirect
4. C₂₃×#501	C₂	semidirect
5. C₂₃×#499	C₂	semidirect
6. C₂₃×#479	C₂	semidirect
7. C₂₃×#479	C₂	semidirect
11. C₂₃×#475	C₂	semidirect
12. C₂₃×#446	C₂	semidirect
67. SD₁₆×C₂₃	Q₈	semidirect
132. #3388	C₂₃	direct (N × H)
152. #557	C₂×C₂₃	semidirect
154. #501	C₂×C₂₃	semidirect
155. #501	C₂×C₂₃	semidirect
156. #499	C₂×C₂₃	semidirect
157. #479	C₂×C₂₃	semidirect
158. #479	C₂×C₂₃	semidirect
162. #475	C₂×C₂₃	semidirect
163. #446	C₂×C₂₃	semidirect
214. C₂₃	#3388	direct (N × H)
223. SD₁₆	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#557	2
C₂₃×#554	2
C₂₃×#501	2
C₂₃×#499	2
C₂₃×#479	2
C₂₃×#478	2
C₂₃×#476	2
C₂₃×#475	2
C₂₃×#446	2
C₂₃×#440	2
C₄×SD₁₆×C₂₃	2
#3388	23

Subgroups — C23×#3388 #15671007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3388 #15671007