Subgroups — C₂₃×#8936 #19156562

Order 5888.

356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2986	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1994	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1574	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1574	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#533	C₂²	1	1	yes	characteristic
9	C₂₃×#446	—	2	1	no	—
10	C₂₃×#446	—	2	1	no	—
11	C₈×Q₈×C₂₃	—	2	1	no	—
12	C₈×Q₈×C₂₃	—	2	1	no	—
13	C₂₃×#433	—	2	2	no	—
14	C₂₃×#431	C₂²	1	1	yes	characteristic
15	C₂₃×#431	C₂²	1	1	yes	characteristic
16	C₂₃×#431	C₂²	1	1	yes	characteristic
17	C₂₃×#431	C₂²	1	1	yes	characteristic
18	C₂₃×#423	C₂²	1	1	yes	characteristic
19	C₄³×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#339	—	2	2	no	—
21	C₂₃×#335	—	2	1	no	—
22	C₂₃×#335	—	2	1	no	—
23	C₂₃×#334	—	2	1	no	—
24	C₂₃×#334	—	2	1	no	—
25	C₂₃×#332	C₄	1	1	yes	characteristic, complemented
26	C₂₃×#332	C₄	1	1	yes	characteristic, complemented
27	C₂₃×#332	C₄	1	1	yes	characteristic, complemented
28	C₂₃×#332	C₄	1	1	yes	characteristic, complemented
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂₃×#125	Q₈	1	2	yes	—
33	C₂₃×C₂²⋊Q₈	—	2	1	no	—
34	C₂₃×C₂²⋊Q₈	—	2	1	no	—
35	C₄×Q₈×C₂₃	—	2	1	no	—
36	C₄×Q₈×C₂₃	—	2	1	no	—
37	C₂₃×#117	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	—	2	1	no	—
39	C₂×C₄²×C₂₃	—	2	1	no	—
40	C₂×C₄²×C₂₃	—	2	1	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	—	2	1	no	—
45	C₂₃×C₄⋊C₈	—	2	1	no	—
46	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
51	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
52	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
53	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₂₃×C₂²⋊C₈	—	2	1	no	—
55	C₂₃×C₂²⋊C₈	—	2	1	no	—
56	C₂₃×C₂²⋊C₈	—	2	1	no	—
57	C₂₃×C₂²⋊C₈	—	2	1	no	—
58	C₂₃×#97	—	2	2	no	—
59	C₄×C₈×C₂₃	—	2	2	no	—
60	C₄×C₈×C₂₃	—	2	1	no	—
61	C₄×C₈×C₂₃	—	2	1	no	—
62	C₄×C₈×C₂₃	—	2	1	no	—
63	C₄×C₈×C₂₃	—	2	1	no	—
64	C₂×Q₈×C₂₃	—	2	1	no	—
65	C₂×Q₈×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	4	1	no	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
70	M₁₆×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	1	no	—
76	C₂×C₈×C₂₃	—	2	1	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
85	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
86	C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₄²×C₂₃	—	2	2	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	—	2	2	no	—
90	C₄²×C₂₃	—	2	2	no	—
91	C₄²×C₂₃	—	4	1	no	—
92	C₄²×C₂₃	—	4	1	no	—
93	C₄²×C₂₃	—	4	1	no	—
94	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
98	#8936	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	#123	1	1	yes	characteristic
100	Q₈×C₂₃	—	4	1	no	—
101	Q₈×C₂₃	—	4	1	no	—
102	Q₈×C₂₃	—	4	1	no	—
103	Q₈×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₈×C₂₃	—	4	1	no	—
122	C₈×C₂₃	—	4	1	no	—
123	C₈×C₂₃	—	4	1	no	—
124	C₈×C₂₃	—	4	1	no	—
125	C₈×C₂₃	—	4	1	no	—
126	C₈×C₂₃	—	4	1	no	—
127	C₈×C₂₃	—	4	1	no	—
128	C₈×C₂₃	—	4	1	no	—
129	C₈×C₂₃	—	8	1	no	—
130	#2986	C₂×C₂₃	1	1	yes	characteristic
131	#1994	C₂×C₂₃	1	1	yes	characteristic
132	#1574	C₂×C₂₃	1	1	yes	characteristic
133	#1574	C₂×C₂₃	1	1	yes	characteristic
134	#1544	C₂×C₂₃	1	1	yes	characteristic
135	#1276	C₂×C₂₃	1	1	yes	characteristic
136	#1276	C₂×C₂₃	1	1	yes	characteristic
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	#385	1	1	yes	characteristic
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	4	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	#443	1	1	yes	characteristic
148	C₄×C₂₃	#443	1	1	yes	characteristic
149	C₄×C₂₃	#444	1	1	yes	characteristic
150	C₄×C₂₃	#444	1	1	yes	characteristic
151	C₄×C₂₃	#454	1	1	yes	characteristic
152	C₄×C₂₃	#456	1	1	yes	characteristic
153	#533	C₂²×C₂₃	1	1	yes	characteristic
154	#446	—	2	1	no	—
155	#446	—	2	1	no	—
156	C₈×Q₈	—	2	1	no	—
157	C₈×Q₈	—	2	1	no	—
158	#433	—	2	2	no	—
159	#431	C₂²×C₂₃	1	1	yes	characteristic
160	#431	C₂²×C₂₃	1	1	yes	characteristic
161	#431	C₂²×C₂₃	1	1	yes	characteristic
162	#431	C₂²×C₂₃	1	1	yes	characteristic
163	#423	C₂²×C₂₃	1	1	yes	characteristic
164	C₄³	C₂²×C₂₃	1	1	yes	characteristic
165	#339	—	2	2	no	—
166	#335	—	2	1	no	—
167	#335	—	2	1	no	—
168	#334	—	2	1	no	—
169	#334	—	2	1	no	—
170	#332	C₄×C₂₃	1	1	yes	characteristic, complemented
171	#332	C₄×C₂₃	1	1	yes	characteristic, complemented
172	#332	C₄×C₂₃	1	1	yes	characteristic, complemented
173	#332	C₄×C₂₃	1	1	yes	characteristic, complemented
174	C₂×C₂₃	—	4	1	no	—
175	C₂×C₂₃	#1857	1	1	yes	characteristic
176	C₂×C₂₃	#1866	1	1	yes	characteristic
177	C₂×C₂₃	#1866	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
181	#125	Q₈×C₂₃	1	2	yes	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	#117	D₄×C₂₃	1	2	yes	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
196	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
198	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
200	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
201	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
202	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₂₃	#8936	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
235	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
248	C₂³	C₂₃×#123	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	1	no	—
252	Q₈	—	4	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
268	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	8	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	C₂₃×#385	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	C₂₃×#443	1	1	yes	characteristic
290	C₄	C₂₃×#443	1	1	yes	characteristic
291	C₄	C₂₃×#444	1	1	yes	characteristic
292	C₄	C₂₃×#444	1	1	yes	characteristic
293	C₄	C₂₃×#454	1	1	yes	characteristic
294	C₄	C₂₃×#456	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	C₂₃×#1857	1	1	yes	characteristic
297	C₂	C₂₃×#1866	1	1	yes	characteristic
298	C₂	C₂₃×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#385
prime radical	C₂³×C₂₃	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₂₃×#332	C₄	semidirect
26. C₂₃×#332	C₄	semidirect
27. C₂₃×#332	C₄	semidirect
28. C₂₃×#332	C₄	semidirect
98. #8936	C₂₃	direct (N × H)
170. #332	C₄×C₂₃	semidirect
171. #332	C₄×C₂₃	semidirect
172. #332	C₄×C₂₃	semidirect
173. #332	C₄×C₂₃	semidirect
213. C₂₃	#8936	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×#2986	2
C₂₃×#1994	2
C₂₃×#1574	2
C₂₃×#1544	2
C₂₃×#1276	2
#8936	23

Subgroups — C23×#8936 #19156562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8936 #19156562