Subgroups — C₂₃×#9099 #19156725

Order 5888.

566 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2972	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2326	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1479	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1460	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1274	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₂₃	—	2	4	no	—
10	C₂₃×#431	C₂²	1	2	yes	—
11	C₂₃×#431	C₂²	1	2	yes	—
12	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
13	C₂₃×#391	C₄	1	1	yes	characteristic
14	C₂₃×#390	C₄	1	1	yes	characteristic
15	C₂₃×#388	—	2	2	no	—
16	C₂₃×#385	—	2	2	no	—
17	C₂₃×#384	C₄	1	1	yes	characteristic
18	C₂₃×#376	C₂²	1	1	yes	characteristic
19	C₂₃×Q₈⋊C₈	—	2	4	no	—
20	C₂₃×#323	—	2	2	no	—
21	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₈×C₂₃	—	2	4	no	—
25	C₂₃×#124	—	2	2	no	—
26	C₂₃×#124	C₈	1	4	yes	complemented
27	C₄×Q₈×C₂₃	—	2	2	no	—
28	C₄×Q₈×C₂₃	—	2	2	no	—
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	—	2	1	no	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
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×C₂²⋊C₈	—	2	4	no	—
45	C₂₃×C₂²⋊C₈	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₈×C₂₃	—	2	2	no	—
48	C₄×C₈×C₂₃	D₄	1	4	yes	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	1	no	—
51	C₂₃×#95	—	2	1	no	—
52	C₂⁴×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂×Q₈×C₂₃	—	2	1	no	—
54	C₂×Q₈×C₂₃	—	2	2	no	—
55	C₂×Q₈×C₂₃	—	2	2	no	—
56	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	complemented
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	8	2	no	—
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	—	8	4	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	4	1	no	—
78	C₄⋊C₄×C₂₃	—	4	1	no	—
79	C₄⋊C₄×C₂₃	—	8	2	no	—
80	C₂²⋊C₄×C₂₃	—	2	8	no	—
81	C₂²⋊C₄×C₂₃	—	2	4	no	—
82	C₂²⋊C₄×C₂₃	—	8	2	no	—
83	C₄²×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	2	2	no	—
85	C₄²×C₂₃	—	4	2	no	—
86	C₄²×C₂₃	C₂×C₈	1	2	yes	complemented
87	C₄²×C₂₃	C₂×D₄	1	2	yes	—
88	C₄²×C₂₃	#41	1	2	yes	—
89	#9099	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	4	2	no	—
93	C₂³×C₂₃	—	4	2	no	—
94	C₂³×C₂₃	C₂²⋊C₈	1	4	yes	—
95	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	4	no	—
97	Q₈×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	2	4	no	—
99	D₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	8	2	no	—
117	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#121	1	2	yes	—
120	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₂₃	#131	1	2	yes	—
124	C₈×C₂₃	—	4	4	no	—
125	C₈×C₂₃	—	8	2	no	—
126	#2972	C₂×C₂₃	1	1	yes	characteristic
127	#2326	C₂×C₂₃	1	1	yes	characteristic
128	#1479	C₂×C₂₃	1	2	yes	complemented
129	#1460	C₂×C₂₃	1	1	yes	characteristic
130	#1274	C₂×C₂₃	1	2	yes	complemented
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	4	no	—
134	C₂²×C₂₃	—	4	4	no	—
135	C₂²×C₂₃	—	4	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	8	2	no	—
138	C₂²×C₂₃	#386	1	1	yes	characteristic
139	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₂₃	#408	1	1	yes	characteristic
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	2	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	8	1	no	—
148	C₄×C₂₃	—	8	1	no	—
149	C₄×C₂₃	C₈×D₄	1	2	yes	—
150	C₄×C₂₃	#435	1	2	yes	—
151	C₂×#124	C₄×C₂₃	1	1	yes	characteristic
152	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₈×D₄	—	2	4	no	—
155	#431	C₂²×C₂₃	1	2	yes	—
156	#431	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂²⋊C₈	—	2	2	no	—
158	#391	C₄×C₂₃	1	1	yes	characteristic
159	#390	C₄×C₂₃	1	1	yes	characteristic
160	#388	—	2	2	no	—
161	#385	—	2	2	no	—
162	#384	C₄×C₂₃	1	1	yes	characteristic
163	#376	C₂²×C₂₃	1	1	yes	characteristic
164	Q₈⋊C₈	—	2	4	no	—
165	#323	—	2	2	no	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	1	no	—
168	C₂×C₂₃	—	4	4	no	—
169	C₂×C₂₃	#1869	1	1	yes	characteristic
170	C₂×C₂₃	#1909	1	1	yes	characteristic
171	C₂×C₂₃	#1911	1	1	yes	characteristic
172	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
173	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₂₃	1	2	yes	—
175	C₂²×C₈	—	2	4	no	—
176	#124	—	2	2	no	—
177	#124	C₈×C₂₃	1	4	yes	complemented
178	C₄×Q₈	—	2	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	D₄×C₂₃	1	4	yes	—
200	#95	—	2	2	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	C₂₃	#9099	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	C₂²⋊C₄×C₂₃	1	2	yes	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	complemented
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
212	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	8	2	no	—
220	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	8	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	8	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₂×C₈×C₂₃	1	2	yes	complemented
239	C₄²	C₂×D₄×C₂₃	1	2	yes	—
240	C₄²	#41×C₂₃	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂₃×C₂²⋊C₈	1	4	yes	—
246	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
269	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
270	C₂×C₄	C₂₃×#121	1	2	yes	—
271	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂₃×#131	1	2	yes	—
275	C₈	—	4	4	no	—
276	C₈	—	8	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₂₃×#386	1	1	yes	characteristic
285	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
286	C₂²	C₂₃×#408	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	1	no	—
294	C₄	—	8	1	no	—
295	C₄	C₈×D₄×C₂₃	1	2	yes	—
296	C₄	C₂₃×#435	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	4	4	no	—
300	C₂	C₂₃×#1869	1	1	yes	characteristic
301	C₂	C₂₃×#1909	1	1	yes	characteristic
302	C₂	C₂₃×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#386
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1479	C₂	semidirect
5. C₂₃×#1274	C₂	semidirect
26. C₂₃×#124	C₈	semidirect
56. C₂×Q₈×C₂₃	C₂×C₈	semidirect
86. C₄²×C₂₃	C₂×C₈	semidirect
89. #9099	C₂₃	direct (N × H)
128. #1479	C₂×C₂₃	semidirect
130. #1274	C₂×C₂₃	semidirect
177. #124	C₈×C₂₃	semidirect
203. C₂₃	#9099	direct (N × H)
208. C₂×Q₈	C₂×C₈×C₂₃	semidirect
238. C₄²	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2972	2
C₂₃×#2326	2
C₂₃×#1479	2
C₂₃×#1460	2
C₂₃×#1274	2
#9099	23

Subgroups — C23×#9099 #19156725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9099 #19156725