Subgroups — C₂₃×#12634 #19160260

Order 5888.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2932	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2132	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2126	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1757	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1395	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₄	1	1	yes	characteristic, complemented
7	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
8	C₂₃×#532	C₂²	1	1	yes	characteristic
9	C₂₃×#435	—	2	2	no	—
10	C₂₃×#435	—	2	2	no	—
11	C₂₃×#423	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#413	C₂²	1	2	yes	—
13	C₂₃×#410	C₂²	1	2	yes	—
14	C₂₃×#408	C₄	1	1	yes	characteristic, complemented
15	C₂₃×#408	C₂²	1	1	yes	characteristic
16	C₂₃×#405	—	2	2	no	—
17	C₂×C₂₃×#97	C₄	1	1	yes	characteristic, complemented
18	C₂₃×#356	C₄	1	2	yes	complemented
19	C₂₃×#355	C₄	1	2	yes	complemented
20	C₂₃×#354	C₄	1	2	yes	—
21	C₂₃×#353	C₄	1	2	yes	—
22	C₂₃×#344	—	2	2	no	—
23	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
25	C₂₃×#131	—	4	2	no	—
26	C₂₃×#131	—	4	2	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
32	C₂²×C₈×C₂₃	—	2	1	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	D₄	1	2	yes	—
36	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
38	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
39	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
40	C₂₃×#121	—	2	2	no	—
41	C₄×D₄×C₂₃	—	2	2	no	—
42	C₂₃×#117	—	2	2	no	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×#101	C₂×C₄	1	4	yes	—
46	C₂₃×#100	—	2	2	no	—
47	C₂₃×#99	—	2	2	no	—
48	C₂₃×#99	C₂×C₄	1	4	yes	—
49	C₂₃×C₂²⋊C₈	—	2	2	no	—
50	C₂₃×C₂²⋊C₈	—	2	2	no	—
51	C₂₃×#97	—	2	1	no	—
52	C₂₃×#97	—	2	1	no	—
53	C₂₃×#97	—	4	2	no	—
54	C₄×C₈×C₂₃	—	4	2	no	—
55	#41×C₂₃	—	4	1	no	—
56	#41×C₂₃	—	4	1	no	—
57	#41×C₂₃	—	4	1	no	—
58	#41×C₂₃	—	4	1	no	—
59	C₂×Q₈×C₂₃	C₄²	1	1	yes	characteristic
60	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	C₄²	1	1	yes	characteristic
64	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×D₄×C₂₃	#41	1	2	yes	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	4	1	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	—	4	2	no	—
83	C₂×C₈×C₂₃	—	4	1	no	—
84	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₂²⋊C₄×C₂₃	—	2	2	no	—
90	C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂²⋊C₄×C₂₃	—	2	4	no	—
92	C₄²×C₂₃	—	2	1	no	—
93	C₄²×C₂₃	—	4	2	no	—
94	C₄²×C₂₃	C₄²	1	1	yes	characteristic
95	C₄²×C₂₃	C₄²	1	1	yes	characteristic
96	#12634	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	1	no	—
98	C₂³×C₂₃	#117	1	1	yes	characteristic
99	C₂³×C₂₃	C₄×D₄	1	2	yes	—
100	Q₈×C₂₃	—	4	1	no	—
101	Q₈×C₂₃	—	4	1	no	—
102	D₄×C₂₃	—	4	1	no	—
103	D₄×C₂₃	—	4	2	no	—
104	D₄×C₂₃	—	4	1	no	—
105	D₄×C₂₃	—	4	2	no	—
106	D₄×C₂₃	—	8	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	8	1	no	—
118	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
121	C₈×C₂₃	—	2	1	no	—
122	C₈×C₂₃	—	2	2	no	—
123	C₈×C₂₃	—	4	4	no	—
124	C₈×C₂₃	—	4	1	no	—
125	C₈×C₂₃	—	4	1	no	—
126	C₈×C₂₃	#99	1	2	yes	—
127	#2932	C₂×C₂₃	1	1	yes	characteristic
128	#2132	C₂×C₂₃	1	1	yes	characteristic
129	#2126	C₂×C₂₃	1	1	yes	characteristic
130	#1757	C₂×C₂₃	1	2	yes	complemented
131	#1395	C₂×C₂₃	1	2	yes	complemented
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	4	1	no	—
135	C₂²×C₂₃	—	4	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₂₃	—	2	2	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	4	1	no	—
142	C₄×C₂₃	—	4	1	no	—
143	C₄×C₂₃	—	4	1	no	—
144	C₄×C₂₃	—	8	4	no	—
145	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
146	C₄×C₂₃	#410	1	1	yes	characteristic
147	C₂×#131	C₄×C₂₃	1	1	yes	characteristic, complemented
148	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
149	#532	C₂²×C₂₃	1	1	yes	characteristic
150	#435	—	2	2	no	—
151	#435	—	2	2	no	—
152	#423	C₄×C₂₃	1	1	yes	characteristic, complemented
153	#413	C₂²×C₂₃	1	2	yes	—
154	#410	C₂²×C₂₃	1	2	yes	—
155	#408	C₄×C₂₃	1	1	yes	characteristic, complemented
156	#408	C₂²×C₂₃	1	1	yes	characteristic
157	#405	—	2	2	no	—
158	C₂×#97	C₄×C₂₃	1	1	yes	characteristic, complemented
159	#356	C₄×C₂₃	1	2	yes	complemented
160	#355	C₄×C₂₃	1	2	yes	complemented
161	#354	C₄×C₂₃	1	2	yes	—
162	#353	C₄×C₂₃	1	2	yes	—
163	#344	—	2	2	no	—
164	C₂×C₂₃	—	2	1	no	—
165	C₂×C₂₃	—	4	2	no	—
166	C₂×C₂₃	—	4	1	no	—
167	C₂×C₂₃	—	8	1	no	—
168	C₂×C₂₃	C₄×#99	1	1	yes	characteristic
169	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
171	#131	—	4	2	no	—
172	#131	—	4	2	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₂₃	1	2	yes	—
182	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
183	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
184	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
185	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
186	#121	—	2	2	no	—
187	C₄×D₄	—	2	2	no	—
188	#117	—	2	2	no	—
189	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	#101	C₂×C₄×C₂₃	1	4	yes	—
192	#100	—	2	2	no	—
193	#99	—	2	2	no	—
194	#99	C₂×C₄×C₂₃	1	4	yes	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	#97	—	2	1	no	—
198	#97	—	2	1	no	—
199	#97	—	4	2	no	—
200	C₄×C₈	—	4	2	no	—
201	C₂₃	#12634	1	1	yes	characteristic, direct product factor, Sylow
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	C₄²×C₂₃	1	1	yes	characteristic
207	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₄²×C₂₃	1	1	yes	characteristic
211	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂×D₄	#41×C₂₃	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
233	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₄²×C₂₃	1	1	yes	characteristic
242	C₄²	C₄²×C₂₃	1	1	yes	characteristic
243	C₂³	—	2	1	no	—
244	C₂³	C₂₃×#117	1	1	yes	characteristic
245	C₂³	C₄×D₄×C₂₃	1	2	yes	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	4	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
267	C₈	—	2	1	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	C₂₃×#99	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	8	4	no	—
286	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
287	C₄	C₂₃×#410	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	8	1	no	—
292	C₂	C₄×C₂₃×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×#41	C₂³×C₂₃
socle	C₂×C₂₃	C₄×#99
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1757	C₂	semidirect
5. C₂₃×#1395	C₂	semidirect
6. C₂×C₂₃×#131	C₄	semidirect
11. C₂₃×#423	C₄	semidirect
14. C₂₃×#408	C₄	semidirect
17. C₂×C₂₃×#97	C₄	semidirect
18. C₂₃×#356	C₄	semidirect
19. C₂₃×#355	C₄	semidirect
96. #12634	C₂₃	direct (N × H)
130. #1757	C₂×C₂₃	semidirect
131. #1395	C₂×C₂₃	semidirect
147. C₂×#131	C₄×C₂₃	semidirect
152. #423	C₄×C₂₃	semidirect
155. #408	C₄×C₂₃	semidirect
158. C₂×#97	C₄×C₂₃	semidirect
159. #356	C₄×C₂₃	semidirect
160. #355	C₄×C₂₃	semidirect
201. C₂₃	#12634	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×#2932	2
C₂₃×#2132	2
C₂₃×#2126	2
C₂₃×#1757	2
C₂₃×#1395	2
#12634	23

Subgroups — C23×#12634 #19160260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12634 #19160260