Subgroups — C₂₃×#12633 #19160259

Order 5888.

460 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×#2931	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×#436	—	2	2	no	—
10	C₈×D₄×C₂₃	—	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×C₄×C₈×C₂₃	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	1	no	—
26	C₂₃×#131	—	4	1	no	—
27	C₂₃×#131	—	4	1	no	—
28	C₂₃×#131	—	4	1	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	D₄	1	2	yes	—
38	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
40	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
41	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
42	C₂₃×#121	—	2	2	no	—
43	C₄×D₄×C₂₃	—	2	2	no	—
44	C₂₃×#117	—	2	2	no	—
45	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂₃×C₄⋊C₈	—	2	2	no	—
47	C₂₃×#101	C₂×C₄	1	4	yes	—
48	C₂₃×#100	—	2	2	no	—
49	C₂₃×#99	—	2	2	no	—
50	C₂₃×#99	C₂×C₄	1	4	yes	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×C₂²⋊C₈	—	2	2	no	—
53	C₂₃×#97	—	4	2	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	—	4	2	no	—
57	#41×C₂₃	—	4	1	no	—
58	#41×C₂₃	—	4	1	no	—
59	#41×C₂₃	—	4	1	no	—
60	#41×C₂₃	—	4	1	no	—
61	C₂×Q₈×C₂₃	C₄²	1	1	yes	characteristic
62	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	—	2	2	no	—
65	C₂×D₄×C₂₃	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₄×C₂₃	#41	1	2	yes	—
68	C₂²×C₄×C₂₃	—	2	1	no	—
69	C₂²×C₄×C₂₃	—	2	1	no	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	—	4	1	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	4	2	no	—
76	M₁₆×C₂₃	—	4	2	no	—
77	M₁₆×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	4	2	no	—
85	C₂×C₈×C₂₃	—	4	1	no	—
86	C₂×C₈×C₂₃	—	4	1	no	—
87	C₂×C₈×C₂₃	—	4	1	no	—
88	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	4	2	no	—
93	C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂²⋊C₄×C₂₃	—	2	4	no	—
96	C₄²×C₂₃	—	2	1	no	—
97	C₄²×C₂₃	—	4	2	no	—
98	C₄²×C₂₃	C₄²	1	1	yes	characteristic
99	C₄²×C₂₃	C₄²	1	1	yes	characteristic
100	#12633	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	2	1	no	—
102	C₂³×C₂₃	#117	1	1	yes	characteristic
103	C₂³×C₂₃	C₄×D₄	1	2	yes	—
104	Q₈×C₂₃	—	4	1	no	—
105	Q₈×C₂₃	—	4	1	no	—
106	D₄×C₂₃	—	4	1	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	1	no	—
109	D₄×C₂₃	—	4	4	no	—
110	D₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	2	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	—	4	1	no	—
120	C₂×C₄×C₂₃	—	4	1	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
125	C₈×C₂₃	—	2	2	no	—
126	C₈×C₂₃	—	2	1	no	—
127	C₈×C₂₃	—	4	4	no	—
128	C₈×C₂₃	—	4	1	no	—
129	C₈×C₂₃	—	4	1	no	—
130	C₈×C₂₃	#99	1	1	yes	characteristic
131	C₈×C₂₃	#99	1	1	yes	characteristic
132	#2931	C₂×C₂₃	1	1	yes	characteristic
133	#2132	C₂×C₂₃	1	1	yes	characteristic
134	#2126	C₂×C₂₃	1	1	yes	characteristic
135	#1757	C₂×C₂₃	1	2	yes	complemented
136	#1395	C₂×C₂₃	1	2	yes	complemented
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	4	1	no	—
140	C₂²×C₂₃	—	4	2	no	—
141	C₂²×C₂₃	—	4	2	no	—
142	C₂²×C₂₃	—	4	1	no	—
143	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	—	8	4	no	—
150	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
151	C₄×C₂₃	#410	1	1	yes	characteristic
152	C₂×#131	C₄×C₂₃	1	1	yes	characteristic, complemented
153	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
154	#532	C₂²×C₂₃	1	1	yes	characteristic
155	#436	—	2	2	no	—
156	C₈×D₄	—	2	2	no	—
157	#423	C₄×C₂₃	1	1	yes	characteristic, complemented
158	#413	C₂²×C₂₃	1	2	yes	—
159	#410	C₂²×C₂₃	1	2	yes	—
160	#408	C₄×C₂₃	1	1	yes	characteristic, complemented
161	#408	C₂²×C₂₃	1	1	yes	characteristic
162	#405	—	2	2	no	—
163	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic, complemented
164	#356	C₄×C₂₃	1	2	yes	complemented
165	#355	C₄×C₂₃	1	2	yes	complemented
166	#354	C₄×C₂₃	1	2	yes	—
167	#353	C₄×C₂₃	1	2	yes	—
168	#344	—	2	2	no	—
169	C₂×C₂₃	—	2	1	no	—
170	C₂×C₂₃	—	4	2	no	—
171	C₂×C₂₃	—	4	1	no	—
172	C₂×C₂₃	—	8	1	no	—
173	C₂×C₂₃	C₄×#99	1	1	yes	characteristic
174	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
176	#131	—	4	1	no	—
177	#131	—	4	1	no	—
178	#131	—	4	1	no	—
179	#131	—	4	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₂₃	1	2	yes	—
189	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
190	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
191	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
192	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
193	#121	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	#117	—	2	2	no	—
196	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	#101	C₂×C₄×C₂₃	1	4	yes	—
199	#100	—	2	2	no	—
200	#99	—	2	2	no	—
201	#99	C₂×C₄×C₂₃	1	4	yes	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	#97	—	4	2	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	4	2	no	—
208	C₂₃	#12633	1	1	yes	characteristic, direct product factor, Sylow
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	C₂×Q₈	C₄²×C₂₃	1	1	yes	characteristic
214	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₄²×C₂₃	1	1	yes	characteristic
218	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
219	C₂×D₄	#41×C₂₃	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
242	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₄²×C₂₃	1	1	yes	characteristic
251	C₄²	C₄²×C₂₃	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	C₂₃×#117	1	1	yes	characteristic
254	C₂³	C₄×D₄×C₂₃	1	2	yes	—
255	Q₈	—	4	1	no	—
256	Q₈	—	4	1	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
276	C₈	—	2	1	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	4	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	C₂₃×#99	1	1	yes	characteristic
282	C₈	C₂₃×#99	1	1	yes	characteristic
283	C₂²	—	2	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	8	4	no	—
296	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
297	C₄	C₂₃×#410	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	4	2	no	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₄×C₂₃×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#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 ◁ C₈×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×C₄×C₈×C₂₃	C₄	semidirect
18. C₂₃×#356	C₄	semidirect
19. C₂₃×#355	C₄	semidirect
100. #12633	C₂₃	direct (N × H)
135. #1757	C₂×C₂₃	semidirect
136. #1395	C₂×C₂₃	semidirect
152. C₂×#131	C₄×C₂₃	semidirect
157. #423	C₄×C₂₃	semidirect
160. #408	C₄×C₂₃	semidirect
163. C₂×C₄×C₈	C₄×C₂₃	semidirect
164. #356	C₄×C₂₃	semidirect
165. #355	C₄×C₂₃	semidirect
208. C₂₃	#12633	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×#2931	2
C₂₃×#2132	2
C₂₃×#2126	2
C₂₃×#1757	2
C₂₃×#1395	2
#12633	23

Subgroups — C23×#12633 #19160259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12633 #19160259