Subgroups — C₂₃×#33600 #19181226

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3050	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2217	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2216	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2215	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#135	C₂²	1	2	yes	—
10	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#525	—	2	2	no	—
12	C₂₃×#508	—	2	4	no	—
13	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₂₃	—	2	4	no	—
17	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#467	C₂²	1	4	yes	—
22	C₂₃×#466	C₂²	1	4	yes	—
23	C₂₃×#465	—	2	4	no	—
24	C₂₃×#427	C₂²	1	2	yes	—
25	C₂₃×#416	—	2	2	no	—
26	C₂₃×#367	C₂²	1	2	yes	complemented
27	C₂₃×#366	C₂²	1	2	yes	complemented
28	C₂₃×#358	C₂²	1	4	yes	complemented
29	C₂₃×#357	C₂²	1	4	yes	complemented
30	C₂×#41×C₂₃	—	2	2	no	—
31	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂₃×#135	—	2	4	no	—
33	C₂₃×#135	—	2	4	no	—
34	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₂₃	—	2	2	no	—
36	C₂×D₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	—	2	2	no	—
39	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	D₄	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×#123	—	4	2	no	—
44	C₂₃×C₂²⋊Q₈	—	2	4	no	—
45	C₂₃×#121	—	2	4	no	—
46	C₂₃×#120	—	4	2	no	—
47	C₄×D₄×C₂₃	—	4	4	no	—
48	C₂₃×#117	—	2	2	no	—
49	C₂₃×Q₃₂	—	2	2	no	—
50	C₂₃×SD₃₂	—	2	4	no	—
51	C₂₃×D₁₆	—	2	2	no	—
52	C₂×C₁₆×C₂₃	—	2	2	no	—
53	C₂×C₁₆×C₂₃	—	2	4	no	—
54	C₂×C₁₆×C₂₃	—	2	4	no	—
55	C₂×C₁₆×C₂₃	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
57	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#107	C₂³	1	4	yes	—
60	C₂₃×#106	—	2	2	no	—
61	C₂₃×#103	—	2	4	no	—
62	C₂₃×#102	—	2	4	no	—
63	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
64	#41×C₂₃	—	4	4	no	—
65	#41×C₂₃	—	4	4	no	—
66	C₂×Q₈×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	—	2	2	no	—
68	C₂×D₄×C₂₃	—	4	4	no	—
69	C₂×D₄×C₂₃	—	4	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	D₈	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₂₃	—	2	2	no	—
78	Q₁₆×C₂₃	—	2	2	no	—
79	SD₁₆×C₂₃	—	4	4	no	—
80	D₈×C₂₃	—	2	2	no	—
81	D₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₂₃	#41	1	2	yes	—
90	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	4	4	no	—
93	C₂²⋊C₄×C₂₃	—	4	2	no	—
94	C₂²⋊C₄×C₂₃	—	4	4	no	—
95	C₂²⋊C₄×C₂₃	—	4	4	no	—
96	C₄²×C₂₃	—	4	2	no	—
97	C₁₆×C₂₃	—	2	2	no	—
98	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
99	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
100	#33600	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	2	2	no	—
102	C₂³×C₂₃	—	2	4	no	—
103	C₂³×C₂₃	—	4	2	no	—
104	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
105	C₂³×C₂₃	C₂×D₈	1	2	yes	—
106	Q₈×C₂₃	—	4	2	no	—
107	Q₈×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	8	4	no	—
111	D₄×C₂₃	—	8	8	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	8	4	no	—
119	C₂×C₄×C₂₃	#121	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#135	1	2	yes	—
123	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₂₃	—	2	2	no	—
125	C₈×C₂₃	#121	1	2	yes	—
126	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₂×#508	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#3050	C₂×C₂₃	1	2	yes	complemented
131	#2217	C₂×C₂₃	1	4	yes	complemented
132	#2216	C₂×C₂₃	1	2	yes	complemented
133	#2215	C₂×C₂₃	1	2	yes	complemented
134	#2158	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#2138	C₂×C₂₃	1	2	yes	complemented
136	C₂²×C₂₃	—	2	8	no	—
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	—	2	2	no	—
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	8	4	no	—
141	C₂²×C₂₃	—	8	2	no	—
142	C₂²×C₂₃	#466	1	2	yes	—
143	C₂²×C₂₃	#508	1	4	yes	complemented
144	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	8	4	no	—
147	C₄×C₂₃	—	8	2	no	—
148	C₄×C₂₃	#466	1	2	yes	—
149	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
151	C₂×#135	C₂²×C₂₃	1	2	yes	—
152	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	#525	—	2	2	no	—
154	#508	—	2	4	no	—
155	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
156	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	complemented
157	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
158	C₂²×C₁₆	—	2	4	no	—
159	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
161	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
162	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
163	#467	C₂²×C₂₃	1	4	yes	—
164	#466	C₂²×C₂₃	1	4	yes	—
165	#465	—	2	4	no	—
166	#427	C₂²×C₂₃	1	2	yes	—
167	#416	—	2	2	no	—
168	#367	C₂²×C₂₃	1	2	yes	complemented
169	#366	C₂²×C₂₃	1	2	yes	complemented
170	#358	C₂²×C₂₃	1	4	yes	complemented
171	#357	C₂²×C₂₃	1	4	yes	complemented
172	C₂×C₂₃	—	2	2	no	—
173	C₂×C₂₃	—	2	4	no	—
174	C₂×C₂₃	—	16	2	no	—
175	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
176	C₂×C₂₃	C₂×#508	1	2	yes	—
177	C₂×#41	—	2	2	no	—
178	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
179	#135	—	2	4	no	—
180	#135	—	2	4	no	—
181	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×D₈	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₂₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
190	#123	—	4	2	no	—
191	C₂²⋊Q₈	—	2	4	no	—
192	#121	—	2	4	no	—
193	#120	—	4	2	no	—
194	C₄×D₄	—	4	4	no	—
195	#117	—	2	2	no	—
196	Q₃₂	—	2	2	no	—
197	SD₃₂	—	2	4	no	—
198	D₁₆	—	2	2	no	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	4	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	D₄×C₂₃	1	4	yes	complemented
203	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
204	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
205	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
206	#107	C₂³×C₂₃	1	4	yes	—
207	#106	—	2	2	no	—
208	#103	—	2	4	no	—
209	#102	—	2	4	no	—
210	C₂₃	#33600	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	D₈×C₂₃	1	2	yes	—
222	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
237	C₂×C₈	#41×C₂₃	1	2	yes	—
238	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	4	2	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
247	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
252	C₂³	C₂×D₈×C₂₃	1	2	yes	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	C₂₃×#121	1	2	yes	—
267	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂₃×#135	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	C₂₃×#121	1	2	yes	—
273	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
274	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	C₂₃×#466	1	2	yes	—
282	C₂²	C₂₃×#508	1	4	yes	complemented
283	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₂₃×#466	1	2	yes	—
288	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
289	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	4	no	—
292	C₂	—	16	2	no	—
293	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
294	C₂	C₂×C₂₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#508	C₂	semidirect
2. C₂³×C₁₆×C₂₃	C₂	semidirect
3. C₂₃×#3050	C₂	semidirect
4. C₂₃×#2217	C₂	semidirect
5. C₂₃×#2216	C₂	semidirect
6. C₂₃×#2215	C₂	semidirect
7. C₂₃×#2158	C₂	semidirect
8. C₂₃×#2138	C₂	semidirect
13. C₂×C₂₃×Q₃₂	C₂²	semidirect
14. C₂×C₂₃×SD₃₂	C₂²	semidirect
15. C₂×C₂₃×D₁₆	C₂²	semidirect
26. C₂₃×#367	C₂²	semidirect
27. C₂₃×#366	C₂²	semidirect
28. C₂₃×#358	C₂²	semidirect
29. C₂₃×#357	C₂²	semidirect
55. C₂×C₁₆×C₂₃	D₄	semidirect
100. #33600	C₂₃	direct (N × H)
128. C₂×#508	C₂×C₂₃	semidirect
129. C₂³×C₁₆	C₂×C₂₃	semidirect
130. #3050	C₂×C₂₃	semidirect
131. #2217	C₂×C₂₃	semidirect
132. #2216	C₂×C₂₃	semidirect
133. #2215	C₂×C₂₃	semidirect
134. #2158	C₂×C₂₃	semidirect
135. #2138	C₂×C₂₃	semidirect
143. C₂²×C₂₃	#508	semidirect
155. C₂×Q₃₂	C₂²×C₂₃	semidirect
156. C₂×SD₃₂	C₂²×C₂₃	semidirect
157. C₂×D₁₆	C₂²×C₂₃	semidirect
168. #367	C₂²×C₂₃	semidirect
169. #366	C₂²×C₂₃	semidirect
170. #358	C₂²×C₂₃	semidirect
171. #357	C₂²×C₂₃	semidirect
202. C₂×C₁₆	D₄×C₂₃	semidirect
210. C₂₃	#33600	direct (N × H)
282. C₂²	C₂₃×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#508	2
C₂³×C₁₆×C₂₃	2
C₂₃×#3050	2
C₂₃×#2217	2
C₂₃×#2216	2
C₂₃×#2215	2
C₂₃×#2158	2
C₂₃×#2138	2
#33600	23

Subgroups — C23×#33600 #19181226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33600 #19181226