Subgroups — C₂₃×#33457 #19181083

Order 5888.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2948	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2175	C₂	1	4	yes	maximal, complemented
5	C₄×C₂₃×SD₃₂	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2140	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#519	—	2	1	no	—
13	C₂₃×#518	—	2	1	no	—
14	C₂×C₂₃×SD₃₂	C₄	1	8	yes	complemented
15	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#442	—	2	2	no	—
20	C₂₃×#441	—	2	2	no	—
21	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
22	C₄×D₈×C₂₃	C₂²	1	4	yes	—
23	C₂₃×#427	C₂²	1	1	yes	characteristic
24	C₂₃×#427	C₂²	1	1	yes	characteristic
25	C₂₃×#419	—	2	1	no	—
26	C₂₃×#418	—	2	1	no	—
27	C₂₃×#405	C₂²	1	1	yes	characteristic
28	C₂₃×#367	C₂²	1	2	yes	complemented
29	C₂₃×#366	C₂²	1	2	yes	complemented
30	C₂₃×#358	C₂²	1	4	yes	complemented
31	C₂₃×#357	C₂²	1	4	yes	—
32	C₂₃×#346	C₂²	1	2	yes	complemented
33	C₄×C₁₆×C₂₃	C₂²	1	2	yes	complemented
34	C₂²×Q₈×C₂₃	—	2	1	no	—
35	C₂²×D₄×C₂₃	—	2	1	no	—
36	C₂×Q₁₆×C₂₃	—	2	1	no	—
37	C₂×Q₁₆×C₂₃	—	2	2	no	—
38	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
39	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
40	C₂×D₈×C₂₃	—	2	1	no	—
41	C₂×D₈×C₂₃	—	2	2	no	—
42	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×D₈×C₂₃	C₂³	1	2	yes	—
44	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₂₃	—	2	4	no	—
47	C₄×Q₈×C₂₃	—	4	2	no	—
48	C₄×D₄×C₂₃	—	2	4	no	—
49	C₄×D₄×C₂₃	—	4	2	no	—
50	C₂₃×#117	—	2	1	no	—
51	C₂₃×#117	—	2	1	no	—
52	C₂₃×#117	—	4	2	no	—
53	C₂₃×#117	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
55	C₂₃×SD₃₂	C₂×C₄	1	8	yes	complemented
56	C₂₃×SD₃₂	C₂×C₄	1	8	yes	complemented
57	C₂₃×M₃₂	—	2	2	no	—
58	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
60	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×#107	C₂³	1	2	yes	—
63	C₂₃×#107	C₂³	1	2	yes	—
64	C₂₃×#106	—	2	1	no	—
65	C₂₃×#106	—	2	1	no	—
66	C₂₃×#103	—	2	4	no	—
67	C₂₃×#102	—	2	4	no	—
68	C₂₃×#97	D₄	1	2	yes	—
69	C₄×C₈×C₂₃	C₂³	1	2	yes	—
70	C₂⁴×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	2	4	no	—
72	C₂×Q₈×C₂₃	—	2	2	no	—
73	C₂×Q₈×C₂₃	—	4	1	no	—
74	C₂×Q₈×C₂₃	—	4	2	no	—
75	C₂×D₄×C₂₃	—	2	4	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	4	2	no	—
79	C₂²×C₄×C₂₃	—	4	1	no	—
80	C₂²×C₄×C₂₃	—	4	4	no	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₂₃	—	2	4	no	—
83	Q₁₆×C₂₃	—	2	2	no	—
84	Q₁₆×C₂₃	C₂²×C₄	1	4	yes	—
85	D₈×C₂₃	—	2	4	no	—
86	D₈×C₂₃	—	2	2	no	—
87	D₈×C₂₃	C₂²×C₄	1	4	yes	—
88	M₁₆×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	#41	1	2	yes	—
93	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
98	C₂²⋊C₄×C₂₃	—	4	1	no	—
99	C₂²⋊C₄×C₂₃	—	4	1	no	—
100	C₂²⋊C₄×C₂₃	—	4	4	no	—
101	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
102	C₄²×C₂₃	—	4	1	no	—
103	C₄²×C₂₃	—	4	1	no	—
104	C₄²×C₂₃	—	4	4	no	—
105	C₄²×C₂₃	C₂×D₄	1	2	yes	—
106	C₁₆×C₂₃	—	2	2	no	—
107	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
108	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
109	#33457	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₂₃	—	4	2	no	—
111	C₂³×C₂₃	—	4	4	no	—
112	C₂³×C₂₃	—	16	2	no	—
113	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₂₃	—	2	4	no	—
115	Q₈×C₂₃	—	4	2	no	—
116	Q₈×C₂₃	—	4	4	no	—
117	D₄×C₂₃	—	2	4	no	—
118	D₄×C₂₃	—	4	2	no	—
119	D₄×C₂₃	—	4	4	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	16	4	no	—
125	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
127	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₂₃	#135	1	2	yes	—
129	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₂₃	—	2	2	no	—
131	C₈×C₂₃	C₄×D₄	1	2	yes	—
132	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
134	C₂²×SD₃₂	C₂×C₂₃	1	1	yes	characteristic
135	#2950	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#2948	C₂×C₂₃	1	1	yes	characteristic
137	#2175	C₂×C₂₃	1	4	yes	complemented
138	C₄×SD₃₂	C₂×C₂₃	1	4	yes	complemented
139	#2158	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#2141	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#2140	C₂×C₂₃	1	1	yes	characteristic
142	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	2	2	no	—
144	C₂²×C₂₃	—	4	4	no	—
145	C₂²×C₂₃	—	8	4	no	—
146	C₂²×C₂₃	—	16	4	no	—
147	C₂²×C₂₃	C₄×D₈	1	2	yes	—
148	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₂₃	—	2	4	no	—
150	C₄×C₂₃	—	4	4	no	—
151	C₄×C₂₃	—	8	2	no	—
152	C₄×C₂₃	—	8	2	no	—
153	C₄×C₂₃	C₄×D₈	1	2	yes	—
154	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
156	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
157	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
158	#519	—	2	1	no	—
159	#518	—	2	1	no	—
160	C₂×SD₃₂	C₄×C₂₃	1	8	yes	complemented
161	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
162	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
163	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
164	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
165	#442	—	2	2	no	—
166	#441	—	2	2	no	—
167	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
168	C₄×D₈	C₂²×C₂₃	1	4	yes	—
169	#427	C₂²×C₂₃	1	1	yes	characteristic
170	#427	C₂²×C₂₃	1	1	yes	characteristic
171	#419	—	2	1	no	—
172	#418	—	2	1	no	—
173	#405	C₂²×C₂₃	1	1	yes	characteristic
174	#367	C₂²×C₂₃	1	2	yes	complemented
175	#366	C₂²×C₂₃	1	2	yes	complemented
176	#358	C₂²×C₂₃	1	4	yes	complemented
177	#357	C₂²×C₂₃	1	4	yes	—
178	#346	C₂²×C₂₃	1	2	yes	complemented
179	C₄×C₁₆	C₂²×C₂₃	1	2	yes	complemented
180	C₂×C₂₃	—	2	2	no	—
181	C₂×C₂₃	—	8	4	no	—
182	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₂₃	#3416	1	2	yes	—
184	C₂²×Q₈	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
189	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
193	C₂×D₈	C₂³×C₂₃	1	2	yes	—
194	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	4	2	no	—
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	4	2	no	—
200	#117	—	2	1	no	—
201	#117	—	2	1	no	—
202	#117	—	4	2	no	—
203	#117	D₄×C₂₃	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	4	2	no	—
205	SD₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
206	SD₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
207	M₃₂	—	2	2	no	—
208	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
209	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
210	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
211	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
212	#107	C₂³×C₂₃	1	2	yes	—
213	#107	C₂³×C₂₃	1	2	yes	—
214	#106	—	2	1	no	—
215	#106	—	2	1	no	—
216	#103	—	2	4	no	—
217	#102	—	2	4	no	—
218	#97	D₄×C₂₃	1	2	yes	—
219	C₄×C₈	C₂³×C₂₃	1	2	yes	—
220	C₂₃	#33457	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	C₂²×C₄×C₂₃	1	4	yes	—
236	D₈	—	2	4	no	—
237	D₈	—	2	2	no	—
238	D₈	C₂²×C₄×C₂₃	1	4	yes	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₈	#41×C₂₃	1	2	yes	—
244	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	4	no	—
256	C₄²	C₂×D₄×C₂₃	1	2	yes	—
257	C₁₆	—	2	2	no	—
258	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
259	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	16	4	no	—
275	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
276	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
277	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂₃×#135	1	2	yes	—
279	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×D₄×C₂₃	1	2	yes	—
282	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
283	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	C₄×D₈×C₂₃	1	2	yes	—
289	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₄×D₈×C₂₃	1	2	yes	—
295	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
296	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic
300	C₂	C₂₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2950	C₂	semidirect
4. C₂₃×#2175	C₂	semidirect
5. C₄×C₂₃×SD₃₂	C₂	semidirect
6. C₂₃×#2158	C₂	semidirect
7. C₂₃×#2141	C₂	semidirect
9. C₂₃×#2108	C₂	semidirect
14. C₂×C₂₃×SD₃₂	C₄	semidirect
21. C₄×Q₁₆×C₂₃	C₂²	semidirect
28. C₂₃×#367	C₂²	semidirect
29. C₂₃×#366	C₂²	semidirect
30. C₂₃×#358	C₂²	semidirect
32. C₂₃×#346	C₂²	semidirect
33. C₄×C₁₆×C₂₃	C₂²	semidirect
38. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
55. C₂₃×SD₃₂	C₂×C₄	semidirect
56. C₂₃×SD₃₂	C₂×C₄	semidirect
58. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
109. #33457	C₂₃	direct (N × H)
135. #2950	C₂×C₂₃	semidirect
137. #2175	C₂×C₂₃	semidirect
138. C₄×SD₃₂	C₂×C₂₃	semidirect
139. #2158	C₂×C₂₃	semidirect
140. #2141	C₂×C₂₃	semidirect
142. #2108	C₂×C₂₃	semidirect
160. C₂×SD₃₂	C₄×C₂₃	semidirect
167. C₄×Q₁₆	C₂²×C₂₃	semidirect
174. #367	C₂²×C₂₃	semidirect
175. #366	C₂²×C₂₃	semidirect
176. #358	C₂²×C₂₃	semidirect
178. #346	C₂²×C₂₃	semidirect
179. C₄×C₁₆	C₂²×C₂₃	semidirect
188. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
205. SD₃₂	C₂×C₄×C₂₃	semidirect
206. SD₃₂	C₂×C₄×C₂₃	semidirect
208. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
220. C₂₃	#33457	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×SD₃₂	2
C₂₃×#2950	2
C₂₃×#2948	2
C₂₃×#2175	2
C₄×C₂₃×SD₃₂	2
C₂₃×#2158	2
C₂₃×#2141	2
C₂₃×#2140	2
C₂₃×#2108	2
#33457	23

Subgroups — C23×#33457 #19181083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33457 #19181083