Subgroups — C₂₃×#2452 #15670071

Order 2944.

620 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#396	C₂	1	2	yes	maximal, complemented
5	C₂₃×#393	C₂	1	2	yes	maximal, complemented
6	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#385	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊Q₈	—	2	4	no	—
16	C₂₃×C₂²⋊Q₈	—	2	8	no	—
17	C₄×D₄×C₂₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#95	C₂²	1	1	yes	characteristic
35	C₂₃×#95	C₂²	1	1	yes	characteristic
36	C₂₃×#95	C₂²	1	2	yes	—
37	C₂₃×#95	C₂²	1	2	yes	—
38	C₂⁴×C₂₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₂₃	—	2	4	no	—
40	C₂×Q₈×C₂₃	—	2	4	no	—
41	C₂×Q₈×C₂₃	—	4	2	no	—
42	C₂×D₄×C₂₃	—	2	2	no	—
43	C₂×D₄×C₂₃	—	2	2	no	—
44	C₂×D₄×C₂₃	Q₈	1	4	yes	complemented
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	—	2	4	no	—
47	C₂²×C₄×C₂₃	—	4	2	no	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	4	no	—
60	C₄⋊C₄×C₂₃	—	2	4	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
65	C₄⋊C₄×C₂₃	D₄	1	4	yes	complemented
66	C₂²⋊C₄×C₂₃	—	2	4	no	—
67	C₂²⋊C₄×C₂₃	—	2	4	no	—
68	C₂²⋊C₄×C₂₃	—	2	4	no	—
69	C₂²⋊C₄×C₂₃	—	2	4	no	—
70	C₂²⋊C₄×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	2	2	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	2	2	no	—
78	C₂³×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	2	2	no	—
80	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
81	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
82	Q₈×C₂₃	—	4	8	no	—
83	D₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	4	8	no	—
107	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	#41	1	2	yes	—
114	C₂×C₄×C₂₃	#41	1	2	yes	—
115	#2452	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₂₃	—	2	4	no	—
117	C₂²×C₂₃	—	2	4	no	—
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
127	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	C₂²⋊Q₈	1	4	yes	complemented
140	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂×C₂²⋊Q₈	C₂×C₂₃	1	2	yes	complemented
142	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
143	#396	C₂×C₂₃	1	2	yes	complemented
144	#393	C₂×C₂₃	1	2	yes	complemented
145	#390	C₂×C₂₃	1	1	yes	characteristic
146	#390	C₂×C₂₃	1	1	yes	characteristic
147	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#385	C₂×C₂₃	1	2	yes	complemented
150	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
151	C₂×C₂₃	—	4	4	no	—
152	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₂₃	#532	1	1	yes	characteristic
154	C₂×C₂₃	#542	1	1	yes	characteristic
155	C₂×C₂₃	D₄²	1	1	yes	characteristic
156	C₂×C₂₃	#548	1	1	yes	characteristic
157	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
158	C₂×C₂₃	#554	1	1	yes	characteristic
159	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
160	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
161	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
162	C₄⋊Q₈	—	2	4	no	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₄×D₄	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
178	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
181	#95	C₂²×C₂₃	1	1	yes	characteristic
182	#95	C₂²×C₂₃	1	1	yes	characteristic
183	#95	C₂²×C₂₃	1	2	yes	—
184	#95	C₂²×C₂₃	1	2	yes	—
185	C₂₃	#2452	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₂³×C₂₃	1	2	yes	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	Q₈×C₂₃	1	4	yes	complemented
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	D₄×C₂₃	1	4	yes	—
213	C₄⋊C₄	D₄×C₂₃	1	4	yes	complemented
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	C₂×Q₈×C₂₃	1	4	yes	—
229	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	D₄	—	2	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
256	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
258	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
259	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
260	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
261	C₂×C₄	#41×C₂₃	1	2	yes	—
262	C₂×C₄	#41×C₂₃	1	2	yes	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
272	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
273	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
274	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
275	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	2	no	—
286	C₄	C₂₃×C₂²⋊Q₈	1	4	yes	complemented
287	C₂	—	4	4	no	—
288	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂	C₂₃×#532	1	1	yes	characteristic
290	C₂	C₂₃×#542	1	1	yes	characteristic
291	C₂	D₄²×C₂₃	1	1	yes	characteristic
292	C₂	C₂₃×#548	1	1	yes	characteristic
293	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
294	C₂	C₂₃×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
2. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
4. C₂₃×#396	C₂	semidirect
5. C₂₃×#393	C₂	semidirect
8. C₂₃×#389	C₂	semidirect
9. C₂₃×#389	C₂	semidirect
10. C₂₃×#385	C₂	semidirect
11. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
44. C₂×D₄×C₂₃	Q₈	semidirect
65. C₄⋊C₄×C₂₃	D₄	semidirect
115. #2452	C₂₃	direct (N × H)
139. C₄×C₂₃	C₂²⋊Q₈	semidirect
140. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
141. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
143. #396	C₂×C₂₃	semidirect
144. #393	C₂×C₂₃	semidirect
147. #389	C₂×C₂₃	semidirect
148. #389	C₂×C₂₃	semidirect
149. #385	C₂×C₂₃	semidirect
150. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
185. C₂₃	#2452	direct (N × H)
192. C₂×D₄	Q₈×C₂₃	semidirect
213. C₄⋊C₄	D₄×C₂₃	semidirect
286. C₄	C₂₃×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#396	2
C₂₃×#393	2
C₂₃×#390	2
C₂₃×#389	2
C₂₃×#385	2
C₄×C₄⋊C₄×C₂₃	2
#2452	23

Subgroups — C23×#2452 #15670071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2452 #15670071