Subgroups — #15682106

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625078	C₂	1	2	yes	maximal, complemented
2	#13092964	C₂	1	1	yes	maximal, characteristic
3	#13092496	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092493	C₂	1	2	yes	maximal, complemented
5	D₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092482	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₂₃)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#13092439	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092382	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	2	yes	—
13	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	#10625111	C₄	1	1	yes	characteristic
16	#10625109	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
18	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
19	#10625078	—	2	8	no	—
20	C₂×#67262	C₂²	1	2	yes	—
21	C₂×#67262	C₂²	1	2	yes	—
22	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₂₃	C₂²	1	2	yes	—
24	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
25	#10625066	C₄	1	1	yes	characteristic
26	#10625059	C₄	1	1	yes	characteristic
27	#10625056	C₄	1	2	yes	—
28	#10625055	C₄	1	2	yes	—
29	C₄×D₉₂	C₂²	1	1	yes	characteristic
30	C₄²×D₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
36	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
37	Dic₂₃⋊C₈	C₂²	1	2	yes	—
38	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
39	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
42	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
47	#67288	—	2	4	no	—
48	#67288	—	2	4	no	—
49	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
50	C₂×D₉₂	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	C₂³	1	2	yes	—
54	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
56	M₁₆×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
62	C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
64	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
65	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
67	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₂₃⋊C₈	—	2	4	no	—
72	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	#67262	—	2	4	no	—
76	#67262	—	2	4	no	—
77	C₈×D₂₃	—	2	2	no	—
78	C₈×D₂₃	—	2	4	no	—
79	C₈×D₂₃	D₄	1	4	yes	complemented
80	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₂₃	—	2	4	no	—
91	C₂²⋊D₂₃	—	2	4	no	—
92	C₂²⋊D₂₃	—	2	4	no	—
93	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
97	D₉₂	—	2	4	no	—
98	D₉₂	—	2	2	no	—
99	C₄×D₂₃	—	2	2	no	—
100	C₄×D₂₃	—	2	2	no	—
101	C₄×D₂₃	—	2	2	no	—
102	C₄×D₂₃	—	2	2	no	—
103	C₄×D₂₃	C₂×D₄	1	2	yes	—
104	C₄×D₂₃	#41	1	2	yes	—
105	Dic₄₆	—	2	4	no	—
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	—	2	1	no	—
108	C₈×C₂₃	C₂×D₄	1	2	yes	—
109	C₂₃⋊C₈	—	2	2	no	—
110	C₂₃⋊C₈	—	2	1	no	—
111	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
112	#2931	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×D₂₃	—	2	1	no	—
118	C₂×D₂₃	—	2	1	no	—
119	C₂×D₂₃	—	2	2	no	—
120	C₂×D₂₃	—	2	2	no	—
121	C₂×D₂₃	—	2	2	no	—
122	C₂×D₂₃	C₄×D₄	1	2	yes	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	#131	1	2	yes	complemented
126	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	Dic₂₃	—	2	2	no	—
129	Dic₂₃	—	2	1	no	—
130	Dic₂₃	C₄×D₄	1	2	yes	—
131	Dic₂₃	#131	1	2	yes	complemented
132	C₂×#131	—	23	2	no	—
133	#532	—	23	1	no	—
134	#436	—	23	2	no	—
135	#436	—	23	1	no	—
136	#436	D₂₃	1	1	yes	characteristic, complemented
137	C₈×D₄	—	23	1	no	—
138	C₈×D₄	—	23	1	no	—
139	C₈×D₄	—	23	2	no	—
140	#423	—	23	1	no	—
141	#408	—	23	2	no	—
142	C₂×C₄×C₈	—	23	1	no	—
143	C₂×C₂₃	—	4	2	no	—
144	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
146	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
147	D₂₃	—	2	2	no	—
148	D₂₃	—	4	2	no	—
149	C₂×#41	—	23	2	no	—
150	#131	—	46	8	no	—
151	C₂×M₁₆	—	23	2	no	—
152	C₂×M₁₆	—	23	2	no	—
153	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
154	C₂²×C₈	—	23	1	no	—
155	C₂²×C₈	—	23	2	no	—
156	C₂²×C₈	—	23	2	no	—
157	C₂²×C₈	—	23	1	no	—
158	C₄⋊Q₈	—	23	1	no	—
159	#127	—	23	1	no	—
160	#124	—	23	2	no	—
161	#121	—	23	1	no	—
162	#121	—	23	1	no	—
163	#121	—	23	2	no	—
164	C₄×D₄	—	23	2	no	—
165	C₄×D₄	—	23	1	no	—
166	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
167	C₂×C₄²	—	23	1	no	—
168	C₄⋊C₈	—	23	2	no	—
169	C₄⋊C₈	—	23	1	no	—
170	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
171	C₂²⋊C₈	—	23	2	no	—
172	C₂²⋊C₈	—	23	2	no	—
173	C₂²⋊C₈	—	23	2	no	—
174	C₂²⋊C₈	C₂×D₂₃	1	2	yes	—
175	C₄×C₈	—	23	1	no	—
176	C₄×C₈	—	23	1	no	—
177	C₄×C₈	—	23	1	no	—
178	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
179	C₂₃	#2931	1	1	yes	characteristic, complemented, Sylow
180	#41	—	46	4	no	—
181	#41	—	46	4	no	—
182	C₂×Q₈	—	23	2	no	—
183	C₂×D₄	—	23	2	no	—
184	C₂×D₄	—	23	2	no	—
185	C₂×D₄	—	23	2	no	—
186	C₂×D₄	—	23	1	no	—
187	C₂×D₄	—	23	2	no	—
188	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
189	C₂²×C₄	—	23	1	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	23	2	no	—
193	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	46	4	no	—
196	M₁₆	—	46	4	no	—
197	C₂×C₈	—	23	1	no	—
198	C₂×C₈	—	23	4	no	—
199	C₂×C₈	—	23	1	no	—
200	C₂×C₈	—	23	2	no	—
201	C₂×C₈	—	46	4	no	—
202	C₂×C₈	—	46	2	no	—
203	C₂×C₈	—	46	4	no	—
204	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₂₃	1	2	yes	—
206	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
207	C₄⋊C₄	—	23	2	no	—
208	C₄⋊C₄	—	23	1	no	—
209	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
210	C₂²⋊C₄	—	23	2	no	—
211	C₂²⋊C₄	—	23	2	no	—
212	C₂²⋊C₄	—	23	2	no	—
213	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
214	C₄²	—	23	1	no	—
215	C₄²	—	23	1	no	—
216	C₄²	—	23	1	no	—
217	C₄²	C₂²×D₂₃	1	1	yes	characteristic
218	C₂³	—	23	1	no	—
219	C₂³	—	23	2	no	—
220	C₂³	C₂×C₄×D₂₃	1	2	yes	—
221	Q₈	—	46	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	46	4	no	—
224	D₄	—	46	4	no	—
225	D₄	—	46	4	no	—
226	D₄	—	46	4	no	—
227	D₄	—	46	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	23	1	no	—
231	C₂×C₄	—	23	1	no	—
232	C₂×C₄	—	23	2	no	—
233	C₂×C₄	—	23	2	no	—
234	C₂×C₄	—	23	1	no	—
235	C₂×C₄	—	23	2	no	—
236	C₂×C₄	—	46	2	no	—
237	C₂×C₄	—	46	2	no	—
238	C₂×C₄	—	46	2	no	—
239	C₂×C₄	—	46	2	no	—
240	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	23	2	no	—
247	C₈	—	46	2	no	—
248	C₈	—	46	1	no	—
249	C₈	D₄×D₂₃	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	23	2	no	—
254	C₂²	—	46	1	no	—
255	C₂²	—	46	1	no	—
256	C₂²	—	46	2	no	—
257	C₂²	—	46	2	no	—
258	C₂²	—	46	2	no	—
259	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	23	2	no	—
263	C₄	—	23	2	no	—
264	C₄	—	46	2	no	—
265	C₄	—	46	1	no	—
266	C₄	#10625078	1	2	yes	complemented
267	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
268	C₄	#41×D₂₃	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	46	2	no	—
271	C₂	—	92	2	no	—
272	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
273	C₂	#131×D₂₃	1	1	yes	characteristic
274	C₂	C₂×#10625078	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#436	C₂
prime radical	#13092964	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#436	D₂₃
hypocenter	C₂₃	#2931
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#10625078	C₂	semidirect
3. #13092496	C₂	semidirect
4. #13092493	C₂	semidirect
5. D₄×C₂₃⋊C₈	C₂	semidirect
6. #13092482	C₂	semidirect
7. (C₂²⋊D₂₃)⋊C₈	C₂	semidirect
8. #13092439	C₂	semidirect
9. #13092382	C₂	semidirect
10. C₄×C₈×D₂₃	C₂	semidirect
11. C₂₃×#436	C₂	semidirect
79. C₈×D₂₃	D₄	semidirect
125. C₄×C₂₃	#131	semidirect
131. Dic₂₃	#131	semidirect
136. #436	D₂₃	semidirect
179. C₂₃	#2931	semidirect
266. C₄	#10625078	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625078	2
#13092964	2
#13092496	2
#13092493	2
D₄×C₂₃⋊C₈	2
#13092482	2
(C₂²⋊D₂₃)⋊C₈	2
#13092439	2
#13092382	2
C₄×C₈×D₂₃	2
C₂₃×#436	2
#2931	23