Subgroups — #15671230

Order 2944.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092470	C₂	1	2	yes	maximal
2	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₂₃	—	2	2	no	—
4	C₂×C₄²×C₂₃	C₄	1	2	yes	—
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
7	C₄⋊(C₂₃⋊C₈)	C₄	1	4	yes	complemented
8	C₄×C₂₃⋊C₈	—	2	2	no	—
9	C₂²×C₄×C₂₃	—	2	2	no	—
10	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	D₄	1	2	yes	—
12	C₄²×C₂₃	—	2	8	no	—
13	C₄²×C₂₃	—	2	2	no	—
14	C₄²×C₂₃	C₈	1	4	yes	complemented
15	C₄²×C₂₃	C₂×C₄	1	2	yes	—
16	C₄²×C₂₃	D₄	1	1	yes	characteristic
17	C₄²×C₂₃	Q₈	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	—	2	4	no	—
19	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	—	2	8	no	—
21	C₂×C₄×C₂₃	—	2	4	no	—
22	C₂×C₄×C₂₃	—	2	4	no	—
23	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
29	C₂₃⋊C₈	—	4	4	no	—
30	#1276	—	23	1	no	maximal, Sylow
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	#95	1	1	yes	characteristic
33	C₂²×C₂₃	C₂²⋊C₈	1	2	yes	—
34	C₄×C₂₃	—	2	8	no	—
35	C₄×C₂₃	—	2	2	no	—
36	C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
37	C₄×C₂₃	#97	1	1	yes	characteristic
38	C₄×C₂₃	#104	1	4	yes	—
39	C₄×C₂₃	C₄⋊C₈	1	2	yes	—
40	#431	—	23	2	no	—
41	C₄³	D₂₃	1	1	yes	characteristic
42	C₂×C₂₃	—	2	2	no	—
43	C₂×C₂₃	#336	1	1	yes	characteristic
44	C₂×C₂₃	#339	1	2	yes	—
45	C₂×C₄²	—	2	2	no	—
46	C₂×C₄²	Dic₂₃	1	2	yes	—
47	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
48	C₄⋊C₈	—	23	4	no	—
49	C₂²⋊C₈	—	46	2	no	—
50	C₄×C₈	—	46	2	no	—
51	C₂₃	#1276	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
55	C₂×C₈	—	46	4	no	—
56	C₄²	—	2	8	no	—
57	C₄²	—	2	2	no	—
58	C₄²	C₂₃⋊C₈	1	4	yes	complemented
59	C₄²	Dic₄₆	1	1	yes	characteristic
60	C₄²	C₄×D₂₃	1	2	yes	—
61	C₄²	D₉₂	1	1	yes	characteristic
62	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
67	C₂×C₄	#67267	1	2	yes	—
68	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
69	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
71	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
72	C₈	—	92	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
75	C₂²	C₂₃⋊#95	1	1	yes	characteristic
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×C₂₃⋊C₈	1	1	yes	characteristic
79	C₄	#10624985	1	1	yes	characteristic
80	C₄	C₄⋊(C₂₃⋊C₈)	1	2	yes	—
81	C₄	#10624987	1	4	yes	—
82	C₂	—	2	2	no	—
83	C₂	#13092172	1	1	yes	characteristic
84	C₂	#13092177	1	2	yes	—

	subgroup	G/H
center	C₄²	D₉₂
commutator	C₄×C₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₄³×C₂₃	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	C₄³	D₂₃
hypocenter	C₂₃	#1276
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
7. C₄⋊(C₂₃⋊C₈)	C₄	semidirect
14. C₄²×C₂₃	C₈	semidirect
51. C₂₃	#1276	semidirect
58. C₄²	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#13092470	2
C₄³×C₂₃	2
#1276	23