Subgroups — #13092356

Order 1472.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃⋊#95	C₂	1	2	yes	maximal
4	C₂₃⋊#95	C₂	1	2	yes	maximal
5	C₂₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₃	C₂²	1	2	yes	—
10	C₂²×Dic₂₃	C₂²	1	2	yes	—
11	C₄⋊C₄×C₂₃	—	2	2	no	—
12	C₄⋊Dic₂₃	—	2	2	no	—
13	C₂³×C₂₃	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₂₃	—	2	1	no	—
15	C₂×C₄×C₂₃	—	2	1	no	—
16	C₂×C₄×C₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	—	2	1	no	—
18	C₂×C₄×C₂₃	—	2	2	no	—
19	C₂×C₄×C₂₃	D₄	1	2	yes	—
20	C₂×C₄×C₂₃	Q₈	1	2	yes	—
21	C₂×Dic₂₃	—	2	2	no	—
22	C₂×Dic₂₃	—	2	2	no	—
23	C₂×Dic₂₃	—	2	2	no	—
24	C₂×Dic₂₃	—	2	4	no	—
25	C₂×Dic₂₃	—	2	2	no	—
26	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₂₃	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₂₃	#41	1	1	yes	characteristic
29	C₂²×C₂₃	#41	1	2	yes	—
30	C₂²×C₂₃	#41	1	2	yes	—
31	C₄×C₂₃	—	2	2	no	—
32	C₄×C₂₃	—	2	2	no	—
33	C₄×C₂₃	—	4	1	no	—
34	Dic₂₃	—	4	2	no	—
35	Dic₂₃	—	4	2	no	—
36	#400	—	23	1	no	maximal, Sylow
37	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₂₃	#123	1	1	yes	characteristic
39	C₂×C₂₃	#124	1	1	yes	characteristic
40	C₂×C₂₃	#125	1	2	yes	—
41	C₂×C₂₃	#126	1	2	yes	—
42	C₂×C₄⋊C₄	—	23	1	no	—
43	C₂×C₄⋊C₄	D₂₃	1	1	yes	characteristic
44	#95	—	23	1	no	—
45	#95	—	23	2	no	—
46	#95	—	23	2	no	—
47	C₂₃	#400	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₄	—	23	2	no	—
49	C₂²×C₄	—	23	2	no	—
50	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	46	2	no	—
55	C₂³	C₂²×D₂₃	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	46	2	no	—
62	C₂×C₄	—	46	2	no	—
63	C₂×C₄	—	46	2	no	—
64	C₂×C₄	—	46	4	no	—
65	C₂×C₄	—	46	2	no	—
66	C₂×C₄	Dic₄₆	1	2	yes	—
67	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
68	C₂²	C₂×Dic₄₆	1	1	yes	characteristic
69	C₂²	#67288	1	1	yes	characteristic
70	C₂²	#67290	1	2	yes	—
71	C₂²	#67292	1	2	yes	—
72	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	1	no	—
76	C₄	—	92	2	no	—
77	C₄	—	92	2	no	—
78	C₂	#10625061	1	2	yes	—
79	C₂	#10625069	1	2	yes	—
80	C₂	#10625095	1	1	yes	characteristic
81	C₂	#10625106	1	1	yes	characteristic
82	C₂	#10625118	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator, socle, prime radical, cosocle	C₂³×C₂₃	C₂³
Fitting	C₂×C₄⋊C₄×C₂₃	C₂
hypercenter	C₂×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#400
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
47. C₂₃	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₃	2
C₂×C₄⋊Dic₂₃	2
C₂₃⋊#95	2
#400	23