Subgroups — #13092974

Order 1472.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	2	yes	maximal
3	#10625095	C₂	1	4	yes	maximal
4	#10625041	C₂	1	2	yes	maximal, complemented
5	#10625040	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₄₆	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₆	C₂²	1	4	yes	—
10	C₄²×C₂₃	C₂²	1	2	yes	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
14	C₄⋊Dic₂₃	C₂²	1	4	yes	—
15	C₄⋊Dic₂₃	C₂²	1	4	yes	—
16	Dic₂₃⋊C₄	C₂²	1	8	yes	—
17	C₄×Dic₂₃	C₂²	1	4	yes	—
18	C₂³×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₃	—	2	4	no	—
20	C₂×C₄×C₂₃	Q₈	1	4	yes	complemented
21	C₂×C₄×C₂₃	C₂³	1	2	yes	—
22	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂³	1	2	yes	—
24	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₃	C₂³	1	8	yes	—
26	Dic₄₆	—	2	8	no	—
27	C₂²×C₂₃	—	2	2	no	—
28	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
29	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
30	C₄×C₂₃	—	2	2	no	—
31	C₄×C₂₃	C₂×Q₈	1	2	yes	—
32	C₄×C₂₃	C₂×Q₈	1	2	yes	—
33	C₄×C₂₃	#41	1	4	yes	—
34	Dic₂₃	—	2	8	no	—
35	#533	—	23	1	no	maximal, Sylow
36	C₂×C₂₃	—	2	2	no	—
37	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₂₃	C₂×#41	1	2	yes	—
39	C₄⋊Q₈	—	23	2	no	—
40	#125	—	23	2	no	—
41	C₂²⋊Q₈	—	23	4	no	—
42	C₄×Q₈	—	23	4	no	—
43	#117	—	23	2	no	—
44	C₂×C₄²	D₂₃	1	1	yes	characteristic
45	C₂₃	#533	1	1	yes	characteristic, complemented, Sylow
46	C₂×Q₈	—	23	4	no	—
47	C₂²×C₄	C₂×D₂₃	1	2	yes	—
48	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
49	C₄⋊C₄	—	23	4	no	—
50	C₄⋊C₄	—	23	8	no	—
51	C₄⋊C₄	—	23	4	no	—
52	C₂²⋊C₄	—	23	4	no	—
53	C₄²	—	23	4	no	—
54	C₄²	C₂×D₂₃	1	1	yes	characteristic
55	C₄²	C₂×D₂₃	1	1	yes	characteristic
56	C₄²	C₂×D₂₃	1	2	yes	—
57	C₂³	C₂²×D₂₃	1	1	yes	characteristic
58	Q₈	—	46	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	23	8	no	—
61	C₂×C₄	Dic₄₆	1	4	yes	complemented
62	C₂×C₄	C₂²×D₂₃	1	2	yes	—
63	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₂₃	1	2	yes	—
65	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Dic₄₆	1	2	yes	—
68	C₂²	C₂³×D₂₃	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	46	8	no	—
71	C₄	C₂×Dic₄₆	1	2	yes	—
72	C₄	C₂×Dic₄₆	1	2	yes	—
73	C₄	#67288	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₄₆	1	1	yes	characteristic
76	C₂	C₂×#67288	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Frattini	C₂²	C₂³×D₂₃
Fitting	C₂×C₄²×C₂₃	C₂
prime radical	C₂³×C₂₃	C₂³
hypercenter	C₂×C₄²	D₂₃
hypocenter	C₂₃	#533
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10625041	C₂	semidirect
5. #10625040	C₂	semidirect
6. C₄×Dic₄₆	C₂	semidirect
20. C₂×C₄×C₂₃	Q₈	semidirect
45. C₂₃	#533	semidirect
61. C₂×C₄	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₂₃	2
#10625097	2
#10625095	2
#10625041	2
#10625040	2
C₄×Dic₄₆	2
#533	23