Subgroups — #13092708

Order 1472.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₉₂	C₂	1	1	yes	maximal, characteristic
2	#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625003	C₂	1	2	yes	maximal
5	#10624999	C₂	1	2	yes	maximal, complemented
6	C₂×D₉₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₂₃	—	2	1	no	—
8	C₂×Dic₄₆	—	2	1	no	—
9	C₂×C₈×C₂₃	C₂²	1	2	yes	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	(C₂×D₂₃)⋊C₄	—	2	1	no	—
12	C₄⋊Dic₂₃	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₃	—	2	1	no	—
16	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₃	—	2	2	no	—
20	C₂×Dic₂₃	—	2	1	no	—
21	D₉₂	—	2	1	no	—
22	D₉₂	Q₈	1	2	yes	complemented
23	C₄×D₂₃	—	4	2	no	—
24	Dic₄₆	—	4	2	no	—
25	C₈×C₂₃	—	2	2	no	—
26	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₃	—	2	2	no	—
28	C₂×D₂₃	—	4	1	no	—
29	C₂×D₂₃	—	4	1	no	—
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	SD₁₆	1	2	yes	—
32	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₃	#41	1	1	yes	characteristic
34	Dic₂₃	—	4	1	no	—
35	Dic₂₃	—	4	2	no	—
36	#476	—	23	1	no	maximal, Sylow
37	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
40	D₂₃	—	4	2	no	—
41	C₄⋊Q₈	—	23	1	no	—
42	C₄×D₄	—	23	1	no	—
43	#106	—	23	2	no	—
44	C₄⋊C₈	D₂₃	1	1	yes	characteristic, complemented
45	#102	—	23	2	no	—
46	C₂₃	#476	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₈	—	46	1	no	—
48	C₂×D₄	—	23	1	no	—
49	C₂²×C₄	—	46	1	no	—
50	C₂×C₈	C₂×D₂₃	1	2	yes	—
51	C₄⋊C₄	—	23	2	no	—
52	C₄⋊C₄	—	23	1	no	—
53	C₄⋊C₄	—	46	1	no	—
54	C₂²⋊C₄	—	46	1	no	—
55	C₄²	C₂×D₂₃	1	1	yes	characteristic
56	C₂³	—	46	1	no	—
57	Q₈	—	92	2	no	—
58	D₄	—	23	2	no	—
59	D₄	—	46	1	no	—
60	C₂×C₄	—	46	1	no	—
61	C₂×C₄	—	46	2	no	—
62	C₂×C₄	—	92	2	no	—
63	C₂×C₄	D₉₂	1	1	yes	characteristic
64	C₂×C₄	D₉₂	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	46	2	no	—
68	C₂²	—	92	1	no	—
69	C₂²	—	92	1	no	—
70	C₂²	C₂×D₉₂	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	92	1	no	—
73	C₄	—	92	2	no	—
74	C₄	#67263	1	2	yes	—
75	C₄	#67290	1	1	yes	characteristic
76	C₄	Q₈×D₂₃	1	1	yes	characteristic
77	C₂	—	92	2	no	—
78	C₂	#10625068	1	1	yes	characteristic
79	C₂	C₂×#67263	1	1	yes	characteristic
80	C₂	#10625079	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×C₄⋊C₈	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₉₂	C₂²
hypercenter	C₄⋊C₈	D₂₃
hypocenter	C₂₃	#476
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #10625040	C₂	semidirect
3. C₂₃×C₄⋊C₈	C₂	semidirect
5. #10624999	C₂	semidirect
22. D₉₂	Q₈	semidirect
44. C₄⋊C₈	D₂₃	semidirect
46. C₂₃	#476	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₉₂	2
#10625040	2
C₂₃×C₄⋊C₈	2
#10625003	2
#10624999	2
#476	23