Subgroups — #13092765

Order 1472.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625045	C₂	1	1	yes	maximal, characteristic
3	#10624991	C₂	1	4	yes	maximal, complemented
4	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₂₃	—	2	1	no	—
6	C₂×D₉₂	—	2	1	no	—
7	C₂×D₉₂	—	2	1	no	—
8	C₂×D₉₂	C₂²	1	2	yes	—
9	C₄⋊C₄×C₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂²×D₂₃	—	2	2	no	—
14	Q₈×C₂₃	—	4	2	no	—
15	C₂×C₄×C₂₃	—	2	2	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	D₉₂	—	2	4	no	—
20	D₉₂	—	4	2	no	—
21	D₉₂	—	4	2	no	—
22	C₂₃⋊C₈	—	2	2	no	—
23	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₃	—	4	2	no	—
25	C₂×D₂₃	—	4	2	no	—
26	C₂×D₂₃	—	4	2	no	—
27	C₄×C₂₃	—	4	2	no	—
28	C₄×C₂₃	D₈	1	2	yes	—
29	C₄×C₂₃	SD₁₆	1	2	yes	—
30	C₄×C₂₃	#41	1	2	yes	—
31	#486	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	#124	1	1	yes	characteristic
33	C₂×C₂₃	C₂×D₈	1	1	yes	characteristic
34	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
35	D₂₃	—	8	2	no	—
36	C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
37	#127	—	23	1	no	—
38	#102	—	23	4	no	—
39	C₄×C₈	—	23	1	no	—
40	C₂₃	#486	1	1	yes	characteristic, complemented, Sylow
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	23	2	no	—
43	C₂×D₄	—	46	1	no	—
44	C₂×D₄	—	46	1	no	—
45	C₂×C₈	—	23	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
48	C₄²	C₂×D₂₃	1	1	yes	characteristic
49	C₂³	—	46	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	46	4	no	—
52	D₄	—	92	2	no	—
53	D₄	—	92	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
58	C₈	—	46	2	no	—
59	C₂²	—	92	2	no	—
60	C₂²	—	92	2	no	—
61	C₂²	—	92	2	no	—
62	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#67272	1	2	yes	—
65	C₄	#67274	1	2	yes	—
66	C₄	#67292	1	2	yes	—
67	C₂	—	184	2	no	—
68	C₂	C₂×#67272	1	1	yes	characteristic
69	C₂	C₂×#67274	1	1	yes	characteristic
70	C₂	#10625118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×C₄⋊Q₈	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	#10625045	C₂
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#486
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₃×C₄⋊Q₈	C₂	semidirect
3. #10624991	C₂	semidirect
4. C₄×C₂₃⋊C₈	C₂	semidirect
36. C₄⋊Q₈	D₂₃	semidirect
40. C₂₃	#486	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
#10625045	2
#10624991	2
C₄×C₂₃⋊C₈	2
#486	23