Subgroups — #13093133

Order 1472.

236 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	#10625118	C₂	1	4	yes	maximal, complemented
3	#10625066	C₂	1	4	yes	maximal, complemented
4	#10625063	C₂	1	4	yes	maximal, complemented
5	C₄×D₉₂	C₂	1	2	yes	maximal, complemented
6	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂×D₉₂	C₂²	1	2	yes	—
8	C₂×D₉₂	C₂²	1	4	yes	—
9	C₂×C₄×D₂₃	C₂²	1	4	yes	—
10	C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	—
14	C₄⋊Dic₂₃	C₂²	1	2	yes	—
15	C₄×Dic₂₃	C₂²	1	4	yes	—
16	C₂²×D₂₃	C₂³	1	4	yes	—
17	Q₈×C₂₃	—	4	2	no	—
18	C₂×C₄×C₂₃	C₂³	1	4	yes	—
19	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂³	1	2	yes	—
21	C₂×Dic₂₃	C₂³	1	4	yes	—
22	D₉₂	—	2	4	no	—
23	D₉₂	—	4	4	no	—
24	C₄×D₂₃	—	2	8	no	—
25	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
26	C₂×D₂₃	—	2	4	no	—
27	C₂×D₂₃	—	2	4	no	—
28	C₂×D₂₃	—	2	4	no	—
29	C₄×C₂₃	—	2	4	no	—
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	#41	1	4	yes	—
32	Dic₂₃	—	2	4	no	—
33	#555	—	23	1	no	maximal, Sylow
34	C₂×C₂₃	C₂×#41	1	2	yes	—
35	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
36	D₂₃	—	4	4	no	—
37	C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
38	#124	—	23	4	no	—
39	#121	—	23	4	no	—
40	C₄×D₄	—	23	2	no	—
41	#117	—	23	4	no	—
42	C₂₃	#555	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₈	C₂×D₂₃	1	2	yes	—
44	C₂×D₄	—	23	2	no	—
45	C₂×D₄	—	23	4	no	—
46	C₂²×C₄	—	23	4	no	—
47	C₄⋊C₄	—	23	2	no	—
48	C₄⋊C₄	C₂×D₂₃	1	4	yes	—
49	C₂²⋊C₄	—	23	8	no	—
50	C₂²⋊C₄	—	23	4	no	—
51	C₄²	—	23	4	no	—
52	C₄²	C₂×D₂₃	1	1	yes	characteristic
53	C₂³	—	23	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	46	4	no	—
56	D₄	—	92	4	no	—
57	C₂×C₄	—	23	4	no	—
58	C₂×C₄	—	46	8	no	—
59	C₂×C₄	C₂²×D₂₃	1	4	yes	—
60	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₂₃	1	2	yes	—
62	C₂²	—	46	4	no	—
63	C₂²	—	46	4	no	—
64	C₂²	—	46	4	no	—
65	C₂²	C₂³×D₂₃	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	46	4	no	—
69	C₄	#67292	1	4	yes	—
70	C₂	—	92	4	no	—
71	C₂	#10625155	1	1	yes	characteristic
72	C₂	C₂×#67292	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#555
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₃×C₄⋊Q₈	C₂	semidirect
2. #10625118	C₂	semidirect
3. #10625066	C₂	semidirect
4. #10625063	C₂	semidirect
5. C₄×D₉₂	C₂	semidirect
37. C₄⋊Q₈	D₂₃	semidirect
42. C₂₃	#555	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
#10625118	2
#10625066	2
#10625063	2
C₄×D₉₂	2
#555	23