Subgroups — #28383317

Order 14272.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂₂₃⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	—	2	1	no	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₂×Q₈×C₂₂₃	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₂₂₃	—	2	1	no	—
9	C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic, complemented
10	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
11	C₂×Dic₂₂₃	—	2	2	no	—
12	C₂×Dic₂₂₃	—	4	1	no	—
13	C₂×Dic₂₂₃	—	4	1	no	—
14	C₂₂₃⋊C₈	—	4	1	no	—
15	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
16	Q₈×C₂₂₃	—	4	1	no	—
17	C₂×C₄×C₂₂₃	—	2	1	no	—
18	C₂×C₄×C₂₂₃	—	2	1	no	—
19	C₂×C₄×C₂₂₃	—	4	1	no	—
20	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₂₂₃	—	2	1	no	—
23	C₂²×C₂₂₃	—	2	1	no	—
24	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
26	C₂²×C₂₂₃	Q₁₆	1	1	yes	characteristic
27	C₄×C₂₂₃	—	2	1	no	—
28	C₄×C₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	—	4	1	no	—
30	C₄×C₂₂₃	—	4	1	no	—
31	Dic₂₂₃	—	4	2	no	—
32	C₂×C₂₂₃	—	2	1	no	—
33	C₂×C₂₂₃	—	2	1	no	—
34	C₂×C₂₂₃	#99	1	1	yes	characteristic
35	C₂×C₂₂₃	#103	1	1	yes	characteristic
36	C₂×C₂₂₃	#104	1	1	yes	characteristic
37	C₂₂₃	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	223	1	no	maximal, Sylow
39	C₂²⋊Q₈	D₂₂₃	1	1	yes	characteristic
40	C₂²⋊C₈	—	223	1	no	—
41	#95	—	223	1	no	—
42	C₂×Q₈	Dic₂₂₃	1	1	yes	characteristic, complemented
43	C₂²×C₄	—	446	1	no	—
44	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂×C₈	—	446	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	Dic₂₂₃	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	2	1	no	—
49	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
50	Q₈	—	4	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	892	1	no	—
55	C₂×C₄	—	892	1	no	—
56	C₂×C₄	—	446	2	no	—
57	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
59	C₈	—	892	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
63	C₂²	#16813157	1	1	yes	characteristic
64	C₂²	#16813160	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	1	no	—
68	C₄	—	4	1	no	—
69	C₄	—	892	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#20297219	1	1	yes	characteristic
73	C₂	#20297228	1	1	yes	characteristic
74	C₂	#20297231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₂₃
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂²⋊Q₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂²⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#328
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₂×Q₈×C₂₂₃	C₄	semidirect
9. C₄⋊C₄×C₂₂₃	C₄	semidirect
37. C₂₂₃	#328	semidirect
42. C₂×Q₈	Dic₂₂₃	semidirect
47. C₄⋊C₄	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₂₂₃⋊C₈)	2
C₂₂₃⋊#95	2
C₂²⋊Q₈×C₂₂₃	2
#328	223