Subgroups — #28383847

Order 14272.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297330	C₂	1	2	yes	maximal, complemented
2	#20297310	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297240	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297221	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813170	C₂²	1	1	yes	characteristic
7	#16813158	—	2	2	no	—
8	#16813155	—	2	2	no	—
9	#16813154	C₂²	1	1	yes	characteristic
10	#16813154	C₂²	1	2	yes	—
11	#16813153	—	2	1	no	—
12	C₈×D₂₂₃	—	2	1	no	—
13	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
14	D₈×C₂₂₃	—	2	1	no	—
15	D₈×C₂₂₃	—	2	1	no	—
16	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	Dic₄₄₆	D₄	1	1	yes	characteristic
18	D₈₉₂	D₄	1	1	yes	characteristic
19	C₂²⋊D₂₂₃	—	2	1	no	—
20	C₄×D₂₂₃	—	2	1	no	—
21	C₂₂₃⋊C₈	—	2	1	no	—
22	C₂₂₃⋊C₈	—	2	2	no	—
23	C₂³×C₂₂₃	—	2	2	no	—
24	D₄×C₂₂₃	—	2	2	no	—
25	D₄×C₂₂₃	—	2	2	no	—
26	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
27	C₈×C₂₂₃	D₄	1	1	yes	characteristic
28	C₈×C₂₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₂₂₃	—	4	2	no	—
30	C₂²×C₂₂₃	—	8	2	no	—
31	C₂²×C₂₂₃	#41	1	1	yes	characteristic
32	C₂×D₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	Dic₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	—	8	2	no	—
38	C₂×C₂₂₃	#121	1	1	yes	characteristic
39	D₂₂₃	—	4	1	no	—
40	C₂₂₃	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	223	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	223	2	no	—
43	C₂×D₈	D₂₂₃	1	1	yes	characteristic, complemented
44	#131	—	223	1	no	—
45	#108	—	223	1	no	—
46	#100	—	223	2	no	—
47	#41	—	223	1	no	—
48	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
49	SD₁₆	—	446	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	446	2	no	—
53	M₁₆	—	446	1	no	—
54	M₁₆	—	223	1	no	—
55	M₁₆	—	223	2	no	—
56	C₂×C₈	—	446	1	no	—
57	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	223	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	446	1	no	—
63	D₄	—	223	1	no	—
64	C₂×C₄	—	446	1	no	—
65	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
66	C₈	—	446	2	no	—
67	C₈	—	446	1	no	—
68	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
69	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	446	1	no	—
73	C₂²	#16813169	1	1	yes	characteristic
74	C₄	—	446	1	no	—
75	C₄	D₄×D₂₂₃	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	892	1	no	—
80	C₂	#20297275	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297275
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂×D₈×C₂₂₃	C₂
socle	C₂×C₂₂₃	#121
hypercenter	C₂×D₈	D₂₂₃
hypocenter	C₂₂₃	#472
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #20297330	C₂	semidirect
2. #20297310	C₂	semidirect
3. #20297240	C₂	semidirect
4. #20297221	C₂	semidirect
5. C₂×D₈×C₂₂₃	C₂	semidirect
40. C₂₂₃	#472	semidirect
43. C₂×D₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297330	2
#20297310	2
#20297240	2
#20297221	2
C₂×D₈×C₂₂₃	2
#472	223