Subgroups — #28383871

Order 14272.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297304	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₈₉₂	C₂	1	1	yes	maximal, characteristic
3	#20297236	C₂	1	2	yes	maximal, complemented
4	#20297226	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₄₆	—	2	1	no	—
7	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	—	2	1	no	—
10	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
11	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
12	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
13	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₂₃	—	2	1	no	—
16	Dic₄₄₆	—	4	2	no	—
17	D₈₉₂	—	2	1	no	—
18	D₈₉₂	Q₈	1	2	yes	complemented
19	C₂×Dic₂₂₃	—	2	1	no	—
20	C₂×Dic₂₂₃	—	2	2	no	—
21	C₄×D₂₂₃	—	4	2	no	—
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₈×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₂₃	—	2	2	no	—
28	C₂×D₂₂₃	—	4	1	no	—
29	C₂×D₂₂₃	—	4	1	no	—
30	C₄×C₂₂₃	—	2	1	no	—
31	C₄×C₂₂₃	SD₁₆	1	2	yes	—
32	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₂₃	#41	1	1	yes	characteristic
34	Dic₂₂₃	—	4	1	no	—
35	Dic₂₂₃	—	4	2	no	—
36	C₂×C₂₂₃	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₂₂₃	—	4	2	no	—
40	C₂₂₃	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	223	1	no	maximal, Sylow
42	C₄⋊Q₈	—	223	1	no	—
43	C₄×D₄	—	223	1	no	—
44	#106	—	223	2	no	—
45	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
46	#102	—	223	2	no	—
47	C₂×Q₈	—	446	1	no	—
48	C₂×D₄	—	223	1	no	—
49	C₂²×C₄	—	446	1	no	—
50	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
51	C₄⋊C₄	—	446	1	no	—
52	C₄⋊C₄	—	223	2	no	—
53	C₄⋊C₄	—	223	1	no	—
54	C₂²⋊C₄	—	446	1	no	—
55	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
56	C₂³	—	446	1	no	—
57	Q₈	—	892	2	no	—
58	D₄	—	446	1	no	—
59	D₄	—	223	2	no	—
60	C₂×C₄	—	892	2	no	—
61	C₂×C₄	—	446	1	no	—
62	C₂×C₄	—	446	2	no	—
63	C₂×C₄	D₈₉₂	1	1	yes	characteristic
64	C₂×C₄	D₈₉₂	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	892	1	no	—
68	C₂²	—	892	1	no	—
69	C₂²	—	446	2	no	—
70	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	892	1	no	—
73	C₄	—	892	2	no	—
74	C₄	#16813159	1	2	yes	—
75	C₄	Q₈×D₂₂₃	1	1	yes	characteristic
76	C₄	#16813169	1	1	yes	characteristic
77	C₂	—	892	2	no	—
78	C₂	#20297284	1	1	yes	characteristic
79	C₂	C₂×#16813159	1	1	yes	characteristic
80	C₂	#20297332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₉₂
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₈₉₂	C₂²
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#476
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #20297304	C₂	semidirect
3. #20297236	C₂	semidirect
5. C₄⋊C₈×C₂₂₃	C₂	semidirect
18. D₈₉₂	Q₈	semidirect
40. C₂₂₃	#476	semidirect
45. C₄⋊C₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297304	2
C₄×D₈₉₂	2
#20297236	2
#20297226	2
C₄⋊C₈×C₂₂₃	2
#476	223