Subgroups — #13056336

Order 1344.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄	C₂	1	4	yes	maximal
2	#10621352	C₂	1	2	yes	maximal
3	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	#89471	—	3	1	no	maximal, Hall
5	C₂×Dic₄₂	C₂²	1	4	yes	—
6	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
7	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₁	—	2	4	no	—
9	Dic₈₄	—	2	8	no	—
10	C₂×Dic₂₈	—	3	4	no	—
11	#6432	—	3	2	no	—
12	C₄×C₇×C₈	D₃	1	1	yes	characteristic
13	#4699	—	7	1	no	maximal, Hall
14	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
15	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
16	C₂×Dic₂₁	—	2	4	no	—
17	Dic₄₂	—	2	8	no	—
18	C₃×C₇×C₈	D₄	1	4	yes	—
19	C₂×Dic₁₄	—	3	4	no	—
20	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
21	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
22	C₄⋊Dic₇	—	6	4	no	—
23	Dic₂₈	—	6	8	no	—
24	C₂×Dic₁₂	—	7	4	no	—
25	#869	—	7	2	no	—
26	C₃×C₄×C₈	D₇	1	1	yes	characteristic
27	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
28	C₃×C₄×C₇	Q₁₆	1	4	yes	complemented
29	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
30	Dic₂₁	—	4	4	no	—
31	#494	—	21	1	no	Sylow
32	C₂×C₄×C₇	D₁₂	1	2	yes	—
33	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
34	C₂×Dic₇	—	6	4	no	—
35	Dic₁₄	—	6	8	no	—
36	C₇×C₈	D₁₂	1	4	yes	—
37	C₂×Dic₆	—	7	4	no	—
38	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
39	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
40	C₄⋊Dic₃	—	14	4	no	—
41	Dic₁₂	—	14	8	no	—
42	C₂×C₃×C₇	#127	1	1	yes	characteristic
43	C₂×C₃×C₇	C₂×Q₁₆	1	2	yes	—
44	C₂×Q₁₆	—	21	4	no	—
45	C₄⋊Q₈	—	21	2	no	—
46	C₄×C₈	D₂₁	1	1	yes	characteristic
47	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
48	C₄×C₇	Dic₁₂	1	4	yes	complemented
49	C₄×C₇	C₂×D₁₂	1	2	yes	—
50	Dic₇	—	12	4	no	—
51	C₂×C₃×C₄	D₂₈	1	2	yes	—
52	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
53	C₂×Dic₃	—	14	4	no	—
54	Dic₆	—	14	8	no	—
55	C₃×C₈	D₂₈	1	4	yes	—
56	C₃×C₇	#494	1	1	yes	characteristic, complemented, Hall
57	C₂×Q₈	—	21	4	no	—
58	Q₁₆	—	42	8	no	—
59	C₂×C₈	C₂×D₂₁	1	2	yes	—
60	C₄⋊C₄	—	42	4	no	—
61	C₄²	C₂×D₂₁	1	1	yes	characteristic
62	C₂×C₇	#874	1	1	yes	characteristic
63	C₂×C₇	C₂×Dic₁₂	1	2	yes	—
64	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
65	C₃×C₄	Dic₂₈	1	4	yes	complemented
66	C₃×C₄	C₂×D₂₈	1	2	yes	—
67	Dic₃	—	28	4	no	—
68	Q₈	—	42	8	no	—
69	C₂×C₄	—	42	4	no	—
70	C₂×C₄	D₈₄	1	2	yes	—
71	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
72	C₈	D₈₄	1	4	yes	—
73	C₇	#4699	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	#6437	1	1	yes	characteristic
75	C₂×C₃	C₂×Dic₂₈	1	2	yes	—
76	C₂²	C₂×D₈₄	1	1	yes	characteristic
77	C₄	—	84	4	no	—
78	C₄	Dic₈₄	1	4	yes	complemented
79	C₄	C₂×D₈₄	1	2	yes	—
80	C₃	#89471	1	1	yes	characteristic, complemented, Sylow
81	C₂	#10621357	1	1	yes	characteristic
82	C₂	C₂×Dic₈₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₁
Fitting	C₃×C₄×C₇×C₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₄×C₈	D₂₁
hypocenter	C₃×C₇	#494
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
28. C₃×C₄×C₇	Q₁₆	semidirect
48. C₄×C₇	Dic₁₂	semidirect
56. C₃×C₇	#494	semidirect
65. C₃×C₄	Dic₂₈	semidirect
73. C₇	#4699	semidirect
78. C₄	Dic₈₄	semidirect
80. C₃	#89471	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄	2
#10621352	2
C₃×C₄×C₇×C₈	2
#89471	3
#4699	7