Subgroups — #10621427

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂	C₂	1	1	yes	maximal, characteristic
3	#66629	C₂	1	4	yes	maximal
4	C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic
5	#6507	—	3	1	no	maximal, Hall
6	C₃×C₇×Q₈	—	2	2	no	—
7	C₂×C₃×C₄×C₇	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁	C₂²	1	2	yes	—
10	C₂×Dic₂₁	C₂²	1	2	yes	—
11	Dic₄₂	—	2	2	no	—
12	C₂×C₇×Q₈	D₃	1	1	yes	characteristic
13	C₂×Dic₁₄	—	3	1	no	—
14	Dic₇⋊C₄	—	3	4	no	—
15	C₄×Dic₇	—	3	1	no	—
16	#944	—	7	1	no	maximal, Hall
17	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
18	C₃×C₄×C₇	—	2	2	no	—
19	C₃×C₄×C₇	D₄	1	2	yes	—
20	Dic₂₁	—	2	2	no	—
21	Dic₂₁	Q₈	1	4	yes	complemented
22	C₇×Q₈	—	2	2	no	—
23	C₂×C₄×C₇	C₂×D₃	1	2	yes	—
24	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
25	C₂×Dic₇	—	3	2	no	—
26	C₂×Dic₇	—	3	2	no	—
27	Dic₁₄	—	6	2	no	—
28	C₂×C₃×Q₈	D₇	1	1	yes	characteristic
29	C₂×Dic₆	—	7	1	no	—
30	Dic₃⋊C₄	—	7	4	no	—
31	C₄×Dic₃	—	7	1	no	—
32	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃×C₇	C₂×Q₈	1	2	yes	—
34	C₄⋊Q₈	—	21	1	no	Sylow
35	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	#67	1	2	yes	—
38	Dic₇	—	3	4	no	—
39	Dic₇	—	6	2	no	—
40	C₃×Q₈	—	2	2	no	—
41	C₂×C₃×C₄	C₂×D₇	1	2	yes	—
42	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
43	C₂×Dic₃	—	7	2	no	—
44	C₂×Dic₃	—	7	2	no	—
45	Dic₆	—	14	2	no	—
46	C₃×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
47	C₂×Q₈	—	21	1	no	—
48	C₂×Q₈	D₂₁	1	1	yes	characteristic
49	C₄⋊C₄	—	21	4	no	—
50	C₄²	—	21	1	no	—
51	C₂×C₇	D₃×Q₈	1	2	yes	—
52	C₂×C₇	C₂×#67	1	1	yes	characteristic
53	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
54	C₃×C₄	—	2	2	no	—
55	C₃×C₄	C₂²⋊D₇	1	2	yes	—
56	Dic₃	—	7	4	no	—
57	Dic₃	—	14	2	no	—
58	Q₈	—	2	2	no	—
59	Q₈	—	42	2	no	—
60	C₂×C₄	—	21	2	no	—
61	C₂×C₄	—	21	2	no	—
62	C₂×C₄	C₂×D₂₁	1	2	yes	—
63	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
64	C₇	#944	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	Q₈×D₇	1	2	yes	—
66	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
67	C₂²	C₂²×D₂₁	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	21	4	no	—
70	C₄	—	42	2	no	—
71	C₄	#4268	1	2	yes	—
72	C₃	#6507	1	1	yes	characteristic, complemented, Sylow
73	C₂	Q₈×D₂₁	1	2	yes	—
74	C₂	C₂×#4268	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₁
commutator, socle, prime radical, cosocle	C₂²×C₃×C₇	C₂³
Fitting	C₂×C₃×C₇×Q₈	C₂
hypercenter	C₂×Q₈	D₂₁
hypocenter	C₃×C₇	C₄⋊Q₈
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
21. Dic₂₁	Q₈	semidirect
46. C₃×C₇	C₄⋊Q₈	semidirect
64. C₇	#944	semidirect
72. C₃	#6507	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×Q₈	2
C₂×Dic₄₂	2
#66629	2
C₄×Dic₂₁	2
#6507	3
#944	7