Subgroups — #10620948

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#66577	C₂	1	2	yes	maximal, complemented
5	#66575	C₂	1	2	yes	maximal
6	#6435	—	3	1	no	maximal, Hall
7	C₂²×D₂₁	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁	C₂²	1	1	yes	characteristic
10	C₄×D₂₁	C₄	1	4	yes	complemented
11	C₂×C₇×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×Dic₇	C₂²	1	2	yes	—
13	C₂×C₄×D₇	—	3	1	no	—
14	C₄²×C₇	—	3	1	no	—
15	(C₂×D₇)⋊C₄	—	3	2	no	—
16	Dic₇⋊C₄	—	3	2	no	—
17	C₄×Dic₇	D₃	1	1	yes	characteristic, complemented
18	#872	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
20	C₂×D₂₁	—	2	2	no	—
21	C₂×D₂₁	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
23	Dic₂₁	C₂×C₄	1	2	yes	—
24	C₃×Dic₇	—	2	2	no	—
25	C₇×Dic₃	—	2	2	no	—
26	C₂²×D₇	—	3	1	no	—
27	C₂×C₄×C₇	—	3	2	no	—
28	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₇	—	3	1	no	—
30	C₂×Dic₇	C₂×D₃	1	2	yes	—
31	C₄×D₇	—	3	4	no	—
32	C₂×C₄×D₃	—	7	1	no	—
33	C₃×C₄²	—	7	1	no	—
34	(C₂×D₃)⋊C₄	—	7	2	no	—
35	Dic₃⋊C₄	—	7	2	no	—
36	C₄×Dic₃	D₇	1	1	yes	characteristic, complemented
37	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃×C₇	#41	1	2	yes	—
39	D₂₁	—	2	2	no	—
40	#117	—	21	1	no	Sylow
41	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
42	C₂×D₇	—	3	2	no	—
43	C₂×D₇	—	6	2	no	—
44	C₄×C₇	—	6	2	no	—
45	C₄×C₇	C₄×D₃	1	2	yes	—
46	Dic₇	—	2	2	no	—
47	Dic₇	—	3	2	no	—
48	C₂²×D₃	—	7	1	no	—
49	C₂×C₃×C₄	—	7	2	no	—
50	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
51	C₂×Dic₃	—	7	1	no	—
52	C₂×Dic₃	C₂×D₇	1	2	yes	—
53	C₄×D₃	—	7	4	no	—
54	C₃×C₇	#117	1	1	yes	characteristic, complemented, Hall
55	C₂²×C₄	—	21	1	no	—
56	C₄⋊C₄	—	21	2	no	—
57	C₂²⋊C₄	—	21	2	no	—
58	C₄²	—	3	1	no	—
59	C₄²	—	7	1	no	—
60	C₂×C₇	C₂×C₄×D₃	1	1	yes	characteristic
61	C₂×C₇	#235	1	2	yes	—
62	D₇	—	6	2	no	—
63	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
64	C₂×D₃	—	7	2	no	—
65	C₂×D₃	—	14	2	no	—
66	C₃×C₄	—	14	2	no	—
67	C₃×C₄	C₄×D₇	1	2	yes	—
68	Dic₃	—	2	2	no	—
69	Dic₃	—	7	2	no	—
70	C₂³	—	21	1	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	7	2	no	—
73	C₂×C₄	—	21	1	no	—
74	C₂×C₄	—	21	4	no	—
75	C₂×C₄	D₃×D₇	1	1	yes	characteristic
76	C₇	#872	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₃	C₂×C₄×D₇	1	1	yes	characteristic
78	C₂×C₃	#1157	1	2	yes	—
79	D₃	—	14	2	no	—
80	C₂²	—	21	2	no	—
81	C₂²	—	42	2	no	—
82	C₂²	C₂×D₃×D₇	1	1	yes	characteristic
83	C₄	—	6	2	no	—
84	C₄	—	14	2	no	—
85	C₄	—	21	2	no	—
86	C₄	D₂₁⋊C₄	1	2	yes	—
87	C₃	#6435	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	42	2	no	—
89	C₂	#66675	1	2	yes	—
90	C₂	C₂×D₂₁⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₃×D₇
commutator	C₂×C₃×C₇	C₂²×C₄
Frattini	C₂²	C₂×D₃×D₇
Fitting	C₂×C₃×C₄×C₇	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂³
prime radical	C₂²×D₂₁	C₂²
hypocenter	C₃×C₇	#117
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×Dic₃	C₂	semidirect
3. C₃×C₄×Dic₇	C₂	semidirect
4. #66577	C₂	semidirect
10. C₄×D₂₁	C₄	semidirect
17. C₄×Dic₇	D₃	semidirect
36. C₄×Dic₃	D₇	semidirect
54. C₃×C₇	#117	semidirect
76. C₇	#872	semidirect
87. C₃	#6435	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₂₁	2
C₄×C₇×Dic₃	2
C₃×C₄×Dic₇	2
#66577	2
#66575	2
#6435	3
#872	7