Subgroups — #10621090

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₇	C₂	1	1	yes	maximal, characteristic
2	#66636	C₂	1	1	yes	maximal, characteristic
3	C₇×#217	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊Dic₃	C₂	1	2	yes	maximal, complemented
5	Dic₃×Dic₇	C₂	1	2	yes	maximal, complemented
6	#6440	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₁	C₂²	1	2	yes	—
9	C₂×C₇×Dic₃	C₂²	1	2	yes	—
10	C₂×C₃×Dic₇	C₄	1	4	yes	complemented
11	C₂×C₃×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₇	D₃	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	—	3	1	no	—
15	C₂²⋊Dic₇	—	3	1	no	—
16	Dic₇⋊C₄	—	3	2	no	—
17	C₄×Dic₇	—	3	2	no	—
18	#926	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	—	2	2	no	—
20	C₂²×C₃×C₇	C₂×C₄	1	2	yes	—
21	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
22	Dic₂₁	—	2	2	no	—
23	C₃×Dic₇	C₂×C₄	1	2	yes	—
24	C₃×Dic₇	C₂×C₄	1	2	yes	—
25	C₇×Dic₃	—	2	2	no	—
26	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
27	C₂×C₄×C₇	—	3	2	no	—
28	C₂×Dic₇	—	3	2	no	—
29	C₂×Dic₇	Dic₃	1	4	yes	complemented
30	C₂×Dic₇	C₂×D₃	1	1	yes	characteristic
31	C₂×Dic₇	C₂×D₃	1	1	yes	characteristic
32	C₂²×C₃×C₄	—	7	1	no	—
33	#217	—	7	1	no	—
34	#217	D₇	1	1	yes	characteristic
35	C₄⋊Dic₃	—	7	2	no	—
36	C₄×Dic₃	—	7	2	no	—
37	C₂×C₃×C₇	—	2	2	no	—
38	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃×C₇	#41	1	2	yes	—
40	#117	—	21	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	C₂×Dic₃	1	2	yes	—
43	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
44	C₄×C₇	—	6	2	no	—
45	Dic₇	—	6	2	no	—
46	Dic₇	C₂×Dic₃	1	2	yes	—
47	Dic₇	C₂×Dic₃	1	2	yes	—
48	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
49	C₂×C₃×C₄	—	7	4	no	—
50	C₂×C₃×C₄	—	7	1	no	—
51	C₂×C₃×C₄	—	7	1	no	—
52	C₂×Dic₃	—	7	2	no	—
53	C₂×Dic₃	C₂×D₇	1	2	yes	—
54	C₃×C₇	#117	1	1	yes	characteristic, complemented, Hall
55	C₂²×C₄	—	7	1	no	—
56	C₄⋊C₄	—	21	2	no	—
57	C₂²⋊C₄	—	3	1	no	—
58	C₂²⋊C₄	—	21	1	no	—
59	C₄²	—	21	2	no	—
60	C₂×C₇	—	2	2	no	—
61	C₂×C₇	#235	1	2	yes	—
62	C₂×C₇	C₂²×Dic₃	1	1	yes	characteristic
63	C₂²×C₃	—	2	2	no	—
64	C₂²×C₃	C₄×D₇	1	2	yes	—
65	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
66	C₃×C₄	—	7	2	no	—
67	C₃×C₄	—	7	2	no	—
68	Dic₃	—	2	2	no	—
69	Dic₃	—	14	2	no	—
70	C₂³	D₃×D₇	1	1	yes	characteristic
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	7	4	no	—
73	C₂×C₄	—	7	1	no	—
74	C₂×C₄	—	7	1	no	—
75	C₂×C₄	—	21	2	no	—
76	C₇	#926	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₃	—	2	2	no	—
78	C₂×C₃	C₂×C₄×D₇	1	1	yes	characteristic
79	C₂×C₃	#1159	1	2	yes	—
80	C₂²	—	2	2	no	—
81	C₂²	Dic₃×D₇	1	2	yes	—
82	C₂²	C₂×D₃×D₇	1	1	yes	characteristic
83	C₄	—	6	2	no	—
84	C₄	—	7	2	no	—
85	C₄	—	7	2	no	—
86	C₄	—	42	2	no	—
87	C₃	#6440	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	2	no	—
89	C₂	C₂×Dic₃×D₇	1	1	yes	characteristic
90	C₂	#66686	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. Dic₇⋊Dic₃	C₂	semidirect
5. Dic₃×Dic₇	C₂	semidirect
10. C₂×C₃×Dic₇	C₄	semidirect
29. C₂×Dic₇	Dic₃	semidirect
54. C₃×C₇	#117	semidirect
76. C₇	#926	semidirect
87. C₃	#6440	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₇	2
#66636	2
C₇×#217	2
Dic₇⋊Dic₃	2
Dic₃×Dic₇	2
#6440	3
#926	7