Subgroups — #66694

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#67	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#4247	C₂	1	1	yes	maximal, characteristic, complemented
6	#4246	C₂	1	1	yes	maximal, characteristic, complemented
7	D₂₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×D₇	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₂₁	—	2	2	no	—
11	C₂×D₂₁	C₂²	1	1	yes	characteristic
12	C₂×D₂₁	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₃×C₇	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×D₇	C₂²	1	1	yes	characteristic, complemented
15	D₃×D₇	—	2	2	no	—
16	C₃×Dic₇	C₂²	1	1	yes	characteristic, complemented
17	C₇×Dic₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₇	—	3	1	no	—
19	C₂²×D₇	—	3	1	no	—
20	C₇×D₄	—	3	1	no	—
21	C₂²⋊D₇	—	3	1	no	—
22	C₂²⋊D₇	D₃	1	1	yes	characteristic, complemented
23	D₂₈	—	3	1	no	—
24	C₄×D₇	—	3	1	no	—
25	D₃×D₄	—	7	1	no	maximal, Hall
26	C₂×C₃×C₇	—	2	1	no	—
27	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
28	D₂₁	—	2	1	no	—
29	D₂₁	D₄	1	2	yes	complemented
30	C₃×D₇	—	2	1	no	—
31	D₃×C₇	—	2	1	no	—
32	C₂²×C₇	—	3	1	no	—
33	C₂²×C₇	C₂×D₃	1	1	yes	characteristic, complemented
34	C₂×D₇	—	3	1	no	—
35	C₂×D₇	—	3	1	no	—
36	C₂×D₇	—	6	2	no	—
37	C₂×D₇	—	6	2	no	—
38	C₂×D₇	C₂×D₃	1	1	yes	characteristic, complemented
39	C₄×C₇	—	3	1	no	—
40	Dic₇	C₂×D₃	1	1	yes	characteristic, complemented
41	C₂²×D₃	—	7	1	no	—
42	C₂²×D₃	—	7	1	no	—
43	C₃×D₄	—	7	1	no	—
44	#67	—	7	1	no	—
45	#67	D₇	1	1	yes	characteristic, complemented
46	D₁₂	—	7	1	no	—
47	C₄×D₃	—	7	1	no	—
48	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
49	C₂×D₄	—	21	1	no	Sylow
50	C₂×C₇	—	2	1	no	—
51	C₂×C₇	—	6	1	no	—
52	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
53	D₇	—	2	1	no	—
54	D₇	—	3	2	no	—
55	D₇	—	6	1	no	—
56	C₂²×C₃	—	7	1	no	—
57	C₂²×C₃	C₂×D₇	1	1	yes	characteristic, complemented
58	C₂×D₃	—	7	1	no	—
59	C₂×D₃	—	7	1	no	—
60	C₂×D₃	—	14	2	no	—
61	C₂×D₃	—	14	2	no	—
62	C₂×D₃	C₂×D₇	1	1	yes	characteristic, complemented
63	C₃×C₄	—	7	1	no	—
64	Dic₃	C₂×D₇	1	1	yes	characteristic, complemented
65	C₂³	—	21	1	no	—
66	C₂³	—	21	1	no	—
67	D₄	—	3	1	no	—
68	D₄	—	7	1	no	—
69	D₄	—	21	1	no	—
70	D₄	—	21	1	no	—
71	C₂×C₄	—	21	1	no	—
72	C₇	D₃×D₄	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	14	1	no	—
75	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
76	D₃	—	2	1	no	—
77	D₃	—	7	2	no	—
78	D₃	—	14	1	no	—
79	C₂²	—	3	1	no	—
80	C₂²	—	7	1	no	—
81	C₂²	—	21	1	no	—
82	C₂²	—	21	1	no	—
83	C₂²	—	42	2	no	—
84	C₂²	—	42	2	no	—
85	C₂²	D₃×D₇	1	1	yes	characteristic, complemented
86	C₄	—	3	1	no	—
87	C₄	—	7	1	no	—
88	C₃	D₄×D₇	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	2	1	no	—
90	C₂	—	6	1	no	—
91	C₂	—	14	1	no	—
92	C₂	—	21	2	no	—
93	C₂	—	42	1	no	—
94	C₂	C₂×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₇
commutator, socle, cosocle	C₂×C₃×C₇	C₂³
Fitting	C₂²×C₃×C₇	C₂²
hypercenter	C₂²	D₃×D₇
hypocenter	C₃×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₁	C₂	semidirect
2. C₂×D₃×D₇	C₂	semidirect
3. C₇×#67	C₂	semidirect
4. C₃×C₂²⋊D₇	C₂	semidirect
5. #4247	C₂	semidirect
6. #4246	C₂	semidirect
7. D₂₁⋊C₄	C₂	semidirect
9. C₂²×C₃×C₇	C₂²	semidirect
12. C₂×D₂₁	C₂²	semidirect
13. C₂×D₃×C₇	C₂²	semidirect
14. C₂×C₃×D₇	C₂²	semidirect
16. C₃×Dic₇	C₂²	semidirect
17. C₇×Dic₃	C₂²	semidirect
22. C₂²⋊D₇	D₃	semidirect
29. D₂₁	D₄	semidirect
33. C₂²×C₇	C₂×D₃	semidirect
38. C₂×D₇	C₂×D₃	semidirect
40. Dic₇	C₂×D₃	semidirect
45. #67	D₇	semidirect
48. C₃×C₇	C₂×D₄	semidirect
57. C₂²×C₃	C₂×D₇	semidirect
62. C₂×D₃	C₂×D₇	semidirect
64. Dic₃	C₂×D₇	semidirect
72. C₇	D₃×D₄	semidirect
85. C₂²	D₃×D₇	semidirect
88. C₃	D₄×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₁	2
C₂×D₃×D₇	2
C₇×#67	2
C₃×C₂²⋊D₇	2
#4247	2
#4246	2
D₂₁⋊C₄	2
D₄×D₇	3
D₃×D₄	7