Subgroups — #90208

Order 448.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#117	C₂	1	1	yes	maximal, characteristic
2	#6489	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#6453	C₂	1	2	yes	maximal, complemented
5	#6451	C₂	1	2	yes	maximal, complemented
6	#6441	C₂	1	4	yes	maximal
7	C₄×Dic₁₄	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	C₂²	1	2	yes	—
10	C₂×Dic₁₄	C₂²	1	4	yes	—
11	C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
13	C₄²×C₇	C₂²	1	2	yes	—
14	C₂²⋊Dic₇	C₂²	1	2	yes	—
15	C₄⋊Dic₇	C₂²	1	4	yes	—
16	C₄⋊Dic₇	C₂²	1	2	yes	—
17	Dic₇⋊C₄	C₂²	1	4	yes	—
18	Dic₇⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₇	C₂²	1	2	yes	—
20	C₄×Dic₇	C₂²	1	2	yes	—
21	C₄×Dic₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₇	C₂²	1	1	yes	characteristic
23	#533	—	7	1	no	maximal, Sylow
24	C₂³×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₇	Q₈	1	4	yes	complemented
26	C₂×C₄×C₇	C₂³	1	4	yes	—
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₇	—	2	4	no	—
30	C₂×Dic₇	C₂³	1	2	yes	—
31	C₂×Dic₇	C₂³	1	4	yes	—
32	C₂×Dic₇	C₂³	1	2	yes	—
33	Dic₁₄	—	2	8	no	—
34	C₄⋊Q₈	—	7	2	no	—
35	#125	—	7	2	no	—
36	C₂²⋊Q₈	—	7	4	no	—
37	C₄×Q₈	—	7	4	no	—
38	#117	—	7	1	no	—
39	#117	D₇	1	1	yes	characteristic
40	C₂×C₄²	—	7	1	no	—
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	C₂×Q₈	1	2	yes	—
43	C₂²×C₇	C₂⁴	1	1	yes	characteristic
44	C₄×C₇	—	2	4	no	—
45	C₄×C₇	C₂×Q₈	1	2	yes	—
46	C₄×C₇	C₂×Q₈	1	2	yes	—
47	Dic₇	—	2	4	no	—
48	Dic₇	—	2	2	no	—
49	Dic₇	#41	1	4	yes	—
50	C₂×Q₈	—	7	4	no	—
51	C₂²×C₄	—	7	2	no	—
52	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
53	C₄⋊C₄	—	7	4	no	—
54	C₄⋊C₄	—	7	4	no	—
55	C₄⋊C₄	—	7	2	no	—
56	C₄⋊C₄	—	7	4	no	—
57	C₄⋊C₄	C₂×D₇	1	2	yes	—
58	C₂²⋊C₄	—	7	2	no	—
59	C₂²⋊C₄	C₂×D₇	1	2	yes	—
60	C₄²	—	7	1	no	—
61	C₄²	—	7	1	no	—
62	C₄²	—	7	2	no	—
63	C₄²	—	7	2	no	—
64	C₄²	C₂×D₇	1	2	yes	—
65	C₂×C₇	—	2	2	no	—
66	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₇	C₂×#41	1	2	yes	—
68	C₂³	C₂²×D₇	1	1	yes	characteristic
69	Q₈	—	14	8	no	—
70	C₂×C₄	—	7	2	no	—
71	C₂×C₄	—	7	2	no	—
72	C₂×C₄	—	7	4	no	—
73	C₂×C₄	—	14	4	no	—
74	C₂×C₄	Dic₁₄	1	4	yes	complemented
75	C₂×C₄	C₂²×D₇	1	4	yes	—
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
78	C₇	#533	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	C₂×Dic₁₄	1	2	yes	—
81	C₂²	C₂³×D₇	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	7	4	no	—
84	C₄	—	14	2	no	—
85	C₄	—	14	4	no	—
86	C₄	C₂×Dic₁₄	1	2	yes	—
87	C₄	C₂×Dic₁₄	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₁₄	1	1	yes	characteristic
90	C₂	D₇×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Frattini	C₂²	C₂³×D₇
Fitting	C₇×#117	C₂
prime radical	C₂³×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#533
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #6453	C₂	semidirect
5. #6451	C₂	semidirect
7. C₄×Dic₁₄	C₂	semidirect
25. C₂×C₄×C₇	Q₈	semidirect
74. C₂×C₄	Dic₁₄	semidirect
78. C₇	#533	semidirect

Maximal subgroups

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

subgroup	index
C₇×#117	2
#6489	2
C₂×C₄×Dic₇	2
#6453	2
#6451	2
#6441	2
C₄×Dic₁₄	2
#533	7