Subgroups — #89428

Order 448.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	2	yes	maximal
3	C₂×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#95	C₂	1	1	yes	maximal, characteristic
5	C₇⋊#95	C₂	1	2	yes	maximal
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×Dic₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	C₂²	1	2	yes	—
10	C₂²×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₄	—	2	2	no	—
12	C₂×Dic₁₄	—	2	4	no	—
13	C₄⋊Dic₇	—	2	4	no	—
14	Dic₇⋊C₄	—	2	2	no	—
15	#395	—	7	1	no	maximal, Sylow
16	C₂³×C₇	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₄×C₇	Q₈	1	4	yes	complemented
22	C₂×Dic₇	—	2	1	no	—
23	C₂×Dic₇	—	2	2	no	—
24	C₂×Dic₇	—	2	2	no	—
25	C₂×Dic₇	—	2	1	no	—
26	C₂×Dic₇	—	2	1	no	—
27	C₂×Dic₇	—	2	2	no	—
28	C₂×Dic₇	D₄	1	4	yes	—
29	C₂×Dic₇	Q₈	1	2	yes	complemented
30	Dic₁₄	—	2	8	no	—
31	C₂²×Q₈	—	7	1	no	—
32	C₂×C₄⋊C₄	—	7	1	no	—
33	C₂×C₄⋊C₄	—	7	2	no	—
34	#95	—	7	2	no	—
35	#95	D₇	1	1	yes	characteristic
36	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₇	C₂×D₄	1	2	yes	—
38	C₂²×C₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₇	#41	1	1	yes	characteristic
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	—	2	4	no	—
43	Dic₇	—	2	4	no	—
44	Dic₇	—	2	2	no	—
45	Dic₇	—	4	1	no	—
46	C₂×Q₈	—	14	4	no	—
47	C₂×Q₈	—	14	2	no	—
48	C₂²×C₄	—	7	1	no	—
49	C₂²×C₄	—	7	2	no	—
50	C₂²×C₄	—	7	1	no	—
51	C₂²×C₄	C₂×D₇	1	2	yes	—
52	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
53	C₄⋊C₄	—	14	4	no	—
54	C₄⋊C₄	—	14	2	no	—
55	C₂×C₇	#120	1	1	yes	characteristic
56	C₂×C₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₇	C₄⋊Q₈	1	2	yes	—
60	C₂³	C₂²×D₇	1	1	yes	characteristic
61	Q₈	—	14	8	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	7	4	no	—
66	C₂×C₄	—	7	2	no	—
67	C₂×C₄	—	14	1	no	—
68	C₂×C₄	—	14	2	no	—
69	C₂×C₄	—	14	2	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	—	14	2	no	—
73	C₂×C₄	Dic₁₄	1	4	yes	complemented
74	C₂×C₄	D₂₈	1	2	yes	—
75	C₇	#395	1	1	yes	characteristic, complemented, Sylow
76	C₂²	C₂×Dic₁₄	1	2	yes	—
77	C₂²	C₂×D₂₈	1	1	yes	characteristic
78	C₂²	D₄×D₇	1	2	yes	—
79	C₂²	#1159	1	1	yes	characteristic
80	C₂²	Q₈×D₇	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	2	2	no	—
83	C₄	—	14	4	no	—
84	C₄	—	14	2	no	—
85	C₄	—	28	1	no	—
86	C₂	#6432	1	1	yes	characteristic
87	C₂	#6441	1	2	yes	—
88	C₂	#6445	1	1	yes	characteristic
89	C₂	#6451	1	2	yes	—
90	C₂	#6460	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂³
Fitting	C₇×#95	C₂
hypercenter	#95	D₇
hypocenter	C₇	#395
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
21. C₂×C₄×C₇	Q₈	semidirect
29. C₂×Dic₇	Q₈	semidirect
73. C₂×C₄	Dic₁₄	semidirect
75. C₇	#395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₄	2
C₂×C₄⋊Dic₇	2
C₂×Dic₇⋊C₄	2
C₇×#95	2
C₇⋊#95	2
#395	7