Subgroups — #89758

Order 448.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₇⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₇⋊#95	C₂	1	2	yes	maximal
5	C₇⋊#95	C₂	1	2	yes	maximal
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	C₂²	1	2	yes	—
10	C₂²×Dic₇	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	—	2	2	no	—
12	C₄⋊Dic₇	—	2	2	no	—
13	#400	—	7	1	no	maximal, Sylow
14	C₂³×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	—	2	1	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₄×C₇	Q₈	1	2	yes	—
22	C₂×Dic₇	—	2	2	no	—
23	C₂×Dic₇	—	2	2	no	—
24	C₂×Dic₇	—	2	2	no	—
25	C₂×Dic₇	—	2	4	no	—
26	C₂×Dic₇	—	2	2	no	—
27	C₂×C₄⋊C₄	—	7	1	no	—
28	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
29	#95	—	7	1	no	—
30	#95	—	7	2	no	—
31	#95	—	7	2	no	—
32	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₇	#41	1	1	yes	characteristic
35	C₂²×C₇	#41	1	2	yes	—
36	C₂²×C₇	#41	1	2	yes	—
37	C₄×C₇	—	2	2	no	—
38	C₄×C₇	—	2	2	no	—
39	C₄×C₇	—	4	1	no	—
40	Dic₇	—	4	2	no	—
41	Dic₇	—	4	2	no	—
42	C₂²×C₄	—	7	2	no	—
43	C₂²×C₄	—	7	2	no	—
44	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
45	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	14	2	no	—
49	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂×C₇	#123	1	1	yes	characteristic
51	C₂×C₇	#124	1	1	yes	characteristic
52	C₂×C₇	#125	1	2	yes	—
53	C₂×C₇	#126	1	2	yes	—
54	C₂³	C₂²×D₇	1	1	yes	characteristic
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	14	2	no	—
61	C₂×C₄	—	14	2	no	—
62	C₂×C₄	—	14	2	no	—
63	C₂×C₄	—	14	4	no	—
64	C₂×C₄	—	14	2	no	—
65	C₂×C₄	Dic₁₄	1	2	yes	—
66	C₂×C₄	C₂²⋊D₇	1	2	yes	—
67	C₇	#400	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₂×Dic₁₄	1	1	yes	characteristic
69	C₂²	#1157	1	1	yes	characteristic
70	C₂²	#1159	1	2	yes	—
71	C₂²	#1161	1	2	yes	—
72	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	1	no	—
76	C₄	—	28	2	no	—
77	C₄	—	28	2	no	—
78	C₂	#6453	1	2	yes	—
79	C₂	#6461	1	2	yes	—
80	C₂	#6487	1	1	yes	characteristic
81	C₂	#6498	1	1	yes	characteristic
82	C₂	#6510	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂³
Fitting	C₂×C₇×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#400
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
67. C₇	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₄	2
C₂×C₄⋊Dic₇	2
C₇⋊#95	2
#400	7