Subgroups — #90363

Order 448.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#124	C₂	1	1	yes	maximal, characteristic
2	#6507	C₂	1	2	yes	maximal, complemented
3	#6498	C₂	1	2	yes	maximal
4	#6442	C₂	1	4	yes	maximal, complemented
5	#6441	C₂	1	4	yes	maximal, complemented
6	#6433	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	C₂²	1	2	yes	—
10	C₂×Dic₁₄	C₂²	1	2	yes	—
11	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₇	C₂²	1	4	yes	—
14	C₂²⋊Dic₇	C₂²	1	2	yes	—
15	C₄⋊Dic₇	C₂²	1	2	yes	—
16	C₄⋊Dic₇	C₂²	1	2	yes	—
17	Dic₇⋊C₄	C₂²	1	4	yes	—
18	Dic₇⋊C₄	C₂²	1	4	yes	—
19	Dic₇⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₇	C₂²	1	2	yes	—
21	#563	—	7	1	no	maximal, Sylow
22	C₂³×C₇	C₂³	1	2	yes	—
23	C₇×Q₈	—	4	1	no	—
24	C₇×D₄	—	4	1	no	—
25	C₂×C₄×C₇	C₂³	1	2	yes	—
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₇	C₂³	1	2	yes	—
28	C₂×Dic₇	—	4	2	no	—
29	C₂×Dic₇	C₂³	1	4	yes	—
30	C₂×Dic₇	C₂³	1	4	yes	—
31	Dic₁₄	—	4	2	no	—
32	C₄⋊Q₈	—	7	2	no	—
33	#126	—	7	4	no	—
34	#125	—	7	2	no	—
35	#124	D₇	1	1	yes	characteristic
36	#123	—	7	2	no	—
37	C₂²⋊Q₈	—	7	4	no	—
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	—	2	4	no	—
40	C₂²×C₇	C₂⁴	1	1	yes	characteristic
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	—	2	1	no	—
43	C₄×C₇	—	2	2	no	—
44	Dic₇	—	2	4	no	—
45	Dic₇	—	2	4	no	—
46	C₂×Q₈	—	7	2	no	—
47	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
48	C₂×D₄	C₂×D₇	1	1	yes	characteristic
49	C₂²×C₄	—	7	2	no	—
50	C₄⋊C₄	—	7	4	no	—
51	C₄⋊C₄	—	7	4	no	—
52	C₄⋊C₄	—	7	4	no	—
53	C₄⋊C₄	—	7	2	no	—
54	C₄⋊C₄	—	7	2	no	—
55	C₂²⋊C₄	—	7	4	no	—
56	C₂²⋊C₄	—	7	2	no	—
57	C₂²⋊C₄	C₂×D₇	1	4	yes	—
58	C₄²	—	7	2	no	—
59	C₄²	C₂×D₇	1	1	yes	characteristic
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂×C₇	2₋¹⁺⁴	1	2	yes	—
63	C₂³	C₂²×D₇	1	2	yes	—
64	Q₈	—	4	1	no	—
65	Q₈	—	28	2	no	—
66	D₄	—	4	1	no	—
67	C₂×C₄	—	7	4	no	—
68	C₂×C₄	—	7	4	no	—
69	C₂×C₄	—	28	2	no	—
70	C₂×C₄	C₂²×D₇	1	2	yes	—
71	C₂×C₄	C₂²×D₇	1	2	yes	—
72	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
73	C₇	#563	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₇	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	14	4	no	—
81	C₄	—	14	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#6548	1	1	yes	characteristic
84	C₂	#6554	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#124	C₂
prime radical	C₂×C₇×D₄	C₂²
hypercenter	#124	D₇
hypocenter	C₇	#563
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #6507	C₂	semidirect
4. #6442	C₂	semidirect
5. #6441	C₂	semidirect
6. #6433	C₂	semidirect
73. C₇	#563	semidirect

Maximal subgroups

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

subgroup	index
C₇×#124	2
#6507	2
#6498	2
#6442	2
#6441	2
#6433	2
#563	7