Subgroups — #90389

Order 448.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6458	C₂	1	4	yes	maximal, complemented
3	#6458	C₂	1	2	yes	maximal, complemented
4	#6457	C₂	1	4	yes	maximal, complemented
5	#6456	C₂	1	2	yes	maximal, complemented
6	#6437	C₂	1	1	yes	maximal, characteristic, complemented
7	#6435	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₈	C₂²	1	2	yes	—
9	C₂×D₂₈	C₂²	1	4	yes	—
10	C₂×D₂₈	C₂²	1	4	yes	—
11	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₇	C₂²	1	4	yes	—
13	C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	C₄²×C₇	C₂²	1	1	yes	characteristic
16	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
17	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
19	Dic₇⋊C₄	C₂²	1	2	yes	—
20	C₄×Dic₇	C₂²	1	1	yes	characteristic
21	#540	—	7	1	no	maximal, Sylow
22	C₂²×D₇	C₂³	1	1	yes	characteristic
23	C₂²×D₇	C₂³	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	2	yes	—
25	C₂×C₄×C₇	C₂³	1	4	yes	—
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇	C₂³	1	2	yes	—
28	C₂×Dic₇	C₂³	1	1	yes	characteristic
29	D₂₈	—	2	4	no	—
30	D₂₈	—	4	4	no	—
31	D₂₈	—	4	4	no	—
32	C₄×D₇	—	2	2	no	—
33	C₄×D₇	—	4	4	no	—
34	#127	—	7	1	no	—
35	#125	D₇	1	1	yes	characteristic, complemented
36	#123	—	7	4	no	—
37	#121	—	7	4	no	—
38	#121	—	7	2	no	—
39	C₄×D₄	—	7	2	no	—
40	#117	—	7	1	no	—
41	C₂²×C₇	C₂⁴	1	1	yes	characteristic
42	C₂×D₇	—	2	1	no	—
43	C₂×D₇	—	2	4	no	—
44	C₂×D₇	—	2	4	no	—
45	C₂×D₇	—	2	2	no	—
46	C₂×D₇	—	2	4	no	—
47	C₄×C₇	—	2	4	no	—
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	#41	1	2	yes	—
50	Dic₇	—	2	1	no	—
51	Dic₇	—	2	2	no	—
52	C₂×D₄	—	7	2	no	—
53	C₂×D₄	—	7	4	no	—
54	C₂×D₄	—	7	4	no	—
55	C₂²×C₄	—	7	1	no	—
56	C₂²×C₄	—	7	4	no	—
57	C₄⋊C₄	—	7	2	no	—
58	C₄⋊C₄	C₂×D₇	1	2	yes	—
59	C₄⋊C₄	C₂×D₇	1	4	yes	—
60	C₂²⋊C₄	—	7	2	no	—
61	C₂²⋊C₄	—	7	4	no	—
62	C₂²⋊C₄	—	7	4	no	—
63	C₄²	—	7	1	no	—
64	C₄²	C₂×D₇	1	1	yes	characteristic
65	C₂×C₇	C₂×#41	1	1	yes	characteristic
66	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
67	D₇	—	4	1	no	—
68	D₇	—	4	4	no	—
69	C₂³	—	7	1	no	—
70	C₂³	—	7	4	no	—
71	D₄	—	14	4	no	—
72	D₄	—	28	4	no	—
73	D₄	—	28	4	no	—
74	C₂×C₄	—	7	2	no	—
75	C₂×C₄	—	7	1	no	—
76	C₂×C₄	—	14	2	no	—
77	C₂×C₄	—	28	4	no	—
78	C₂×C₄	C₂²×D₇	1	2	yes	—
79	C₂×C₄	C₂²×D₇	1	4	yes	—
80	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
81	C₇	#540	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	14	1	no	—
83	C₂²	—	14	4	no	—
84	C₂²	—	14	4	no	—
85	C₂²	—	14	2	no	—
86	C₂²	—	14	4	no	—
87	C₂²	C₂³×D₇	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	14	1	no	—
91	C₄	—	14	2	no	—
92	C₄	#1161	1	2	yes	—
93	C₂	—	28	1	no	—
94	C₂	—	28	4	no	—
95	C₂	C₂×#1161	1	1	yes	characteristic
96	C₂	#6553	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#125	C₂
hypercenter	#125	D₇
hypocenter	C₇	#540
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₇×#125	C₂	semidirect
2. #6458	C₂	semidirect
3. #6458	C₂	semidirect
4. #6457	C₂	semidirect
5. #6456	C₂	semidirect
6. #6437	C₂	semidirect
7. #6435	C₂	semidirect
35. #125	D₇	semidirect
81. C₇	#540	semidirect

Maximal subgroups

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

subgroup	index
C₇×#125	2
#6458	2
#6457	2
#6456	2
#6437	2
#6435	2
#540	7