Subgroups — D₇×#125 #90378

Order 448.

226 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	D₇×C₄⋊C₄	C₂	1	4	yes	maximal
3	D₇×C₄⋊C₄	C₂	1	2	yes	maximal
4	#6453	C₂	1	2	yes	maximal, complemented
5	#6452	C₂	1	4	yes	maximal, complemented
6	C₄²×D₇	C₂	1	1	yes	maximal, characteristic
7	#6433	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇	C₂²	1	2	yes	—
9	C₂×C₄×D₇	C₂²	1	4	yes	—
10	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₇	C₂²	1	2	yes	—
15	C₄⋊Dic₇	C₂²	1	4	yes	—
16	Dic₇⋊C₄	C₂²	1	4	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	1	yes	characteristic
21	C₂×#125	—	7	1	no	maximal, Sylow
22	C₂²×D₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	2	yes	—
24	C₂×C₄×C₇	C₂³	1	4	yes	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₇	C₂³	1	2	yes	—
27	C₂×Dic₇	C₂³	1	4	yes	—
28	C₂×Dic₇	C₂³	1	1	yes	characteristic
29	C₄×D₇	—	2	4	no	—
30	C₄×D₇	—	2	8	no	—
31	C₄×D₇	Q₈	1	4	yes	—
32	#125	—	7	4	no	—
33	#125	—	7	2	no	—
34	#125	—	7	1	no	—
35	#125	D₇	1	1	yes	characteristic, direct product factor
36	C₂×C₄⋊C₄	—	7	2	no	—
37	C₂×C₄⋊C₄	—	7	4	no	—
38	C₂×C₄²	—	7	1	no	—
39	C₂²×C₇	C₂⁴	1	1	yes	characteristic
40	C₂×D₇	C₂×Q₈	1	2	yes	—
41	C₂×D₇	#41	1	4	yes	—
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	—	2	4	no	—
44	C₄×C₇	C₂×Q₈	1	2	yes	—
45	Dic₇	—	2	2	no	—
46	Dic₇	—	2	4	no	—
47	Dic₇	C₂×Q₈	1	2	yes	—
48	C₂²×C₄	—	7	1	no	—
49	C₂²×C₄	—	7	2	no	—
50	C₂²×C₄	—	7	4	no	—
51	C₄⋊C₄	—	7	4	no	—
52	C₄⋊C₄	—	7	4	no	—
53	C₄⋊C₄	—	7	4	no	—
54	C₄⋊C₄	—	7	4	no	—
55	C₄⋊C₄	—	7	2	no	—
56	C₄⋊C₄	C₂×D₇	1	4	yes	—
57	C₄⋊C₄	C₂×D₇	1	2	yes	—
58	C₄²	—	7	2	no	—
59	C₄²	—	7	1	no	—
60	C₄²	C₂×D₇	1	1	yes	characteristic
61	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₇	C₂×#41	1	2	yes	—
63	D₇	#125	1	4	yes	direct product factor
64	C₂³	—	7	1	no	—
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	7	4	no	—
67	C₂×C₄	—	7	2	no	—
68	C₂×C₄	—	7	4	no	—
69	C₂×C₄	—	14	4	no	—
70	C₂×C₄	—	14	8	no	—
71	C₂×C₄	C₂²×D₇	1	2	yes	—
72	C₂×C₄	C₂²×D₇	1	4	yes	—
73	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
74	C₇	C₂×#125	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	7	2	no	—
76	C₂²	—	7	4	no	—
77	C₂²	C₂³×D₇	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	4	no	—
80	C₄	—	7	2	no	—
81	C₄	—	14	2	no	—
82	C₄	—	14	4	no	—
83	C₄	Q₈×D₇	1	2	yes	—
84	C₂	—	7	4	no	—
85	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
86	C₂	D₇×#41	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#125	C₂
prime radical	C₂²×D₇	C₂³
hypercenter	#125	D₇
hypocenter	C₇	C₂×#125
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
4. #6453	C₂	semidirect
5. #6452	C₂	semidirect
7. #6433	C₂	semidirect
35. #125	D₇	direct (N × H)
63. D₇	#125	direct (N × H)
74. C₇	C₂×#125	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
D₇×C₄⋊C₄	2
#6453	2
#6452	2
C₄²×D₇	2
#6433	2
C₂×#125	7

Subgroups — D7×#125 #90378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#125 #90378