Subgroups — D₇×#126 #90394

Order 448.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	#6461	C₂	1	3	yes	maximal, complemented
3	D₇×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
4	D₇×C₂²⋊C₄	C₂	1	3	yes	maximal
5	#6442	C₂	1	3	yes	maximal, complemented
6	#6439	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×D₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₇	C₂²	1	3	yes	—
10	C₂×C₄×D₇	C₂²	1	3	yes	—
11	C₇×C₄⋊C₄	C₂²	1	3	yes	complemented
12	C₇×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²⋊Dic₇	C₂²	1	3	yes	—
15	(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
17	C₄⋊Dic₇	C₂²	1	3	yes	complemented
18	Dic₇⋊C₄	C₂²	1	3	yes	complemented
19	Dic₇⋊C₄	C₂²	1	3	yes	complemented
20	C₄×Dic₇	C₂²	1	3	yes	complemented
21	C₂×#126	—	7	1	no	maximal, Sylow
22	C₂³×C₇	C₂³	1	1	yes	characteristic
23	C₂²×D₇	—	2	6	no	—
24	C₂²×D₇	C₂³	1	1	yes	characteristic
25	C₂²×D₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂³	1	3	yes	—
27	C₂×C₄×C₇	C₂³	1	3	yes	—
28	C₂×Dic₇	C₂³	1	3	yes	—
29	C₂×Dic₇	C₂³	1	3	yes	—
30	C₄×D₇	—	2	6	no	—
31	C₄×D₇	—	2	6	no	—
32	#126	—	7	3	no	—
33	#126	—	7	3	no	—
34	#126	—	7	1	no	—
35	#126	D₇	1	1	yes	characteristic, direct product factor
36	C₂×C₄⋊C₄	—	7	3	no	—
37	C₂×C₂²⋊C₄	—	7	3	no	—
38	C₂×C₄²	—	7	1	no	—
39	C₂²×C₇	—	2	3	no	—
40	C₂²×C₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₇	—	2	3	no	—
42	C₂×D₇	—	4	4	no	—
43	C₂×D₇	#41	1	6	yes	—
44	C₄×C₇	—	2	3	no	—
45	C₄×C₇	—	2	3	no	—
46	Dic₇	—	2	3	no	—
47	Dic₇	—	2	3	no	—
48	C₂⁴	—	7	1	no	—
49	C₂²×C₄	—	7	3	no	—
50	C₂²×C₄	—	7	3	no	—
51	C₄⋊C₄	—	7	3	no	—
52	C₄⋊C₄	—	7	3	no	—
53	C₄⋊C₄	—	7	3	no	—
54	C₄⋊C₄	C₂×D₇	1	3	yes	complemented
55	C₂²⋊C₄	—	7	3	no	—
56	C₂²⋊C₄	—	7	3	no	—
57	C₂²⋊C₄	—	7	3	no	—
58	C₂²⋊C₄	C₂×D₇	1	3	yes	—
59	C₄²	—	7	3	no	—
60	C₄²	C₂×D₇	1	1	yes	characteristic, complemented
61	C₂×C₇	—	4	1	no	—
62	C₂×C₇	C₂×#41	1	3	yes	—
63	D₇	—	4	1	no	—
64	D₇	#126	1	4	yes	direct product factor
65	C₂³	—	7	1	no	—
66	C₂³	—	7	1	no	—
67	C₂³	—	14	6	no	—
68	C₂³	C₂²×D₇	1	1	yes	characteristic
69	C₂×C₄	—	7	3	no	—
70	C₂×C₄	—	7	3	no	—
71	C₂×C₄	—	14	6	no	—
72	C₂×C₄	—	14	6	no	—
73	C₂×C₄	C₂²×D₇	1	3	yes	—
74	C₂×C₄	C₂²×D₇	1	3	yes	—
75	C₇	C₂×#126	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	2	3	no	—
77	C₂²	—	7	6	no	—
78	C₂²	—	14	3	no	—
79	C₂²	—	28	4	no	—
80	C₂²	C₂³×D₇	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	3	no	—
83	C₄	—	14	3	no	—
84	C₄	—	14	3	no	—
85	C₂	—	4	1	no	—
86	C₂	—	7	4	no	—
87	C₂	—	28	1	no	—
88	C₂	D₇×#41	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#126	C₂
prime radical	C₂³×D₇	C₂²
hypercenter	#126	D₇
hypocenter	C₇	C₂×#126
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. C₇×#126	C₂	semidirect
2. #6461	C₂	semidirect
3. D₇×C₄⋊C₄	C₂	semidirect
5. #6442	C₂	semidirect
6. #6439	C₂	semidirect
7. C₄²×D₇	C₂	semidirect
11. C₇×C₄⋊C₄	C₂²	semidirect
13. C₄²×C₇	C₂²	semidirect
17. C₄⋊Dic₇	C₂²	semidirect
18. Dic₇⋊C₄	C₂²	semidirect
19. Dic₇⋊C₄	C₂²	semidirect
20. C₄×Dic₇	C₂²	semidirect
35. #126	D₇	direct (N × H)
54. C₄⋊C₄	C₂×D₇	semidirect
60. C₄²	C₂×D₇	semidirect
64. D₇	#126	direct (N × H)
75. C₇	C₂×#126	semidirect

Maximal subgroups

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

subgroup	index
C₇×#126	2
#6461	2
D₇×C₄⋊C₄	2
D₇×C₂²⋊C₄	2
#6442	2
#6439	2
C₄²×D₇	2
C₂×#126	7

Subgroups — D7×#126 #90394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#126 #90394