Group #90556 — subgroups

Subgroups — C₇×#535 #90556

Order 448.

334 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#127	C₂	1	2	yes	maximal, complemented
4	C₇×#124	C₂	1	2	yes	maximal, complemented
5	C₇×#121	C₂	1	4	yes	maximal, complemented
6	C₇×#120	C₂	1	4	yes	maximal, complemented
7	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×C₇	C₂²	1	2	yes	—
9	C₇×#41	—	2	4	no	—
10	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₇×D₄	—	2	4	no	—
12	C₂×C₇×D₄	C₂²	1	2	yes	—
13	C₂×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₇×D₄	C₂²	1	2	yes	—
15	C₂×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₇×D₄	C₂²	1	8	yes	complemented
17	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₇	C₂²	1	2	yes	complemented
19	C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₇×C₂²⋊C₄	C₂²	1	8	yes	complemented
22	C₄²×C₇	C₂²	1	2	yes	complemented
23	#535	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₇	—	2	4	no	—
25	C₂³×C₇	—	4	4	no	—
26	C₂³×C₇	C₂³	1	4	yes	—
27	C₂³×C₇	C₂³	1	2	yes	—
28	C₂³×C₇	C₂³	1	1	yes	characteristic
29	C₇×Q₈	—	2	2	no	—
30	C₇×D₄	—	2	4	no	—
31	C₇×D₄	—	2	4	no	—
32	C₇×D₄	—	2	4	no	—
33	C₇×D₄	—	2	2	no	—
34	C₇×D₄	—	4	8	no	—
35	C₂×C₄×C₇	—	2	4	no	—
36	C₂×C₄×C₇	D₄	1	4	yes	complemented
37	C₂×C₄×C₇	C₂³	1	4	yes	—
38	C₂×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×#41	C₂×C₇	1	1	yes	characteristic, complemented
42	C₂²×D₄	C₂×C₇	1	1	yes	characteristic, complemented
43	#127	C₂×C₇	1	2	yes	complemented
44	#124	C₂×C₇	1	2	yes	complemented
45	#121	C₂×C₇	1	4	yes	complemented
46	#120	C₂×C₇	1	4	yes	complemented
47	#117	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂²×C₇	—	2	2	no	—
49	C₂²×C₇	—	2	2	no	—
50	C₂²×C₇	—	2	4	no	—
51	C₂²×C₇	—	2	4	no	—
52	C₂²×C₇	—	2	8	no	—
53	C₂²×C₇	—	4	8	no	—
54	C₂²×C₇	C₂×D₄	1	2	yes	—
55	C₂²×C₇	C₂⁴	1	1	yes	characteristic
56	C₄×C₇	—	2	2	no	—
57	C₄×C₇	—	2	4	no	—
58	C₄×C₇	C₂×D₄	1	2	yes	—
59	C₄×C₇	C₂×D₄	1	2	yes	—
60	C₂⁴	C₂²×C₇	1	2	yes	—
61	#41	—	2	4	no	—
62	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
63	C₂×D₄	—	2	4	no	—
64	C₂×D₄	C₂²×C₇	1	2	yes	—
65	C₂×D₄	C₂²×C₇	1	2	yes	—
66	C₂×D₄	C₂²×C₇	1	2	yes	—
67	C₂×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
68	C₂×D₄	C₂²×C₇	1	8	yes	complemented
69	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×C₇	1	2	yes	complemented
71	C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
72	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
73	C₂²⋊C₄	C₂²×C₇	1	8	yes	complemented
74	C₄²	C₂²×C₇	1	2	yes	complemented
75	C₂×C₇	—	2	2	no	—
76	C₂×C₇	—	4	2	no	—
77	C₂×C₇	—	4	4	no	—
78	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
80	C₂³	—	2	4	no	—
81	C₂³	—	4	4	no	—
82	C₂³	C₂³×C₇	1	4	yes	—
83	C₂³	C₂³×C₇	1	2	yes	—
84	C₂³	C₂³×C₇	1	1	yes	characteristic
85	Q₈	—	2	2	no	—
86	D₄	—	2	4	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	4	no	—
89	D₄	—	2	2	no	—
90	D₄	—	4	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₇×D₄	1	4	yes	complemented
93	C₂×C₄	C₂³×C₇	1	4	yes	—
94	C₂×C₄	C₂³×C₇	1	2	yes	—
95	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
96	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
97	C₇	#535	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	2	4	no	—
102	C₂²	—	2	8	no	—
103	C₂²	—	4	8	no	—
104	C₂²	C₂×C₇×D₄	1	2	yes	—
105	C₂²	C₂⁴×C₇	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	4	no	—
108	C₄	C₂×C₇×D₄	1	2	yes	—
109	C₄	C₂×C₇×D₄	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	4	no	—
112	C₂	—	4	2	no	—
113	C₂	C₂²×C₇×D₄	1	1	yes	characteristic
114	C₂	C₇×H(5,𝔽₂)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#41

C₂

yes

maximal, characteristic, complemented

C₂²×C₇×D₄

C₂

yes

maximal, characteristic, complemented

C₇×#127

C₂

yes

maximal, complemented

C₇×#124

C₂

yes

maximal, complemented

C₇×#121

C₂

yes

maximal, complemented

C₇×#120

C₂

yes

maximal, complemented

C₇×#117

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₇

C₂²

yes

—

C₇×#41

—

C₂×C₇×Q₈

C₂²

yes

characteristic, complemented

C₂×C₇×D₄

—

C₂×C₇×D₄

C₂²

yes

—

C₂×C₇×D₄

C₂²

yes

—

C₂×C₇×D₄

C₂²

yes

—

C₂×C₇×D₄

C₂²

yes

characteristic, complemented

C₂×C₇×D₄

C₂²

yes

complemented

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

complemented

C₇×C₄⋊C₄

C₂²

yes

complemented

C₇×C₂²⋊C₄

C₂²

yes

—

C₇×C₂²⋊C₄

C₂²

yes

complemented

C₄²×C₇

C₂²

yes

complemented

#535

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂³

yes

—

C₂³×C₇

C₂³

yes

—

C₂³×C₇

C₂³

yes

characteristic

C₇×Q₈

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

D₄

yes

complemented

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×#41

C₂×C₇

yes

characteristic, complemented

C₂²×D₄

C₂×C₇

yes

characteristic, complemented

#127

C₂×C₇

yes

complemented

#124

C₂×C₇

yes

complemented

#121

C₂×C₇

yes

complemented

#120

C₂×C₇

yes

complemented

#117

C₂×C₇

yes

characteristic, complemented

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×D₄

yes

—

C₂²×C₇

C₂⁴

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

—

C₄×C₇

C₂×D₄

yes

—

C₂⁴

C₂²×C₇

yes

—

#41

—

C₂×Q₈

C₂²×C₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

C₂²×C₇

yes

—

C₂×D₄

C₂²×C₇

yes

—

C₂×D₄

C₂²×C₇

yes

—

C₂×D₄

C₂²×C₇

yes

characteristic, complemented

C₂×D₄

C₂²×C₇

yes

complemented

C₂²×C₄

C₂²×C₇

yes

characteristic

C₂²×C₄

C₂²×C₇

yes

complemented

C₄⋊C₄

C₂²×C₇

yes

complemented

C₂²⋊C₄

C₂²×C₇

yes

—

C₂²⋊C₄

C₂²×C₇

yes

complemented

C₄²

C₂²×C₇

yes

complemented

C₂×C₇

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂²×D₄

yes

characteristic

C₂×C₇

H(5,𝔽₂)

yes

—

C₂³

—

C₂³

—

C₂³

C₂³×C₇

yes

—

C₂³

C₂³×C₇

yes

—

C₂³

C₂³×C₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₇×D₄

yes

complemented

C₂×C₄

C₂³×C₇

yes

—

C₂×C₄

C₂³×C₇

yes

—

C₂×C₄

C₂³×C₇

yes

characteristic

C₂×C₄

C₂³×C₇

yes

characteristic

C₇

#535

yes

characteristic, direct product factor, Sylow

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₇×D₄

yes

—

105

C₂²

C₂⁴×C₇

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×C₇×D₄

yes

—

109

C₄

C₂×C₇×D₄

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

113

C₂

C₂²×C₇×D₄

yes

characteristic

114

C₂

C₇×H(5,𝔽₂)

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

C₂⁴

commutator, Frattini, cosocle

C₂²

C₂⁴×C₇

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇ ◁ G

N	H ≅ G/N	type
1. C₂×C₇×#41	C₂	semidirect
2. C₂²×C₇×D₄	C₂	semidirect
3. C₇×#127	C₂	semidirect
4. C₇×#124	C₂	semidirect
5. C₇×#121	C₂	semidirect
6. C₇×#120	C₂	semidirect
7. C₇×#117	C₂	semidirect
10. C₂×C₇×Q₈	C₂²	semidirect
15. C₂×C₇×D₄	C₂²	semidirect
16. C₂×C₇×D₄	C₂²	semidirect
18. C₂²×C₄×C₇	C₂²	semidirect
19. C₇×C₄⋊C₄	C₂²	semidirect
21. C₇×C₂²⋊C₄	C₂²	semidirect
22. C₄²×C₇	C₂²	semidirect
23. #535	C₇	direct (N × H)
36. C₂×C₄×C₇	D₄	semidirect
41. C₂×#41	C₂×C₇	semidirect
42. C₂²×D₄	C₂×C₇	semidirect
43. #127	C₂×C₇	semidirect
44. #124	C₂×C₇	semidirect
45. #121	C₂×C₇	semidirect
46. #120	C₂×C₇	semidirect
47. #117	C₂×C₇	semidirect
62. C₂×Q₈	C₂²×C₇	semidirect
67. C₂×D₄	C₂²×C₇	semidirect
68. C₂×D₄	C₂²×C₇	semidirect
70. C₂²×C₄	C₂²×C₇	semidirect
71. C₄⋊C₄	C₂²×C₇	semidirect
73. C₂²⋊C₄	C₂²×C₇	semidirect
74. C₄²	C₂²×C₇	semidirect
92. C₂×C₄	C₇×D₄	semidirect
97. C₇	#535	direct (N × H)

H ≅ G/N

type

1. C₂×C₇×#41

C₂

semidirect

2. C₂²×C₇×D₄

C₂

semidirect

3. C₇×#127

C₂

semidirect

4. C₇×#124

C₂

semidirect

5. C₇×#121

C₂

semidirect

6. C₇×#120

C₂

semidirect

7. C₇×#117

C₂

semidirect

10. C₂×C₇×Q₈

C₂²

semidirect

15. C₂×C₇×D₄

C₂²

semidirect

16. C₂×C₇×D₄

C₂²

semidirect

18. C₂²×C₄×C₇

C₂²

semidirect

19. C₇×C₄⋊C₄

C₂²

semidirect

21. C₇×C₂²⋊C₄

C₂²

semidirect

22. C₄²×C₇

C₂²

semidirect

23. #535

C₇

direct (N × H)

36. C₂×C₄×C₇

D₄

semidirect

41. C₂×#41

C₂×C₇

semidirect

42. C₂²×D₄

C₂×C₇

semidirect

43. #127

C₂×C₇

semidirect

44. #124

C₂×C₇

semidirect

45. #121

C₂×C₇

semidirect

46. #120

C₂×C₇

semidirect

47. #117

C₂×C₇

semidirect

62. C₂×Q₈

C₂²×C₇

semidirect

67. C₂×D₄

C₂²×C₇

semidirect

68. C₂×D₄

C₂²×C₇

semidirect

70. C₂²×C₄

C₂²×C₇

semidirect

71. C₄⋊C₄

C₂²×C₇

semidirect

73. C₂²⋊C₄

C₂²×C₇

semidirect

74. C₄²

C₂²×C₇

semidirect

92. C₂×C₄

C₇×D₄

semidirect

97. C₇

#535

direct (N × H)

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

subgroup	index
C₂×C₇×#41	2
C₂²×C₇×D₄	2
C₇×#127	2
C₇×#124	2
C₇×#121	2
C₇×#120	2
C₇×#117	2
#535	7

subgroup

index

C₂×C₇×#41

C₂²×C₇×D₄

Subgroups — C₇×#535 #90556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#535 #90556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#535 #90556