Group #90279 — subgroups

Subgroups — D₇×#120 #90279

Order 448.

662 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂⁴×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₇	C₂	1	3	yes	maximal, complemented
3	C₇×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	#6516	C₂	1	1	yes	maximal, characteristic, complemented
5	#6500	C₂	1	3	yes	maximal, complemented
6	#6445	C₂	1	3	yes	maximal, complemented
7	D₇×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₇	C₂²	1	1	yes	characteristic, complemented
9	C₂³×D₇	—	2	12	no	—
10	C₂³×D₇	C₂²	1	3	yes	—
11	C₂³×D₇	C₂²	1	3	yes	complemented
12	C₂³×D₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₇×D₄	C₂²	1	3	yes	complemented
14	C₂×C₂²⋊D₇	C₂²	1	3	yes	complemented
15	C₂×C₂²⋊D₇	C₂²	1	3	yes	complemented
16	D₄×D₇	—	2	12	no	—
17	C₂×D₂₈	C₂²	1	3	yes	complemented
18	C₂×C₄×D₇	C₂²	1	3	yes	complemented
19	C₇×C₂²⋊C₄	C₂²	1	3	yes	complemented
20	C₂²⋊Dic₇	C₂²	1	3	yes	complemented
21	(C₂×D₇)⋊C₄	C₂²	1	6	yes	complemented
22	C₂×#120	—	7	1	no	maximal, Sylow
23	C₂³×C₇	—	2	6	no	—
24	C₂³×C₇	C₂³	1	3	yes	—
25	C₂³×C₇	C₂³	1	1	yes	characteristic, complemented
26	C₂²×D₇	—	2	32	no	—
27	C₂²×D₇	—	2	6	no	—
28	C₂²×D₇	—	2	12	no	—
29	C₂²×D₇	—	2	12	no	—
30	C₂²×D₇	—	2	6	no	—
31	C₂²×D₇	D₄	1	12	yes	complemented
32	C₂²×D₇	C₂³	1	3	yes	—
33	C₂²×D₇	C₂³	1	1	yes	characteristic
34	C₂²×D₇	C₂³	1	1	yes	characteristic, complemented
35	C₇×D₄	—	2	6	no	—
36	C₂×C₄×C₇	C₂³	1	3	yes	complemented
37	C₂²⋊D₇	—	2	6	no	—
38	C₂²⋊D₇	—	2	6	no	—
39	C₂×Dic₇	C₂³	1	3	yes	complemented
40	D₂₈	—	2	6	no	—
41	C₄×D₇	—	2	6	no	—
42	C₂⁵	—	7	1	no	—
43	C₂²×D₄	—	7	3	no	—
44	#120	—	7	1	no	—
45	#120	—	7	3	no	—
46	#120	—	7	3	no	—
47	#120	D₇	1	1	yes	characteristic, direct product factor
48	C₂×C₂²⋊C₄	—	7	3	no	—
49	C₂²×C₇	—	2	8	no	—
50	C₂²×C₇	—	2	6	no	—
51	C₂²×C₇	—	2	3	no	—
52	C₂²×C₇	C₂×D₄	1	6	yes	complemented
53	C₂²×C₇	C₂⁴	1	1	yes	characteristic
54	C₂×D₇	—	2	24	no	—
55	C₂×D₇	—	2	6	no	—
56	C₂×D₇	—	2	24	no	—
57	C₂×D₇	—	2	3	no	—
58	C₂×D₇	—	4	4	no	—
59	C₂×D₇	C₂×D₄	1	6	yes	—
60	C₂×D₇	C₂×D₄	1	6	yes	complemented
61	C₄×C₇	—	2	3	no	—
62	Dic₇	—	2	3	no	—
63	C₂⁴	—	7	3	no	—
64	C₂⁴	—	7	3	no	—
65	C₂⁴	—	7	1	no	—
66	C₂⁴	—	14	12	no	—
67	C₂⁴	C₂×D₇	1	1	yes	characteristic, complemented
68	C₂×D₄	—	7	3	no	—
69	C₂×D₄	—	7	3	no	—
70	C₂×D₄	—	7	3	no	—
71	C₂×D₄	—	14	12	no	—
72	C₂×D₄	C₂×D₇	1	3	yes	complemented
73	C₂²×C₄	—	7	3	no	—
74	C₂²⋊C₄	—	7	3	no	—
75	C₂²⋊C₄	—	7	6	no	—
76	C₂²⋊C₄	C₂×D₇	1	3	yes	complemented
77	C₂×C₇	—	2	6	no	—
78	C₂×C₇	—	4	1	no	—
79	C₂×C₇	C₂²×D₄	1	3	yes	—
80	D₇	—	2	6	no	—
81	D₇	—	4	1	no	—
82	D₇	#120	1	4	yes	direct product factor
83	C₂³	—	2	6	no	—
84	C₂³	—	7	1	no	—
85	C₂³	—	7	12	no	—
86	C₂³	—	7	3	no	—
87	C₂³	—	7	1	no	—
88	C₂³	—	14	12	no	—
89	C₂³	—	14	12	no	—
90	C₂³	—	14	6	no	—
91	C₂³	—	14	32	no	—
92	C₂³	—	14	6	no	—
93	C₂³	C₂²×D₇	1	3	yes	—
94	C₂³	C₂²×D₇	1	1	yes	characteristic, complemented
95	D₄	—	2	6	no	—
96	D₄	—	14	6	no	—
97	D₄	—	14	6	no	—
98	D₄	—	14	6	no	—
99	C₂×C₄	—	7	3	no	—
100	C₂×C₄	—	14	6	no	—
101	C₂×C₄	C₂²×D₇	1	3	yes	complemented
102	C₇	C₂×#120	1	1	yes	characteristic, complemented, Sylow
103	C₂²	—	2	8	no	—
104	C₂²	—	2	6	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	7	6	no	—
107	C₂²	—	7	6	no	—
108	C₂²	—	14	24	no	—
109	C₂²	—	14	24	no	—
110	C₂²	—	14	6	no	—
111	C₂²	—	14	3	no	—
112	C₂²	—	28	4	no	—
113	C₂²	D₄×D₇	1	6	yes	complemented
114	C₂²	C₂³×D₇	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	14	3	no	—
117	C₂	—	2	6	no	—
118	C₂	—	4	1	no	—
119	C₂	—	7	4	no	—
120	C₂	—	14	6	no	—
121	C₂	—	28	1	no	—
122	C₂	C₂×D₄×D₇	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×D₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₇

C₂

yes

maximal, complemented

C₇×#120

C₂

yes

maximal, characteristic, complemented

#6516

C₂

yes

maximal, characteristic, complemented

#6500

C₂

yes

maximal, complemented

#6445

C₂

yes

maximal, complemented

D₇×C₂²⋊C₄

C₂

yes

maximal, complemented

C₂⁴×C₇

C₂²

yes

characteristic, complemented

C₂³×D₇

—

C₂³×D₇

C₂²

yes

—

C₂³×D₇

C₂²

yes

complemented

C₂³×D₇

C₂²

yes

characteristic, complemented

C₂×C₇×D₄

C₂²

yes

complemented

C₂×C₂²⋊D₇

C₂²

yes

complemented

C₂×C₂²⋊D₇

C₂²

yes

complemented

D₄×D₇

—

C₂×D₂₈

C₂²

yes

complemented

C₂×C₄×D₇

C₂²

yes

complemented

C₇×C₂²⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₇

C₂²

yes

complemented

(C₂×D₇)⋊C₄

C₂²

yes

complemented

C₂×#120

—

maximal, Sylow

C₂³×C₇

—

C₂³×C₇

C₂³

yes

—

C₂³×C₇

C₂³

yes

characteristic, complemented

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

D₄

yes

complemented

C₂²×D₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

characteristic

C₂²×D₇

C₂³

yes

characteristic, complemented

C₇×D₄

—

C₂×C₄×C₇

C₂³

yes

complemented

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

C₂³

yes

complemented

D₂₈

—

C₄×D₇

—

C₂⁵

—

C₂²×D₄

—

—

—

—

D₇

yes

characteristic, direct product factor

C₂×C₂²⋊C₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×D₄

yes

complemented

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×D₄

yes

—

C₂×D₇

C₂×D₄

yes

complemented

C₄×C₇

—

Dic₇

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×D₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

complemented

C₂²×C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

complemented

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂²×D₄

yes

—

D₇

—

D₇

—

D₇

#120

yes

direct product factor

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₇

yes

—

C₂³

C₂²×D₇

yes

characteristic, complemented

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²×D₇

yes

complemented

102

C₇

C₂×#120

yes

characteristic, complemented, Sylow

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

D₄×D₇

yes

complemented

114

C₂²

C₂³×D₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

C₂×D₄×D₇

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#120	C₂
hypercenter	#120	D₇
hypocenter	C₇	C₂×#120
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
1. C₂⁴×D₇	C₂	semidirect
2. C₂×D₄×D₇	C₂	semidirect
3. C₇×#120	C₂	semidirect
4. #6516	C₂	semidirect
5. #6500	C₂	semidirect
6. #6445	C₂	semidirect
7. D₇×C₂²⋊C₄	C₂	semidirect
8. C₂⁴×C₇	C₂²	semidirect
11. C₂³×D₇	C₂²	semidirect
12. C₂³×D₇	C₂²	semidirect
13. C₂×C₇×D₄	C₂²	semidirect
14. C₂×C₂²⋊D₇	C₂²	semidirect
15. C₂×C₂²⋊D₇	C₂²	semidirect
17. C₂×D₂₈	C₂²	semidirect
18. C₂×C₄×D₇	C₂²	semidirect
19. C₇×C₂²⋊C₄	C₂²	semidirect
20. C₂²⋊Dic₇	C₂²	semidirect
21. (C₂×D₇)⋊C₄	C₂²	semidirect
25. C₂³×C₇	C₂³	semidirect
31. C₂²×D₇	D₄	semidirect
34. C₂²×D₇	C₂³	semidirect
36. C₂×C₄×C₇	C₂³	semidirect
39. C₂×Dic₇	C₂³	semidirect
47. #120	D₇	direct (N × H)
52. C₂²×C₇	C₂×D₄	semidirect
60. C₂×D₇	C₂×D₄	semidirect
67. C₂⁴	C₂×D₇	semidirect
72. C₂×D₄	C₂×D₇	semidirect
76. C₂²⋊C₄	C₂×D₇	semidirect
82. D₇	#120	direct (N × H)
94. C₂³	C₂²×D₇	semidirect
101. C₂×C₄	C₂²×D₇	semidirect
102. C₇	C₂×#120	semidirect
113. C₂²	D₄×D₇	semidirect

H ≅ G/N

type

1. C₂⁴×D₇

C₂

semidirect

2. C₂×D₄×D₇

C₂

semidirect

3. C₇×#120

C₂

semidirect

4. #6516

C₂

semidirect

5. #6500

C₂

semidirect

6. #6445

C₂

semidirect

7. D₇×C₂²⋊C₄

C₂

semidirect

8. C₂⁴×C₇

C₂²

semidirect

11. C₂³×D₇

C₂²

semidirect

12. C₂³×D₇

C₂²

semidirect

13. C₂×C₇×D₄

C₂²

semidirect

14. C₂×C₂²⋊D₇

C₂²

semidirect

15. C₂×C₂²⋊D₇

C₂²

semidirect

17. C₂×D₂₈

C₂²

semidirect

18. C₂×C₄×D₇

C₂²

semidirect

19. C₇×C₂²⋊C₄

C₂²

semidirect

20. C₂²⋊Dic₇

C₂²

semidirect

21. (C₂×D₇)⋊C₄

C₂²

semidirect

25. C₂³×C₇

C₂³

semidirect

31. C₂²×D₇

D₄

semidirect

34. C₂²×D₇

C₂³

semidirect

36. C₂×C₄×C₇

C₂³

semidirect

39. C₂×Dic₇

C₂³

semidirect

47. #120

D₇

direct (N × H)

52. C₂²×C₇

C₂×D₄

semidirect

60. C₂×D₇

C₂×D₄

semidirect

67. C₂⁴

C₂×D₇

semidirect

72. C₂×D₄

C₂×D₇

semidirect

76. C₂²⋊C₄

C₂×D₇

semidirect

82. D₇

#120

direct (N × H)

94. C₂³

C₂²×D₇

semidirect

101. C₂×C₄

C₂²×D₇

semidirect

102. C₇

C₂×#120

semidirect

113. C₂²

D₄×D₇

semidirect

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

subgroup	index
C₂⁴×D₇	2
C₂×D₄×D₇	2
C₇×#120	2
#6516	2
#6500	2
#6445	2
D₇×C₂²⋊C₄	2
C₂×#120	7

subgroup

index

C₂⁴×D₇

C₂×D₄×D₇

D₇×C₂²⋊C₄

Subgroups — D₇×#120 #90279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×#120 #90279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#120 #90279