Group #90215 — subgroups

Subgroups — #90215

Order 448.

330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 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×#1157	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#6460	C₂	1	2	yes	maximal, complemented
5	#6458	C₂	1	2	yes	maximal, complemented
6	#6449	C₂	1	2	yes	maximal, complemented
7	#6445	C₂	1	2	yes	maximal, complemented
8	C₄×D₂₈	C₂	1	4	yes	maximal, complemented
9	C₂³×D₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
12	C₂²×Dic₇	C₂²	1	1	yes	characteristic
13	#1157	—	2	4	no	—
14	C₂×D₂₈	C₂²	1	4	yes	—
15	C₂×D₂₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₄×D₇	—	2	4	no	—
17	C₂×C₄×D₇	C₂²	1	2	yes	—
18	C₂×C₄×D₇	C₂²	1	2	yes	—
19	C₂×C₄×D₇	C₂²	1	2	yes	complemented
20	C₂×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
21	C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
22	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₄²×C₇	C₂²	1	2	yes	complemented
24	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
25	(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
26	C₄⋊Dic₇	C₂²	1	4	yes	complemented
27	#525	—	7	1	no	maximal, Sylow
28	C₂³×C₇	C₂³	1	1	yes	characteristic
29	C₂²×D₇	—	2	2	no	—
30	C₂²×D₇	—	2	4	no	—
31	C₂²×D₇	C₂³	1	2	yes	—
32	C₂²×D₇	C₂³	1	2	yes	—
33	C₂×C₄×C₇	D₄	1	4	yes	complemented
34	C₂×C₄×C₇	C₂³	1	4	yes	—
35	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²⋊D₇	—	2	4	no	—
38	C₂×Dic₇	—	2	2	no	—
39	C₂×Dic₇	C₂³	1	2	yes	—
40	C₂×Dic₇	C₂³	1	2	yes	—
41	D₂₈	—	2	8	no	—
42	D₂₈	—	2	2	no	—
43	C₄×D₇	—	2	4	no	—
44	C₄×D₇	—	2	4	no	—
45	C₄×D₇	—	2	4	no	—
46	Dic₁₄	—	2	2	no	—
47	C₂×#41	—	7	1	no	—
48	C₂³×C₄	—	7	1	no	—
49	#123	—	7	2	no	—
50	C₂²⋊Q₈	—	7	2	no	—
51	#121	—	7	2	no	—
52	#120	—	7	2	no	—
53	C₄×D₄	—	7	4	no	—
54	#117	D₇	1	1	yes	characteristic, complemented
55	C₂²×C₇	—	2	2	no	—
56	C₂²×C₇	C₂×D₄	1	2	yes	—
57	C₂²×C₇	C₂⁴	1	1	yes	characteristic
58	C₂×D₇	—	2	8	no	—
59	C₂×D₇	—	2	2	no	—
60	C₂×D₇	—	2	2	no	—
61	C₂×D₇	—	2	2	no	—
62	C₂×D₇	—	2	4	no	—
63	C₂×D₇	#41	1	4	yes	complemented
64	C₄×C₇	—	2	4	no	—
65	C₄×C₇	C₂×D₄	1	2	yes	—
66	C₄×C₇	C₂×D₄	1	2	yes	—
67	Dic₇	—	2	2	no	—
68	Dic₇	—	2	2	no	—
69	C₂⁴	—	7	1	no	—
70	#41	—	14	4	no	—
71	C₂×Q₈	—	7	1	no	—
72	C₂×D₄	—	7	2	no	—
73	C₂×D₄	—	7	4	no	—
74	C₂×D₄	—	7	1	no	—
75	C₂²×C₄	—	7	1	no	—
76	C₂²×C₄	—	7	2	no	—
77	C₂²×C₄	—	7	2	no	—
78	C₂²×C₄	—	7	2	no	—
79	C₂²×C₄	—	14	4	no	—
80	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
81	C₄⋊C₄	—	7	4	no	—
82	C₄⋊C₄	C₂×D₇	1	2	yes	complemented
83	C₂²⋊C₄	—	7	4	no	—
84	C₂²⋊C₄	—	7	4	no	—
85	C₂²⋊C₄	C₂×D₇	1	2	yes	—
86	C₄²	C₂×D₇	1	2	yes	complemented
87	C₂×C₇	—	2	2	no	—
88	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₇	C₂×#41	1	2	yes	—
90	D₇	—	2	4	no	—
91	D₇	—	4	2	no	—
92	C₂³	—	7	2	no	—
93	C₂³	—	7	2	no	—
94	C₂³	—	14	4	no	—
95	C₂³	—	14	2	no	—
96	C₂³	C₂²×D₇	1	1	yes	characteristic
97	Q₈	—	14	2	no	—
98	D₄	—	14	8	no	—
99	D₄	—	14	4	no	—
100	D₄	—	14	2	no	—
101	C₂×C₄	—	7	2	no	—
102	C₂×C₄	—	7	2	no	—
103	C₂×C₄	—	14	2	no	—
104	C₂×C₄	—	14	4	no	—
105	C₂×C₄	—	14	4	no	—
106	C₂×C₄	—	14	4	no	—
107	C₂×C₄	D₂₈	1	4	yes	complemented
108	C₂×C₄	C₂²×D₇	1	4	yes	—
109	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
111	C₇	#525	1	1	yes	characteristic, complemented, Sylow
112	C₂²	—	2	2	no	—
113	C₂²	—	7	4	no	—
114	C₂²	—	14	8	no	—
115	C₂²	—	14	2	no	—
116	C₂²	—	14	2	no	—
117	C₂²	—	14	2	no	—
118	C₂²	—	14	4	no	—
119	C₂²	C₂×D₂₈	1	2	yes	—
120	C₂²	C₂³×D₇	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	—	14	2	no	—
123	C₄	—	14	2	no	—
124	C₄	C₂×D₂₈	1	2	yes	—
125	C₄	C₂×D₂₈	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	14	4	no	—
128	C₂	—	28	2	no	—
129	C₂	C₂²×D₂₈	1	1	yes	characteristic
130	C₂	D₇×#41	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1157

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇

C₂

yes

maximal, characteristic, complemented

C₇×#117

C₂

yes

maximal, characteristic, complemented

#6460

C₂

yes

maximal, complemented

#6458

C₂

yes

maximal, complemented

#6449

C₂

yes

maximal, complemented

#6445

C₂

yes

maximal, complemented

C₄×D₂₈

C₂

yes

maximal, complemented

C₂³×D₇

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

characteristic

#1157

—

C₂×D₂₈

C₂²

yes

—

C₂×D₂₈

C₂²

yes

characteristic, complemented

C₂×C₄×D₇

—

C₂×C₄×D₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

complemented

C₂×Dic₁₄

C₂²

yes

characteristic, complemented

C₇×C₄⋊C₄

C₂²

yes

complemented

C₇×C₂²⋊C₄

C₂²

yes

—

C₄²×C₇

C₂²

yes

complemented

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

complemented

C₄⋊Dic₇

C₂²

yes

complemented

#525

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

—

C₂×C₄×C₇

D₄

yes

complemented

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

D₂₈

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

C₂×#41

—

C₂³×C₄

—

#123

—

C₂²⋊Q₈

—

#121

—

#120

—

C₄×D₄

—

#117

D₇

yes

characteristic, complemented

C₂²×C₇

—

C₂²×C₇

C₂×D₄

yes

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

#41

yes

complemented

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

—

C₄×C₇

C₂×D₄

yes

—

Dic₇

—

Dic₇

—

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

—

C₄²

C₂×D₇

yes

complemented

C₂×C₇

—

C₂×C₇

C₂²×D₄

yes

characteristic

C₂×C₇

C₂×#41

yes

—

D₇

—

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₂₈

yes

complemented

108

C₂×C₄

C₂²×D₇

yes

—

109

C₂×C₄

C₂²×D₇

yes

characteristic

110

C₂×C₄

C₂²×D₇

yes

characteristic

111

C₇

#525

yes

characteristic, complemented, Sylow

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂×D₂₈

yes

—

120

C₂²

C₂³×D₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×D₂₈

yes

—

125

C₄

C₂×D₂₈

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

C₂²×D₂₈

yes

characteristic

130

C₂

D₇×#41

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Frattini	C₂²	C₂³×D₇
Fitting	C₇×#117	C₂
hypercenter	#117	D₇
hypocenter	C₇	#525
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center

C₂×C₄

C₂²×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Frattini

C₂²

C₂³×D₇

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
1. C₂×#1157	C₂	semidirect
2. C₂²×C₄×D₇	C₂	semidirect
3. C₇×#117	C₂	semidirect
4. #6460	C₂	semidirect
5. #6458	C₂	semidirect
6. #6449	C₂	semidirect
7. #6445	C₂	semidirect
8. C₄×D₂₈	C₂	semidirect
15. C₂×D₂₈	C₂²	semidirect
19. C₂×C₄×D₇	C₂²	semidirect
20. C₂×Dic₁₄	C₂²	semidirect
21. C₇×C₄⋊C₄	C₂²	semidirect
23. C₄²×C₇	C₂²	semidirect
25. (C₂×D₇)⋊C₄	C₂²	semidirect
26. C₄⋊Dic₇	C₂²	semidirect
33. C₂×C₄×C₇	D₄	semidirect
54. #117	D₇	semidirect
63. C₂×D₇	#41	semidirect
82. C₄⋊C₄	C₂×D₇	semidirect
86. C₄²	C₂×D₇	semidirect
107. C₂×C₄	D₂₈	semidirect
111. C₇	#525	semidirect

H ≅ G/N

type

1. C₂×#1157

C₂

semidirect

2. C₂²×C₄×D₇

C₂

semidirect

3. C₇×#117

C₂

semidirect

4. #6460

C₂

semidirect

5. #6458

C₂

semidirect

6. #6449

C₂

semidirect

7. #6445

C₂

semidirect

8. C₄×D₂₈

C₂

semidirect

15. C₂×D₂₈

C₂²

semidirect

19. C₂×C₄×D₇

C₂²

semidirect

20. C₂×Dic₁₄

C₂²

semidirect

21. C₇×C₄⋊C₄

C₂²

semidirect

23. C₄²×C₇

C₂²

semidirect

25. (C₂×D₇)⋊C₄

C₂²

semidirect

26. C₄⋊Dic₇

C₂²

semidirect

33. C₂×C₄×C₇

D₄

semidirect

54. #117

D₇

semidirect

63. C₂×D₇

#41

semidirect

82. C₄⋊C₄

C₂×D₇

semidirect

86. C₄²

C₂×D₇

semidirect

107. C₂×C₄

D₂₈

semidirect

111. C₇

#525

semidirect

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

subgroup	index
C₂×#1157	2
C₂²×C₄×D₇	2
C₇×#117	2
#6460	2
#6458	2
#6449	2
#6445	2
C₄×D₂₈	2
#525	7

subgroup

index

C₂²×C₄×D₇

C₄×D₂₈

Subgroups — #90215

Conjugacy classes of subgroups

Complements

Maximal subgroups