Group #11731305 — subgroups

Subgroups — #11731305

Order 896.

712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6516	C₂	1	2	yes	maximal, complemented
2	C₇×#379	C₂	1	1	yes	maximal, characteristic, complemented
3	#89733	C₂	1	6	yes	maximal, complemented
4	#89732	C₂	1	3	yes	maximal, complemented
5	#89725	C₂	1	3	yes	maximal, complemented
6	C₂⁵×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊D₇	C₂²	1	6	yes	—
8	C₂²×D₂₈	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
10	C₂×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
11	#6516	—	2	8	no	—
12	C₂×C₂²⋊Dic₇	C₂²	1	6	yes	—
13	C₂×(C₂×D₇)⋊C₄	C₂²	1	6	yes	complemented
14	C₂×C₄⋊Dic₇	C₂²	1	3	yes	complemented
15	C₇⋊#95	C₂²	1	3	yes	complemented
16	C₇⋊#95	C₂²	1	3	yes	complemented
17	#2812	—	7	1	no	maximal, Sylow
18	C₂⁴×C₇	—	4	3	no	—
19	C₂⁴×C₇	—	4	3	no	—
20	C₂⁴×C₇	D₄	1	4	yes	complemented
21	C₂⁴×C₇	C₂³	1	3	yes	—
22	C₂³×D₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₇	C₂³	1	3	yes	—
25	C₂×C₂²⋊D₇	—	2	12	no	—
26	C₂×C₂²⋊D₇	—	4	6	no	—
27	C₂×C₂²⋊D₇	—	4	6	no	—
28	C₂²×Dic₇	C₂³	1	6	yes	—
29	C₂×D₂₈	—	4	3	no	—
30	C₇×C₂²⋊C₄	—	4	3	no	—
31	C₇×C₂²⋊C₄	—	4	3	no	—
32	C₂²⋊Dic₇	—	2	12	no	—
33	(C₂×D₇)⋊C₄	—	4	6	no	—
34	C₄⋊Dic₇	—	4	3	no	—
35	C₂×#120	—	7	2	no	—
36	#397	—	7	3	no	—
37	#394	—	7	6	no	—
38	#392	—	7	3	no	—
39	#379	D₇	1	1	yes	characteristic, complemented
40	C₂³×C₇	—	2	3	no	—
41	C₂³×C₇	—	2	3	no	—
42	C₂³×C₇	—	2	6	no	—
43	C₂³×C₇	—	2	6	no	—
44	C₂³×C₇	—	4	12	no	—
45	C₂³×C₇	—	4	12	no	—
46	C₂³×C₇	—	4	4	no	—
47	C₂³×C₇	C₂×D₄	1	6	yes	—
48	C₂³×C₇	C₂⁴	1	1	yes	characteristic
49	C₂²×D₇	—	2	2	no	—
50	C₂²×D₇	—	2	6	no	—
51	C₂²×D₇	—	2	6	no	—
52	C₂×C₄×C₇	—	2	3	no	—
53	C₂×C₄×C₇	—	2	6	no	—
54	C₂×C₄×C₇	—	2	3	no	—
55	C₂²⋊D₇	—	4	24	no	—
56	C₂×Dic₇	—	2	6	no	—
57	C₂×Dic₇	—	2	6	no	—
58	C₂×Dic₇	—	2	6	no	—
59	D₂₈	—	4	4	no	—
60	C₂⁵	C₂×D₇	1	1	yes	characteristic
61	C₂²×D₄	—	7	6	no	—
62	C₂²×D₄	—	7	1	no	—
63	#120	—	14	8	no	—
64	C₂×C₄⋊C₄	—	7	3	no	—
65	C₂×C₂²⋊C₄	—	7	6	no	—
66	C₂×C₂²⋊C₄	—	7	6	no	—
67	C₂×C₂²⋊C₄	C₂×D₇	1	3	yes	—
68	C₂×C₂²⋊C₄	C₂×D₇	1	3	yes	—
69	#95	—	7	3	no	—
70	#95	—	7	3	no	—
71	C₂²×C₇	—	2	12	no	—
72	C₂²×C₇	—	2	6	no	—
73	C₂²×C₇	—	4	16	no	—
74	C₂²×C₇	—	4	6	no	—
75	C₂²×C₇	—	4	3	no	—
76	C₂²×C₇	—	4	3	no	—
77	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
79	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
80	C₂×D₇	—	4	6	no	—
81	C₂×D₇	—	4	6	no	—
82	C₂×D₇	—	4	2	no	—
83	C₄×C₇	—	4	3	no	—
84	C₄×C₇	—	4	1	no	—
85	Dic₇	—	4	6	no	—
86	C₂⁴	—	4	3	no	—
87	C₂⁴	—	4	3	no	—
88	C₂⁴	—	7	2	no	—
89	C₂⁴	D₂₈	1	4	yes	complemented
90	C₂⁴	C₂²×D₇	1	3	yes	—
91	C₂×D₄	—	14	12	no	—
92	C₂×D₄	—	28	3	no	—
93	C₂×D₄	—	28	6	no	—
94	C₂×D₄	—	28	6	no	—
95	C₂²×C₄	—	7	6	no	—
96	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₇	1	3	yes	—
98	C₄⋊C₄	—	28	3	no	—
99	C₂²⋊C₄	—	4	3	no	—
100	C₂²⋊C₄	—	4	3	no	—
101	C₂²⋊C₄	—	14	12	no	—
102	C₂²⋊C₄	—	28	6	no	—
103	C₂×C₇	—	4	6	no	—
104	C₂×C₇	#534	1	3	yes	—
105	C₂×C₇	#560	1	3	yes	—
106	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
107	D₇	—	8	2	no	—
108	C₂³	—	2	3	no	—
109	C₂³	—	2	3	no	—
110	C₂³	—	2	6	no	—
111	C₂³	—	2	6	no	—
112	C₂³	—	4	12	no	—
113	C₂³	—	4	4	no	—
114	C₂³	—	4	12	no	—
115	C₂³	—	14	2	no	—
116	C₂³	—	14	6	no	—
117	C₂³	—	14	6	no	—
118	C₂³	C₂×D₂₈	1	6	yes	—
119	C₂³	C₂³×D₇	1	1	yes	characteristic
120	D₄	—	28	4	no	—
121	D₄	—	28	24	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	14	6	no	—
126	C₂×C₄	—	14	6	no	—
127	C₂×C₄	—	14	6	no	—
128	C₇	#2812	1	1	yes	characteristic, complemented, Sylow
129	C₂²	—	2	12	no	—
130	C₂²	—	2	6	no	—
131	C₂²	—	4	16	no	—
132	C₂²	—	4	3	no	—
133	C₂²	—	4	3	no	—
134	C₂²	—	4	6	no	—
135	C₂²	—	28	6	no	—
136	C₂²	—	28	2	no	—
137	C₂²	—	28	6	no	—
138	C₂²	C₂²×D₂₈	1	1	yes	characteristic
139	C₂²	#6548	1	3	yes	—
140	C₂²	#6548	1	3	yes	—
141	C₄	—	4	1	no	—
142	C₄	—	4	3	no	—
143	C₄	—	28	6	no	—
144	C₂	—	4	6	no	—
145	C₂	—	56	2	no	—
146	C₂	#90184	1	3	yes	—
147	C₂	#90285	1	1	yes	characteristic
148	C₂	#90286	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6516

C₂

yes

maximal, complemented

C₇×#379

C₂

yes

maximal, characteristic, complemented

#89733

C₂

yes

maximal, complemented

#89732

C₂

yes

maximal, complemented

#89725

C₂

yes

maximal, complemented

C₂⁵×C₇

C₂²

yes

characteristic

C₂²×C₂²⋊D₇

C₂²

yes

—

C₂²×D₂₈

C₂²

yes

characteristic, complemented

C₂×C₇×C₂²⋊C₄

C₂²

yes

—

C₂×C₇×C₂²⋊C₄

C₂²

yes

—

#6516

—

C₂×C₂²⋊Dic₇

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

complemented

C₂×C₄⋊Dic₇

C₂²

yes

complemented

C₇⋊#95

C₂²

yes

complemented

C₇⋊#95

C₂²

yes

complemented

#2812

—

maximal, Sylow

C₂⁴×C₇

—

C₂⁴×C₇

—

C₂⁴×C₇

D₄

yes

complemented

C₂⁴×C₇

C₂³

yes

—

C₂³×D₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂×C₂²⋊D₇

—

C₂×C₂²⋊D₇

—

C₂×C₂²⋊D₇

—

C₂²×Dic₇

C₂³

yes

—

C₂×D₂₈

—

C₇×C₂²⋊C₄

—

C₇×C₂²⋊C₄

—

C₂²⋊Dic₇

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂×D₄

yes

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

C₂⁵

C₂×D₇

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

#120

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

C₂×D₇

yes

—

#95

—

#95

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

D₂₈

yes

complemented

C₂⁴

C₂²×D₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂×C₇

—

104

C₂×C₇

#534

yes

—

105

C₂×C₇

#560

yes

—

106

C₂×C₇

H(3,𝔽₄)

yes

characteristic

107

D₇

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

C₂×D₂₈

yes

—

119

C₂³

C₂³×D₇

yes

characteristic

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₇

#2812

yes

characteristic, complemented, Sylow

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²×D₂₈

yes

characteristic

139

C₂²

#6548

yes

—

140

C₂²

#6548

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₂

—

145

C₂

—

146

C₂

#90184

yes

—

147

C₂

#90285

yes

characteristic

148

C₂

#90286

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#379	C₂
hypercenter	#379	D₇
hypocenter	C₇	#2812
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
1. C₂×#6516	C₂	semidirect
2. C₇×#379	C₂	semidirect
3. #89733	C₂	semidirect
4. #89732	C₂	semidirect
5. #89725	C₂	semidirect
8. C₂²×D₂₈	C₂²	semidirect
13. C₂×(C₂×D₇)⋊C₄	C₂²	semidirect
14. C₂×C₄⋊Dic₇	C₂²	semidirect
15. C₇⋊#95	C₂²	semidirect
16. C₇⋊#95	C₂²	semidirect
20. C₂⁴×C₇	D₄	semidirect
39. #379	D₇	semidirect
89. C₂⁴	D₂₈	semidirect
128. C₇	#2812	semidirect

H ≅ G/N

type

1. C₂×#6516

C₂

semidirect

2. C₇×#379

C₂

semidirect

3. #89733

C₂

semidirect

4. #89732

C₂

semidirect

5. #89725

C₂

semidirect

8. C₂²×D₂₈

C₂²

semidirect

13. C₂×(C₂×D₇)⋊C₄

C₂²

semidirect

14. C₂×C₄⋊Dic₇

C₂²

semidirect

15. C₇⋊#95

C₂²

semidirect

16. C₇⋊#95

C₂²

semidirect

20. C₂⁴×C₇

D₄

semidirect

39. #379

D₇

semidirect

89. C₂⁴

D₂₈

semidirect

128. C₇

#2812

semidirect

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

subgroup	index
C₂×#6516	2
C₇×#379	2
#89733	2
#89732	2
#89725	2
#2812	7

subgroup

index

Subgroups — #11731305

Conjugacy classes of subgroups

Complements

Maximal subgroups