Group #11727395 — subgroups

Subgroups — #11727395

Order 896.

258 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₇×#390	C₂	1	1	yes	maximal, characteristic
2	C₂×#6406	C₂	1	2	yes	maximal
3	#89708	C₂	1	1	yes	maximal, characteristic, complemented
4	#89324	C₂	1	2	yes	maximal, complemented
5	#89319	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
12	#6406	—	2	4	no	—
13	C₇⋊#95	—	2	2	no	—
14	#2073	—	7	1	no	maximal, Sylow
15	C₂⁴×C₇	—	2	1	no	—
16	C₂×C₇×D₄	—	2	2	no	—
17	C₂×C₇×D₄	—	2	2	no	—
18	C₂×C₇×D₄	—	2	1	no	—
19	C₂×C₇×D₄	D₄	1	2	yes	—
20	C₂²×C₄×C₇	—	2	1	no	—
21	C₂²×C₄×C₇	D₄	1	2	yes	—
22	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₇	—	2	2	no	—
24	C₇×C₄⋊C₄	—	2	2	no	—
25	C₇×C₂²⋊C₄	—	4	4	no	—
26	C₄²×C₇	—	2	2	no	—
27	C₄⋊Dic₇	—	2	2	no	—
28	C₄⋊Dic₇	—	2	2	no	—
29	C₂×C₇⋊C₈	—	2	2	no	—
30	C₂×C₇⋊C₈	—	2	2	no	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×#102	—	7	2	no	—
33	#390	D₇	1	1	yes	characteristic
34	#382	—	7	1	no	—
35	#340	—	7	2	no	—
36	#336	—	7	1	no	—
37	C₂³×C₇	—	2	2	no	—
38	C₂³×C₇	—	4	1	no	—
39	C₂³×C₇	—	4	1	no	—
40	C₂³×C₇	—	4	2	no	—
41	C₂³×C₇	—	4	2	no	—
42	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
43	C₇×D₄	—	2	4	no	—
44	C₇×D₄	—	4	2	no	—
45	C₂×C₄×C₇	—	2	2	no	—
46	C₂×C₄×C₇	—	2	2	no	—
47	C₂×C₄×C₇	—	2	2	no	—
48	C₂×C₄×C₇	—	4	1	no	—
49	C₂×C₄×C₇	—	4	2	no	—
50	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₇	#41	1	2	yes	—
53	C₂×C₄×C₇	#41	1	2	yes	—
54	C₂×Dic₇	—	4	2	no	—
55	C₂×Dic₇	—	4	2	no	—
56	C₂×Dic₇	—	4	2	no	—
57	C₇⋊C₈	—	4	2	no	—
58	C₂²×D₄	C₂×D₇	1	1	yes	characteristic
59	C₂²×C₈	—	7	2	no	—
60	C₂×C₄⋊C₄	—	7	2	no	—
61	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
62	C₂×C₂²⋊C₄	—	2	2	no	—
63	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
64	#102	—	14	4	no	—
65	#95	—	14	2	no	—
66	C₂²×C₇	—	4	2	no	—
67	C₂²×C₇	—	4	4	no	—
68	C₂²×C₇	—	8	2	no	—
69	C₂²×C₇	—	8	1	no	—
70	C₂²×C₇	—	8	1	no	—
71	C₂²×C₇	#121	1	1	yes	characteristic
72	C₂²×C₇	#123	1	1	yes	characteristic
73	C₂²×C₇	#124	1	1	yes	characteristic
74	C₂²×C₇	#135	1	2	yes	—
75	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₄×C₇	—	4	2	no	—
77	C₄×C₇	—	8	1	no	—
78	C₄×C₇	#123	1	2	yes	—
79	C₄×C₇	#126	1	2	yes	—
80	Dic₇	—	8	2	no	—
81	C₂⁴	—	2	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	D₂₈	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	14	2	no	—
88	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
90	C₂×C₈	—	14	2	no	—
91	C₂×C₈	—	14	2	no	—
92	C₂×C₈	—	14	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	14	2	no	—
95	C₄⋊C₄	—	14	2	no	—
96	C₂²⋊C₄	—	4	4	no	—
97	C₄²	—	2	2	no	—
98	C₂×C₇	—	8	2	no	—
99	C₂×C₇	#397	1	1	yes	characteristic
100	C₂×C₇	#463	1	2	yes	—
101	C₂×C₇	#482	1	2	yes	—
102	C₂×C₇	#488	1	1	yes	characteristic
103	C₂×C₇	#490	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	4	2	no	—
109	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
110	D₄	—	2	4	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	28	2	no	—
118	C₂×C₄	—	28	2	no	—
119	C₂×C₄	—	28	2	no	—
120	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
121	C₂×C₄	#1157	1	1	yes	characteristic
122	C₂×C₄	#1159	1	2	yes	—
123	C₂×C₄	#1159	1	2	yes	—
124	C₈	—	28	2	no	—
125	C₇	#2073	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	4	2	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	8	1	no	—
131	C₂²	#6493	1	1	yes	characteristic
132	C₂²	#6495	1	2	yes	—
133	C₂²	#6498	1	1	yes	characteristic
134	C₂²	#6499	1	1	yes	characteristic
135	C₂²	#6513	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	56	2	no	—
139	C₄	#6442	1	2	yes	—
140	C₄	#6449	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	#89725	1	1	yes	characteristic
143	C₂	#89788	1	2	yes	—
144	C₂	#89807	1	2	yes	—
145	C₂	#89828	1	1	yes	characteristic
146	C₂	#89843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#390

C₂

yes

maximal, characteristic

C₂×#6406

C₂

yes

maximal

#89708

C₂

yes

maximal, characteristic, complemented

#89324

C₂

yes

maximal, complemented

#89319

C₂

yes

maximal, characteristic, complemented

C₂²×C₇×D₄

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₇×C₂²⋊C₄

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

—

—

—

—

C₂⁴×C₇

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

D₄

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#102

—

#390

D₇

yes

—

—

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇⋊C₈

—

C₂²×D₄

C₂×D₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#102

—

#95

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₂²×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#123

yes

—

C₄×C₇

#126

yes

—

Dic₇

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₂₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

#397

yes

characteristic

100

C₂×C₇

#463

yes

—

101

C₂×C₇

#482

yes

—

102

C₂×C₇

#488

yes

characteristic

103

C₂×C₇

#490

yes

characteristic

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₇

yes

characteristic

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₂₈

yes

characteristic

121

C₂×C₄

#1157

yes

characteristic

122

C₂×C₄

#1159

yes

—

123

C₂×C₄

#1159

yes

—

124

C₈

—

125

C₇

#2073

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#6493

yes

characteristic

132

C₂²

#6495

yes

—

133

C₂²

#6498

yes

characteristic

134

C₂²

#6499

yes

characteristic

135

C₂²

#6513

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#6442

yes

—

140

C₄

#6449

yes

—

141

C₂

—

142

C₂

#89725

yes

characteristic

143

C₂

#89788

yes

—

144

C₂

#89807

yes

—

145

C₂

#89828

yes

characteristic

146

C₂

#89843

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#390	C₂
socle	C₂³×C₇	C₂×D₄
prime radical	C₂²×C₇×D₄	C₂²
hypercenter	#390	D₇
hypocenter	C₇	#2073
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#390

C₂

socle

C₂³×C₇

C₂×D₄

prime radical

C₂²×C₇×D₄

C₂²

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
3. #89708	C₂	semidirect
4. #89324	C₂	semidirect
5. #89319	C₂	semidirect
125. C₇	#2073	semidirect

H ≅ G/N

type

3. #89708

C₂

semidirect

4. #89324

C₂

semidirect

5. #89319

C₂

semidirect

125. C₇

#2073

semidirect

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

subgroup	index
C₇×#390	2
C₂×#6406	2
#89708	2
#89324	2
#89319	2
#2073	7

subgroup

index

Subgroups — #11727395

Conjugacy classes of subgroups

Complements

Maximal subgroups