Group #11727439 — subgroups

Subgroups — #11727439

Order 896.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#391	C₂	1	1	yes	maximal, characteristic
2	C₂×#6409	C₂	1	2	yes	maximal
3	C₂×#6382	C₂	1	2	yes	maximal
4	#89707	C₂	1	1	yes	maximal, characteristic
5	#89319	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	—	2	2	no	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
12	C₇×#95	—	2	2	no	—
13	#6409	—	2	4	no	—
14	#6382	—	2	4	no	—
15	#2096	—	7	1	no	maximal, Sylow
16	C₂×C₇×Q₈	—	2	1	no	—
17	C₂×C₇×Q₈	—	2	2	no	—
18	C₂×C₇×Q₈	—	2	2	no	—
19	C₂×C₇×Q₈	Q₈	1	2	yes	—
20	C₂²×C₄×C₇	—	2	1	no	—
21	C₂²×C₄×C₇	—	2	1	no	—
22	C₂²×C₄×C₇	D₄	1	2	yes	—
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₇	—	2	2	no	—
25	C₇×C₄⋊C₄	—	2	2	no	—
26	C₄²×C₇	—	2	2	no	—
27	C₄⋊Dic₇	—	2	2	no	—
28	C₄⋊Dic₇	—	2	2	no	—
29	C₄⋊Dic₇	—	4	4	no	—
30	C₂×C₇⋊C₈	—	2	2	no	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×C₇⋊C₈	D₄	1	4	yes	—
33	C₂×#106	—	7	2	no	—
34	C₂×#103	—	7	2	no	—
35	#391	D₇	1	1	yes	characteristic
36	#384	—	7	1	no	—
37	#336	—	7	1	no	—
38	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
39	C₇×Q₈	—	2	4	no	—
40	C₇×Q₈	—	4	2	no	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	—	2	2	no	—
43	C₂×C₄×C₇	—	2	2	no	—
44	C₂×C₄×C₇	—	4	1	no	—
45	C₂×C₄×C₇	—	4	1	no	—
46	C₂×C₄×C₇	—	4	1	no	—
47	C₂×C₄×C₇	—	4	2	no	—
48	C₂×C₄×C₇	SD₁₆	1	4	yes	—
49	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₇	#41	1	2	yes	—
53	C₂×Dic₇	—	4	2	no	—
54	C₂×Dic₇	—	4	4	no	—
55	C₇⋊C₈	—	2	4	no	—
56	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
57	C₂²×C₈	—	7	2	no	—
58	C₂×C₄⋊C₄	—	7	2	no	—
59	C₂×C₄⋊C₄	—	14	2	no	—
60	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
61	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
62	#106	—	14	4	no	—
63	#103	—	14	4	no	—
64	#95	—	2	2	no	—
65	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂²×C₇	#123	1	1	yes	characteristic
67	C₂²×C₇	#127	1	1	yes	characteristic
68	C₂²×C₇	C₂×SD₁₆	1	2	yes	—
69	C₂²×C₇	#137	1	2	yes	—
70	C₄×C₇	—	2	4	no	—
71	C₄×C₇	—	4	2	no	—
72	C₄×C₇	—	8	1	no	—
73	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₇	#123	1	2	yes	—
75	Dic₇	—	8	2	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	Dic₁₄	1	2	yes	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	14	2	no	—
83	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
84	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
85	C₂×C₈	—	7	4	no	—
86	C₂×C₈	—	14	2	no	—
87	C₂×C₈	—	14	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	14	2	no	—
90	C₄⋊C₄	—	14	2	no	—
91	C₄⋊C₄	—	28	4	no	—
92	C₄²	—	2	2	no	—
93	C₂×C₇	#399	1	1	yes	characteristic
94	C₂×C₇	#475	1	2	yes	—
95	C₂×C₇	#483	1	2	yes	—
96	C₂×C₇	#492	1	1	yes	characteristic
97	C₂×C₇	#497	1	1	yes	characteristic
98	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
99	Q₈	—	2	4	no	—
100	Q₈	—	4	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	28	2	no	—
109	C₂×C₄	—	28	4	no	—
110	C₂×C₄	#1143	1	4	yes	—
111	C₂×C₄	C₂×Dic₁₄	1	1	yes	characteristic
112	C₂×C₄	#1157	1	1	yes	characteristic
113	C₂×C₄	D₄×D₇	1	2	yes	—
114	C₂×C₄	#1159	1	2	yes	—
115	C₈	—	14	4	no	—
116	C₇	#2096	1	1	yes	characteristic, complemented, Sylow
117	C₂²	#6487	1	1	yes	characteristic
118	C₂²	#6498	1	1	yes	characteristic
119	C₂²	#6503	1	1	yes	characteristic
120	C₂²	C₂×#1143	1	2	yes	—
121	C₂²	#6514	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	56	2	no	—
126	C₄	#6441	1	2	yes	—
127	C₄	#6446	1	2	yes	—
128	C₂	#89721	1	1	yes	characteristic
129	C₂	#89792	1	2	yes	—
130	C₂	#89815	1	2	yes	—
131	C₂	#89832	1	1	yes	characteristic
132	C₂	#89857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#391

C₂

yes

maximal, characteristic

C₂×#6409

C₂

yes

maximal

C₂×#6382

C₂

yes

maximal

#89707

C₂

yes

maximal, characteristic

#89319

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

—

C₇×#95

—

—

—

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

Q₈

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

D₄

yes

—

C₂×#106

—

C₂×#103

—

#391

D₇

yes

characteristic

#384

—

#336

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

SD₁₆

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇⋊C₈

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₄²

C₂×D₇

yes

—

—

—

C₂²×C₇

C₂²⋊Q₈

yes

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

C₂×SD₁₆

yes

—

C₂²×C₇

#137

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

#123

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₄

yes

—

C₂²×C₄

—

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

—

C₄⋊C₄

—

C₄²

—

C₂×C₇

#399

yes

characteristic

C₂×C₇

#475

yes

—

C₂×C₇

#483

yes

—

C₂×C₇

yes

C₂×C₇

yes

C₂³

C₂×C₂²⋊D₇

yes

characteristic

Q₈

—

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#1143

yes

—

111

C₂×C₄

C₂×Dic₁₄

yes

characteristic

112

C₂×C₄

#1157

yes

characteristic

113

C₂×C₄

D₄×D₇

yes

—

114

C₂×C₄

#1159

yes

—

115

C₈

—

116

C₇

#2096

yes

characteristic, complemented, Sylow

117

C₂²

#6487

yes

characteristic

118

C₂²

#6498

yes

characteristic

119

C₂²

#6503

yes

characteristic

120

C₂²

C₂×#1143

yes

—

121

C₂²

#6514

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#6441

yes

—

127

C₄

#6446

yes

—

128

C₂

#89721

yes

characteristic

129

C₂

#89792

yes

—

130

C₂

#89815

yes

—

131

C₂

#89832

yes

characteristic

132

C₂

#89857

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#391	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
hypercenter	#391	D₇
hypocenter	C₇	#2096
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#391

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
116. C₇	#2096	semidirect

H ≅ G/N

type

116. C₇

#2096

semidirect

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

subgroup	index
C₇×#391	2
C₂×#6409	2
C₂×#6382	2
#89707	2
#89319	2
#2096	7

subgroup

index

Subgroups — #11727439

Conjugacy classes of subgroups

Complements

Maximal subgroups