Group #11727423 — subgroups

Subgroups — #11727423

Order 896.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#6433	C₂	1	1	yes	maximal, characteristic
2	C₇×#391	C₂	1	1	yes	maximal, characteristic
3	C₂×#6409	C₂	1	2	yes	maximal
4	#89324	C₂	1	2	yes	maximal
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	C₂²	1	2	yes	—
10	C₂×Dic₇⋊C₄	—	2	2	no	—
11	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
12	#6433	—	2	4	no	—
13	C₇×#95	—	2	2	no	—
14	#6409	—	2	4	no	—
15	#2026	—	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	D₄	1	4	yes	—
29	Dic₇⋊C₄	—	2	8	no	—
30	C₂×C₇⋊C₈	—	2	2	no	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×C₇⋊C₈	—	2	2	no	—
33	C₂×#125	—	7	1	no	—
34	C₂×#103	—	7	2	no	—
35	#391	D₇	1	1	yes	characteristic
36	#340	—	7	2	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	—	2	2	no	—
45	C₂×C₄×C₇	—	4	1	no	—
46	C₂×C₄×C₇	—	4	1	no	—
47	C₂×C₄×C₇	—	4	1	no	—
48	C₂×C₄×C₇	—	4	2	no	—
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	—	2	4	no	—
54	C₂×Dic₇	—	4	2	no	—
55	C₂×Dic₇	—	4	2	no	—
56	C₇⋊C₈	—	4	2	no	—
57	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
58	C₂²×C₈	—	7	2	no	—
59	#125	—	14	4	no	—
60	C₂×C₄⋊C₄	—	7	2	no	—
61	C₂×C₄⋊C₄	—	14	2	no	—
62	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
64	#103	—	14	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₇	#120	1	1	yes	characteristic
67	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₇	#124	1	1	yes	characteristic
69	C₂²×C₇	#135	1	2	yes	—
70	C₂²×C₇	#137	1	2	yes	—
71	C₄×C₇	—	4	2	no	—
72	C₄×C₇	—	4	2	no	—
73	C₄×C₇	—	8	1	no	—
74	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₇	#124	1	2	yes	—
76	Dic₇	—	4	4	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	Dic₁₄	1	2	yes	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	14	2	no	—
84	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
86	C₂×C₈	—	14	2	no	—
87	C₂×C₈	—	14	2	no	—
88	C₂×C₈	—	14	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	7	4	no	—
91	C₄⋊C₄	—	14	8	no	—
92	C₄⋊C₄	—	14	2	no	—
93	C₄²	—	2	2	no	—
94	C₂×C₇	#393	1	1	yes	characteristic
95	C₂×C₇	#452	1	2	yes	—
96	C₂×C₇	#479	1	2	yes	—
97	C₂×C₇	#488	1	1	yes	characteristic
98	C₂×C₇	#491	1	1	yes	characteristic
99	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	14	4	no	—
111	C₂×C₄	—	28	2	no	—
112	C₂×C₄	—	28	2	no	—
113	C₂×C₄	C₂×Dic₁₄	1	1	yes	characteristic
114	C₂×C₄	#1157	1	1	yes	characteristic
115	C₂×C₄	D₄×D₇	1	2	yes	—
116	C₂×C₄	#1159	1	2	yes	—
117	C₈	—	28	2	no	—
118	C₇	#2026	1	1	yes	characteristic, complemented, Sylow
119	C₂²	#6487	1	1	yes	characteristic
120	C₂²	#6499	1	1	yes	characteristic
121	C₂²	#6500	1	1	yes	characteristic
122	C₂²	#6505	1	2	yes	—
123	C₂²	#6513	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	—	8	1	no	—
127	C₄	—	28	4	no	—
128	C₄	#6441	1	2	yes	—
129	C₄	#6448	1	2	yes	—
130	C₂	#89719	1	1	yes	characteristic
131	C₂	#89794	1	2	yes	—
132	C₂	#89819	1	2	yes	—
133	C₂	#89828	1	1	yes	characteristic
134	C₂	#89854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6433

C₂

yes

maximal, characteristic

C₇×#391

C₂

yes

maximal, characteristic

C₂×#6409

C₂

yes

maximal

#89324

C₂

yes

maximal

#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

C₂²

yes

—

C₂×Dic₇⋊C₄

—

C₂²×C₇⋊C₈

C₂²

yes

—

#6433

—

C₇×#95

—

#6409

—

#2026

—

maximal, Sylow

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

D₄

yes

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#125

—

C₂×#103

—

#391

D₇

yes

characteristic

#340

—

#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

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇⋊C₈

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₄²

C₂×D₇

yes

characteristic

#103

—

#95

—

C₂²×C₇

#120

yes

characteristic

C₂²×C₇

C₂²⋊Q₈

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₂²×C₇

#137

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

#124

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

#393

yes

characteristic

C₂×C₇

#452

yes

—

C₂×C₇

#479

yes

—

C₂×C₇

yes

C₂×C₇

yes

C₂³

C₂×C₂²⋊D₇

yes

characteristic

100

Q₈

—

101

Q₈

—

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

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₁₄

yes

characteristic

114

C₂×C₄

#1157

yes

characteristic

115

C₂×C₄

D₄×D₇

yes

—

116

C₂×C₄

#1159

yes

—

117

C₈

—

118

C₇

#2026

yes

characteristic, complemented, Sylow

119

C₂²

#6487

yes

characteristic

120

C₂²

#6499

yes

characteristic

121

C₂²

#6500

yes

characteristic

122

C₂²

#6505

yes

—

123

C₂²

#6513

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#6441

yes

—

129

C₄

#6448

yes

—

130

C₂

#89719

yes

characteristic

131

C₂

#89794

yes

—

132

C₂

#89819

yes

—

133

C₂

#89828

yes

characteristic

134

C₂

#89854

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	#2026
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
118. C₇	#2026	semidirect

H ≅ G/N

type

118. C₇

#2026

semidirect

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

subgroup	index
C₂×#6433	2
C₇×#391	2
C₂×#6409	2
#89324	2
#89319	2
#2026	7

subgroup

index

Subgroups — #11727423

Conjugacy classes of subgroups

Complements

Maximal subgroups