Group #11735985 — subgroups

Subgroups — #11735985

Order 896.

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

normal subgroups only

This group has 358 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	#90161	C₂	1	2	yes	maximal
2	C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#89909	C₂	1	4	yes	maximal
4	#89488	C₂	1	4	yes	maximal, complemented
5	#89471	C₂	1	1	yes	maximal, characteristic, complemented
6	#89453	C₂	1	2	yes	maximal, complemented
7	#89452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	#6489	—	2	2	no	—
11	#6487	—	2	2	no	—
12	#6487	C₂²	1	4	yes	—
13	C₂×Dic₂₈	C₂²	1	4	yes	—
14	#6433	—	2	2	no	—
15	#6432	C₂²	1	2	yes	—
16	#6432	C₂²	1	2	yes	—
17	C₄×Dic₁₄	—	2	2	no	—
18	C₄×Dic₁₄	C₂²	1	4	yes	—
19	C₇×#97	C₂²	1	2	yes	—
20	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	#6391	C₂²	1	4	yes	—
23	#6390	C₂²	1	8	yes	—
24	#3185	—	7	1	no	maximal, Sylow
25	C₂²×C₄×C₇	D₄	1	2	yes	—
26	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₄	—	2	2	no	—
28	C₂×Dic₁₄	C₂³	1	4	yes	—
29	C₇×M₁₆	—	2	4	no	—
30	C₂×C₇×C₈	C₂³	1	2	yes	—
31	C₂×C₇×C₈	C₂³	1	2	yes	—
32	C₄²×C₇	D₄	1	2	yes	—
33	C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₇	—	2	4	no	—
36	C₄⋊Dic₇	—	2	2	no	—
37	C₄⋊Dic₇	—	2	2	no	—
38	C₄⋊Dic₇	—	2	2	no	—
39	C₄⋊Dic₇	C₂³	1	4	yes	—
40	Dic₇⋊C₄	—	2	4	no	—
41	Dic₇⋊C₄	—	2	4	no	—
42	C₄×Dic₇	—	2	4	no	—
43	Dic₂₈	—	2	8	no	—
44	#533	—	7	2	no	—
45	#498	—	7	1	no	—
46	#494	—	7	1	no	—
47	#487	—	7	2	no	—
48	#470	—	7	4	no	—
49	#441	—	7	4	no	—
50	C₄×M₁₆	D₇	1	1	yes	characteristic
51	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇	—	2	4	no	—
53	C₂×C₄×C₇	—	2	2	no	—
54	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₇	—	2	4	no	—
59	C₂×Dic₇	—	2	4	no	—
60	Dic₁₄	—	2	8	no	—
61	Dic₁₄	—	4	4	no	—
62	C₇×C₈	—	2	2	no	—
63	C₇×C₈	#41	1	4	yes	—
64	C₂×Q₁₆	—	7	4	no	—
65	C₂×M₁₆	C₂×D₇	1	2	yes	—
66	C₄⋊Q₈	—	7	2	no	—
67	C₄⋊Q₈	—	7	2	no	—
68	#125	—	14	2	no	—
69	C₂²⋊Q₈	—	7	4	no	—
70	C₂²⋊Q₈	—	14	2	no	—
71	C₄×Q₈	—	7	4	no	—
72	C₄×Q₈	—	14	2	no	—
73	#117	—	14	2	no	—
74	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
75	#106	—	7	4	no	—
76	#103	—	7	8	no	—
77	#97	C₂×D₇	1	2	yes	—
78	C₄×C₈	C₂×D₇	1	1	yes	characteristic
79	C₄×C₈	C₂×D₇	1	1	yes	characteristic
80	C₂²×C₇	—	2	1	no	—
81	C₂²×C₇	—	2	2	no	—
82	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₇	—	2	1	no	—
84	C₄×C₇	—	2	2	no	—
85	C₄×C₇	#137	1	4	yes	—
86	C₄×C₇	C₂×#41	1	2	yes	—
87	Dic₇	—	4	4	no	—
88	Dic₇	—	4	4	no	—
89	C₂×Q₈	—	7	4	no	—
90	C₂×Q₈	—	14	2	no	—
91	C₂²×C₄	D₂₈	1	2	yes	—
92	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
93	Q₁₆	—	14	8	no	—
94	M₁₆	—	2	4	no	—
95	C₂×C₈	C₂²×D₇	1	2	yes	—
96	C₂×C₈	C₂²×D₇	1	2	yes	—
97	C₄⋊C₄	—	7	4	no	—
98	C₄⋊C₄	—	14	4	no	—
99	C₄⋊C₄	—	14	4	no	—
100	C₄⋊C₄	—	14	2	no	—
101	C₄⋊C₄	—	14	2	no	—
102	C₄⋊C₄	—	14	2	no	—
103	C₂²⋊C₄	—	14	4	no	—
104	C₄²	—	14	4	no	—
105	C₄²	D₂₈	1	2	yes	—
106	C₄²	C₂²×D₇	1	1	yes	characteristic
107	C₄²	C₂²×D₇	1	1	yes	characteristic
108	C₂×C₇	—	4	1	no	—
109	C₂×C₇	#532	1	1	yes	characteristic
110	C₂×C₇	C₂×#137	1	2	yes	—
111	C₂³	C₂×D₂₈	1	1	yes	characteristic
112	Q₈	—	14	8	no	—
113	Q₈	—	28	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	14	4	no	—
117	C₂×C₄	—	14	4	no	—
118	C₂×C₄	C₂×D₂₈	1	2	yes	—
119	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
120	C₂×C₄	C₂×D₂₈	1	2	yes	—
121	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	#1157	1	4	yes	—
124	C₇	#3185	1	1	yes	characteristic, complemented, Sylow
125	C₂²	—	2	1	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂²×D₂₈	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	28	4	no	—
131	C₄	—	28	4	no	—
132	C₄	#6472	1	4	yes	—
133	C₄	C₂×#1157	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#90168	1	1	yes	characteristic
136	C₂	C₂×#6472	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#90161

C₂

yes

maximal

C₄×C₇×M₁₆

C₂

yes

maximal, characteristic

#89909

C₂

yes

maximal

#89488

C₂

yes

maximal, complemented

#89471

C₂

yes

maximal, characteristic, complemented

#89453

C₂

yes

maximal, complemented

#89452

C₂

yes

maximal, characteristic, complemented

C₂×C₇×M₁₆

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₂₈

C₂²

yes

—

#6433

—

#6432

C₂²

yes

—

#6432

C₂²

yes

—

C₄×Dic₁₄

—

C₄×Dic₁₄

C₂²

yes

—

C₇×#97

C₂²

yes

—

C₄×C₇×C₈

C₂²

yes

characteristic

C₄×C₇×C₈

C₂²

yes

characteristic

#6391

C₂²

yes

—

#6390

C₂²

yes

—

#3185

—

maximal, Sylow

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

C₂³

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

C₂³

yes

—

C₂×C₇×C₈

C₂³

yes

—

C₄²×C₇

D₄

yes

—

C₄²×C₇

C₂³

yes

characteristic

C₄²×C₇

C₂³

yes

characteristic

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂³

yes

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

Dic₂₈

—

—

—

—

—

—

—

C₄×M₁₆

D₇

yes

characteristic

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

#41

yes

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₇

yes

—

—

C₂×D₇

yes

—

C₄×C₈

C₂×D₇

yes

characteristic

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#137

yes

—

C₄×C₇

C₂×#41

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₂₈

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂²×D₇

yes

—

C₂×C₈

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

D₂₈

yes

—

106

C₄²

C₂²×D₇

yes

characteristic

107

C₄²

C₂²×D₇

yes

characteristic

108

C₂×C₇

—

109

C₂×C₇

#532

yes

characteristic

110

C₂×C₇

C₂×#137

yes

—

111

C₂³

C₂×D₂₈

yes

characteristic

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×D₂₈

yes

—

119

C₂×C₄

C₂×D₂₈

yes

characteristic

120

C₂×C₄

C₂×D₂₈

yes

—

121

C₂×C₄

C₂³×D₇

yes

characteristic

122

C₈

—

123

C₈

#1157

yes

—

124

C₇

#3185

yes

characteristic, complemented, Sylow

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₂₈

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#6472

yes

—

133

C₄

C₂×#1157

yes

—

134

C₂

—

135

C₂

#90168

yes

characteristic

136

C₂

C₂×#6472

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₄×C₇×M₁₆	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂³×C₇	C₂×D₄
hypercenter	C₄×M₁₆	D₇
hypocenter	C₇	#3185
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₂₈

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₄×C₇×M₁₆

C₂

socle

C₂²×C₇

C₂²×D₄

prime radical

C₂³×C₇

C₂×D₄

hypercenter

C₄×M₁₆

D₇

hypocenter

C₇

#3185

LCS

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
4. #89488	C₂	semidirect
5. #89471	C₂	semidirect
6. #89453	C₂	semidirect
7. #89452	C₂	semidirect
124. C₇	#3185	semidirect

H ≅ G/N

type

4. #89488

C₂

semidirect

5. #89471

C₂

semidirect

6. #89453

C₂

semidirect

7. #89452

C₂

semidirect

124. C₇

#3185

semidirect

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

subgroup	index
#90161	2
C₄×C₇×M₁₆	2
#89909	2
#89488	2
#89471	2
#89453	2
#89452	2
#3185	7

subgroup

index

C₄×C₇×M₁₆

Subgroups — #11735985

Conjugacy classes of subgroups

Complements

Maximal subgroups