Group #11726985 — subgroups

Subgroups — #11726985

Order 896.

140 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 47 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	#89960	C₂	1	1	yes	maximal, characteristic
2	D₇×#107	C₂	1	1	yes	maximal, characteristic
3	C₇×#358	C₂	1	1	yes	maximal, characteristic, complemented
4	#89359	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#89300	C₂	1	1	yes	maximal, characteristic, complemented
7	#89285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
9	#6509	—	2	1	no	—
10	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
11	D₇×C₄⋊C₄	—	2	1	no	—
12	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₇×#107	C₂²	1	1	yes	characteristic
14	#6409	—	2	1	no	—
15	#6392	C₂²	1	1	yes	characteristic
16	#6392	C₂²	1	1	yes	characteristic
17	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	#6381	—	2	1	no	—
19	#2235	—	7	1	no	maximal, Sylow
20	C₂×C₇×Q₈	—	2	1	no	—
21	C₂×C₄×D₇	—	4	1	no	—
22	C₂×C₄×D₇	D₄	1	1	yes	characteristic
23	C₇×Q₁₆	—	2	1	no	—
24	C₇×Q₁₆	—	2	1	no	—
25	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₇×C₄⋊C₄	—	2	1	no	—
27	(C₂×D₇)⋊C₄	—	4	1	no	—
28	C₄⋊Dic₇	—	2	1	no	—
29	C₄⋊Dic₇	—	2	1	no	—
30	Dic₇⋊C₄	—	4	1	no	—
31	Dic₇⋊C₄	—	4	1	no	—
32	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
33	C₈×D₇	—	2	1	no	—
34	C₈×D₇	—	2	1	no	—
35	C₇×C₁₆	—	2	1	no	—
36	C₇⋊C₁₆	—	2	1	no	—
37	#467	—	7	1	no	—
38	C₂×#107	—	7	1	no	—
39	#367	—	7	1	no	—
40	#367	—	7	1	no	—
41	#358	—	7	1	no	—
42	#358	D₇	1	1	yes	characteristic, complemented
43	C₂²⋊C₁₆	—	7	1	no	—
44	C₂²×D₇	D₈	1	1	yes	characteristic
45	C₇×Q₈	—	4	1	no	—
46	C₇×Q₈	—	4	1	no	—
47	C₂×C₄×C₇	—	4	1	no	—
48	C₂×C₄×C₇	—	4	1	no	—
49	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₇	—	4	1	no	—
51	C₂×Dic₇	—	4	1	no	—
52	C₂×Dic₇	D₈	1	1	yes	characteristic
53	C₄×D₇	—	2	1	no	—
54	C₄×D₇	—	2	1	no	—
55	C₄×D₇	—	8	2	no	—
56	C₇×C₈	#41	1	1	yes	characteristic
57	C₇×C₈	#41	1	1	yes	characteristic
58	C₇⋊C₈	—	2	1	no	—
59	C₂×Q₁₆	C₂×D₇	1	1	yes	characteristic
60	C₂²×C₈	—	7	1	no	—
61	C₂²⋊Q₈	—	14	1	no	—
62	C₂×C₄⋊C₄	—	14	1	no	—
63	C₂×C₁₆	—	7	1	no	—
64	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
65	#107	—	7	1	no	—
66	#107	—	7	1	no	—
67	#107	—	14	1	no	—
68	#107	C₂×D₇	1	1	yes	characteristic
69	#103	—	14	1	no	—
70	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
71	C₂×D₇	—	2	1	no	—
72	C₂×D₇	—	2	1	no	—
73	C₂×D₇	SD₃₂	1	2	yes	—
74	C₄×C₇	—	8	1	no	—
75	C₄×C₇	—	8	1	no	—
76	C₄×C₇	#123	1	1	yes	characteristic
77	C₄×C₇	#137	1	1	yes	characteristic
78	Dic₇	—	2	1	no	—
79	Dic₇	—	8	1	no	—
80	Dic₇	—	8	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂²×C₄	—	7	1	no	—
83	C₂²×C₄	—	28	1	no	—
84	Q₁₆	—	2	1	no	—
85	Q₁₆	—	2	1	no	—
86	C₂×C₈	—	7	1	no	—
87	C₂×C₈	—	14	1	no	—
88	C₂×C₈	—	14	1	no	—
89	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	14	1	no	—
92	C₄⋊C₄	—	14	1	no	—
93	C₄⋊C₄	—	28	1	no	—
94	C₄⋊C₄	—	28	1	no	—
95	C₂²⋊C₄	—	28	1	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	14	1	no	—
98	C₂×C₇	#480	1	1	yes	characteristic
99	C₂×C₇	C₂×SD₃₂	1	1	yes	characteristic
100	C₂×C₇	#510	1	1	yes	characteristic
101	D₇	—	2	2	no	—
102	C₂³	—	7	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	7	1	no	—
108	C₂×C₄	—	14	1	no	—
109	C₂×C₄	—	14	1	no	—
110	C₂×C₄	—	28	1	no	—
111	C₂×C₄	—	28	1	no	—
112	C₂×C₄	—	56	2	no	—
113	C₂×C₄	D₄×D₇	1	1	yes	characteristic
114	C₈	—	14	1	no	—
115	C₈	#1157	1	1	yes	characteristic
116	C₈	#1159	1	1	yes	characteristic
117	C₇	#2235	1	1	yes	characteristic, complemented, Sylow
118	C₂²	—	7	2	no	—
119	C₂²	—	14	1	no	—
120	C₂²	—	14	1	no	—
121	C₂²	D₇×D₈	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	14	1	no	—
125	C₄	—	56	1	no	—
126	C₄	—	56	1	no	—
127	C₄	#6446	1	1	yes	characteristic
128	C₄	#6478	1	1	yes	characteristic
129	C₂	—	14	2	no	—
130	C₂	#89546	1	1	yes	characteristic
131	C₂	D₇×SD₃₂	1	1	yes	characteristic
132	C₂	#89690	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#89960

C₂

yes

maximal, characteristic

D₇×#107

C₂

yes

maximal, characteristic

C₇×#358

C₂

yes

maximal, characteristic, complemented

#89359

C₂

yes

maximal, characteristic, complemented

(C₂×D₇)⋊C₁₆

C₂

yes

maximal, characteristic

#89300

C₂

yes

maximal, characteristic, complemented

#89285

C₂

yes

maximal, characteristic, complemented

C₂×C₇×Q₁₆

C₂²

yes

characteristic

#6509

—

C₂×C₈×D₇

C₂²

yes

characteristic

D₇×C₄⋊C₄

—

C₂×C₇×C₁₆

C₂²

yes

characteristic

C₇×#107

C₂²

yes

characteristic

#6409

—

#6392

C₂²

yes

characteristic

#6392

C₂²

yes

characteristic

C₂×C₇⋊C₁₆

C₂²

yes

—

—

C₂×C₇×Q₈

—

C₂×C₄×D₇

—

C₂×C₄×D₇

D₄

yes

characteristic

C₇×Q₁₆

—

C₇×Q₁₆

—

C₂×C₇×C₈

C₂³

yes

characteristic

C₇×C₄⋊C₄

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

D₄

yes

characteristic

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂²×D₇

D₈

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

D₈

yes

characteristic

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

yes

C₇×C₈

yes

C₇⋊C₈

—

C₂×Q₁₆

C₂×D₇

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇

yes

—

—

—

C₂×D₇

yes

characteristic

#103

—

C₂²×C₇

C₂×D₈

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

SD₃₂

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₁₆

—

C₂×C₇

#480

yes

characteristic

C₂×C₇

C₂×SD₃₂

yes

characteristic

100

C₂×C₇

#510

yes

characteristic

101

D₇

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

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

D₄×D₇

yes

characteristic

114

C₈

—

115

C₈

#1157

yes

characteristic

116

C₈

#1159

yes

characteristic

117

C₇

#2235

yes

characteristic, complemented, Sylow

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

D₇×D₈

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#6446

yes

characteristic

128

C₄

#6478

yes

characteristic

129

C₂

—

130

C₂

#89546

yes

characteristic

131

C₂

D₇×SD₃₂

yes

characteristic

132

C₂

#89690

yes

characteristic

	subgroup	G/H
center	C₂²	D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#358	C₂
socle	C₂²×C₇	C₂×D₈
prime radical	C₂²×D₇	D₈
hypercenter	#358	D₇
hypocenter	C₇	#2235
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₇×D₈

commutator, cosocle

C₂×C₇×C₈

C₂³

Frattini

C₂×C₈

C₂²×D₇

Fitting

C₇×#358

C₂

socle

C₂²×C₇

C₂×D₈

prime radical

C₂²×D₇

D₈

D₇

C₇

G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
3. C₇×#358	C₂	semidirect
4. #89359	C₂	semidirect
6. #89300	C₂	semidirect
7. #89285	C₂	semidirect
42. #358	D₇	semidirect
117. C₇	#2235	semidirect

H ≅ G/N

type

3. C₇×#358

C₂

semidirect

4. #89359

C₂

semidirect

6. #89300

C₂

semidirect

7. #89285

C₂

semidirect

42. #358

D₇

semidirect

117. C₇

#2235

semidirect

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

subgroup	index
#89960	2
D₇×#107	2
C₇×#358	2
#89359	2
(C₂×D₇)⋊C₁₆	2
#89300	2
#89285	2
#2235	7

subgroup

index

(C₂×D₇)⋊C₁₆

Subgroups — #11726985

Conjugacy classes of subgroups

Complements

Maximal subgroups