Group #11726676 — subgroups

Subgroups — #11726676

Order 896.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#89905	C₂	1	2	yes	maximal
2	#89882	C₂	1	1	yes	maximal, characteristic
3	C₂×#6391	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#341	C₂	1	1	yes	maximal, characteristic, complemented
5	#89349	C₂	1	2	yes	maximal, complemented
6	C₂×#1157	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
10	C₂×#1136	C₂²	1	1	yes	characteristic
11	#6395	—	2	2	no	—
12	(C₂×D₇)⋊C₈	—	2	2	no	—
13	#6391	—	2	2	no	—
14	#6390	—	2	2	no	—
15	#2079	—	7	1	no	maximal, Sylow
16	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×C₂²⋊D₇	—	2	1	no	—
18	C₂²×Dic₇	—	2	2	no	—
19	#1157	—	2	2	no	—
20	#1157	—	2	2	no	—
21	C₂×D₂₈	D₄	1	1	yes	characteristic
22	C₂×C₄×D₇	—	2	1	no	—
23	C₂×Dic₁₄	D₄	1	1	yes	characteristic
24	C₇×M₁₆	—	2	2	no	—
25	C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×C₈	—	2	2	no	—
27	C₂×C₇×C₈	—	2	2	no	—
28	C₂×C₇×C₈	D₄	1	2	yes	—
29	C₄⋊Dic₇	—	2	2	no	—
30	C₄⋊Dic₇	—	2	2	no	—
31	#1136	—	2	2	no	—
32	C₂×C₇⋊C₈	—	2	1	no	—
33	C₂×#106	—	7	1	no	—
34	#417	—	7	2	no	—
35	#408	—	7	1	no	—
36	#343	—	7	2	no	—
37	#341	D₇	1	1	yes	characteristic, complemented
38	C₂³×C₇	#41	1	1	yes	characteristic
39	C₂²×D₇	—	2	1	no	—
40	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇	#41	1	2	yes	—
43	C₂×C₄×C₇	#41	1	2	yes	—
44	C₂²⋊D₇	—	4	2	no	—
45	C₂×Dic₇	—	2	1	no	—
46	C₂×Dic₇	—	4	2	no	—
47	C₂×Dic₇	—	8	2	no	—
48	D₂₈	—	2	2	no	—
49	C₄×D₇	—	4	1	no	—
50	C₄×D₇	—	4	1	no	—
51	Dic₁₄	—	2	2	no	—
52	C₇×C₈	—	2	2	no	—
53	C₇×C₈	—	4	2	no	—
54	C₇⋊C₈	—	4	1	no	—
55	C₂×#41	—	7	1	no	—
56	C₂×M₁₆	—	7	1	no	—
57	C₂×M₁₆	C₂×D₇	1	2	yes	—
58	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
59	C₂×C₄⋊C₄	—	7	2	no	—
60	#106	—	14	2	no	—
61	#103	—	14	2	no	—
62	#102	—	14	2	no	—
63	C₂²⋊C₈	—	14	2	no	—
64	C₂²×C₇	—	2	2	no	—
65	C₂²×C₇	#121	1	1	yes	characteristic
66	C₂²×C₇	#126	1	2	yes	—
67	C₂×D₇	—	4	1	no	—
68	C₂×D₇	—	4	2	no	—
69	C₄×C₇	#123	1	2	yes	—
70	C₄×C₇	#123	1	1	yes	characteristic
71	C₄×C₇	#124	1	1	yes	characteristic
72	Dic₇	—	4	1	no	—
73	Dic₇	—	8	2	no	—
74	#41	—	14	2	no	—
75	#41	—	14	2	no	—
76	C₂×Q₈	—	7	1	no	—
77	C₂×D₄	—	7	1	no	—
78	C₂×D₄	—	14	1	no	—
79	C₂²×C₄	—	14	1	no	—
80	C₂²×C₄	—	14	2	no	—
81	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	2	no	—
84	M₁₆	—	14	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	14	1	no	—
88	C₂×C₈	D₂₈	1	2	yes	—
89	C₄⋊C₄	—	14	2	no	—
90	C₄⋊C₄	—	14	2	no	—
91	C₂×C₇	—	2	2	no	—
92	C₂×C₇	#397	1	1	yes	characteristic
93	C₂×C₇	#471	1	2	yes	—
94	D₇	—	8	1	no	—
95	C₂³	—	14	1	no	—
96	C₂³	#1161	1	1	yes	characteristic
97	Q₈	—	14	2	no	—
98	D₄	—	14	2	no	—
99	D₄	—	28	2	no	—
100	C₂×C₄	—	14	1	no	—
101	C₂×C₄	—	28	1	no	—
102	C₂×C₄	—	28	1	no	—
103	C₂×C₄	—	28	2	no	—
104	C₂×C₄	—	56	2	no	—
105	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
106	C₂×C₄	#1157	1	2	yes	—
107	C₂×C₄	D₄×D₇	1	1	yes	characteristic
108	C₂×C₄	#1159	1	2	yes	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	28	1	no	—
112	C₇	#2079	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	2	2	no	—
114	C₂²	—	28	2	no	—
115	C₂²	—	28	1	no	—
116	C₂²	#6458	1	1	yes	characteristic
117	C₂²	#6461	1	2	yes	—
118	C₄	—	28	1	no	—
119	C₄	—	56	2	no	—
120	C₄	#6438	1	1	yes	characteristic
121	C₄	#6446	1	2	yes	—
122	C₄	#6449	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	56	1	no	—
125	C₂	#89446	1	1	yes	characteristic
126	C₂	#89670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#89905

C₂

yes

maximal

#89882

C₂

yes

maximal, characteristic

C₂×#6391

C₂

yes

maximal, characteristic, complemented

C₇×#341

C₂

yes

maximal, characteristic, complemented

#89349

C₂

yes

maximal, complemented

C₂×#1157

C₂²

yes

characteristic

C₂×C₇×M₁₆

C₂²

yes

—

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×#1136

C₂²

yes

characteristic

#6395

—

(C₂×D₇)⋊C₈

—

—

—

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂×C₂²⋊D₇

—

C₂²×Dic₇

—

#1157

—

#1157

—

C₂×D₂₈

D₄

yes

characteristic

C₂×C₄×D₇

—

C₂×Dic₁₄

D₄

yes

characteristic

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂³×C₇

#41

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇

yes

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

#103

—

#102

—

C₂²⋊C₈

—

C₂²×C₇

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#126

yes

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

#123

yes

—

C₄×C₇

yes

C₄×C₇

yes

Dic₇

—

Dic₇

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂×C₇

—

C₂×C₇

#397

yes

characteristic

C₂×C₇

#471

yes

—

D₇

—

C₂³

—

C₂³

#1161

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×D₂₈

yes

characteristic

106

C₂×C₄

#1157

yes

—

107

C₂×C₄

D₄×D₇

yes

characteristic

108

C₂×C₄

#1159

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₇

#2079

yes

characteristic, complemented, Sylow

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#6458

yes

characteristic

117

C₂²

#6461

yes

—

118

C₄

—

119

C₄

—

120

C₄

#6438

yes

characteristic

121

C₄

#6446

yes

—

122

C₄

#6449

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#89446

yes

characteristic

126

C₂

#89670

yes

—

	subgroup	G/H
center	C₂²	#6458
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#341	C₂
socle	C₂²×C₇	#121
prime radical	C₂×#1157	C₂²
hypercenter	#341	D₇
hypocenter	C₇	#2079
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#6458

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

C₂

C₂²×C₇

C₂²

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
3. C₂×#6391	C₂	semidirect
4. C₇×#341	C₂	semidirect
5. #89349	C₂	semidirect
37. #341	D₇	semidirect
112. C₇	#2079	semidirect

H ≅ G/N

type

3. C₂×#6391

C₂

semidirect

4. C₇×#341

C₂

semidirect

5. #89349

C₂

semidirect

37. #341

D₇

semidirect

112. C₇

#2079

semidirect

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

subgroup	index
#89905	2
#89882	2
C₂×#6391	2
C₇×#341	2
#89349	2
#2079	7

subgroup

index

Subgroups — #11726676

Conjugacy classes of subgroups

Complements

Maximal subgroups