Group #11726688 — subgroups

Subgroups — #11726688

Order 896.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6398	C₂	1	2	yes	maximal
3	#89892	C₂	1	2	yes	maximal
4	#89874	C₂	1	1	yes	maximal, characteristic
5	C₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄	C₂²	1	2	yes	—
7	C₂×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	#6489	C₂²	1	1	yes	characteristic
10	C₂×#1136	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₈	—	2	4	no	—
12	C₂×Dic₂₈	—	2	2	no	—
13	#6398	—	2	4	no	—
14	#6390	—	2	4	no	—
15	Dic₇⋊C₈	—	2	2	no	—
16	#2047	—	7	1	no	maximal, Sylow
17	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₇	—	2	2	no	—
19	C₂×Dic₁₄	—	2	2	no	—
20	C₂×Dic₁₄	—	2	2	no	—
21	C₂×Dic₁₄	—	2	2	no	—
22	C₂×Dic₁₄	—	2	2	no	—
23	C₂×Dic₁₄	—	2	2	no	—
24	C₂×Dic₁₄	D₄	1	4	yes	—
25	C₇×M₁₆	—	2	2	no	—
26	C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×C₈	—	2	2	no	—
28	C₂×C₇×C₈	—	2	2	no	—
29	C₂×C₇×C₈	D₄	1	2	yes	—
30	C₂²⋊Dic₇	—	2	1	no	—
31	C₄⋊Dic₇	D₄	1	2	yes	—
32	C₄×Dic₇	—	2	1	no	—
33	#1136	—	2	2	no	—
34	C₂×C₇⋊C₈	—	2	1	no	—
35	Dic₂₈	—	2	8	no	—
36	C₂²×Q₁₆	—	7	1	no	—
37	#424	—	7	1	no	—
38	#419	—	7	2	no	—
39	C₂×#101	—	7	2	no	—
40	#341	D₇	1	1	yes	characteristic
41	C₂³×C₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇	#41	1	2	yes	—
46	C₂×Dic₇	—	2	2	no	—
47	C₂×Dic₇	—	2	4	no	—
48	C₂×Dic₇	—	4	2	no	—
49	C₂×Dic₇	—	4	2	no	—
50	Dic₁₄	—	2	4	no	—
51	Dic₁₄	—	2	4	no	—
52	Dic₁₄	—	4	2	no	—
53	Dic₁₄	—	4	2	no	—
54	C₇×C₈	—	2	2	no	—
55	C₇×C₈	—	4	2	no	—
56	C₇⋊C₈	—	4	1	no	—
57	C₂²×Q₈	—	7	2	no	—
58	C₂×Q₁₆	—	14	4	no	—
59	C₂×Q₁₆	—	14	2	no	—
60	C₂×M₁₆	—	7	1	no	—
61	C₂×M₁₆	C₂×D₇	1	2	yes	—
62	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
63	#117	—	7	1	no	—
64	C₄⋊C₈	—	14	2	no	—
65	#103	—	14	4	no	—
66	#101	—	14	4	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	#121	1	1	yes	characteristic
69	C₂²×C₇	#123	1	2	yes	—
70	C₄×C₇	#120	1	1	yes	characteristic
71	C₄×C₇	#121	1	2	yes	—
72	C₄×C₇	#124	1	1	yes	characteristic
73	Dic₇	—	4	4	no	—
74	Dic₇	—	4	2	no	—
75	C₂×Q₈	—	7	4	no	—
76	C₂×Q₈	—	14	2	no	—
77	C₂×Q₈	—	14	2	no	—
78	C₂×Q₈	—	14	2	no	—
79	C₂×Q₈	—	14	2	no	—
80	C₂×Q₈	—	14	2	no	—
81	C₂²×C₄	—	14	2	no	—
82	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
83	Q₁₆	—	14	8	no	—
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	14	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	14	1	no	—
90	C₂×C₈	D₂₈	1	2	yes	—
91	C₄⋊C₄	—	7	2	no	—
92	C₂²⋊C₄	—	14	1	no	—
93	C₄²	—	14	1	no	—
94	C₂×C₇	—	2	2	no	—
95	C₂×C₇	#394	1	1	yes	characteristic
96	C₂×C₇	#473	1	2	yes	—
97	C₂³	#1161	1	1	yes	characteristic
98	Q₈	—	14	4	no	—
99	Q₈	—	14	4	no	—
100	Q₈	—	28	2	no	—
101	Q₈	—	28	2	no	—
102	C₂×C₄	—	14	4	no	—
103	C₂×C₄	—	14	2	no	—
104	C₂×C₄	—	28	2	no	—
105	C₂×C₄	—	28	2	no	—
106	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
107	C₂×C₄	#1157	1	2	yes	—
108	C₂×C₄	D₄×D₇	1	2	yes	—
109	C₂×C₄	D₄×D₇	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	28	1	no	—
113	C₇	#2047	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	#6457	1	2	yes	—
116	C₂²	#6458	1	1	yes	characteristic
117	C₄	—	28	2	no	—
118	C₄	—	28	4	no	—
119	C₄	#6438	1	1	yes	characteristic
120	C₄	#6445	1	1	yes	characteristic
121	C₄	#6447	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#89444	1	1	yes	characteristic
124	C₂	#89668	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₈

C₂

yes

maximal, characteristic

C₂×#6398

C₂

yes

maximal

#89892

C₂

yes

maximal

#89874

C₂

yes

maximal, characteristic

C₇×#341

C₂

yes

maximal, characteristic

C₂²×Dic₁₄

C₂²

yes

—

C₂×C₇×M₁₆

C₂²

yes

—

C₂²×C₇×C₈

C₂²

yes

characteristic

#6489

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

characteristic

C₂×Dic₂₈

—

C₂×Dic₂₈

—

#6398

—

#6390

—

Dic₇⋊C₈

—

#2047

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

D₄

yes

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

D₄

yes

—

C₄×Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

Dic₂₈

—

C₂²×Q₁₆

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

#41

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇

yes

—

C₂²×C₈

C₂×D₇

yes

characteristic

#117

—

C₄⋊C₈

—

#103

—

#101

—

C₂²×C₇

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#123

yes

—

C₄×C₇

#120

yes

characteristic

C₄×C₇

#121

yes

—

C₄×C₇

#124

yes

characteristic

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

#394

yes

characteristic

C₂×C₇

#473

yes

—

C₂³

#1161

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×D₂₈

yes

characteristic

107

C₂×C₄

#1157

yes

—

108

C₂×C₄

D₄×D₇

yes

—

109

C₂×C₄

D₄×D₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₇

#2047

yes

characteristic, complemented, Sylow

114

C₂²

—

115

C₂²

#6457

yes

—

116

C₂²

#6458

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#6438

yes

characteristic

120

C₄

#6445

yes

characteristic

121

C₄

#6447

yes

—

122

C₂

—

123

C₂

#89444

yes

characteristic

124

C₂

#89668

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	C₂³×C₇	#41
hypercenter	#341	D₇
hypocenter	C₇	#2047
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

C₂³×C₇

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
113. C₇	#2047	semidirect

H ≅ G/N

type

113. C₇

#2047

semidirect

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

subgroup	index
C₂²×Dic₂₈	2
C₂×#6398	2
#89892	2
#89874	2
C₇×#341	2
#2047	7

subgroup

index

C₂²×Dic₂₈

Subgroups — #11726688

Conjugacy classes of subgroups

Complements

Maximal subgroups