Group #11727728 — subgroups

Subgroups — #11727728

Order 896.

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

normal subgroups only

This group has 272 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×#6489	C₂	1	1	yes	maximal, characteristic
2	C₇×#407	C₂	1	1	yes	maximal, characteristic
3	#89979	C₂	1	1	yes	maximal, characteristic
4	#89349	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₇	C₄	1	2	yes	—
8	#6489	—	2	2	no	—
9	#6489	—	2	2	no	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₇	C₄	1	2	yes	—
12	C₂×#1136	C₂²	1	2	yes	—
13	C₇×C₂²⋊C₈	—	2	2	no	—
14	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
15	#1822	—	7	1	no	maximal, Sylow
16	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₇	—	2	4	no	—
18	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇	D₄	1	2	yes	—
20	C₂²×C₄×C₇	D₄	1	2	yes	—
21	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₇	C₂×C₄	1	4	yes	—
23	C₇×M₁₆	—	2	4	no	—
24	C₂×C₇×C₈	—	2	2	no	—
25	C₂²⋊Dic₇	—	2	4	no	—
26	C₄⋊Dic₇	—	2	2	no	—
27	C₄⋊Dic₇	—	2	2	no	—
28	C₄×Dic₇	—	2	4	no	—
29	#1136	—	2	4	no	—
30	C₂×C₇⋊C₈	—	2	2	no	—
31	C₂×#117	—	7	1	no	—
32	#407	—	7	1	no	—
33	#407	D₇	1	1	yes	characteristic
34	#343	—	7	4	no	—
35	C₂³×C₇	—	2	2	no	—
36	C₂³×C₇	—	4	2	no	—
37	C₂³×C₇	C₂²⋊C₄	1	2	yes	—
38	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇	—	2	2	no	—
40	C₂×C₄×C₇	—	2	4	no	—
41	C₂×C₄×C₇	—	2	4	no	—
42	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇	#41	1	4	yes	—
46	C₂×Dic₇	—	2	4	no	—
47	C₂×Dic₇	—	4	4	no	—
48	C₇×C₈	—	4	2	no	—
49	C₇⋊C₈	—	4	2	no	—
50	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
51	C₂×M₁₆	—	7	2	no	—
52	C₂×M₁₆	C₂×D₇	1	2	yes	—
53	#117	—	14	2	no	—
54	#117	—	14	2	no	—
55	C₂×C₄⋊C₄	—	7	2	no	—
56	C₂×C₂²⋊C₄	—	7	2	no	—
57	C₂×C₄²	—	7	2	no	—
58	C₂²⋊C₈	—	2	2	no	—
59	C₂²⋊C₈	—	14	2	no	—
60	C₂²×C₇	—	2	2	no	—
61	C₂²×C₇	—	2	2	no	—
62	C₂²×C₇	—	2	4	no	—
63	C₂²×C₇	—	4	4	no	—
64	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇	#117	1	2	yes	—
66	C₄×C₇	—	2	2	no	—
67	C₄×C₇	#123	1	2	yes	—
68	C₄×C₇	#123	1	2	yes	—
69	Dic₇	—	4	4	no	—
70	C₂⁴	C₄×D₇	1	1	yes	characteristic
71	C₂²×C₄	—	2	4	no	—
72	C₂²×C₄	—	7	4	no	—
73	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
74	C₂²×C₄	D₂₈	1	2	yes	—
75	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
76	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	M₁₆	—	14	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	14	2	no	—
81	C₄⋊C₄	—	14	2	no	—
82	C₄⋊C₄	—	14	2	no	—
83	C₂²⋊C₄	—	14	4	no	—
84	C₄²	—	14	4	no	—
85	C₂×C₇	—	2	2	no	—
86	C₂×C₇	—	4	2	no	—
87	C₂×C₇	#381	1	1	yes	characteristic
88	C₂×C₇	#413	1	2	yes	—
89	C₂³	—	2	2	no	—
90	C₂³	—	4	2	no	—
91	C₂³	(C₂×D₇)⋊C₄	1	2	yes	—
92	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	14	4	no	—
97	C₂×C₄	—	28	4	no	—
98	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
100	C₂×C₄	#1159	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	28	2	no	—
104	C₇	#1822	1	1	yes	characteristic, complemented, Sylow
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	#6440	1	2	yes	—
110	C₂²	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	28	4	no	—
113	C₄	#6449	1	2	yes	—
114	C₄	#6498	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#89715	1	1	yes	characteristic
118	C₂	#89904	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6489

C₂

yes

maximal, characteristic

C₇×#407

C₂

yes

maximal, characteristic

#89979

C₂

yes

maximal, characteristic

#89349

C₂

yes

maximal, complemented

C₂³×C₄×C₇

C₂²

yes

characteristic

C₂×C₇×M₁₆

C₂²

yes

—

C₂×C₂²⋊Dic₇

C₄

yes

—

#6489

—

#6489

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₄

yes

—

C₂×#1136

C₂²

yes

—

C₇×C₂²⋊C₈

—

C₂²⋊(C₇⋊C₈)

—

#1822

—

maximal, Sylow

C₂⁴×C₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂×C₄

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

C₂×#117

—

#407

—

#407

D₇

yes

characteristic

#343

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂²⋊C₄

yes

—

C₂³×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇⋊C₈

—

C₂³×C₄

C₂×D₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇

#117

yes

—

C₄×C₇

—

C₄×C₇

#123

yes

—

C₄×C₇

#123

yes

—

Dic₇

—

C₂⁴

C₄×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇

yes

characteristic

C₂²×C₄

D₂₈

yes

—

C₂²×C₄

C₂²⋊D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

#381

yes

characteristic

C₂×C₇

#413

yes

—

C₂³

—

C₂³

—

C₂³

(C₂×D₇)⋊C₄

yes

—

C₂³

C₂×C₄×D₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₇)⋊C₄

yes

—

C₂×C₄

C₂×D₂₈

yes

characteristic

100

C₂×C₄

#1159

yes

—

101

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

102

C₈

—

103

C₈

—

104

C₇

#1822

yes

characteristic, complemented, Sylow

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#6440

yes

—

110

C₂²

C₂×(C₂×D₇)⋊C₄

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#6449

yes

—

114

C₄

#6498

yes

—

115

C₂

—

116

C₂

—

117

C₂

#89715

yes

characteristic

118

C₂

#89904

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇)⋊C₄
commutator	C₂³×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#407	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇	C₂×C₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	#407	D₇
hypocenter	C₇	#1822
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×(C₂×D₇)⋊C₄

commutator

C₂³×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#407

C₂

socle

C₂²×C₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₇

C₂×C₄

cosocle

C₂²×C₄×C₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
4. #89349	C₂	semidirect
104. C₇	#1822	semidirect

H ≅ G/N

type

4. #89349

C₂

semidirect

104. C₇

#1822

semidirect

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

subgroup	index
C₂×#6489	2
C₇×#407	2
#89979	2
#89349	2
#1822	7

subgroup

index

Subgroups — #11727728

Conjugacy classes of subgroups

Complements

Maximal subgroups