Group #28334754 — subgroups

Subgroups — #28334754

Order 13696.

196 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20292680	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20292349	C₂	1	2	yes	maximal, complemented
4	#20292348	C₂	1	2	yes	maximal, complemented
5	#431×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₇⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₁₀₇⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₁₀₇⋊C₈)	C₄	1	2	yes	—
9	C₄×C₁₀₇⋊C₈	C₄	1	2	yes	—
10	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
11	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₇	C₄	1	2	yes	—
15	C₂²⋊C₈×C₁₀₇	C₄	1	2	yes	—
16	C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
17	C₂×C₁₀₇⋊C₈	—	2	4	no	—
18	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
25	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂³×C₁₀₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₁₀₇	—	2	2	no	—
31	C₂×C₄×C₁₀₇	—	2	2	no	—
32	C₂×C₄×C₁₀₇	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	M₁₆	1	4	yes	—
36	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₁₀₇	—	2	4	no	—
40	C₁₀₇⋊C₈	—	2	2	no	—
41	C₁₀₇⋊C₈	—	2	2	no	—
42	C₁₀₇⋊C₈	—	2	4	no	—
43	C₂²×C₁₀₇	—	2	1	no	—
44	C₂²×C₁₀₇	—	2	1	no	—
45	C₂²×C₁₀₇	#97	1	2	yes	—
46	C₂²×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₁₀₇	—	2	1	no	—
48	C₄×C₁₀₇	—	2	1	no	—
49	C₄×C₁₀₇	#97	1	2	yes	—
50	C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
51	C₄×C₁₀₇	#131	1	2	yes	—
52	C₄×C₁₀₇	#131	1	2	yes	—
53	C₂×C₁₀₇	—	2	2	no	—
54	C₂×C₁₀₇	C₂×#97	1	1	yes	characteristic
55	C₂×C₁₀₇	#405	1	1	yes	characteristic
56	C₂×C₁₀₇	#405	1	1	yes	characteristic
57	#1452	—	107	1	no	maximal, Sylow
58	C₁₀₇	#1452	1	1	yes	characteristic, complemented, Sylow
59	#431	—	107	1	no	—
60	#431	D₁₀₇	1	1	yes	characteristic
61	C₂×C₄×C₈	—	107	1	no	—
62	#322	—	107	2	no	—
63	#322	—	107	2	no	—
64	C₂²×C₈	—	107	2	no	—
65	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
66	C₄⋊C₈	—	107	2	no	—
67	C₄⋊C₈	Dic₁₀₇	1	2	yes	—
68	C₂²⋊C₈	—	107	2	no	—
69	C₂²⋊C₈	Dic₁₀₇	1	2	yes	—
70	C₄×C₈	—	107	2	no	—
71	C₄×C₈	—	107	1	no	—
72	C₄×C₈	—	107	1	no	—
73	C₄×C₈	—	107	2	no	—
74	C₄×C₈	C₂×D₁₀₇	1	2	yes	—
75	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
78	C₂×C₈	—	107	2	no	—
79	C₂×C₈	—	107	2	no	—
80	C₂×C₈	—	107	4	no	—
81	C₂×C₈	—	214	4	no	—
82	C₂×C₈	C₂×Dic₁₀₇	1	4	yes	—
83	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
84	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
86	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
87	C₂³	C₄×Dic₁₀₇	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	C₄×Dic₁₀₇	1	1	yes	characteristic
91	C₂×C₄	C₄×Dic₁₀₇	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₁₀₇	1	1	yes	characteristic
93	C₂×C₄	#13133801	1	4	yes	—
94	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	214	2	no	—
99	C₈	—	214	2	no	—
100	C₈	—	214	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	#16812019	1	2	yes	—
104	C₂²	C₂×C₄×Dic₁₀₇	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#16812019	1	2	yes	—
108	C₄	C₂×#13133801	1	2	yes	—
109	C₄	#16812116	1	2	yes	—
110	C₄	#16812117	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	C₂×#16812019	1	1	yes	characteristic
113	C₂	#20292583	1	1	yes	characteristic
114	C₂	#20292585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292680

C₂

yes

maximal, characteristic

C₂×C₄×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic

#20292349

C₂

yes

maximal, complemented

#20292348

C₂

yes

maximal, complemented

#431×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₄

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₄

yes

—

C₂²⋊(C₁₀₇⋊C₈)

C₄

yes

—

C₄×C₁₀₇⋊C₈

C₄

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

C₄

yes

—

C₂²⋊C₈×C₁₀₇

C₄

yes

—

C₄×C₈×C₁₀₇

C₂²

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

C₄²

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄²

yes

characteristic

C₂×C₄×C₁₀₇

C₄²

yes

characteristic

C₂×C₄×C₁₀₇

C₄²

yes

characteristic

C₂×C₄×C₁₀₇

M₁₆

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#97

yes

—

C₂²×C₁₀₇

C₂×C₄²

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#97

yes

—

C₄×C₁₀₇

C₂×M₁₆

yes

—

C₄×C₁₀₇

#131

yes

—

C₄×C₁₀₇

#131

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#97

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1452

yes

characteristic, complemented, Sylow

#431

—

107

—

#431

D₁₀₇

yes

characteristic

C₂×C₄×C₈

—

107

—

#322

—

107

—

#322

—

107

—

C₂²×C₈

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₄⋊C₈

Dic₁₀₇

yes

—

C₂²⋊C₈

—

107

—

C₂²⋊C₈

Dic₁₀₇

yes

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

—

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

C₂×Dic₁₀₇

yes

—

C₄²

C₄×D₁₀₇

yes

characteristic

C₄²

C₄×D₁₀₇

yes

characteristic

C₄²

C₂×Dic₁₀₇

yes

characteristic

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

C₄×Dic₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₇

yes

characteristic

C₂×C₄

C₄×Dic₁₀₇

yes

characteristic

C₂×C₄

C₄×Dic₁₀₇

yes

characteristic

C₂×C₄

#13133801

yes

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

C₈

—

C₈

—

214

—

C₈

—

214

—

100

C₈

—

214

—

101

C₂²

—

102

C₂²

—

103

C₂²

#16812019

yes

—

104

C₂²

C₂×C₄×Dic₁₀₇

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

#16812019

yes

—

108

C₄

C₂×#13133801

yes

—

109

C₄

#16812116

yes

—

110

C₄

#16812117

yes

—

111

C₂

—

112

C₂

C₂×#16812019

yes

characteristic

113

C₂

#20292583

yes

characteristic

114

C₂

#20292585

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂×C₄²
Fitting	#431×C₁₀₇	C₂
prime radical	C₂³×C₁₀₇	C₄²
cosocle	C₄²×C₁₀₇	C₂³
hypercenter	#431	D₁₀₇
hypocenter	C₁₀₇	#1452
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂×C₄²

Fitting

#431×C₁₀₇

C₂

prime radical

C₂³×C₁₀₇

C₄²

cosocle

C₄²×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #20292349	C₂	semidirect
4. #20292348	C₂	semidirect
58. C₁₀₇	#1452	semidirect

H ≅ G/N

type

3. #20292349

C₂

semidirect

4. #20292348

C₂

semidirect

58. C₁₀₇

#1452

semidirect

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

subgroup	index
#20292680	2
C₂×C₄×C₁₀₇⋊C₈	2
#20292349	2
#20292348	2
#431×C₁₀₇	2
#1452	107

subgroup

index

C₂×C₄×C₁₀₇⋊C₈

#431×C₁₀₇

107

Subgroups — #28334754

Conjugacy classes of subgroups

Complements

Maximal subgroups