Group #28337242 — subgroups

Subgroups — #28337242

Order 13696.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20292669	C₂	1	2	yes	maximal
2	C₂×#16812041	C₂	1	1	yes	maximal, characteristic
3	#20292582	C₂	1	1	yes	maximal, characteristic
4	#20292395	C₂	1	2	yes	maximal
5	C₂×#108×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	1	yes	characteristic
7	#16812068	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	#16812043	—	2	2	no	—
10	#16812041	—	2	2	no	—
11	#16812041	—	2	1	no	—
12	#16812041	—	2	1	no	—
13	#16812019	—	2	1	no	—
14	C₈×Dic₁₀₇	—	2	1	no	—
15	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	#108×C₁₀₇	C₄	1	4	yes	complemented
18	C₂²×Dic₁₀₇	—	2	2	no	—
19	#13133802	—	2	1	no	—
20	#13133802	Q₈	1	2	yes	—
21	C₂×C₁₀₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₀₇	—	2	2	no	—
23	C₄⋊Dic₁₀₇	—	2	2	no	—
24	C₄⋊Dic₁₀₇	D₄	1	2	yes	—
25	C₂²⋊Dic₁₀₇	—	2	1	no	—
26	C₄×Dic₁₀₇	—	2	1	no	—
27	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₇	—	2	2	no	—
29	M₁₆×C₁₀₇	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₁₀₇	—	2	2	no	—
31	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₇	Q₈	1	2	yes	—
35	C₂³×C₁₀₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
42	C₂×Dic₁₀₇	—	2	2	no	—
43	C₂×Dic₁₀₇	—	4	2	no	—
44	C₂×Dic₁₀₇	—	8	2	no	—
45	C₈×C₁₀₇	—	2	4	no	—
46	C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
47	C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
48	C₁₀₇⋊C₈	—	2	2	no	—
49	C₂²×C₁₀₇	—	2	2	no	—
50	C₂²×C₁₀₇	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₇	#121	1	1	yes	characteristic
52	C₂²×C₁₀₇	#125	1	1	yes	characteristic
53	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
55	C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₁₀₇	—	4	2	no	—
58	Dic₁₀₇	—	8	2	no	—
59	C₂×C₁₀₇	—	2	1	no	—
60	C₂×C₁₀₇	—	2	1	no	—
61	C₂×C₁₀₇	#389	1	1	yes	characteristic
62	C₂×C₁₀₇	#471	1	2	yes	—
63	#1951	—	107	1	no	maximal, Sylow
64	C₁₀₇	#1951	1	1	yes	characteristic, complemented, Sylow
65	C₂×#108	D₁₀₇	1	1	yes	characteristic
66	#428	—	107	2	no	—
67	C₂×#106	—	107	1	no	—
68	#405	—	107	1	no	—
69	#343	—	107	2	no	—
70	C₂×M₁₆	—	107	1	no	—
71	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	—
72	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
73	#117	—	107	1	no	—
74	C₂×C₄⋊C₄	—	107	2	no	—
75	#108	Dic₁₀₇	1	4	yes	complemented
76	#107	—	214	2	no	—
77	#106	—	214	2	no	—
78	#106	—	214	1	no	—
79	#106	—	214	1	no	—
80	#97	—	214	1	no	—
81	C₄×C₈	—	214	1	no	—
82	C₂²×C₄	—	214	2	no	—
83	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	107	2	no	—
86	M₁₆	—	214	1	no	—
87	M₁₆	C₂×Dic₁₀₇	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	214	1	no	—
90	C₂×C₈	Dic₂₁₄	1	2	yes	—
91	C₂×C₈	D₄₂₈	1	1	yes	characteristic
92	C₂×C₈	D₄₂₈	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
94	C₄⋊C₄	—	107	2	no	—
95	C₄⋊C₄	—	214	2	no	—
96	C₄⋊C₄	—	214	2	no	—
97	C₂²⋊C₄	—	214	1	no	—
98	C₄²	—	214	1	no	—
99	C₂³	#13133819	1	1	yes	characteristic
100	C₂×C₄	—	856	2	no	—
101	C₂×C₄	—	428	2	no	—
102	C₂×C₄	—	214	2	no	—
103	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
108	C₂×C₄	#13133817	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	214	2	no	—
111	C₈	C₄⋊Dic₁₀₇	1	2	yes	—
112	C₈	C₄⋊Dic₁₀₇	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₁₀₇	1	1	yes	characteristic
115	C₂²	#16812084	1	1	yes	characteristic
116	C₂²	#16812100	1	1	yes	characteristic
117	C₄	—	856	2	no	—
118	C₄	—	428	2	no	—
119	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
120	C₄	D₄×Dic₁₀₇	1	1	yes	characteristic
121	C₄	#16812089	1	1	yes	characteristic
122	C₄	#16812108	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#20292526	1	1	yes	characteristic
126	C₂	#20292891	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20292669

C₂

yes

maximal

C₂×#16812041

C₂

yes

maximal, characteristic

#20292582

C₂

yes

maximal, characteristic

#20292395

C₂

yes

maximal

C₂×#108×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

characteristic

#16812068

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

—

—

—

—

—

C₈×Dic₁₀₇

—

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

#108×C₁₀₇

C₄

yes

complemented

C₂²×Dic₁₀₇

—

#13133802

—

#13133802

Q₈

yes

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

D₄

yes

—

C₂²⋊Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

D₄

yes

characteristic

C₂×C₈×C₁₀₇

D₄

yes

characteristic

C₂×C₈×C₁₀₇

Q₈

yes

—

C₂³×C₁₀₇

#41

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₄⋊C₄

yes

—

C₈×C₁₀₇

C₄⋊C₄

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×Q₈

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₄×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₇

C₄×D₄

yes

characteristic

C₄×C₁₀₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#389

yes

characteristic

C₂×C₁₀₇

#471

yes

—

#1951

—

107

maximal, Sylow

C₁₀₇

#1951

yes

characteristic, complemented, Sylow

C₂×#108

D₁₀₇

yes

characteristic

#428

—

107

—

C₂×#106

—

107

—

#405

—

107

—

#343

—

107

—

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

—

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

#117

—

107

—

C₂×C₄⋊C₄

—

107

—

#108

Dic₁₀₇

yes

complemented

#107

—

214

—

#106

—

214

—

#106

—

214

—

#106

—

214

—

#97

—

214

—

C₄×C₈

—

214

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

107

—

M₁₆

—

214

—

M₁₆

C₂×Dic₁₀₇

yes

—

C₂×C₈

—

C₂×C₈

—

214

—

C₂×C₈

Dic₂₁₄

yes

—

C₂×C₈

D₄₂₈

yes

characteristic

C₂×C₈

D₄₂₈

yes

characteristic

C₂×C₈

C₂×Dic₁₀₇

yes

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₂²⋊C₄

—

214

—

C₄²

—

214

—

C₂³

#13133819

yes

characteristic

100

C₂×C₄

—

856

—

101

C₂×C₄

—

428

—

102

C₂×C₄

—

214

—

103

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

104

C₂×C₄

D₄×D₁₀₇

yes

characteristic

105

C₂×C₄

C₂×D₄₂₈

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₇

yes

characteristic

107

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

108

C₂×C₄

#13133817

yes

characteristic

109

C₈

—

110

C₈

—

214

—

111

C₈

C₄⋊Dic₁₀₇

yes

—

112

C₈

C₄⋊Dic₁₀₇

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₁₀₇

yes

characteristic

115

C₂²

#16812084

yes

characteristic

116

C₂²

#16812100

yes

characteristic

117

C₄

—

856

—

118

C₄

—

428

—

119

C₄

C₂×C₄⋊Dic₁₀₇

yes

characteristic

120

C₄

D₄×Dic₁₀₇

yes

characteristic

121

C₄

#16812089

yes

characteristic

122

C₄

#16812108

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#20292526

yes

characteristic

126

C₂

#20292891

yes

—

	subgroup	G/H
center	C₂²	#16812084
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×#108×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#121
prime radical	C₂³×C₁₀₇	#41
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×#108	D₁₀₇
hypocenter	C₁₀₇	#1951
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#16812084

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×#108×C₁₀₇

C₂

C₂²×C₁₀₇

C₂³×C₁₀₇

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₁₀₇	C₄	semidirect
64. C₁₀₇	#1951	semidirect
75. #108	Dic₁₀₇	semidirect

H ≅ G/N

type

17. #108×C₁₀₇

C₄

semidirect

64. C₁₀₇

#1951

semidirect

75. #108

Dic₁₀₇

semidirect

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

subgroup	index
#20292669	2
C₂×#16812041	2
#20292582	2
#20292395	2
C₂×#108×C₁₀₇	2
#1951	107

subgroup

index

C₂×#108×C₁₀₇

107

Subgroups — #28337242

Conjugacy classes of subgroups

Complements

Maximal subgroups