Group #28337400 — subgroups

Subgroups — #28337400

Order 13696.

280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
3	#20292381	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₄	C₄	1	1	yes	characteristic
6	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₂₁₄	—	2	2	no	—
8	C₄×Dic₂₁₄	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₇	C₄	1	1	yes	characteristic
10	C₂×Dic₁₀₇⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₀₇	C₂²	1	2	yes	—
12	#16812020	—	2	2	no	—
13	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₇	—	2	2	no	—
16	C₂×Dic₂₁₄	—	2	2	no	—
17	C₂×Dic₂₁₄	—	2	4	no	—
18	C₂²×Dic₁₀₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₀₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₀₇⋊C₈	—	2	2	no	—
21	C₂×C₁₀₇⋊C₈	—	2	2	no	—
22	C₂×C₁₀₇⋊C₈	Q₈	1	4	yes	complemented
23	C₄⋊Dic₁₀₇	—	2	2	no	—
24	Dic₁₀₇⋊C₄	—	2	4	no	—
25	C₄×Dic₁₀₇	—	2	2	no	—
26	C₄×Dic₁₀₇	—	2	2	no	—
27	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₀₇	—	2	2	no	—
31	C₂×C₈×C₁₀₇	—	2	4	no	—
32	C₄²×C₁₀₇	D₄	1	2	yes	—
33	C₄²×C₁₀₇	D₄	1	2	yes	—
34	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	—	2	2	no	—
36	C₂×C₄×C₁₀₇	—	2	2	no	—
37	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₇	#41	1	2	yes	—
43	C₂×Dic₁₀₇	—	2	2	no	—
44	C₂×Dic₁₀₇	—	2	2	no	—
45	C₂×Dic₁₀₇	—	2	2	no	—
46	C₂×Dic₁₀₇	—	2	4	no	—
47	C₂×Dic₁₀₇	M₁₆	1	4	yes	—
48	Dic₂₁₄	—	2	8	no	—
49	C₈×C₁₀₇	—	4	2	no	—
50	C₁₀₇⋊C₈	—	2	4	no	—
51	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₁₀₇	#131	1	2	yes	—
55	C₄×C₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	—	2	2	no	—
57	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₀₇	#124	1	1	yes	characteristic
59	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₀₇	—	2	4	no	—
61	Dic₁₀₇	—	4	2	no	—
62	C₂×C₁₀₇	#391	1	1	yes	characteristic
63	C₂×C₁₀₇	#407	1	1	yes	characteristic
64	C₂×C₁₀₇	#408	1	1	yes	characteristic
65	C₂×C₁₀₇	#446	1	4	yes	—
66	#1985	—	107	1	no	maximal, Sylow
67	C₁₀₇	#1985	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	—	107	1	no	—
69	C₂×C₄⋊C₈	D₁₀₇	1	1	yes	characteristic
70	C₂×#97	—	107	1	no	—
71	#336	—	107	4	no	—
72	C₂²×Q₈	—	107	1	no	—
73	C₂²×C₈	—	107	2	no	—
74	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
75	C₄×Q₈	—	214	2	no	—
76	C₄×Q₈	—	214	2	no	—
77	C₂×C₄⋊C₄	—	107	2	no	—
78	C₂×C₄⋊C₄	—	107	1	no	—
79	C₂×C₄²	—	107	2	no	—
80	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	214	2	no	—
83	C₂×Q₈	—	214	4	no	—
84	C₂×Q₈	—	214	2	no	—
85	C₂²×C₄	—	107	2	no	—
86	C₂²×C₄	—	107	2	no	—
87	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	107	4	no	—
93	C₂×C₈	—	214	2	no	—
94	C₂×C₈	—	214	2	no	—
95	C₄⋊C₄	—	214	4	no	—
96	C₄⋊C₄	—	214	2	no	—
97	C₄²	—	214	2	no	—
98	C₄²	—	214	2	no	—
99	C₄²	D₄₂₈	1	2	yes	—
100	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
101	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
102	Q₈	—	214	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	107	4	no	—
106	C₂×C₄	—	214	2	no	—
107	C₂×C₄	—	214	4	no	—
108	C₂×C₄	—	214	2	no	—
109	C₂×C₄	—	214	2	no	—
110	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
111	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
115	C₂×C₄	#13133817	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	214	4	no	—
118	C₂²	C₂×(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
119	C₂²	#16812074	1	2	yes	—
120	C₂²	M₁₆×D₁₀₇	1	2	yes	—
121	C₂²	#16812115	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	428	2	no	—
125	C₄	—	214	4	no	—
126	C₄	#16812089	1	2	yes	—
127	C₄	#16812096	1	1	yes	characteristic
128	C₄	#16812107	1	1	yes	characteristic
129	C₂	#20292532	1	1	yes	characteristic
130	C₂	#20292590	1	1	yes	characteristic
131	C₂	#20292593	1	1	yes	characteristic
132	C₂	#20292756	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₁₄

C₂

yes

maximal, characteristic

C₂×#16812020

C₂

yes

maximal, characteristic

#20292381

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₄

yes

characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₄×Dic₂₁₄

—

C₄×Dic₂₁₄

—

C₂×C₄⋊Dic₁₀₇

C₄

yes

characteristic

C₂×Dic₁₀₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

#16812020

—

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂²×Dic₁₀₇

C₂×C₄

yes

—

C₂²×Dic₁₀₇

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₀₇

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

M₁₆

yes

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₂²×C₁₀₇

C₂×M₁₆

yes

—

C₂²×C₁₀₇

#131

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

#124

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

#1985

—

107

maximal, Sylow

C₁₀₇

#1985

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

107

—

C₂×C₄⋊C₈

D₁₀₇

yes

characteristic

C₂×#97

—

107

—

#336

—

107

—

C₂²×Q₈

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₄×Q₈

—

214

—

C₄×Q₈

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

#97

—

214

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

—

214

—

C₄²

—

214

—

C₄²

D₄₂₈

yes

—

100

C₄²

C₂²⋊D₁₀₇

yes

—

101

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

102

Q₈

—

214

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

107

—

106

C₂×C₄

—

214

—

107

C₂×C₄

—

214

—

108

C₂×C₄

—

214

—

109

C₂×C₄

—

214

—

110

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

111

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

113

C₂×C₄

C₂×D₄₂₈

yes

characteristic

114

C₂×C₄

Q₈×D₁₀₇

yes

—

115

C₂×C₄

#13133817

yes

—

116

C₈

—

117

C₈

—

214

—

118

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

yes

characteristic

119

C₂²

#16812074

yes

—

120

C₂²

M₁₆×D₁₀₇

yes

—

121

C₂²

#16812115

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

428

—

125

C₄

—

214

—

126

C₄

#16812089

yes

—

127

C₄

#16812096

yes

characteristic

128

C₄

#16812107

yes

characteristic

129

C₂

#20292532

yes

characteristic

130

C₂

#20292590

yes

characteristic

131

C₂

#20292593

yes

characteristic

132

C₂

#20292756

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₇
commutator, socle, prime radical	C₂³×C₁₀₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₇	C₂
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₇
hypocenter	C₁₀₇	#1985
LCS	G ▷ C₂³×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₇

commutator, socle, prime radical

C₂³×C₁₀₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₀₇

C₂

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₇

C₁₀₇

G ▷ C₂³×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₂×C₁₀₇⋊C₈	Q₈	semidirect
67. C₁₀₇	#1985	semidirect

H ≅ G/N

type

22. C₂×C₁₀₇⋊C₈

Q₈

semidirect

67. C₁₀₇

#1985

semidirect

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

subgroup	index
C₂×C₄×Dic₂₁₄	2
C₂×#16812020	2
#20292381	2
C₂×C₄⋊C₈×C₁₀₇	2
#1985	107

subgroup

index

C₂×C₄×Dic₂₁₄

C₂×#16812020

#20292381

C₂×C₄⋊C₈×C₁₀₇

#1985

107

Subgroups — #28337400

Conjugacy classes of subgroups

Complements

Maximal subgroups