Group #28336680 — subgroups

Subgroups — #28336680

Order 13696.

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

normal subgroups only

This group has 252 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	#20293394	C₂	1	1	yes	maximal, characteristic
2	#20292675	C₂	1	2	yes	maximal, complemented
3	#20292611	C₂	1	1	yes	maximal, characteristic
4	#20292392	C₂	1	2	yes	maximal, complemented
5	#408×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	#16812116	—	2	4	no	—
7	C₂×#13133802	—	2	2	no	—
8	C₂×#13133802	C₂²	1	2	yes	—
9	C₂×#13133802	C₂²	1	2	yes	—
10	#16812044	—	2	4	no	—
11	#16812028	C₄	1	2	yes	—
12	#16812026	C₄	1	2	yes	—
13	C₂×#41×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
17	#13133802	—	2	4	no	—
18	#13133802	—	2	2	no	—
19	#13133802	—	2	2	no	—
20	#13133802	C₂×C₄	1	4	yes	—
21	#13133802	D₄	1	4	yes	—
22	C₂×C₁₀₇⋊C₈	—	2	4	no	—
23	C₂×C₁₀₇⋊C₈	—	2	2	no	—
24	C₂×C₁₀₇⋊C₈	—	2	2	no	—
25	#41×C₁₀₇	—	2	2	no	—
26	#41×C₁₀₇	—	2	2	no	—
27	C₂×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₀₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₀₇	—	2	2	no	—
33	C₂×C₈×C₁₀₇	—	2	1	no	—
34	C₂×C₈×C₁₀₇	—	2	1	no	—
35	C₂×C₈×C₁₀₇	—	2	2	no	—
36	C₂³×C₁₀₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₁₀₇	—	2	2	no	—
39	D₄×C₁₀₇	—	2	4	no	—
40	D₄×C₁₀₇	—	2	2	no	—
41	C₂×C₄×C₁₀₇	—	2	2	no	—
42	C₂×C₄×C₁₀₇	—	2	2	no	—
43	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₇	#41	1	2	yes	—
48	C₈×C₁₀₇	—	4	1	no	—
49	C₈×C₁₀₇	—	4	1	no	—
50	C₁₀₇⋊C₈	—	2	4	no	—
51	C₁₀₇⋊C₈	—	2	4	no	—
52	C₂²×C₁₀₇	—	2	2	no	—
53	C₂²×C₁₀₇	—	4	1	no	—
54	C₂²×C₁₀₇	—	4	2	no	—
55	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₀₇	—	2	2	no	—
58	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
59	C₄×C₁₀₇	#120	1	1	yes	characteristic
60	C₄×C₁₀₇	#123	1	1	yes	characteristic
61	C₂×C₁₀₇	—	2	1	no	—
62	C₂×C₁₀₇	—	2	2	no	—
63	C₂×C₁₀₇	—	4	2	no	—
64	C₂×C₁₀₇	#385	1	1	yes	characteristic
65	#1858	—	107	1	no	maximal, Sylow
66	C₁₀₇	#1858	1	1	yes	characteristic, complemented, Sylow
67	#568	—	107	1	no	—
68	#430	—	107	2	no	—
69	#413	—	107	1	no	—
70	#408	D₁₀₇	1	1	yes	characteristic
71	#341	—	107	2	no	—
72	C₂×#41	C₂×D₁₀₇	1	1	yes	characteristic
73	#131	—	214	4	no	—
74	C₂×M₁₆	—	107	2	no	—
75	C₂×M₁₆	—	107	2	no	—
76	C₂×M₁₆	—	214	2	no	—
77	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
79	#108	—	214	4	no	—
80	#101	—	107	2	no	—
81	#100	—	107	2	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₀₇	1	1	yes	characteristic
86	C₂×D₄	Dic₂₁₄	1	2	yes	—
87	C₂×D₄	C₄×D₁₀₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	107	4	no	—
92	M₁₆	—	107	4	no	—
93	M₁₆	—	214	4	no	—
94	M₁₆	—	214	2	no	—
95	M₁₆	—	214	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	214	4	no	—
100	C₂×C₈	—	214	2	no	—
101	C₂×C₈	—	214	2	no	—
102	C₂³	Dic₁₀₇⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₂₁₄	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
113	C₂×C₄	#13133817	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	214	4	no	—
117	C₈	—	214	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×Dic₁₀₇⋊C₄	1	1	yes	characteristic
122	C₂²	#16812086	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	(C₂²⋊D₁₀₇)⋊C₄	1	2	yes	—
125	C₄	#16812077	1	1	yes	characteristic
126	C₄	#16812090	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#20292499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293394

C₂

yes

maximal, characteristic

#20292675

C₂

yes

maximal, complemented

#20292611

C₂

yes

maximal, characteristic

#20292392

C₂

yes

maximal, complemented

#408×C₁₀₇

C₂

yes

maximal, characteristic

#16812116

—

C₂×#13133802

—

C₂×#13133802

C₂²

yes

—

C₂×#13133802

C₂²

yes

—

#16812044

—

#16812028

C₄

yes

—

#16812026

C₄

yes

—

C₂×#41×C₁₀₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₇

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

C₂²⋊C₈×C₁₀₇

—

—

—

—

C₂×C₄

yes

—

#13133802

D₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

#41×C₁₀₇

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂³×C₁₀₇

C₄⋊C₄

yes

—

C₂³×C₁₀₇

C₂×Q₈

yes

characteristic

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#385

yes

characteristic

#1858

—

107

maximal, Sylow

C₁₀₇

#1858

yes

characteristic, complemented, Sylow

#568

—

107

—

#430

—

107

—

#413

—

107

—

#408

D₁₀₇

yes

characteristic

#341

—

107

—

C₂×#41

C₂×D₁₀₇

yes

characteristic

#131

—

214

—

C₂×M₁₆

—

107

—

C₂×M₁₆

—

107

—

C₂×M₁₆

—

214

—

C₂×M₁₆

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

#108

—

214

—

#101

—

107

—

#100

—

107

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₀₇

yes

characteristic

C₂×D₄

Dic₂₁₄

yes

—

C₂×D₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

107

—

M₁₆

—

107

—

M₁₆

—

214

—

M₁₆

—

214

—

M₁₆

—

214

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

214

—

100

C₂×C₈

—

214

—

101

C₂×C₈

—

214

—

102

C₂³

Dic₁₀₇⋊C₄

yes

—

103

C₂³

C₂×Dic₂₁₄

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

111

C₂×C₄

D₄×D₁₀₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

113

C₂×C₄

#13133817

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

214

—

117

C₈

—

214

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×Dic₁₀₇⋊C₄

yes

characteristic

122

C₂²

#16812086

yes

—

123

C₄

—

124

C₄

(C₂²⋊D₁₀₇)⋊C₄

yes

—

125

C₄

#16812077

yes

characteristic

126

C₄

#16812090

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20292499

yes

characteristic

	subgroup	G/H
center	C₄	#16812077
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#408×C₁₀₇	C₂
socle	C₂×C₁₀₇	#385
prime radical	C₂×#41×C₁₀₇	C₂²
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#408	D₁₀₇
hypocenter	C₁₀₇	#1858
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#16812077

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#408×C₁₀₇

C₂

socle

C₂×C₁₀₇

#385

prime radical

C₂×#41×C₁₀₇

C₂²

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #20292675	C₂	semidirect
4. #20292392	C₂	semidirect
66. C₁₀₇	#1858	semidirect

H ≅ G/N

type

2. #20292675

C₂

semidirect

4. #20292392

C₂

semidirect

66. C₁₀₇

#1858

semidirect

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

subgroup	index
#20293394	2
#20292675	2
#20292611	2
#20292392	2
#408×C₁₀₇	2
#1858	107

subgroup

index

#408×C₁₀₇

107

Subgroups — #28336680

Conjugacy classes of subgroups

Complements

Maximal subgroups