Group #28346841 — subgroups

Subgroups — #28346841

Order 13696.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20293184	C₂	1	1	yes	maximal, characteristic
2	#20292995	C₂	1	4	yes	maximal, complemented
3	#20292977	C₂	1	4	yes	maximal, complemented
4	#20292636	C₂	1	4	yes	maximal
5	C₄×#13133802	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133802	C₂²	1	2	yes	—
8	#16812109	C₂²	1	2	yes	—
9	#16812101	C₂²	1	2	yes	—
10	#16812088	—	2	2	no	—
11	C₄×Dic₂₁₄	—	2	2	no	—
12	#16812068	C₂²	1	2	yes	—
13	#16812033	C₂²	1	16	yes	—
14	#16812020	C₂²	1	2	yes	—
15	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
16	C₂²×Q₈×C₁₀₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
18	C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
21	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₁₄	—	2	2	no	—
23	#13133802	—	2	4	no	—
24	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
25	C₄⋊Dic₁₀₇	—	2	2	no	—
26	C₄⋊Dic₁₀₇	C₂³	1	4	yes	—
27	Dic₁₀₇⋊C₄	—	2	4	no	—
28	C₂²⋊Dic₁₀₇	—	2	2	no	—
29	C₄×Dic₁₀₇	—	2	2	no	—
30	C₂×Q₈×C₁₀₇	—	2	2	no	—
31	C₂×Q₈×C₁₀₇	—	2	8	no	—
32	C₂×Q₈×C₁₀₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₀₇	—	2	2	no	—
34	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₇	—	2	4	no	—
37	C₄⋊C₄×C₁₀₇	—	2	4	no	—
38	C₄²×C₁₀₇	D₄	1	2	yes	—
39	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
40	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₀₇	—	2	8	no	—
43	Q₈×C₁₀₇	—	4	4	no	—
44	C₂×C₄×C₁₀₇	—	2	4	no	—
45	C₂×C₄×C₁₀₇	—	2	4	no	—
46	C₂×C₄×C₁₀₇	—	4	4	no	—
47	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₀₇	#41	1	4	yes	—
51	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₁₀₇	—	2	4	no	—
53	Dic₂₁₄	—	4	4	no	—
54	C₁₀₇⋊C₈	—	2	4	no	—
55	C₂²×C₁₀₇	—	2	2	no	—
56	C₂²×C₁₀₇	#124	1	2	yes	—
57	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₇	—	2	2	no	—
59	C₄×C₁₀₇	—	4	4	no	—
60	C₄×C₁₀₇	#124	1	2	yes	—
61	C₄×C₁₀₇	#137	1	4	yes	—
62	C₄×C₁₀₇	C₂×#41	1	2	yes	—
63	Dic₁₀₇	—	4	4	no	—
64	C₂×C₁₀₇	—	2	2	no	—
65	C₂×C₁₀₇	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₀₇	C₂×#137	1	2	yes	—
67	#3139	—	107	1	no	maximal, Sylow
68	C₁₀₇	#3139	1	1	yes	characteristic, complemented, Sylow
69	#533	—	107	1	no	—
70	C₂×C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic
71	#491	—	107	4	no	—
72	#487	—	107	4	no	—
73	#419	—	107	4	no	—
74	C₄×M₁₆	—	107	1	no	—
75	C₂²×Q₈	C₂×D₁₀₇	1	2	yes	—
76	C₂×M₁₆	—	107	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	107	2	no	—
79	C₄⋊Q₈	C₂×D₁₀₇	1	2	yes	—
80	C₄⋊Q₈	C₂×D₁₀₇	1	2	yes	—
81	#125	—	107	2	no	—
82	C₂²⋊Q₈	—	214	2	no	—
83	C₄×Q₈	—	214	2	no	—
84	#117	—	107	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
87	#103	—	107	16	no	—
88	#97	—	107	2	no	—
89	C₄×C₈	—	107	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	214	2	no	—
93	C₂×Q₈	C₂²×D₁₀₇	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
97	M₁₆	—	214	4	no	—
98	C₂×C₈	—	107	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	107	4	no	—
102	C₄⋊C₄	—	214	4	no	—
103	C₄⋊C₄	—	214	2	no	—
104	C₂²⋊C₄	—	214	2	no	—
105	C₄²	—	214	2	no	—
106	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
107	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	428	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	214	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
120	C₂×C₄	#13133817	1	4	yes	—
121	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
122	C₈	—	214	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#16812096	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	428	4	no	—
129	C₄	#16812096	1	2	yes	—
130	C₄	#16812140	1	4	yes	—
131	C₄	C₂×#13133817	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#16812096	1	1	yes	characteristic
134	C₂	C₂×#16812140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293184

C₂

yes

maximal, characteristic

#20292995

C₂

yes

maximal, complemented

#20292977

C₂

yes

maximal, complemented

#20292636

C₂

yes

maximal

C₄×#13133802

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

#16812109

C₂²

yes

—

#16812101

C₂²

yes

—

#16812088

—

C₄×Dic₂₁₄

—

#16812068

C₂²

yes

—

#16812033

C₂²

yes

—

#16812020

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₁₀₇

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

#13133802

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

C₂³

yes

—

Dic₁₀₇⋊C₄

—

C₂²⋊Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂×Dic₁₀₇

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#124

yes

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#124

yes

—

C₄×C₁₀₇

#137

yes

—

C₄×C₁₀₇

C₂×#41

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#124

yes

characteristic

C₂×C₁₀₇

C₂×#137

yes

—

#3139

—

107

maximal, Sylow

C₁₀₇

#3139

yes

characteristic, complemented, Sylow

#533

—

107

—

C₂×C₄⋊Q₈

D₁₀₇

yes

characteristic

#491

—

107

—

#487

—

107

—

#419

—

107

—

C₄×M₁₆

—

107

—

C₂²×Q₈

C₂×D₁₀₇

yes

—

C₂×M₁₆

—

107

—

C₄⋊Q₈

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

—

C₄⋊Q₈

C₂×D₁₀₇

yes

—

#125

—

107

—

C₂²⋊Q₈

—

214

—

C₄×Q₈

—

214

—

#117

—

107

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#103

—

107

—

#97

—

107

—

C₄×C₈

—

107

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

214

—

C₂×Q₈

C₂²×D₁₀₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

214

—

C₂×C₈

—

107

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

107

—

102

C₄⋊C₄

—

214

—

103

C₄⋊C₄

—

214

—

104

C₂²⋊C₄

—

214

—

105

C₄²

—

214

—

106

C₄²

C₂²⋊D₁₀₇

yes

—

107

C₄²

C₂²×D₁₀₇

yes

characteristic

108

C₄²

C₂²×D₁₀₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

428

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

214

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

120

C₂×C₄

#13133817

yes

—

121

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

122

C₈

—

214

—

123

C₂²

—

124

C₂²

#16812096

yes

—

125

C₂²

C₂²×C₂²⋊D₁₀₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

428

—

129

C₄

#16812096

yes

—

130

C₄

#16812140

yes

—

131

C₄

C₂×#13133817

yes

—

132

C₂

—

133

C₂

C₂×#16812096

yes

characteristic

134

C₂

C₂×#16812140

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₇
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₂×C₄⋊Q₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂³×C₁₀₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#3139
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₇

commutator, cosocle

C₂×C₄×C₁₀₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

C₂×C₄⋊Q₈×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂²×D₄

prime radical

C₂³×C₁₀₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₀₇

hypocenter

C₁₀₇

#3139

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20292995	C₂	semidirect
3. #20292977	C₂	semidirect
68. C₁₀₇	#3139	semidirect

H ≅ G/N

type

2. #20292995

C₂

semidirect

3. #20292977

C₂

semidirect

68. C₁₀₇

#3139

semidirect

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

subgroup	index
#20293184	2
#20292995	2
#20292977	2
#20292636	2
C₄×#13133802	2
C₂×C₄⋊Q₈×C₁₀₇	2
#3139	107

subgroup

index

C₂×C₄⋊Q₈×C₁₀₇

107

Subgroups — #28346841

Conjugacy classes of subgroups

Complements

Maximal subgroups