Group #28349128 — subgroups

Subgroups — #28349128

Order 13696.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20292973	C₂	1	3	yes	maximal, complemented
3	#20292917	C₂	1	3	yes	maximal, complemented
4	#20292824	C₂	1	3	yes	maximal, complemented
5	Q₈×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13133806	C₂	1	3	yes	maximal, complemented
7	#550×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133806	C₂²	1	3	yes	—
9	#16812109	C₂²	1	3	yes	—
10	#16812108	—	2	3	no	—
11	Q₈×Dic₁₀₇	—	2	1	no	—
12	C₄×Dic₂₁₄	C₂²	1	3	yes	—
13	#16812040	C₂²	1	3	yes	—
14	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	3	yes	—
15	#16812032	C₂²	1	1	yes	characteristic
16	#16812030	C₂²	1	6	yes	—
17	#16812030	C₂²	1	3	yes	—
18	C₄×C₁₀₇⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₁₀₇	—	2	1	no	—
20	#127×C₁₀₇	C₂²	1	3	yes	—
21	#121×C₁₀₇	—	2	3	no	—
22	C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₀₇	C₂²	1	3	yes	—
24	C₂×Dic₂₁₄	—	2	3	no	—
25	C₂×Dic₂₁₄	C₂³	1	1	yes	characteristic
26	#13133806	—	2	6	no	—
27	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
28	C₂×C₁₀₇⋊C₈	C₂³	1	3	yes	—
29	C₄⋊Dic₁₀₇	—	2	3	no	—
30	C₄⋊Dic₁₀₇	C₂³	1	3	yes	—
31	Dic₁₀₇⋊C₄	—	2	3	no	—
32	C₄×Dic₁₀₇	—	2	3	no	—
33	#41×C₁₀₇	—	4	4	no	—
34	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₀₇	—	2	3	no	—
36	C₂×D₄×C₁₀₇	—	2	3	no	—
37	C₂×D₄×C₁₀₇	C₂³	1	3	yes	—
38	C₂²×C₄×C₁₀₇	—	2	3	no	—
39	C₄⋊C₄×C₁₀₇	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₀₇	C₂³	1	1	yes	characteristic
41	C₂²⋊C₄×C₁₀₇	—	2	3	no	—
42	C₄²×C₁₀₇	C₂³	1	3	yes	—
43	C₂³×C₁₀₇	—	2	3	no	—
44	Q₈×C₁₀₇	SD₁₆	1	4	yes	complemented
45	D₄×C₁₀₇	—	2	3	no	—
46	D₄×C₁₀₇	—	2	3	no	—
47	D₄×C₁₀₇	—	4	6	no	—
48	D₄×C₁₀₇	—	8	3	no	—
49	C₂×C₄×C₁₀₇	—	2	1	no	—
50	C₂×C₄×C₁₀₇	—	4	6	no	—
51	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₁₀₇	—	2	1	no	—
55	C₂×Dic₁₀₇	—	2	3	no	—
56	Dic₂₁₄	—	2	1	no	—
57	Dic₂₁₄	—	8	3	no	—
58	Dic₂₁₄	#41	1	2	yes	complemented
59	C₁₀₇⋊C₈	—	2	3	no	—
60	C₁₀₇⋊C₈	—	2	1	no	—
61	C₂²×C₁₀₇	—	4	3	no	—
62	C₂²×C₁₀₇	—	4	3	no	—
63	C₂²×C₁₀₇	—	4	3	no	—
64	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₀₇	—	2	3	no	—
66	C₄×C₁₀₇	—	4	1	no	—
67	C₄×C₁₀₇	C₂×SD₁₆	1	6	yes	—
68	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
69	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₁₀₇	—	2	2	no	—
71	Dic₁₀₇	—	4	3	no	—
72	C₂×C₁₀₇	—	8	3	no	—
73	C₂×C₁₀₇	#548	1	1	yes	characteristic
74	C₂×C₁₀₇	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₁₀₇	#576	1	1	yes	characteristic
76	#3362	—	107	1	no	maximal, Sylow
77	C₁₀₇	#3362	1	1	yes	characteristic, complemented, Sylow
78	Q₈²	—	107	1	no	—
79	#550	D₁₀₇	1	1	yes	characteristic
80	#486	—	107	3	no	—
81	#476	—	107	3	no	—
82	#460	—	107	3	no	—
83	C₈×Q₈	—	107	1	no	—
84	C₄×SD₁₆	—	107	3	no	—
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	107	3	no	—
87	C₄⋊Q₈	—	107	3	no	—
88	C₄⋊Q₈	—	214	3	no	—
89	#127	C₂×D₁₀₇	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	107	3	no	—
92	C₄×Q₈	—	214	1	no	—
93	C₄×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
94	C₄×D₄	C₂×D₁₀₇	1	3	yes	—
95	#106	—	107	3	no	—
96	C₄⋊C₈	—	107	3	no	—
97	#103	—	107	1	no	—
98	#102	—	107	3	no	—
99	#102	—	107	6	no	—
100	C₄×C₈	—	107	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	107	1	no	—
103	C₂×Q₈	—	214	3	no	—
104	C₂×Q₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₁₀₇	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	214	6	no	—
110	C₂×C₈	—	107	1	no	—
111	C₂×C₈	—	107	3	no	—
112	C₄⋊C₄	—	107	3	no	—
113	C₄⋊C₄	—	214	3	no	—
114	C₄⋊C₄	—	214	3	no	—
115	C₄⋊C₄	C₂²⋊D₁₀₇	1	3	yes	—
116	C₄⋊C₄	C₂²×D₁₀₇	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	214	3	no	—
119	C₄²	C₂²×D₁₀₇	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	856	3	no	—
122	Q₈	—	107	2	no	—
123	Q₈	—	214	1	no	—
124	Q₈	#13133806	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	214	3	no	—
132	C₂×C₄	—	214	1	no	—
133	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	3	yes	—
135	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
136	C₈	—	214	3	no	—
137	C₈	—	214	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	428	3	no	—
145	C₄	—	214	2	no	—
146	C₄	C₂×#13133806	1	6	yes	—
147	C₄	#41×D₁₀₇	1	1	yes	characteristic
148	C₄	#16812156	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#20293284	1	1	yes	characteristic
151	C₂	C₂²×#13133806	1	1	yes	characteristic
152	C₂	#20293430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₁₄

C₂

yes

maximal, characteristic, complemented

#20292973

C₂

yes

maximal, complemented

#20292917

C₂

yes

maximal, complemented

#20292824

C₂

yes

maximal, complemented

Q₈×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#13133806

C₂

yes

maximal, complemented

#550×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133806

C₂²

yes

—

#16812109

C₂²

yes

—

#16812108

—

Q₈×Dic₁₀₇

—

C₄×Dic₂₁₄

C₂²

yes

—

#16812040

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812032

C₂²

yes

characteristic

#16812030

C₂²

yes

—

#16812030

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₂×#41×C₁₀₇

—

#127×C₁₀₇

C₂²

yes

—

#121×C₁₀₇

—

C₄×Q₈×C₁₀₇

C₂²

yes

characteristic

C₄×D₄×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂³

yes

characteristic

#13133806

—

C₂×C₁₀₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

C₂³

yes

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

characteristic

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

—

Q₈×C₁₀₇

SD₁₆

yes

complemented

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

#41

yes

complemented

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

C₂×#41

yes

characteristic

C₄×C₁₀₇

2₋¹⁺⁴

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#548

yes

characteristic

C₂×C₁₀₇

C₂²×SD₁₆

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#3362

yes

characteristic, complemented, Sylow

Q₈²

—

107

—

#550

D₁₀₇

yes

characteristic

#486

—

107

—

#476

—

107

—

#460

—

107

—

C₈×Q₈

—

107

—

C₄×SD₁₆

—

107

—

C₂×#41

—

C₂×SD₁₆

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

214

—

#127

C₂×D₁₀₇

yes

—

#121

—

C₄×Q₈

—

107

—

C₄×Q₈

—

214

—

C₄×Q₈

C₂×D₁₀₇

yes

characteristic

C₄×D₄

C₂×D₁₀₇

yes

—

#106

—

107

—

C₄⋊C₈

—

107

—

#103

—

107

—

#102

—

107

—

#102

—

107

—

100

C₄×C₈

—

107

—

101

#41

—

102

C₂×Q₈

—

107

—

103

C₂×Q₈

—

214

—

104

C₂×Q₈

C₂²⋊D₁₀₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₁₀₇

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

214

—

110

C₂×C₈

—

107

—

111

C₂×C₈

—

107

—

112

C₄⋊C₄

—

107

—

113

C₄⋊C₄

—

214

—

114

C₄⋊C₄

—

214

—

115

C₄⋊C₄

C₂²⋊D₁₀₇

yes

—

116

C₄⋊C₄

C₂²×D₁₀₇

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

214

—

119

C₄²

C₂²×D₁₀₇

yes

—

120

C₂³

—

121

Q₈

—

856

—

122

Q₈

—

107

—

123

Q₈

—

214

—

124

Q₈

#13133806

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

214

—

132

C₂×C₄

—

214

—

133

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

135

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

136

C₈

—

214

—

137

C₈

—

214

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₁₀₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

428

—

145

C₄

—

214

—

146

C₄

C₂×#13133806

yes

—

147

C₄

#41×D₁₀₇

yes

characteristic

148

C₄

#16812156

yes

characteristic

149

C₂

—

150

C₂

#20293284

yes

characteristic

151

C₂

C₂²×#13133806

yes

characteristic

152

C₂

#20293430

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₇
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting, prime radical	#550×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
hypercenter	#550	D₁₀₇
hypocenter	C₁₀₇	#3362
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₇

commutator, cosocle

C₂×C₄×C₁₀₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting, prime radical

#550×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂²×D₄

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₂₁₄	C₂	semidirect
2. #20292973	C₂	semidirect
3. #20292917	C₂	semidirect
4. #20292824	C₂	semidirect
5. Q₈×C₁₀₇⋊C₈	C₂	semidirect
6. C₄×#13133806	C₂	semidirect
44. Q₈×C₁₀₇	SD₁₆	semidirect
58. Dic₂₁₄	#41	semidirect
77. C₁₀₇	#3362	semidirect
124. Q₈	#13133806	semidirect

H ≅ G/N

type

1. Q₈×Dic₂₁₄

C₂

semidirect

2. #20292973

C₂

semidirect

3. #20292917

C₂

semidirect

4. #20292824

C₂

semidirect

5. Q₈×C₁₀₇⋊C₈

C₂

semidirect

6. C₄×#13133806

C₂

semidirect

44. Q₈×C₁₀₇

SD₁₆

semidirect

58. Dic₂₁₄

#41

semidirect

77. C₁₀₇

#3362

semidirect

124. Q₈

#13133806

semidirect

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

subgroup	index
Q₈×Dic₂₁₄	2
#20292973	2
#20292917	2
#20292824	2
Q₈×C₁₀₇⋊C₈	2
C₄×#13133806	2
#550×C₁₀₇	2
#3362	107

subgroup

index

Q₈×Dic₂₁₄

Q₈×C₁₀₇⋊C₈

#550×C₁₀₇

107

Subgroups — #28349128

Conjugacy classes of subgroups

Complements

Maximal subgroups