Group #28336091 — subgroups

Subgroups — #28336091

Order 13696.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#20293023	C₂	1	1	yes	maximal, characteristic, complemented
2	#20293003	C₂	1	1	yes	maximal, characteristic
3	#20293001	C₂	1	1	yes	maximal, characteristic, complemented
4	#20293000	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292973	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292349	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133807	—	2	2	no	—
9	C₂×#13133805	—	2	2	no	—
10	C₂×#13133803	—	2	2	no	—
11	#16812109	C₂²	1	1	yes	characteristic
12	#16812106	C₂²	1	1	yes	characteristic
13	#16812040	—	2	2	no	—
14	#16812030	—	2	2	no	—
15	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
18	#127×C₁₀₇	C₂²	1	1	yes	characteristic
19	#102×C₁₀₇	—	2	2	no	—
20	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₄	—	2	1	no	—
22	C₂×D₄₂₈	—	2	1	no	—
23	C₂×D₄₂₈	—	2	1	no	—
24	C₂×D₄₂₈	—	2	1	no	—
25	#13133807	—	4	4	no	—
26	#13133805	—	4	4	no	—
27	#13133803	—	4	4	no	—
28	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
29	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
30	C₄⋊Dic₁₀₇	—	2	1	no	—
31	C₄⋊Dic₁₀₇	—	2	1	no	—
32	C₂×Q₈×C₁₀₇	—	2	1	no	—
33	C₂×D₄×C₁₀₇	—	2	1	no	—
34	C₂×D₄×C₁₀₇	—	2	1	no	—
35	C₂×D₄×C₁₀₇	—	2	1	no	—
36	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₀₇	—	2	1	no	—
38	C₄⋊C₄×C₁₀₇	—	2	1	no	—
39	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
40	C₂³×C₁₀₇	—	4	1	no	—
41	C₂²×D₁₀₇	—	4	1	no	—
42	Q₈×C₁₀₇	—	4	2	no	—
43	D₄×C₁₀₇	—	4	2	no	—
44	D₄×C₁₀₇	—	4	2	no	—
45	D₄×C₁₀₇	—	4	2	no	—
46	C₂×C₄×C₁₀₇	—	4	1	no	—
47	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₀₇	—	4	1	no	—
51	D₄₂₈	—	4	2	no	—
52	D₄₂₈	—	4	2	no	—
53	D₄₂₈	—	4	2	no	—
54	Dic₂₁₄	—	4	2	no	—
55	C₈×C₁₀₇	—	2	2	no	—
56	C₁₀₇⋊C₈	—	2	2	no	—
57	C₁₀₇⋊C₈	SD₁₆	1	4	yes	complemented
58	C₂²×C₁₀₇	—	8	1	no	—
59	C₂²×C₁₀₇	—	8	1	no	—
60	C₂²×C₁₀₇	—	8	1	no	—
61	C₂²×C₁₀₇	#127	1	1	yes	characteristic
62	C₂×D₁₀₇	—	8	1	no	—
63	C₂×D₁₀₇	—	8	1	no	—
64	C₂×D₁₀₇	—	8	1	no	—
65	C₄×C₁₀₇	—	8	1	no	—
66	C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
67	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₁₀₇	—	8	1	no	—
70	C₂×C₁₀₇	—	16	1	no	—
71	C₂×C₁₀₇	#492	1	1	yes	characteristic
72	C₂×C₁₀₇	#496	1	1	yes	characteristic
73	C₂×C₁₀₇	#496	1	1	yes	characteristic
74	D₁₀₇	—	16	1	no	—
75	#1715	—	107	1	no	maximal, Sylow
76	C₁₀₇	#1715	1	1	yes	characteristic, complemented, Sylow
77	#498	—	107	1	no	—
78	#493	—	107	1	no	—
79	#492	—	107	1	no	—
80	#492	—	107	1	no	—
81	#486	—	107	1	no	—
82	#486	D₁₀₇	1	1	yes	characteristic, complemented
83	#322	—	107	1	no	—
84	C₂×SD₁₆	—	214	2	no	—
85	C₂×SD₁₆	—	214	2	no	—
86	C₂×D₈	—	214	2	no	—
87	C₄⋊Q₈	—	107	1	no	—
88	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
89	#127	—	107	1	no	—
90	#127	C₂×D₁₀₇	1	1	yes	characteristic
91	#106	—	214	2	no	—
92	#102	—	2	2	no	—
93	#102	—	214	2	no	—
94	C₄×C₈	—	107	1	no	—
95	C₄×C₈	—	107	1	no	—
96	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	214	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	214	1	no	—
103	C₂×D₄	—	214	1	no	—
104	C₂×D₄	—	214	1	no	—
105	SD₁₆	—	428	4	no	—
106	SD₁₆	—	428	4	no	—
107	D₈	—	428	4	no	—
108	C₂×C₈	—	107	2	no	—
109	C₂×C₈	—	107	2	no	—
110	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	214	1	no	—
114	C₄⋊C₄	—	214	1	no	—
115	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	428	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	428	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	428	2	no	—
124	D₄	—	428	2	no	—
125	D₄	—	428	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	428	1	no	—
128	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	107	4	no	—
133	C₈	—	214	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	856	1	no	—
138	C₂²	—	856	1	no	—
139	C₂²	—	856	1	no	—
140	C₂²	#16812105	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	856	1	no	—
143	C₄	SD₁₆×D₁₀₇	1	2	yes	—
144	C₄	#16812133	1	2	yes	—
145	C₄	#16812134	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	1712	1	no	—
148	C₂	#20292998	1	1	yes	characteristic
149	C₂	#20293015	1	1	yes	characteristic
150	C₂	#20293017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293023

C₂

yes

maximal, characteristic, complemented

#20293003

C₂

yes

maximal, characteristic

#20293001

C₂

yes

maximal, characteristic, complemented

#20293000

C₂

yes

maximal, characteristic, complemented

#20292973

C₂

yes

maximal, characteristic, complemented

#20292349

C₂

yes

maximal, characteristic, complemented

#486×C₁₀₇

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#16812106

C₂²

yes

characteristic

#16812040

—

#16812030

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

#127×C₁₀₇

C₂²

yes

characteristic

#102×C₁₀₇

—

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

#13133807

—

#13133805

—

#13133803

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#127

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

D₁₀₇

—

#1715

—

107

maximal, Sylow

C₁₀₇

#1715

yes

characteristic, complemented, Sylow

#498

—

107

—

#493

—

107

—

#492

—

107

—

#492

—

107

—

#486

—

107

—

#486

D₁₀₇

yes

characteristic, complemented

#322

—

107

—

C₂×SD₁₆

—

214

—

C₂×SD₁₆

—

214

—

C₂×D₈

—

214

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#127

—

107

—

#127

C₂×D₁₀₇

yes

characteristic

#106

—

214

—

#102

—

#102

—

214

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

214

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

214

—

103

C₂×D₄

—

214

—

104

C₂×D₄

—

214

—

105

SD₁₆

—

428

—

106

SD₁₆

—

428

—

107

D₈

—

428

—

108

C₂×C₈

—

107

—

109

C₂×C₈

—

107

—

110

C₂×C₈

C₂²⋊D₁₀₇

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

214

—

114

C₄⋊C₄

—

214

—

115

C₄²

C₂²×D₁₀₇

yes

characteristic

116

C₂³

—

117

C₂³

—

428

—

118

Q₈

—

119

Q₈

—

428

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

428

—

124

D₄

—

428

—

125

D₄

—

428

—

126

C₂×C₄

—

127

C₂×C₄

—

428

—

128

C₂×C₄

D₄×D₁₀₇

yes

characteristic

129

C₂×C₄

D₄×D₁₀₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

131

C₈

—

132

C₈

—

107

—

133

C₈

—

214

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

856

—

138

C₂²

—

856

—

139

C₂²

—

856

—

140

C₂²

#16812105

yes

characteristic

141

C₄

—

142

C₄

—

856

—

143

C₄

SD₁₆×D₁₀₇

yes

—

144

C₄

#16812133

yes

—

145

C₄

#16812134

yes

—

146

C₂

—

147

C₂

—

1712

—

148

C₂

#20292998

yes

characteristic

149

C₂

#20293015

yes

characteristic

150

C₂

#20293017

yes

characteristic

	subgroup	G/H
center	C₂²	#16812105
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#486×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#127
prime radical	#20293003	C₂
hypercenter	#486	D₁₀₇
hypocenter	C₁₀₇	#1715
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#16812105

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#486×C₁₀₇

C₂

C₂²×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #20293023	C₂	semidirect
3. #20293001	C₂	semidirect
4. #20293000	C₂	semidirect
5. #20292973	C₂	semidirect
6. #20292349	C₂	semidirect
7. #486×C₁₀₇	C₂	semidirect
57. C₁₀₇⋊C₈	SD₁₆	semidirect
76. C₁₀₇	#1715	semidirect
82. #486	D₁₀₇	semidirect

H ≅ G/N

type

1. #20293023

C₂

semidirect

3. #20293001

C₂

semidirect

4. #20293000

C₂

semidirect

5. #20292973

C₂

semidirect

6. #20292349

C₂

semidirect

7. #486×C₁₀₇

C₂

semidirect

57. C₁₀₇⋊C₈

SD₁₆

semidirect

76. C₁₀₇

#1715

semidirect

82. #486

D₁₀₇

semidirect

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

subgroup	index
#20293023	2
#20293003	2
#20293001	2
#20293000	2
#20292973	2
#20292349	2
#486×C₁₀₇	2
#1715	107

subgroup

index

#486×C₁₀₇

107

Subgroups — #28336091

Conjugacy classes of subgroups

Complements

Maximal subgroups