Group #28336080 — subgroups

Subgroups — #28336080

Order 13696.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20293032	C₂	1	1	yes	maximal, characteristic, complemented
2	#20293023	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292999	C₂	1	1	yes	maximal, characteristic
4	#20292979	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292973	C₂	1	1	yes	maximal, characteristic
6	#20292348	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133806	—	2	2	no	—
9	#16812109	C₂²	1	1	yes	characteristic
10	#16812109	C₂²	1	1	yes	characteristic
11	#16812043	—	2	2	no	—
12	#16812040	—	2	2	no	—
13	#16812032	—	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	C₂×SD₁₆×C₁₀₇	—	2	2	no	—
18	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
19	#127×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₄	—	2	1	no	—
22	C₂×Dic₂₁₄	—	2	1	no	—
23	#13133806	—	4	4	no	—
24	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₀₇	—	2	1	no	—
27	C₄⋊Dic₁₀₇	—	2	1	no	—
28	C₄⋊Dic₁₀₇	—	2	1	no	—
29	C₄⋊Dic₁₀₇	—	2	1	no	—
30	C₂×Q₈×C₁₀₇	—	2	1	no	—
31	C₂×D₄×C₁₀₇	—	2	1	no	—
32	C₂×D₄×C₁₀₇	—	2	1	no	—
33	C₂×D₄×C₁₀₇	—	2	1	no	—
34	SD₁₆×C₁₀₇	—	4	4	no	—
35	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
36	C₄⋊C₄×C₁₀₇	—	2	1	no	—
37	C₄⋊C₄×C₁₀₇	—	2	1	no	—
38	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
39	C₂³×C₁₀₇	—	4	1	no	—
40	Q₈×C₁₀₇	—	4	2	no	—
41	D₄×C₁₀₇	—	4	2	no	—
42	D₄×C₁₀₇	—	4	2	no	—
43	D₄×C₁₀₇	—	4	2	no	—
44	C₂×C₄×C₁₀₇	—	4	1	no	—
45	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₀₇	—	4	1	no	—
49	C₂×Dic₁₀₇	—	4	1	no	—
50	Dic₂₁₄	—	4	2	no	—
51	Dic₂₁₄	—	4	2	no	—
52	C₈×C₁₀₇	SD₁₆	1	4	yes	complemented
53	C₁₀₇⋊C₈	—	2	2	no	—
54	C₁₀₇⋊C₈	—	2	2	no	—
55	C₂²×C₁₀₇	—	8	1	no	—
56	C₂²×C₁₀₇	—	8	1	no	—
57	C₂²×C₁₀₇	—	8	1	no	—
58	C₂²×C₁₀₇	#127	1	1	yes	characteristic
59	C₄×C₁₀₇	—	8	1	no	—
60	C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
61	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₀₇	#137	1	2	yes	—
63	Dic₁₀₇	—	8	1	no	—
64	Dic₁₀₇	—	8	1	no	—
65	C₂×C₁₀₇	—	16	1	no	—
66	C₂×C₁₀₇	#492	1	1	yes	characteristic
67	C₂×C₁₀₇	#496	1	1	yes	characteristic
68	C₂×C₁₀₇	#497	1	1	yes	characteristic
69	#1714	—	107	1	no	maximal, Sylow
70	C₁₀₇	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	—	107	1	no	—
72	#498	—	107	1	no	—
73	#492	—	107	1	no	—
74	#492	D₁₀₇	1	1	yes	characteristic
75	#487	—	107	1	no	—
76	#486	—	107	1	no	—
77	#322	—	107	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	214	2	no	—
80	C₄⋊Q₈	—	107	1	no	—
81	C₄⋊Q₈	—	107	1	no	—
82	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
83	#127	C₂×D₁₀₇	1	1	yes	characteristic
84	#107	—	214	2	no	—
85	#106	—	214	2	no	—
86	#103	—	214	2	no	—
87	#102	—	214	2	no	—
88	C₄×C₈	—	107	1	no	—
89	C₄×C₈	—	107	1	no	—
90	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	214	1	no	—
93	C₂×Q₈	—	214	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	428	4	no	—
99	C₂×C₈	—	107	2	no	—
100	C₂×C₈	—	107	2	no	—
101	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	214	1	no	—
105	C₄⋊C₄	—	214	1	no	—
106	C₄⋊C₄	—	214	1	no	—
107	C₄⋊C₄	—	214	1	no	—
108	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	428	2	no	—
112	Q₈	—	428	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	428	1	no	—
118	C₂×C₄	—	428	1	no	—
119	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
122	C₈	—	214	2	no	—
123	C₈	—	214	2	no	—
124	C₈	#13133806	1	4	yes	complemented
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#16812105	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	856	1	no	—
131	C₄	—	856	1	no	—
132	C₄	C₂×#13133806	1	2	yes	—
133	C₄	#16812136	1	2	yes	—
134	C₄	#16812138	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#20292999	1	1	yes	characteristic
137	C₂	#20293013	1	1	yes	characteristic
138	C₂	#20293018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293032

C₂

yes

maximal, characteristic, complemented

#20293023

C₂

yes

maximal, characteristic, complemented

#20292999

C₂

yes

maximal, characteristic

#20292979

C₂

yes

maximal, characteristic, complemented

#20292973

C₂

yes

maximal, characteristic

#20292348

C₂

yes

maximal, characteristic, complemented

#492×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133806

—

#16812109

C₂²

yes

characteristic

#16812109

C₂²

yes

—

—

—

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

#127×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

#13133806

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

SD₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

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

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

SD₁₆

yes

complemented

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#127

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₇

#137

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1714

yes

characteristic, complemented, Sylow

#500

—

107

—

#498

—

107

—

#492

—

107

—

#492

D₁₀₇

yes

characteristic

#487

—

107

—

#486

—

107

—

#322

—

107

—

C₂×SD₁₆

—

C₂×SD₁₆

—

214

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#127

C₂×D₁₀₇

yes

characteristic

#107

—

214

—

#106

—

214

—

#103

—

214

—

#102

—

214

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

428

—

C₂×C₈

—

107

—

100

C₂×C₈

—

107

—

101

C₂×C₈

C₂²⋊D₁₀₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

214

—

105

C₄⋊C₄

—

214

—

106

C₄⋊C₄

—

214

—

107

C₄⋊C₄

—

214

—

108

C₄²

C₂²×D₁₀₇

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

428

—

112

Q₈

—

428

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

428

—

118

C₂×C₄

—

428

—

119

C₂×C₄

D₄×D₁₀₇

yes

characteristic

120

C₂×C₄

D₄×D₁₀₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

122

C₈

—

214

—

123

C₈

—

214

—

124

C₈

#13133806

yes

complemented

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#16812105

yes

characteristic

129

C₄

—

130

C₄

—

856

—

131

C₄

—

856

—

132

C₄

C₂×#13133806

yes

—

133

C₄

#16812136

yes

—

134

C₄

#16812138

yes

—

135

C₂

—

136

C₂

#20292999

yes

characteristic

137

C₂

#20293013

yes

characteristic

138

C₂

#20293018

yes

characteristic

	subgroup	G/H
center	C₂²	#16812105
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#492×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#127
prime radical	#127×C₁₀₇	C₂²
hypercenter	#492	D₁₀₇
hypocenter	C₁₀₇	#1714
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#16812105

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#492×C₁₀₇

C₂

socle

C₂²×C₁₀₇

#127

prime radical

#127×C₁₀₇

C₂²

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20293032	C₂	semidirect
2. #20293023	C₂	semidirect
4. #20292979	C₂	semidirect
6. #20292348	C₂	semidirect
52. C₈×C₁₀₇	SD₁₆	semidirect
70. C₁₀₇	#1714	semidirect
124. C₈	#13133806	semidirect

H ≅ G/N

type

1. #20293032

C₂

semidirect

2. #20293023

C₂

semidirect

4. #20292979

C₂

semidirect

6. #20292348

C₂

semidirect

52. C₈×C₁₀₇

SD₁₆

semidirect

70. C₁₀₇

#1714

semidirect

124. C₈

#13133806

semidirect

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

subgroup	index
#20293032	2
#20293023	2
#20292999	2
#20292979	2
#20292973	2
#20292348	2
#492×C₁₀₇	2
#1714	107

subgroup

index

#492×C₁₀₇

107

Subgroups — #28336080

Conjugacy classes of subgroups

Complements

Maximal subgroups