Group #28349139 — subgroups

Subgroups — #28349139

Order 13696.

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

normal subgroups only

This group has 336 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	#20293356	C₂	1	1	yes	maximal, characteristic
2	#20292979	C₂	1	3	yes	maximal
3	#20292919	C₂	1	3	yes	maximal
4	#20292837	C₂	1	3	yes	maximal
5	Q₈×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#13133808	C₂	1	3	yes	maximal
7	Q₈²×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133808	C₂²	1	3	yes	—
9	#16812109	C₂²	1	3	yes	—
10	#16812100	—	2	3	no	—
11	Q₈×Dic₁₀₇	—	2	1	no	—
12	C₄×Dic₂₁₄	C₂²	1	3	yes	—
13	#16812042	C₂²	1	3	yes	—
14	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	3	yes	—
15	#16812033	C₂²	1	1	yes	characteristic
16	#16812032	C₂²	1	6	yes	—
17	#16812032	C₂²	1	3	yes	—
18	C₄×C₁₀₇⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₀₇	—	2	3	no	—
20	C₄⋊Q₈×C₁₀₇	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₀₇	—	2	1	no	—
22	C₄×Q₈×C₁₀₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₁₄	C₂³	1	3	yes	—
25	#13133808	—	2	6	no	—
26	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₂×C₁₀₇⋊C₈	C₂³	1	3	yes	—
28	C₄⋊Dic₁₀₇	—	2	3	no	—
29	C₄⋊Dic₁₀₇	—	2	3	no	—
30	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
31	Dic₁₀₇⋊C₄	—	2	3	no	—
32	C₄×Dic₁₀₇	—	2	3	no	—
33	C₂×Q₈×C₁₀₇	—	2	3	no	—
34	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₇	—	2	3	no	—
37	C₄⋊C₄×C₁₀₇	—	2	3	no	—
38	C₄⋊C₄×C₁₀₇	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₀₇	C₂³	1	3	yes	—
40	C₄²×C₁₀₇	—	2	3	no	—
41	C₄²×C₁₀₇	C₂³	1	3	yes	—
42	Q₈×C₁₀₇	—	2	1	no	—
43	Q₈×C₁₀₇	—	8	3	no	—
44	Q₈×C₁₀₇	Q₁₆	1	4	yes	complemented
45	Q₈×C₁₀₇	#41	1	2	yes	—
46	C₂×C₄×C₁₀₇	—	2	1	no	—
47	C₂×C₄×C₁₀₇	—	2	3	no	—
48	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₇	—	2	1	no	—
52	C₂×Dic₁₀₇	—	2	3	no	—
53	Dic₂₁₄	—	2	3	no	—
54	Dic₂₁₄	—	2	3	no	—
55	C₁₀₇⋊C₈	—	2	3	no	—
56	C₁₀₇⋊C₈	—	2	1	no	—
57	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₇	—	2	2	no	—
59	C₄×C₁₀₇	—	2	3	no	—
60	C₄×C₁₀₇	—	4	3	no	—
61	C₄×C₁₀₇	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₀₇	—	4	3	no	—
65	Dic₁₀₇	—	4	1	no	—
66	C₂×C₁₀₇	#548	1	1	yes	characteristic
67	C₂×C₁₀₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₀₇	#577	1	1	yes	characteristic
69	#3363	—	107	1	no	maximal, Sylow
70	C₁₀₇	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	D₁₀₇	1	1	yes	characteristic
72	#557	—	107	1	no	—
73	#487	—	107	3	no	—
74	#477	—	107	3	no	—
75	#462	—	107	3	no	—
76	C₈×Q₈	—	107	1	no	—
77	C₄×Q₁₆	—	107	3	no	—
78	C₂×Q₁₆	—	107	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	107	3	no	—
81	C₄⋊Q₈	C₂×D₁₀₇	1	3	yes	—
82	#125	—	214	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	107	3	no	—
85	C₄×Q₈	—	214	1	no	—
86	C₄×Q₈	C₂×D₁₀₇	1	3	yes	—
87	C₄×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
88	#107	—	107	3	no	—
89	C₄⋊C₈	—	107	3	no	—
90	#103	—	107	1	no	—
91	#103	—	107	3	no	—
92	#103	—	107	6	no	—
93	C₄×C₈	—	107	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	107	3	no	—
96	C₂×Q₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₀₇	1	1	yes	characteristic
98	Q₁₆	—	214	6	no	—
99	C₂×C₈	—	107	1	no	—
100	C₂×C₈	—	107	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	107	1	no	—
104	C₄⋊C₄	—	214	3	no	—
105	C₄⋊C₄	—	214	3	no	—
106	C₄⋊C₄	—	214	3	no	—
107	C₄⋊C₄	C₂²⋊D₁₀₇	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₀₇	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	214	3	no	—
111	C₄²	C₂²×D₁₀₇	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	214	3	no	—
115	Q₈	—	214	3	no	—
116	Q₈	#13133808	1	4	yes	complemented
117	Q₈	#13133819	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	214	1	no	—
121	C₂×C₄	—	214	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	3	yes	—
124	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
125	C₈	—	214	3	no	—
126	C₈	—	214	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	428	1	no	—
132	C₄	—	428	3	no	—
133	C₄	C₂×#13133808	1	6	yes	—
134	C₄	C₂×#13133819	1	1	yes	characteristic
135	C₄	#16812157	1	1	yes	characteristic
136	C₂	#20293288	1	1	yes	characteristic
137	C₂	C₂²×#13133808	1	1	yes	characteristic
138	C₂	#20293438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293356

C₂

yes

maximal, characteristic

#20292979

C₂

yes

maximal

#20292919

C₂

yes

maximal

#20292837

C₂

yes

maximal

Q₈×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic

C₄×#13133808

C₂

yes

maximal

Q₈²×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133808

C₂²

yes

—

#16812109

C₂²

yes

—

#16812100

—

Q₈×Dic₁₀₇

—

C₄×Dic₂₁₄

C₂²

yes

—

#16812042

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812033

C₂²

yes

characteristic

#16812032

C₂²

yes

—

#16812032

C₂²

yes

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

—

C₄×Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

C₂²

yes

—

C₄×Q₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

C₂³

yes

—

#13133808

—

C₂×C₁₀₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

C₂³

yes

characteristic

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

characteristic

C₂×Q₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

Q₁₆

yes

complemented

Q₈×C₁₀₇

#41

yes

—

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

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×Q₁₆

yes

—

C₄×C₁₀₇

C₂×#41

yes

characteristic

C₄×C₁₀₇

2₋¹⁺⁴

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

#548

yes

characteristic

C₂×C₁₀₇

C₂²×Q₁₆

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#3363

yes

characteristic, complemented, Sylow

Q₈²

D₁₀₇

yes

characteristic

#557

—

107

—

#487

—

107

—

#477

—

107

—

#462

—

107

—

C₈×Q₈

—

107

—

C₄×Q₁₆

—

107

—

C₂×Q₁₆

—

107

—

C₄⋊Q₈

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

—

#125

—

214

—

C₄×Q₈

—

C₄×Q₈

—

107

—

C₄×Q₈

—

214

—

C₄×Q₈

C₂×D₁₀₇

yes

—

C₄×Q₈

C₂×D₁₀₇

yes

characteristic

#107

—

107

—

C₄⋊C₈

—

107

—

#103

—

107

—

#103

—

107

—

#103

—

107

—

C₄×C₈

—

107

—

C₂×Q₈

—

C₂×Q₈

—

107

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

characteristic

C₂×Q₈

C₂²×D₁₀₇

yes

characteristic

Q₁₆

—

214

—

C₂×C₈

—

107

—

100

C₂×C₈

—

107

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

107

—

104

C₄⋊C₄

—

214

—

105

C₄⋊C₄

—

214

—

106

C₄⋊C₄

—

214

—

107

C₄⋊C₄

C₂²⋊D₁₀₇

yes

—

108

C₄⋊C₄

C₂²×D₁₀₇

yes

—

109

C₄²

—

110

C₄²

—

214

—

111

C₄²

C₂²×D₁₀₇

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

214

—

115

Q₈

—

214

—

116

Q₈

#13133808

yes

complemented

117

Q₈

#13133819

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

214

—

121

C₂×C₄

—

214

—

122

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

124

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

125

C₈

—

214

—

126

C₈

—

214

—

127

C₂²

C₂²×C₂²⋊D₁₀₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

428

—

132

C₄

—

428

—

133

C₄

C₂×#13133808

yes

—

134

C₄

C₂×#13133819

yes

characteristic

135

C₄

#16812157

yes

characteristic

136

C₂

#20293288

yes

characteristic

137

C₂

C₂²×#13133808

yes

characteristic

138

C₂

#20293438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₇
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	Q₈²×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂²×D₄
hypercenter	Q₈²	D₁₀₇
hypocenter	C₁₀₇	#3363
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₇

commutator, cosocle

C₂×C₄×C₁₀₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

Q₈²×C₁₀₇

C₂

socle, prime radical

C₂²×C₁₀₇

C₂²×D₄

hypercenter

Q₈²

D₁₀₇

hypocenter

C₁₀₇

#3363

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
44. Q₈×C₁₀₇	Q₁₆	semidirect
70. C₁₀₇	#3363	semidirect
116. Q₈	#13133808	semidirect

H ≅ G/N

type

44. Q₈×C₁₀₇

Q₁₆

semidirect

70. C₁₀₇

#3363

semidirect

116. Q₈

#13133808

semidirect

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

subgroup	index
#20293356	2
#20292979	2
#20292919	2
#20292837	2
Q₈×C₁₀₇⋊C₈	2
C₄×#13133808	2
Q₈²×C₁₀₇	2
#3363	107

subgroup

index

Q₈×C₁₀₇⋊C₈

Q₈²×C₁₀₇

107

Subgroups — #28349139

Conjugacy classes of subgroups

Complements

Maximal subgroups