Group #20293254 — subgroups

Subgroups — #20293254

Order 6848.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#16812108	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812107	C₂	1	2	yes	maximal, complemented
3	#16812101	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812100	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812099	C₂	1	2	yes	maximal, complemented
6	#16812087	C₂	1	2	yes	maximal
7	#16812085	C₂	1	2	yes	maximal
8	#16812067	C₂	1	1	yes	maximal, characteristic
9	#16812066	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×D₁₀₇	C₂	1	1	yes	maximal, characteristic
11	C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₁₄	C₂²	1	2	yes	—
13	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
18	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
19	Dic₁₀₇⋊C₄	C₂²	1	1	yes	characteristic
20	Dic₁₀₇⋊C₄	C₂²	1	1	yes	characteristic
21	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
22	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
23	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
24	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₀₇)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₀₇)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₀₇	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₀₇	C₂²	1	2	yes	—
31	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
34	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
35	C₂²×D₁₀₇	C₂³	1	1	yes	characteristic
36	Q₈×C₁₀₇	—	4	2	no	—
37	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
40	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
41	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₀₇	C₂³	1	2	yes	—
44	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₀₇	C₂³	1	2	yes	—
46	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
47	C₄×D₁₀₇	—	4	2	no	—
48	C₄×D₁₀₇	Q₈	1	4	yes	complemented
49	Dic₂₁₄	—	4	2	no	—
50	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₀₇	—	2	1	no	—
52	C₂×D₁₀₇	—	2	1	no	—
53	C₂×D₁₀₇	C₂×Q₈	1	2	yes	—
54	C₄×C₁₀₇	—	2	2	no	—
55	C₄×C₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	—	2	1	no	—
57	C₄×C₁₀₇	—	2	1	no	—
58	C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
59	Dic₁₀₇	—	2	1	no	—
60	Dic₁₀₇	—	2	2	no	—
61	Dic₁₀₇	—	2	1	no	—
62	Dic₁₀₇	—	2	2	no	—
63	Dic₁₀₇	C₂×Q₈	1	2	yes	—
64	C₂×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
67	D₁₀₇	—	2	2	no	—
68	C₁₀₇	#544	1	1	yes	characteristic, complemented, Sylow
69	#544	—	107	1	no	maximal, Sylow
70	C₄⋊Q₈	—	107	2	no	—
71	C₄⋊Q₈	—	107	1	no	—
72	C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic, complemented
73	#125	—	107	2	no	—
74	#125	—	107	1	no	—
75	#125	—	107	1	no	—
76	C₂²⋊Q₈	—	107	2	no	—
77	C₂²⋊Q₈	—	107	2	no	—
78	#117	—	107	1	no	—
79	#117	—	107	1	no	—
80	C₂×C₄⋊C₄	—	107	1	no	—
81	C₂×Q₈	—	107	2	no	—
82	C₂×Q₈	C₂×D₁₀₇	1	2	yes	—
83	C₂²×C₄	—	107	1	no	—
84	C₂²×C₄	—	107	2	no	—
85	C₄⋊C₄	—	107	2	no	—
86	C₄⋊C₄	—	107	1	no	—
87	C₄⋊C₄	—	107	1	no	—
88	C₄⋊C₄	—	107	1	no	—
89	C₄⋊C₄	—	107	1	no	—
90	C₄⋊C₄	—	107	2	no	—
91	C₄⋊C₄	—	107	2	no	—
92	C₄⋊C₄	—	107	2	no	—
93	C₄⋊C₄	—	107	2	no	—
94	C₄⋊C₄	—	107	2	no	—
95	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
98	C₂²⋊C₄	—	107	2	no	—
99	C₂²⋊C₄	—	107	1	no	—
100	C₂²⋊C₄	—	107	1	no	—
101	C₄²	—	107	2	no	—
102	C₄²	—	107	1	no	—
103	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
104	C₂³	—	107	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	428	2	no	—
107	C₂×C₄	—	107	1	no	—
108	C₂×C₄	—	107	4	no	—
109	C₂×C₄	—	107	1	no	—
110	C₂×C₄	—	107	2	no	—
111	C₂×C₄	—	107	1	no	—
112	C₂×C₄	—	107	2	no	—
113	C₂×C₄	—	428	2	no	—
114	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
115	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
118	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
119	C₂²	—	107	2	no	—
120	C₂²	—	214	1	no	—
121	C₂²	—	214	1	no	—
122	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	107	2	no	—
128	C₄	—	214	2	no	—
129	C₄	—	214	1	no	—
130	C₄	—	214	1	no	—
131	C₄	—	214	2	no	—
132	C₄	Q₈×D₁₀₇	1	2	yes	—
133	C₂	—	214	2	no	—
134	C₂	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
135	C₂	#16812154	1	1	yes	characteristic
136	C₂	#16812157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812108

C₂

yes

maximal, characteristic, complemented

#16812107

C₂

yes

maximal, complemented

#16812101

C₂

yes

maximal, characteristic, complemented

#16812100

C₂

yes

maximal, characteristic, complemented

#16812099

C₂

yes

maximal, complemented

#16812087

C₂

yes

maximal

#16812085

C₂

yes

maximal

#16812067

C₂

yes

maximal, characteristic

#16812066

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₀₇

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₄

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

characteristic

Dic₁₀₇⋊C₄

C₂²

yes

characteristic

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₀₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

characteristic

C₄×Dic₁₀₇

C₂²

yes

—

C₂×Q₈×C₁₀₇

C₂²

yes

—

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₄²×C₁₀₇

C₂²

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

characteristic

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₄×D₁₀₇

—

C₄×D₁₀₇

Q₈

yes

complemented

Dic₂₁₄

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

C₂×Q₈

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×Q₈

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

C₂×Q₈

yes

—

C₂×C₁₀₇

C₂²×Q₈

yes

characteristic

C₂×C₁₀₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

D₁₀₇

—

C₁₀₇

#544

yes

characteristic, complemented, Sylow

#544

—

107

maximal, Sylow

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

D₁₀₇

yes

characteristic, complemented

#125

—

107

—

#125

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

#117

—

107

—

#117

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

C₂×D₁₀₇

yes

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

100

C₂²⋊C₄

—

107

—

101

C₄²

—

107

—

102

C₄²

—

107

—

103

C₄²

C₂×D₁₀₇

yes

characteristic

104

C₂³

—

107

—

105

Q₈

—

106

Q₈

—

428

—

107

C₂×C₄

—

107

—

108

C₂×C₄

—

107

—

109

C₂×C₄

—

107

—

110

C₂×C₄

—

107

—

111

C₂×C₄

—

107

—

112

C₂×C₄

—

107

—

113

C₂×C₄

—

428

—

114

C₂×C₄

C₂²×D₁₀₇

yes

—

115

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

116

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

117

C₂×C₄

C₂²×D₁₀₇

yes

—

118

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

119

C₂²

—

107

—

120

C₂²

—

214

—

121

C₂²

—

214

—

122

C₂²

C₂³×D₁₀₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

107

—

128

C₄

—

214

—

129

C₄

—

214

—

130

C₄

—

214

—

131

C₄

—

214

—

132

C₄

Q₈×D₁₀₇

yes

—

133

C₂

—

214

—

134

C₂

C₂×Q₈×D₁₀₇

yes

characteristic

135

C₂

#16812154

yes

characteristic

136

C₂

#16812157

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₇	C₂
hypercenter	C₄⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#544
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₀₇

C₂

hypercenter

C₄⋊Q₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #16812108	C₂	semidirect
2. #16812107	C₂	semidirect
3. #16812101	C₂	semidirect
4. #16812100	C₂	semidirect
5. #16812099	C₂	semidirect
11. C₄⋊Q₈×C₁₀₇	C₂	semidirect
48. C₄×D₁₀₇	Q₈	semidirect
68. C₁₀₇	#544	semidirect
72. C₄⋊Q₈	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812108

C₂

semidirect

2. #16812107

C₂

semidirect

3. #16812101

C₂

semidirect

4. #16812100

C₂

semidirect

5. #16812099

C₂

semidirect

11. C₄⋊Q₈×C₁₀₇

C₂

semidirect

48. C₄×D₁₀₇

Q₈

semidirect

68. C₁₀₇

#544

semidirect

72. C₄⋊Q₈

D₁₀₇

semidirect

Subgroups — #20293254

Conjugacy classes of subgroups

Complements