Group #20293224 — subgroups

Subgroups — #20293224

Order 6848.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#16812104	C₂	1	2	yes	maximal, complemented
2	#16812099	C₂	1	2	yes	maximal, complemented
3	#16812094	C₂	1	2	yes	maximal, complemented
4	#16812092	C₂	1	2	yes	maximal, complemented
5	#16812088	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812087	C₂	1	2	yes	maximal, complemented
7	#16812083	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊D₁₀₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic
11	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₀₇	C₂²	1	2	yes	—
13	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
14	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
16	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
17	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
18	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
19	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
20	C₂²⋊Dic₁₀₇	C₂²	1	2	yes	—
21	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₁₀₇	C₂²	1	2	yes	—
26	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
30	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
31	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₇	C₂³	1	2	yes	—
33	C₂×C₄×C₁₀₇	—	2	2	no	—
34	C₂×C₄×C₁₀₇	—	4	1	no	—
35	C₂×C₄×C₁₀₇	—	4	1	no	—
36	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
37	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
40	C₂²⋊D₁₀₇	—	2	4	no	—
41	C₂×Dic₁₀₇	C₂³	1	2	yes	—
42	C₂×Dic₁₀₇	C₂³	1	2	yes	—
43	C₂×Dic₁₀₇	C₂³	1	2	yes	—
44	D₄₂₈	—	4	1	no	—
45	C₄×D₁₀₇	—	4	2	no	—
46	Dic₂₁₄	—	4	1	no	—
47	C₂²×C₁₀₇	—	2	1	no	—
48	C₂²×C₁₀₇	—	2	1	no	—
49	C₂²×C₁₀₇	#41	1	2	yes	—
50	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₀₇	—	2	2	no	—
52	C₂×D₁₀₇	—	2	2	no	—
53	C₂×D₁₀₇	—	2	2	no	—
54	C₄×C₁₀₇	—	2	2	no	—
55	C₄×C₁₀₇	—	2	1	no	—
56	C₄×C₁₀₇	—	2	2	no	—
57	C₄×C₁₀₇	—	2	1	no	—
58	Dic₁₀₇	—	2	2	no	—
59	Dic₁₀₇	—	2	2	no	—
60	Dic₁₀₇	—	2	2	no	—
61	C₂×C₁₀₇	—	2	2	no	—
62	C₂×C₁₀₇	C₂×#41	1	1	yes	characteristic
63	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
65	D₁₀₇	—	4	2	no	—
66	C₁₀₇	#539	1	1	yes	characteristic, complemented, Sylow
67	#539	—	107	1	no	maximal, Sylow
68	#126	—	107	2	no	—
69	#125	—	107	2	no	—
70	#123	—	107	2	no	—
71	#123	—	107	2	no	—
72	C₂²⋊Q₈	—	107	1	no	—
73	C₂²⋊Q₈	—	107	2	no	—
74	#121	—	107	1	no	—
75	C₄×D₄	—	107	2	no	—
76	C₂×C₄⋊C₄	D₁₀₇	1	1	yes	characteristic, complemented
77	C₂×Q₈	—	107	1	no	—
78	C₂×D₄	—	107	2	no	—
79	C₂×D₄	—	107	1	no	—
80	C₂²×C₄	—	107	2	no	—
81	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
84	C₄⋊C₄	—	107	2	no	—
85	C₄⋊C₄	—	107	1	no	—
86	C₄⋊C₄	—	107	2	no	—
87	C₄⋊C₄	—	107	2	no	—
88	C₄⋊C₄	—	107	1	no	—
89	C₄⋊C₄	—	107	2	no	—
90	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
91	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
92	C₂²⋊C₄	—	107	2	no	—
93	C₂²⋊C₄	—	107	2	no	—
94	C₂²⋊C₄	—	107	2	no	—
95	C₂²⋊C₄	—	107	2	no	—
96	C₂²⋊C₄	—	107	2	no	—
97	C₄²	—	107	2	no	—
98	C₂³	—	107	2	no	—
99	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
100	Q₈	—	428	1	no	—
101	D₄	—	428	1	no	—
102	D₄	—	214	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	107	2	no	—
107	C₂×C₄	—	107	2	no	—
108	C₂×C₄	—	107	2	no	—
109	C₂×C₄	—	428	2	no	—
110	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
111	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
112	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	214	2	no	—
117	C₂²	—	214	2	no	—
118	C₂²	—	214	2	no	—
119	C₂²	#13133818	1	2	yes	—
120	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	214	2	no	—
126	C₄	—	214	2	no	—
127	C₄	—	214	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	428	2	no	—
130	C₂	C₂×#13133818	1	1	yes	characteristic
131	C₂	#16812154	1	1	yes	characteristic
132	C₂	#16812157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812104

C₂

yes

maximal, complemented

#16812099

C₂

yes

maximal, complemented

#16812094

C₂

yes

maximal, complemented

#16812092

C₂

yes

maximal, complemented

#16812088

C₂

yes

maximal, characteristic, complemented

#16812087

C₂

yes

maximal, complemented

#16812083

C₂

yes

maximal, characteristic

C₄×C₂²⋊D₁₀₇

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₄

C₂²

yes

characteristic

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#41

yes

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

D₁₀₇

—

C₁₀₇

#539

yes

characteristic, complemented, Sylow

#539

—

107

maximal, Sylow

#126

—

107

—

#125

—

107

—

#123

—

107

—

#123

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

#121

—

107

—

C₄×D₄

—

107

—

C₂×C₄⋊C₄

D₁₀₇

yes

characteristic, complemented

C₂×Q₈

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂×D₁₀₇

yes

—

C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₄²

—

107

—

C₂³

—

107

—

C₂³

C₂²×D₁₀₇

yes

characteristic

100

Q₈

—

428

—

101

D₄

—

428

—

102

D₄

—

214

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

—

107

C₂×C₄

—

107

—

108

C₂×C₄

—

107

—

109

C₂×C₄

—

428

—

110

C₂×C₄

C₂²×D₁₀₇

yes

—

111

C₂×C₄

C₂²×D₁₀₇

yes

—

112

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

113

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

214

—

117

C₂²

—

214

—

118

C₂²

—

214

—

119

C₂²

#13133818

yes

—

120

C₂²

C₂³×D₁₀₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

214

—

126

C₄

—

214

—

127

C₄

—

214

—

128

C₂

—

129

C₂

—

428

—

130

C₂

C₂×#13133818

yes

characteristic

131

C₂

#16812154

yes

characteristic

132

C₂

#16812157

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₁₀₇	C₂
hypercenter	C₂×C₄⋊C₄	D₁₀₇
hypocenter	C₁₀₇	#539
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

C₂×C₄⋊C₄×C₁₀₇

C₂

hypercenter

C₂×C₄⋊C₄

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #16812104	C₂	semidirect
2. #16812099	C₂	semidirect
3. #16812094	C₂	semidirect
4. #16812092	C₂	semidirect
5. #16812088	C₂	semidirect
6. #16812087	C₂	semidirect
8. C₄×C₂²⋊D₁₀₇	C₂	semidirect
9. C₂×C₄⋊C₄×C₁₀₇	C₂	semidirect
66. C₁₀₇	#539	semidirect
76. C₂×C₄⋊C₄	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812104

C₂

semidirect

2. #16812099

C₂

semidirect

3. #16812094

C₂

semidirect

4. #16812092

C₂

semidirect

5. #16812088

C₂

semidirect

6. #16812087

C₂

semidirect

8. C₄×C₂²⋊D₁₀₇

C₂

semidirect

9. C₂×C₄⋊C₄×C₁₀₇

C₂

semidirect

66. C₁₀₇

#539

semidirect

76. C₂×C₄⋊C₄

D₁₀₇

semidirect

Subgroups — #20293224

Conjugacy classes of subgroups

Complements