Group #20293205 — subgroups

Subgroups — #20293205

Order 6848.

294 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#13133818	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812088	C₂	1	2	yes	maximal, complemented
3	#16812085	C₂	1	2	yes	maximal, complemented
4	#16812083	C₂	1	2	yes	maximal, complemented
5	#16812081	C₂	1	4	yes	maximal, complemented
6	#16812080	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13133818	—	2	4	no	—
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×C₂²⋊D₁₀₇	C₂²	1	4	yes	—
14	C₂²×Dic₁₀₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
18	Dic₁₀₇⋊C₄	C₂²	1	4	yes	—
19	C₂²⋊Dic₁₀₇	C₂²	1	4	yes	—
20	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	—
21	#41×C₁₀₇	—	2	4	no	—
22	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
24	C₂×D₄×C₁₀₇	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂²	1	2	yes	—
26	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
27	C₂³×C₁₀₇	C₂³	1	2	yes	—
28	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂²×D₁₀₇	C₂³	1	2	yes	—
30	Q₈×C₁₀₇	—	2	2	no	—
31	D₄×C₁₀₇	—	2	4	no	—
32	D₄×C₁₀₇	—	2	2	no	—
33	C₂×C₄×C₁₀₇	—	2	4	no	—
34	C₂×C₄×C₁₀₇	D₄	1	4	yes	complemented
35	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
36	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
38	C₂²⋊D₁₀₇	—	2	4	no	—
39	C₂²⋊D₁₀₇	—	4	4	no	—
40	C₂×Dic₁₀₇	—	4	2	no	—
41	C₂×Dic₁₀₇	C₂³	1	4	yes	—
42	C₂×Dic₁₀₇	C₂³	1	2	yes	—
43	D₄₂₈	—	2	2	no	—
44	C₄×D₁₀₇	—	2	4	no	—
45	Dic₂₁₄	—	2	2	no	—
46	C₂²×C₁₀₇	—	2	1	no	—
47	C₂²×C₁₀₇	—	2	1	no	—
48	C₂²×C₁₀₇	—	2	2	no	—
49	C₂²×C₁₀₇	—	2	2	no	—
50	C₂²×C₁₀₇	—	2	2	no	—
51	C₂²×C₁₀₇	C₂×D₄	1	2	yes	—
52	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₀₇	—	2	2	no	—
54	C₂×D₁₀₇	—	2	2	no	—
55	C₂×D₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	—	2	2	no	—
57	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
58	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
59	Dic₁₀₇	—	2	2	no	—
60	Dic₁₀₇	—	2	4	no	—
61	C₂×C₁₀₇	—	2	2	no	—
62	C₂×C₁₀₇	—	4	2	no	—
63	C₂×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₀₇	—	4	2	no	—
67	C₁₀₇	#537	1	1	yes	characteristic, complemented, Sylow
68	#537	—	107	1	no	maximal, Sylow
69	C₂×#41	—	107	1	no	—
70	C₂×#41	D₁₀₇	1	1	yes	characteristic, complemented
71	C₂²⋊Q₈	—	107	2	no	—
72	C₂²⋊Q₈	—	107	2	no	—
73	#121	—	107	2	no	—
74	#121	—	107	4	no	—
75	#121	—	107	2	no	—
76	C₂×C₄⋊C₄	—	107	1	no	—
77	#41	—	2	4	no	—
78	#41	—	214	4	no	—
79	C₂×Q₈	—	107	1	no	—
80	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
81	C₂×D₄	—	107	4	no	—
82	C₂×D₄	—	107	2	no	—
83	C₂×D₄	—	107	1	no	—
84	C₂×D₄	C₂×D₁₀₇	1	2	yes	—
85	C₂×D₄	C₂×D₁₀₇	1	1	yes	characteristic
86	C₂²×C₄	—	107	2	no	—
87	C₂²×C₄	—	107	2	no	—
88	C₂²×C₄	C₂×D₁₀₇	1	2	yes	—
89	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
90	C₄⋊C₄	—	107	2	no	—
91	C₄⋊C₄	—	107	2	no	—
92	C₄⋊C₄	—	107	4	no	—
93	C₂²⋊C₄	—	107	4	no	—
94	C₂²⋊C₄	—	107	4	no	—
95	C₂³	—	107	2	no	—
96	C₂³	C₂²×D₁₀₇	1	2	yes	—
97	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	Q₈	—	214	2	no	—
100	D₄	—	2	4	no	—
101	D₄	—	2	2	no	—
102	D₄	—	428	4	no	—
103	D₄	—	214	4	no	—
104	D₄	—	214	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	107	4	no	—
107	C₂×C₄	—	107	2	no	—
108	C₂×C₄	—	428	2	no	—
109	C₂×C₄	—	214	4	no	—
110	C₂×C₄	C₂²⋊D₁₀₇	1	4	yes	complemented
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	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	214	2	no	—
120	C₂²	—	214	2	no	—
121	C₂²	—	214	2	no	—
122	C₂²	C₂×C₂²⋊D₁₀₇	1	2	yes	—
123	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	214	2	no	—
126	C₄	—	214	4	no	—
127	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
128	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	428	2	no	—
132	C₂	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
133	C₂	#16812155	1	1	yes	characteristic
134	C₂	#16812156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133818

C₂

yes

maximal, characteristic, complemented

#16812088

C₂

yes

maximal, complemented

#16812085

C₂

yes

maximal, complemented

#16812083

C₂

yes

maximal, complemented

#16812081

C₂

yes

maximal, complemented

#16812080

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#13133818

—

C₂×Dic₂₁₄

C₂²

yes

characteristic

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂²×Dic₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂²

yes

characteristic

C₂×D₄×C₁₀₇

C₂²

yes

characteristic

C₂×D₄×C₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂³×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

—

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

D₄

yes

complemented

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×D₄

yes

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×D₄

yes

—

C₄×C₁₀₇

C₂×D₄

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂²×D₄

yes

characteristic

C₂×C₁₀₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

D₁₀₇

—

C₁₀₇

#537

yes

characteristic, complemented, Sylow

#537

—

107

maximal, Sylow

C₂×#41

—

107

—

C₂×#41

D₁₀₇

yes

characteristic, complemented

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

#121

—

107

—

#121

—

107

—

#121

—

107

—

C₂×C₄⋊C₄

—

107

—

#41

—

#41

—

214

—

C₂×Q₈

—

107

—

C₂×Q₈

C₂×D₁₀₇

yes

characteristic

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂×D₄

C₂×D₁₀₇

yes

—

C₂×D₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂³

—

107

—

C₂³

C₂²×D₁₀₇

yes

—

C₂³

C₂²×D₁₀₇

yes

characteristic

Q₈

—

Q₈

—

214

—

100

D₄

—

101

D₄

—

102

D₄

—

428

—

103

D₄

—

214

—

104

D₄

—

214

—

105

C₂×C₄

—

106

C₂×C₄

—

107

—

107

C₂×C₄

—

107

—

108

C₂×C₄

—

428

—

109

C₂×C₄

—

214

—

110

C₂×C₄

C₂²⋊D₁₀₇

yes

complemented

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

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

214

—

120

C₂²

—

214

—

121

C₂²

—

214

—

122

C₂²

C₂×C₂²⋊D₁₀₇

yes

—

123

C₂²

C₂³×D₁₀₇

yes

characteristic

124

C₄

—

125

C₄

—

214

—

126

C₄

—

214

—

127

C₄

C₂×C₂²⋊D₁₀₇

yes

—

128

C₄

C₂×C₂²⋊D₁₀₇

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

428

—

132

C₂

C₂²×C₂²⋊D₁₀₇

yes

characteristic

133

C₂

#16812155

yes

characteristic

134

C₂

#16812156

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₂×#41×C₁₀₇	C₂
hypercenter	C₂×#41	D₁₀₇
hypocenter	C₁₀₇	#537
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

C₂×#41×C₁₀₇

C₂

hypercenter

C₂×#41

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#13133818	C₂	semidirect
2. #16812088	C₂	semidirect
3. #16812085	C₂	semidirect
4. #16812083	C₂	semidirect
5. #16812081	C₂	semidirect
6. #16812080	C₂	semidirect
7. C₂×C₄⋊Dic₁₀₇	C₂	semidirect
8. C₂×#41×C₁₀₇	C₂	semidirect
34. C₂×C₄×C₁₀₇	D₄	semidirect
67. C₁₀₇	#537	semidirect
70. C₂×#41	D₁₀₇	semidirect
110. C₂×C₄	C₂²⋊D₁₀₇	semidirect

H ≅ G/N

type

1. C₂×#13133818

C₂

semidirect

2. #16812088

C₂

semidirect

3. #16812085

C₂

semidirect

4. #16812083

C₂

semidirect

5. #16812081

C₂

semidirect

6. #16812080

C₂

semidirect

7. C₂×C₄⋊Dic₁₀₇

C₂

semidirect

8. C₂×#41×C₁₀₇

C₂

semidirect

34. C₂×C₄×C₁₀₇

D₄

semidirect

67. C₁₀₇

#537

semidirect

70. C₂×#41

D₁₀₇

semidirect

110. C₂×C₄

C₂²⋊D₁₀₇

semidirect

Subgroups — #20293205

Conjugacy classes of subgroups

Complements