Group #20293345 — subgroups

Subgroups — #20293345

Order 6848.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16812103	C₂	1	2	yes	maximal, complemented
3	#16812102	C₂	1	2	yes	maximal, complemented
4	#16812095	C₂	1	2	yes	maximal
5	#16812089	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₂₁₄	C₂	1	2	yes	maximal, complemented
8	#16812074	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊D₁₀₇	C₂	1	1	yes	maximal, characteristic
10	#16812067	C₂	1	1	yes	maximal, characteristic, complemented
11	#117×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₀₇	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	1	yes	characteristic
19	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
20	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
21	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₁₀₇	C₂²	1	2	yes	—
24	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₀₇)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
27	C₄×Dic₁₀₇	C₂²	1	2	yes	—
28	C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₀₇	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×C₁₀₇	C₂²	1	2	yes	—
34	C₄²×C₁₀₇	C₂²	1	2	yes	—
35	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
36	C₂²×D₁₀₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	—	2	2	no	—
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	C₂²⋊D₁₀₇	—	2	2	no	—
43	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₀₇	C₂³	1	2	yes	—
46	C₂×Dic₁₀₇	C₂³	1	2	yes	—
47	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
48	C₄×D₁₀₇	—	2	2	no	—
49	Dic₂₁₄	—	2	2	no	—
50	Dic₂₁₄	—	2	4	no	—
51	C₂²×C₁₀₇	—	2	1	no	—
52	C₂²×C₁₀₇	—	2	1	no	—
53	C₂²×C₁₀₇	—	2	1	no	—
54	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₀₇	—	2	1	no	—
56	C₂×D₁₀₇	—	2	1	no	—
57	C₂×D₁₀₇	—	2	1	no	—
58	C₄×C₁₀₇	—	2	2	no	—
59	C₄×C₁₀₇	—	2	1	no	—
60	C₄×C₁₀₇	—	2	2	no	—
61	C₄×C₁₀₇	#41	1	2	yes	—
62	Dic₁₀₇	—	2	2	no	—
63	Dic₁₀₇	—	2	2	no	—
64	Dic₁₀₇	—	2	1	no	—
65	Dic₁₀₇	—	2	1	no	—
66	Dic₁₀₇	#41	1	2	yes	—
67	C₂×C₁₀₇	—	4	1	no	—
68	C₂×C₁₀₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₀₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
71	D₁₀₇	—	4	1	no	—
72	C₁₀₇	#556	1	1	yes	characteristic, complemented, Sylow
73	#556	—	107	1	no	maximal, Sylow
74	C₄⋊Q₈	—	107	1	no	—
75	#126	—	107	2	no	—
76	#126	—	107	2	no	—
77	#124	—	107	2	no	—
78	C₂²⋊Q₈	—	107	1	no	—
79	C₂²⋊Q₈	—	107	1	no	—
80	C₄×Q₈	—	107	1	no	—
81	C₄×Q₈	—	107	2	no	—
82	C₄×D₄	—	107	1	no	—
83	#117	—	107	1	no	—
84	#117	D₁₀₇	1	1	yes	characteristic, complemented
85	C₂×Q₈	—	107	1	no	—
86	C₂×Q₈	—	107	2	no	—
87	C₂×D₄	—	107	1	no	—
88	C₂²×C₄	—	107	1	no	—
89	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
90	C₄⋊C₄	—	107	1	no	—
91	C₄⋊C₄	—	107	1	no	—
92	C₄⋊C₄	—	107	2	no	—
93	C₄⋊C₄	—	107	2	no	—
94	C₄⋊C₄	—	107	2	no	—
95	C₄⋊C₄	—	107	2	no	—
96	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
98	C₂²⋊C₄	—	107	1	no	—
99	C₂²⋊C₄	—	107	2	no	—
100	C₂²⋊C₄	—	107	2	no	—
101	C₂²⋊C₄	—	107	1	no	—
102	C₂²⋊C₄	—	107	2	no	—
103	C₂²⋊C₄	C₂×D₁₀₇	1	2	yes	—
104	C₄²	—	107	2	no	—
105	C₄²	—	107	1	no	—
106	C₄²	—	107	2	no	—
107	C₄²	C₂×D₁₀₇	1	2	yes	—
108	C₂³	—	107	1	no	—
109	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
110	Q₈	—	214	4	no	—
111	Q₈	—	214	2	no	—
112	D₄	—	214	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	107	1	no	—
115	C₂×C₄	—	107	1	no	—
116	C₂×C₄	—	107	1	no	—
117	C₂×C₄	—	107	2	no	—
118	C₂×C₄	—	107	2	no	—
119	C₂×C₄	—	214	2	no	—
120	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
121	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
123	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	214	1	no	—
128	C₂²	—	214	1	no	—
129	C₂²	—	214	1	no	—
130	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	107	2	no	—
135	C₄	—	214	2	no	—
136	C₄	—	214	1	no	—
137	C₄	—	214	2	no	—
138	C₄	—	214	1	no	—
139	C₄	#13133818	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	428	1	no	—
142	C₂	#41×D₁₀₇	1	1	yes	characteristic
143	C₂	C₂×#13133818	1	1	yes	characteristic
144	C₂	#16812156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812108

C₂

yes

maximal, characteristic, complemented

#16812103

C₂

yes

maximal, complemented

#16812102

C₂

yes

maximal, complemented

#16812095

C₂

yes

maximal

#16812089

C₂

yes

maximal, characteristic, complemented

#16812088

C₂

yes

maximal, characteristic, complemented

C₄×Dic₂₁₄

C₂

yes

maximal, complemented

#16812074

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₀₇

C₂

yes

maximal, characteristic

#16812067

C₂

yes

maximal, characteristic, complemented

#117×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₄

C₂²

yes

characteristic

C₂×Dic₂₁₄

C₂²

yes

—

C₂×C₂²⋊D₁₀₇

C₂²

yes

characteristic

C₂×C₄×D₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

characteristic

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

characteristic

C₂²⋊Dic₁₀₇

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

characteristic

C₄×Dic₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₇

C₂²

yes

—

C₄²×C₁₀₇

C₂²

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₄×D₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#41

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

#41

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

D₁₀₇

—

C₁₀₇

#556

yes

characteristic, complemented, Sylow

#556

—

107

maximal, Sylow

C₄⋊Q₈

—

107

—

#126

—

107

—

#126

—

107

—

#124

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

C₄×Q₈

—

107

—

C₄×Q₈

—

107

—

C₄×D₄

—

107

—

#117

—

107

—

#117

D₁₀₇

yes

characteristic, complemented

C₂×Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

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

characteristic

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

100

C₂²⋊C₄

—

107

—

101

C₂²⋊C₄

—

107

—

102

C₂²⋊C₄

—

107

—

103

C₂²⋊C₄

C₂×D₁₀₇

yes

—

104

C₄²

—

107

—

105

C₄²

—

107

—

106

C₄²

—

107

—

107

C₄²

C₂×D₁₀₇

yes

—

108

C₂³

—

107

—

109

C₂³

C₂²×D₁₀₇

yes

characteristic

110

Q₈

—

214

—

111

Q₈

—

214

—

112

D₄

—

214

—

113

C₂×C₄

—

114

C₂×C₄

—

107

—

115

C₂×C₄

—

107

—

116

C₂×C₄

—

107

—

117

C₂×C₄

—

107

—

118

C₂×C₄

—

107

—

119

C₂×C₄

—

214

—

120

C₂×C₄

C₂²×D₁₀₇

yes

—

121

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

122

C₂×C₄

C₂²×D₁₀₇

yes

—

123

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

214

—

128

C₂²

—

214

—

129

C₂²

—

214

—

130

C₂²

C₂³×D₁₀₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

107

—

135

C₄

—

214

—

136

C₄

—

214

—

137

C₄

—

214

—

138

C₄

—

214

—

139

C₄

#13133818

yes

—

140

C₂

—

141

C₂

—

428

—

142

C₂

#41×D₁₀₇

yes

characteristic

143

C₂

C₂×#13133818

yes

characteristic

144

C₂

#16812156

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	#117×C₁₀₇	C₂
hypercenter	#117	D₁₀₇
hypocenter	C₁₀₇	#556
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

#117×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #16812108	C₂	semidirect
2. #16812103	C₂	semidirect
3. #16812102	C₂	semidirect
5. #16812089	C₂	semidirect
6. #16812088	C₂	semidirect
7. C₄×Dic₂₁₄	C₂	semidirect
8. #16812074	C₂	semidirect
10. #16812067	C₂	semidirect
11. #117×C₁₀₇	C₂	semidirect
72. C₁₀₇	#556	semidirect
84. #117	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812108

C₂

semidirect

2. #16812103

C₂

semidirect

3. #16812102

C₂

semidirect

5. #16812089

C₂

semidirect

6. #16812088

C₂

semidirect

7. C₄×Dic₂₁₄

C₂

semidirect

8. #16812074

C₂

semidirect

10. #16812067

C₂

semidirect

11. #117×C₁₀₇

C₂

semidirect

72. C₁₀₇

#556

semidirect

84. #117

D₁₀₇

semidirect

Subgroups — #20293345

Conjugacy classes of subgroups

Complements