Group #20293335 — subgroups

Subgroups — #20293335

Order 6848.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#16812099	C₂	1	2	yes	maximal, complemented
3	#16812089	C₂	1	2	yes	maximal, complemented
4	#16812086	C₂	1	2	yes	maximal, complemented
5	#16812085	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₁₀₇	C₂	1	1	yes	maximal, characteristic
8	(C₂²⋊D₁₀₇)⋊C₄	C₂	1	2	yes	maximal
9	C₂×C₄⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₄×D₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₁₄	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₀₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
20	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
21	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
22	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
23	Dic₁₀₇⋊C₄	C₂²	1	1	yes	characteristic
24	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
26	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₀₇)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₀₇	C₂²	1	2	yes	—
29	C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
31	C₂²×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	C₂²⋊C₄×C₁₀₇	C₂²	1	2	yes	—
35	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
36	C₂²×D₁₀₇	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₇	—	2	2	no	—
38	C₂×C₄×C₁₀₇	—	2	2	no	—
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×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
43	C₂²⋊D₁₀₇	Q₈	1	4	yes	complemented
44	C₂×Dic₁₀₇	—	4	2	no	—
45	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀₇	C₂³	1	2	yes	—
47	C₂×Dic₁₀₇	C₂³	1	2	yes	—
48	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₀₇	C₂³	1	1	yes	characteristic
50	C₄×D₁₀₇	—	2	2	no	—
51	C₄×D₁₀₇	—	4	2	no	—
52	Dic₂₁₄	—	4	2	no	—
53	C₂²×C₁₀₇	—	2	1	no	—
54	C₂²×C₁₀₇	—	2	1	no	—
55	C₂²×C₁₀₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
57	C₂×D₁₀₇	—	2	1	no	—
58	C₂×D₁₀₇	—	2	1	no	—
59	C₂×D₁₀₇	C₂×Q₈	1	2	yes	—
60	C₄×C₁₀₇	—	2	1	no	—
61	C₄×C₁₀₇	—	2	2	no	—
62	C₄×C₁₀₇	—	2	2	no	—
63	C₄×C₁₀₇	#41	1	2	yes	—
64	Dic₁₀₇	—	2	1	no	—
65	Dic₁₀₇	—	2	1	no	—
66	Dic₁₀₇	—	2	2	no	—
67	Dic₁₀₇	—	2	2	no	—
68	Dic₁₀₇	C₂×Q₈	1	2	yes	—
69	C₂×C₁₀₇	—	2	2	no	—
70	C₂×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₁₀₇	C₂×#41	1	1	yes	characteristic
72	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
73	D₁₀₇	—	2	2	no	—
74	C₁₀₇	#554	1	1	yes	characteristic, complemented, Sylow
75	#554	—	107	1	no	maximal, Sylow
76	C₄⋊Q₈	—	107	1	no	—
77	#125	—	107	2	no	—
78	C₂²⋊Q₈	—	107	1	no	—
79	C₂²⋊Q₈	—	107	2	no	—
80	C₂²⋊Q₈	—	107	2	no	—
81	C₂²⋊Q₈	D₁₀₇	1	1	yes	characteristic, complemented
82	C₄×Q₈	—	107	1	no	—
83	C₄×D₄	—	107	1	no	—
84	C₄×D₄	—	107	2	no	—
85	C₂×C₄⋊C₄	—	107	1	no	—
86	C₂×C₄⋊C₄	—	107	1	no	—
87	C₂×Q₈	—	107	2	no	—
88	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
89	C₂×D₄	—	107	1	no	—
90	C₂²×C₄	—	107	2	no	—
91	C₂²×C₄	—	107	2	no	—
92	C₂²×C₄	—	107	1	no	—
93	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
94	C₄⋊C₄	—	107	1	no	—
95	C₄⋊C₄	—	107	2	no	—
96	C₄⋊C₄	—	107	2	no	—
97	C₄⋊C₄	—	107	1	no	—
98	C₄⋊C₄	—	107	1	no	—
99	C₄⋊C₄	—	107	2	no	—
100	C₄⋊C₄	—	107	2	no	—
101	C₄⋊C₄	—	107	2	no	—
102	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
103	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
104	C₂²⋊C₄	—	107	1	no	—
105	C₂²⋊C₄	—	107	1	no	—
106	C₂²⋊C₄	—	107	2	no	—
107	C₂²⋊C₄	C₂×D₁₀₇	1	2	yes	—
108	C₄²	—	107	2	no	—
109	C₄²	—	107	1	no	—
110	C₂³	—	107	1	no	—
111	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	Q₈	—	428	2	no	—
114	D₄	—	107	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	107	1	no	—
117	C₂×C₄	—	107	1	no	—
118	C₂×C₄	—	107	2	no	—
119	C₂×C₄	—	107	1	no	—
120	C₂×C₄	—	107	2	no	—
121	C₂×C₄	—	428	2	no	—
122	C₂×C₄	—	428	2	no	—
123	C₂×C₄	—	214	2	no	—
124	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
125	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
127	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	107	2	no	—
131	C₂²	—	214	1	no	—
132	C₂²	—	214	1	no	—
133	C₂²	Q₈×D₁₀₇	1	2	yes	—
134	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	107	2	no	—
139	C₄	—	214	1	no	—
140	C₄	—	214	2	no	—
141	C₄	—	214	2	no	—
142	C₄	—	214	1	no	—
143	C₄	#13133817	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	214	2	no	—
146	C₂	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
147	C₂	C₂×#13133817	1	1	yes	characteristic
148	C₂	#16812155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812108

C₂

yes

maximal, characteristic, complemented

#16812099

C₂

yes

maximal, complemented

#16812089

C₂

yes

maximal, complemented

#16812086

C₂

yes

maximal, complemented

#16812085

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₀₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₀₇

C₂

yes

maximal, characteristic

(C₂²⋊D₁₀₇)⋊C₄

C₂

yes

maximal

C₂×C₄⋊Dic₁₀₇

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₄

C₂²

yes

—

C₂×C₂²⋊D₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

characteristic

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

characteristic

Dic₁₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

characteristic

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₀₇

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

characteristic

C₂×Q₈×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²⋊C₄×C₁₀₇

C₂²

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

characteristic

Q₈×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

Q₈

yes

complemented

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

characteristic

C₄×D₁₀₇

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×Q₈

yes

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

C₂×Q₈

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#41

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

C₂×Q₈

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂²×Q₈

yes

characteristic

C₂×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

H(5,𝔽₂)

yes

characteristic

D₁₀₇

—

C₁₀₇

#554

yes

characteristic, complemented, Sylow

#554

—

107

maximal, Sylow

C₄⋊Q₈

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

D₁₀₇

yes

characteristic, complemented

C₄×Q₈

—

107

—

C₄×D₄

—

107

—

C₄×D₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

C₂×D₁₀₇

yes

characteristic

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

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

—

100

C₄⋊C₄

—

107

—

101

C₄⋊C₄

—

107

—

102

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

103

C₄⋊C₄

C₂×D₁₀₇

yes

—

104

C₂²⋊C₄

—

107

—

105

C₂²⋊C₄

—

107

—

106

C₂²⋊C₄

—

107

—

107

C₂²⋊C₄

C₂×D₁₀₇

yes

—

108

C₄²

—

107

—

109

C₄²

—

107

—

110

C₂³

—

107

—

111

C₂³

C₂²×D₁₀₇

yes

characteristic

112

Q₈

—

113

Q₈

—

428

—

114

D₄

—

107

—

115

C₂×C₄

—

116

C₂×C₄

—

107

—

117

C₂×C₄

—

107

—

118

C₂×C₄

—

107

—

119

C₂×C₄

—

107

—

120

C₂×C₄

—

107

—

121

C₂×C₄

—

428

—

122

C₂×C₄

—

428

—

123

C₂×C₄

—

214

—

124

C₂×C₄

C₂²×D₁₀₇

yes

—

125

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

126

C₂×C₄

C₂²×D₁₀₇

yes

—

127

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

107

—

131

C₂²

—

214

—

132

C₂²

—

214

—

133

C₂²

Q₈×D₁₀₇

yes

—

134

C₂²

C₂³×D₁₀₇

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

107

—

139

C₄

—

214

—

140

C₄

—

214

—

141

C₄

—

214

—

142

C₄

—

214

—

143

C₄

#13133817

yes

—

144

C₂

—

145

C₂

—

214

—

146

C₂

C₂×Q₈×D₁₀₇

yes

characteristic

147

C₂

C₂×#13133817

yes

characteristic

148

C₂

#16812155

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₂²⋊Q₈×C₁₀₇	C₂
hypercenter	C₂²⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#554
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

C₂²⋊Q₈×C₁₀₇

C₂

hypercenter

C₂²⋊Q₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #16812108	C₂	semidirect
2. #16812099	C₂	semidirect
3. #16812089	C₂	semidirect
4. #16812086	C₂	semidirect
5. #16812085	C₂	semidirect
6. Q₈×Dic₁₀₇	C₂	semidirect
9. C₂×C₄⋊Dic₁₀₇	C₂	semidirect
10. C₄⋊C₄×D₁₀₇	C₂	semidirect
11. C₂²⋊Q₈×C₁₀₇	C₂	semidirect
43. C₂²⋊D₁₀₇	Q₈	semidirect
74. C₁₀₇	#554	semidirect
81. C₂²⋊Q₈	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812108

C₂

semidirect

2. #16812099

C₂

semidirect

3. #16812089

C₂

semidirect

4. #16812086

C₂

semidirect

5. #16812085

C₂

semidirect

6. Q₈×Dic₁₀₇

C₂

semidirect

9. C₂×C₄⋊Dic₁₀₇

C₂

semidirect

10. C₄⋊C₄×D₁₀₇

C₂

semidirect

11. C₂²⋊Q₈×C₁₀₇

C₂

semidirect

43. C₂²⋊D₁₀₇

Q₈

semidirect

74. C₁₀₇

#554

semidirect

81. C₂²⋊Q₈

D₁₀₇

semidirect

Subgroups — #20293335

Conjugacy classes of subgroups

Complements