Group #20268345 — subgroups

Subgroups — #20268345

Order 6592.

222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 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	#15693020	C₂	1	1	yes	maximal, characteristic, complemented
2	#15693019	C₂	1	1	yes	maximal, characteristic, complemented
3	#15693011	C₂	1	2	yes	maximal, complemented
4	#15693001	C₂	1	1	yes	maximal, characteristic
5	#15692999	C₂	1	2	yes	maximal
6	#15692997	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₂₀₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₂₀₆	C₂	1	2	yes	maximal, complemented
9	#15692978	C₂	1	2	yes	maximal
10	C₄²×D₁₀₃	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₀₆	C₂²	1	2	yes	—
13	C₂×Dic₂₀₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
19	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
20	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
21	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
22	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
23	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
26	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₀₃	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
32	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
33	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
34	C₄²×C₁₀₃	C₂²	1	2	yes	—
35	C₂²×D₁₀₃	C₂³	1	1	yes	characteristic
36	Q₈×C₁₀₃	—	2	2	no	—
37	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
39	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
40	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₀₃	C₂³	1	2	yes	—
44	C₂×Dic₁₀₃	C₂³	1	2	yes	—
45	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
47	C₄×D₁₀₃	—	2	2	no	—
48	C₄×D₁₀₃	—	2	2	no	—
49	C₄×D₁₀₃	Q₈	1	4	yes	complemented
50	Dic₂₀₆	—	2	2	no	—
51	Dic₂₀₆	—	2	4	no	—
52	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₀₃	—	2	1	no	—
54	C₂×D₁₀₃	—	2	1	no	—
55	C₂×D₁₀₃	C₂×Q₈	1	2	yes	—
56	C₄×C₁₀₃	—	2	2	no	—
57	C₄×C₁₀₃	—	2	1	no	—
58	C₄×C₁₀₃	—	2	2	no	—
59	C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
60	C₄×C₁₀₃	#41	1	2	yes	—
61	Dic₁₀₃	—	2	2	no	—
62	Dic₁₀₃	—	2	2	no	—
63	Dic₁₀₃	—	2	1	no	—
64	Dic₁₀₃	C₂×Q₈	1	2	yes	—
65	Dic₁₀₃	#41	1	2	yes	—
66	C₂×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
69	D₁₀₃	—	2	2	no	—
70	C₁₀₃	#533	1	1	yes	characteristic, complemented, Sylow
71	#533	—	103	1	no	maximal, Sylow
72	C₄⋊Q₈	—	103	1	no	—
73	C₄⋊Q₈	—	103	1	no	—
74	#125	—	103	2	no	—
75	C₂²⋊Q₈	—	103	1	no	—
76	C₂²⋊Q₈	—	103	2	no	—
77	C₂²⋊Q₈	—	103	1	no	—
78	C₄×Q₈	—	103	2	no	—
79	C₄×Q₈	—	103	1	no	—
80	C₄×Q₈	D₁₀₃	1	1	yes	characteristic, complemented
81	#117	—	103	2	no	—
82	C₂×C₄²	—	103	1	no	—
83	C₂×Q₈	—	103	2	no	—
84	C₂×Q₈	—	103	1	no	—
85	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
86	C₂²×C₄	—	103	1	no	—
87	C₂²×C₄	—	103	1	no	—
88	C₂²×C₄	—	103	1	no	—
89	C₄⋊C₄	—	103	2	no	—
90	C₄⋊C₄	—	103	2	no	—
91	C₄⋊C₄	—	103	2	no	—
92	C₄⋊C₄	—	103	2	no	—
93	C₄⋊C₄	—	103	2	no	—
94	C₄⋊C₄	—	103	2	no	—
95	C₄⋊C₄	—	103	1	no	—
96	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
97	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
98	C₂²⋊C₄	—	103	2	no	—
99	C₂²⋊C₄	—	103	2	no	—
100	C₄²	—	103	1	no	—
101	C₄²	—	103	1	no	—
102	C₄²	—	103	1	no	—
103	C₄²	—	103	2	no	—
104	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
105	C₄²	C₂×D₁₀₃	1	2	yes	—
106	C₂³	—	103	1	no	—
107	Q₈	—	2	2	no	—
108	Q₈	—	206	4	no	—
109	Q₈	—	206	2	no	—
110	C₂×C₄	—	103	1	no	—
111	C₂×C₄	—	103	1	no	—
112	C₂×C₄	—	103	4	no	—
113	C₂×C₄	—	103	1	no	—
114	C₂×C₄	—	103	2	no	—
115	C₂×C₄	—	103	2	no	—
116	C₂×C₄	—	206	2	no	—
117	C₂×C₄	—	206	2	no	—
118	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
120	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
121	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
123	C₂²	—	103	2	no	—
124	C₂²	—	206	1	no	—
125	C₂²	—	206	1	no	—
126	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	—	103	2	no	—
131	C₄	—	103	2	no	—
132	C₄	—	206	2	no	—
133	C₄	—	206	1	no	—
134	C₄	—	206	2	no	—
135	C₄	Q₈×D₁₀₃	1	2	yes	—
136	C₄	#13110105	1	2	yes	—
137	C₂	—	206	2	no	—
138	C₂	C₂×Q₈×D₁₀₃	1	1	yes	characteristic
139	C₂	#41×D₁₀₃	1	1	yes	characteristic
140	C₂	C₂×#13110105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693020

C₂

yes

maximal, characteristic, complemented

#15693019

C₂

yes

maximal, characteristic, complemented

#15693011

C₂

yes

maximal, complemented

#15693001

C₂

yes

maximal, characteristic

#15692999

C₂

yes

maximal

#15692997

C₂

yes

maximal, characteristic

C₄×Dic₂₀₆

C₂

yes

maximal, characteristic, complemented

C₄×Dic₂₀₆

C₂

yes

maximal, complemented

#15692978

C₂

yes

maximal

C₄²×D₁₀₃

C₂

yes

maximal, characteristic

C₄×Q₈×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₆

C₂²

yes

—

C₂×Dic₂₀₆

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

—

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₄²×C₁₀₃

C₂²

yes

characteristic

C₄²×C₁₀₃

C₂²

yes

—

C₂²×D₁₀₃

C₂³

yes

characteristic

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₄×D₁₀₃

—

C₄×D₁₀₃

—

C₄×D₁₀₃

Q₈

yes

complemented

Dic₂₀₆

—

Dic₂₀₆

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

C₂×Q₈

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×Q₈

yes

—

C₄×C₁₀₃

#41

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

C₂×Q₈

yes

—

Dic₁₀₃

#41

yes

—

C₂×C₁₀₃

C₂²×Q₈

yes

characteristic

C₂×C₁₀₃

C₂×#41

yes

characteristic

C₂×C₁₀₃

C₂×#41

yes

characteristic

D₁₀₃

—

C₁₀₃

#533

yes

characteristic, complemented, Sylow

#533

—

103

maximal, Sylow

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

#125

—

103

—

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

—

103

—

C₄×Q₈

—

103

—

C₄×Q₈

—

103

—

C₄×Q₈

D₁₀₃

yes

characteristic, complemented

#117

—

103

—

C₂×C₄²

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

C₂×D₁₀₃

yes

—

C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

100

C₄²

—

103

—

101

C₄²

—

103

—

102

C₄²

—

103

—

103

C₄²

—

103

—

104

C₄²

C₂×D₁₀₃

yes

characteristic

105

C₄²

C₂×D₁₀₃

yes

—

106

C₂³

—

103

—

107

Q₈

—

108

Q₈

—

206

—

109

Q₈

—

206

—

110

C₂×C₄

—

103

—

111

C₂×C₄

—

103

—

112

C₂×C₄

—

103

—

113

C₂×C₄

—

103

—

114

C₂×C₄

—

103

—

115

C₂×C₄

—

103

—

116

C₂×C₄

—

206

—

117

C₂×C₄

—

206

—

118

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

119

C₂×C₄

C₂²×D₁₀₃

yes

—

120

C₂×C₄

C₂²×D₁₀₃

yes

—

121

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

122

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

123

C₂²

—

103

—

124

C₂²

—

206

—

125

C₂²

—

206

—

126

C₂²

C₂³×D₁₀₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

103

—

131

C₄

—

103

—

132

C₄

—

206

—

133

C₄

—

206

—

134

C₄

—

206

—

135

C₄

Q₈×D₁₀₃

yes

—

136

C₄

#13110105

yes

—

137

C₂

—

206

—

138

C₂

C₂×Q₈×D₁₀₃

yes

characteristic

139

C₂

#41×D₁₀₃

yes

characteristic

140

C₂

C₂×#13110105

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Frattini	C₂²	C₂³×D₁₀₃
Fitting	C₄×Q₈×C₁₀₃	C₂
hypercenter	C₄×Q₈	D₁₀₃
hypocenter	C₁₀₃	#533
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂⁴

Frattini

C₂²

C₂³×D₁₀₃

Fitting

C₄×Q₈×C₁₀₃

C₂

hypercenter

C₄×Q₈

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂×C₄ ◁ C₄×Q₈

N	H ≅ G/N	type
1. #15693020	C₂	semidirect
2. #15693019	C₂	semidirect
3. #15693011	C₂	semidirect
7. C₄×Dic₂₀₆	C₂	semidirect
8. C₄×Dic₂₀₆	C₂	semidirect
11. C₄×Q₈×C₁₀₃	C₂	semidirect
49. C₄×D₁₀₃	Q₈	semidirect
70. C₁₀₃	#533	semidirect
80. C₄×Q₈	D₁₀₃	semidirect

H ≅ G/N

type

1. #15693020

C₂

semidirect

2. #15693019

C₂

semidirect

3. #15693011

C₂

semidirect

7. C₄×Dic₂₀₆

C₂

semidirect

8. C₄×Dic₂₀₆

C₂

semidirect

11. C₄×Q₈×C₁₀₃

C₂

semidirect

49. C₄×D₁₀₃

Q₈

semidirect

70. C₁₀₃

#533

semidirect

80. C₄×Q₈

D₁₀₃

semidirect

Subgroups — #20268345

Conjugacy classes of subgroups

Complements