Group #20268421 — subgroups

Subgroups — #20268421

Order 6592.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#15693021	C₂	1	1	yes	maximal, characteristic, complemented
2	#15693020	C₂	1	1	yes	maximal, characteristic, complemented
3	#15693020	C₂	1	2	yes	maximal, complemented
4	#15693012	C₂	1	1	yes	maximal, characteristic, complemented
5	#15693011	C₂	1	2	yes	maximal, complemented
6	#15693001	C₂	1	2	yes	maximal
7	#15692999	C₂	1	2	yes	maximal
8	#15692979	C₂	1	1	yes	maximal, characteristic
9	#15692978	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×D₁₀₃	C₂	1	1	yes	maximal, characteristic
11	#125×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₀₆	C₂²	1	2	yes	—
13	C₂×Dic₂₀₆	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
15	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
19	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
20	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
21	Dic₁₀₃⋊C₄	C₂²	1	1	yes	characteristic
22	Dic₁₀₃⋊C₄	C₂²	1	1	yes	characteristic
23	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
24	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₀₃)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₁₀₃)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₀₃	C₂²	1	2	yes	—
30	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
34	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
35	C₂²×D₁₀₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
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×Dic₁₀₃	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	C₄×D₁₀₃	—	4	2	no	—
47	C₄×D₁₀₃	Q₈	1	4	yes	complemented
48	Dic₂₀₆	—	4	2	no	—
49	Dic₂₀₆	—	4	2	no	—
50	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₀₃	—	2	1	no	—
52	C₂×D₁₀₃	—	2	1	no	—
53	C₂×D₁₀₃	C₂×Q₈	1	2	yes	—
54	C₄×C₁₀₃	—	2	1	no	—
55	C₄×C₁₀₃	—	2	2	no	—
56	C₄×C₁₀₃	—	2	1	no	—
57	C₄×C₁₀₃	—	2	2	no	—
58	C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
59	Dic₁₀₃	—	2	1	no	—
60	Dic₁₀₃	—	2	2	no	—
61	Dic₁₀₃	—	2	2	no	—
62	Dic₁₀₃	—	2	1	no	—
63	Dic₁₀₃	C₂×Q₈	1	2	yes	—
64	C₂×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×C₁₀₃	2₋¹⁺⁴	1	1	yes	characteristic
67	D₁₀₃	—	2	2	no	—
68	C₁₀₃	#544	1	1	yes	characteristic, complemented, Sylow
69	#544	—	103	1	no	maximal, Sylow
70	C₄⋊Q₈	—	103	2	no	—
71	C₄⋊Q₈	—	103	1	no	—
72	C₄⋊Q₈	—	103	1	no	—
73	#125	—	103	2	no	—
74	#125	—	103	1	no	—
75	#125	D₁₀₃	1	1	yes	characteristic, complemented
76	C₂²⋊Q₈	—	103	2	no	—
77	C₂²⋊Q₈	—	103	2	no	—
78	#117	—	103	1	no	—
79	#117	—	103	1	no	—
80	C₂×C₄⋊C₄	—	103	1	no	—
81	C₂×Q₈	—	103	2	no	—
82	C₂×Q₈	—	103	2	no	—
83	C₂²×C₄	—	103	1	no	—
84	C₂²×C₄	—	103	2	no	—
85	C₄⋊C₄	—	103	2	no	—
86	C₄⋊C₄	—	103	1	no	—
87	C₄⋊C₄	—	103	1	no	—
88	C₄⋊C₄	—	103	2	no	—
89	C₄⋊C₄	—	103	2	no	—
90	C₄⋊C₄	—	103	1	no	—
91	C₄⋊C₄	—	103	1	no	—
92	C₄⋊C₄	—	103	2	no	—
93	C₄⋊C₄	—	103	2	no	—
94	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
95	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
97	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
98	C₂²⋊C₄	—	103	1	no	—
99	C₂²⋊C₄	—	103	2	no	—
100	C₂²⋊C₄	—	103	1	no	—
101	C₄²	—	103	2	no	—
102	C₄²	—	103	1	no	—
103	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
104	C₂³	—	103	1	no	—
105	Q₈	—	412	2	no	—
106	Q₈	—	412	2	no	—
107	C₂×C₄	—	103	1	no	—
108	C₂×C₄	—	103	4	no	—
109	C₂×C₄	—	103	1	no	—
110	C₂×C₄	—	103	1	no	—
111	C₂×C₄	—	103	2	no	—
112	C₂×C₄	—	103	2	no	—
113	C₂×C₄	—	412	2	no	—
114	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
115	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
118	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
119	C₂²	—	103	2	no	—
120	C₂²	—	206	1	no	—
121	C₂²	—	206	1	no	—
122	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	103	2	no	—
128	C₄	—	206	2	no	—
129	C₄	—	206	1	no	—
130	C₄	—	206	2	no	—
131	C₄	—	206	1	no	—
132	C₄	Q₈×D₁₀₃	1	2	yes	—
133	C₂	—	206	2	no	—
134	C₂	C₂×Q₈×D₁₀₃	1	1	yes	characteristic
135	C₂	#15693067	1	1	yes	characteristic
136	C₂	#15693068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693021

C₂

yes

maximal, characteristic, complemented

#15693020

C₂

yes

maximal, characteristic, complemented

#15693020

C₂

yes

maximal, complemented

#15693012

C₂

yes

maximal, characteristic, complemented

#15693011

C₂

yes

maximal, complemented

#15693001

C₂

yes

maximal

#15692999

C₂

yes

maximal

#15692979

C₂

yes

maximal, characteristic

#15692978

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₀₃

C₂

yes

maximal, characteristic

#125×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₆

C₂²

yes

—

C₂×Dic₂₀₆

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

characteristic

Dic₁₀₃⋊C₄

C₂²

yes

characteristic

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₀₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

characteristic

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

—

C₄×C₁₀₃

C₂×Q₈

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

C₂×Q₈

yes

—

C₂×C₁₀₃

C₂²×Q₈

yes

characteristic

C₂×C₁₀₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₃

2₋¹⁺⁴

yes

characteristic

D₁₀₃

—

C₁₀₃

#544

yes

characteristic, complemented, Sylow

#544

—

103

maximal, Sylow

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

#125

—

103

—

#125

—

103

—

#125

D₁₀₃

yes

characteristic, complemented

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

—

103

—

#117

—

103

—

#117

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

—

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

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₄⋊C₄

C₂×D₁₀₃

yes

—

C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

100

C₂²⋊C₄

—

103

—

101

C₄²

—

103

—

102

C₄²

—

103

—

103

C₄²

C₂×D₁₀₃

yes

characteristic

104

C₂³

—

103

—

105

Q₈

—

412

—

106

Q₈

—

412

—

107

C₂×C₄

—

103

—

108

C₂×C₄

—

103

—

109

C₂×C₄

—

103

—

110

C₂×C₄

—

103

—

111

C₂×C₄

—

103

—

112

C₂×C₄

—

103

—

113

C₂×C₄

—

412

—

114

C₂×C₄

C₂²×D₁₀₃

yes

—

115

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

116

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

117

C₂×C₄

C₂²×D₁₀₃

yes

—

118

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

119

C₂²

—

103

—

120

C₂²

—

206

—

121

C₂²

—

206

—

122

C₂²

C₂³×D₁₀₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

103

—

128

C₄

—

206

—

129

C₄

—

206

—

130

C₄

—

206

—

131

C₄

—

206

—

132

C₄

Q₈×D₁₀₃

yes

—

133

C₂

—

206

—

134

C₂

C₂×Q₈×D₁₀₃

yes

characteristic

135

C₂

#15693067

yes

characteristic

136

C₂

#15693068

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Fitting	#125×C₁₀₃	C₂
hypercenter	#125	D₁₀₃
hypocenter	C₁₀₃	#544
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂⁴

Fitting

#125×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #15693021	C₂	semidirect
2. #15693020	C₂	semidirect
3. #15693020	C₂	semidirect
4. #15693012	C₂	semidirect
5. #15693011	C₂	semidirect
11. #125×C₁₀₃	C₂	semidirect
47. C₄×D₁₀₃	Q₈	semidirect
68. C₁₀₃	#544	semidirect
75. #125	D₁₀₃	semidirect

H ≅ G/N

type

1. #15693021

C₂

semidirect

2. #15693020

C₂

semidirect

3. #15693020

C₂

semidirect

4. #15693012

C₂

semidirect

5. #15693011

C₂

semidirect

11. #125×C₁₀₃

C₂

semidirect

47. C₄×D₁₀₃

Q₈

semidirect

68. C₁₀₃

#544

semidirect

75. #125

D₁₀₃

semidirect

Subgroups — #20268421

Conjugacy classes of subgroups

Complements