Group #20268390 — subgroups

Subgroups — #20268390

Order 6592.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#15693016	C₂	1	2	yes	maximal, complemented
2	#15693012	C₂	1	2	yes	maximal, complemented
3	#15693006	C₂	1	2	yes	maximal, complemented
4	#15693005	C₂	1	2	yes	maximal, complemented
5	#15692997	C₂	1	2	yes	maximal, complemented
6	#15692995	C₂	1	1	yes	maximal, characteristic
7	(C₂²⋊D₁₀₃)⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₁₀₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₀₃	C₂²	1	2	yes	—
12	C₂²×Dic₁₀₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
15	C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
18	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
19	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
20	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₁₀₃	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×C₁₀₃	C₂²	1	2	yes	—
31	C₂³×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₃	C₂³	1	2	yes	—
33	Q₈×C₁₀₃	—	4	1	no	—
34	C₂×C₄×C₁₀₃	—	4	1	no	—
35	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
36	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
37	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
39	C₂²⋊D₁₀₃	—	2	4	no	—
40	C₂×Dic₁₀₃	—	2	2	no	—
41	C₂×Dic₁₀₃	—	4	1	no	—
42	C₂×Dic₁₀₃	C₂³	1	2	yes	—
43	C₂×Dic₁₀₃	C₂³	1	2	yes	—
44	C₂×Dic₁₀₃	C₂³	1	2	yes	—
45	D₄₁₂	—	4	1	no	—
46	C₄×D₁₀₃	—	4	2	no	—
47	C₂²×C₁₀₃	—	2	1	no	—
48	C₂²×C₁₀₃	—	2	1	no	—
49	C₂²×C₁₀₃	#41	1	2	yes	—
50	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₀₃	—	2	2	no	—
52	C₂×D₁₀₃	—	2	2	no	—
53	C₂×D₁₀₃	—	2	2	no	—
54	C₄×C₁₀₃	—	2	2	no	—
55	C₄×C₁₀₃	—	2	1	no	—
56	C₄×C₁₀₃	—	2	1	no	—
57	C₄×C₁₀₃	—	2	2	no	—
58	Dic₁₀₃	—	2	2	no	—
59	Dic₁₀₃	—	2	2	no	—
60	Dic₁₀₃	—	2	2	no	—
61	C₂×C₁₀₃	—	2	2	no	—
62	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
63	C₂×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₀₃	2₋¹⁺⁴	1	1	yes	characteristic
65	D₁₀₃	—	4	2	no	—
66	C₁₀₃	#539	1	1	yes	characteristic, complemented, Sylow
67	#539	—	103	1	no	maximal, Sylow
68	#126	—	103	2	no	—
69	#125	—	103	2	no	—
70	#123	—	103	2	no	—
71	#123	—	103	2	no	—
72	C₂²⋊Q₈	—	103	2	no	—
73	C₂²⋊Q₈	D₁₀₃	1	1	yes	characteristic, complemented
74	#121	—	103	1	no	—
75	C₄×D₄	—	103	2	no	—
76	C₂×C₄⋊C₄	—	103	1	no	—
77	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
78	C₂×D₄	—	103	2	no	—
79	C₂×D₄	—	103	1	no	—
80	C₂²×C₄	—	103	1	no	—
81	C₂²×C₄	—	103	1	no	—
82	C₂²×C₄	—	103	2	no	—
83	C₂²×C₄	C₂×D₁₀₃	1	1	yes	characteristic
84	C₄⋊C₄	—	103	2	no	—
85	C₄⋊C₄	—	103	2	no	—
86	C₄⋊C₄	—	103	2	no	—
87	C₄⋊C₄	—	103	1	no	—
88	C₄⋊C₄	—	103	2	no	—
89	C₄⋊C₄	—	103	2	no	—
90	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
91	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
92	C₂²⋊C₄	—	103	2	no	—
93	C₂²⋊C₄	—	103	2	no	—
94	C₂²⋊C₄	—	103	2	no	—
95	C₂²⋊C₄	—	103	2	no	—
96	C₂²⋊C₄	C₂×D₁₀₃	1	2	yes	—
97	C₄²	—	103	2	no	—
98	C₂³	—	103	2	no	—
99	C₂³	C₂²×D₁₀₃	1	1	yes	characteristic
100	Q₈	—	4	1	no	—
101	D₄	—	412	1	no	—
102	D₄	—	206	4	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	103	2	no	—
105	C₂×C₄	—	103	2	no	—
106	C₂×C₄	—	103	2	no	—
107	C₂×C₄	—	412	2	no	—
108	C₂×C₄	—	412	1	no	—
109	C₂×C₄	—	206	2	no	—
110	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
111	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
113	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	206	2	no	—
117	C₂²	—	206	2	no	—
118	C₂²	—	206	2	no	—
119	C₂²	#13110106	1	2	yes	—
120	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	206	2	no	—
126	C₄	—	206	2	no	—
127	C₄	—	206	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	412	2	no	—
130	C₂	C₂×#13110106	1	1	yes	characteristic
131	C₂	#15693066	1	1	yes	characteristic
132	C₂	#15693068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693016

C₂

yes

maximal, complemented

#15693012

C₂

yes

maximal, complemented

#15693006

C₂

yes

maximal, complemented

#15693005

C₂

yes

maximal, complemented

#15692997

C₂

yes

maximal, complemented

#15692995

C₂

yes

maximal, characteristic

(C₂²⋊D₁₀₃)⋊C₄

C₂

yes

maximal, complemented

C₂×Dic₁₀₃⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄₁₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₃

C₂²

yes

—

C₂²×Dic₁₀₃

C₂²

yes

characteristic

C₂²×Dic₁₀₃

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

—

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic

C₂²×C₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₃

C₂²

yes

—

C₂³×C₁₀₃

C₂³

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

D₄₁₂

—

C₄×D₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#41

yes

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×#41

yes

characteristic

C₂×C₁₀₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₃

2₋¹⁺⁴

yes

characteristic

D₁₀₃

—

C₁₀₃

#539

yes

characteristic, complemented, Sylow

#539

—

103

maximal, Sylow

#126

—

103

—

#125

—

103

—

#123

—

103

—

#123

—

103

—

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

D₁₀₃

yes

characteristic, complemented

#121

—

103

—

C₄×D₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₂×D₁₀₃

yes

characteristic

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

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

C₂×D₁₀₃

yes

—

C₄²

—

103

—

C₂³

—

103

—

C₂³

C₂²×D₁₀₃

yes

characteristic

100

Q₈

—

101

D₄

—

412

—

102

D₄

—

206

—

103

C₂×C₄

—

104

C₂×C₄

—

103

—

105

C₂×C₄

—

103

—

106

C₂×C₄

—

103

—

107

C₂×C₄

—

412

—

108

C₂×C₄

—

412

—

109

C₂×C₄

—

206

—

110

C₂×C₄

C₂²×D₁₀₃

yes

—

111

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

112

C₂×C₄

C₂²×D₁₀₃

yes

—

113

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

206

—

117

C₂²

—

206

—

118

C₂²

—

206

—

119

C₂²

#13110106

yes

—

120

C₂²

C₂³×D₁₀₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

206

—

126

C₄

—

206

—

127

C₄

—

206

—

128

C₂

—

129

C₂

—

412

—

130

C₂

C₂×#13110106

yes

characteristic

131

C₂

#15693066

yes

characteristic

132

C₂

#15693068

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Fitting	C₂²⋊Q₈×C₁₀₃	C₂
hypercenter	C₂²⋊Q₈	D₁₀₃
hypocenter	C₁₀₃	#539
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂⁴

Fitting

C₂²⋊Q₈×C₁₀₃

C₂

hypercenter

C₂²⋊Q₈

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #15693016	C₂	semidirect
2. #15693012	C₂	semidirect
3. #15693006	C₂	semidirect
4. #15693005	C₂	semidirect
5. #15692997	C₂	semidirect
7. (C₂²⋊D₁₀₃)⋊C₄	C₂	semidirect
8. C₂×Dic₁₀₃⋊C₄	C₂	semidirect
9. C₂²⋊Q₈×C₁₀₃	C₂	semidirect
66. C₁₀₃	#539	semidirect
73. C₂²⋊Q₈	D₁₀₃	semidirect

H ≅ G/N

type

1. #15693016

C₂

semidirect

2. #15693012

C₂

semidirect

3. #15693006

C₂

semidirect

4. #15693005

C₂

semidirect

5. #15692997

C₂

semidirect

7. (C₂²⋊D₁₀₃)⋊C₄

C₂

semidirect

8. C₂×Dic₁₀₃⋊C₄

C₂

semidirect

9. C₂²⋊Q₈×C₁₀₃

C₂

semidirect

66. C₁₀₃

#539

semidirect

73. C₂²⋊Q₈

D₁₀₃

semidirect

Subgroups — #20268390

Conjugacy classes of subgroups

Complements