Group #20268298 — subgroups

Subgroups — #20268298

Order 6592.

330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂²×C₄×D₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#15693002	C₂	1	2	yes	maximal, complemented
3	#15692997	C₂	1	2	yes	maximal, complemented
4	#15692992	C₂	1	2	yes	maximal, complemented
5	#15692989	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₁₀₃	C₂	1	4	yes	maximal, complemented
7	#15692980	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₁₀₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₀₃	C₂²	1	4	yes	—
11	C₂²×Dic₁₀₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₃	—	2	4	no	—
13	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
15	C₄⋊Dic₁₀₃	C₂²	1	2	yes	complemented
16	Dic₁₀₃⋊C₄	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₁₀₃	C₂²	1	2	yes	—
18	C₂²⋊Dic₁₀₃	C₂²	1	4	yes	complemented
19	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₁₀₃	C₂²	1	2	yes	complemented
21	#41×C₁₀₃	—	2	4	no	—
22	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₄×C₁₀₃	C₂²	1	2	yes	—
24	C₂×D₄×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₃	C₂²	1	2	yes	complemented
27	C₂³×C₁₀₃	C₂³	1	2	yes	—
28	C₂³×C₁₀₃	C₂³	1	1	yes	characteristic
29	C₂²×D₁₀₃	—	2	2	no	—
30	C₂²×D₁₀₃	—	2	4	no	—
31	C₂²×D₁₀₃	C₂³	1	2	yes	—
32	Q₈×C₁₀₃	—	2	2	no	—
33	D₄×C₁₀₃	—	2	4	no	—
34	D₄×C₁₀₃	—	2	2	no	—
35	C₂×C₄×C₁₀₃	—	2	4	no	—
36	C₂×C₄×C₁₀₃	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
38	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
40	C₂²⋊D₁₀₃	—	2	8	no	—
41	C₂×Dic₁₀₃	—	2	2	no	—
42	C₂×Dic₁₀₃	C₂³	1	4	yes	—
43	C₂×Dic₁₀₃	C₂³	1	2	yes	—
44	C₄×D₁₀₃	—	2	4	no	—
45	C₄×D₁₀₃	—	2	4	no	—
46	C₂²×C₁₀₃	—	2	2	no	—
47	C₂²×C₁₀₃	—	2	2	no	—
48	C₂²×C₁₀₃	—	2	4	no	—
49	C₂²×C₁₀₃	C₂×D₄	1	2	yes	—
50	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₁₀₃	—	2	8	no	—
52	C₂×D₁₀₃	—	2	2	no	—
53	C₂×D₁₀₃	—	2	2	no	—
54	C₂×D₁₀₃	#41	1	4	yes	complemented
55	C₄×C₁₀₃	—	2	2	no	—
56	C₄×C₁₀₃	C₂×D₄	1	2	yes	—
57	C₄×C₁₀₃	C₂×D₄	1	2	yes	—
58	Dic₁₀₃	—	2	2	no	—
59	Dic₁₀₃	—	2	4	no	—
60	C₂×C₁₀₃	—	2	2	no	—
61	C₂×C₁₀₃	—	4	2	no	—
62	C₂×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₁₀₃	C₂×#41	1	2	yes	—
64	D₁₀₃	—	2	4	no	—
65	C₁₀₃	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	103	1	no	maximal, Sylow
67	C₂×#41	D₁₀₃	1	1	yes	characteristic, complemented
68	C₂³×C₄	—	103	1	no	—
69	#123	—	103	2	no	—
70	C₂²⋊Q₈	—	103	2	no	—
71	#121	—	103	2	no	—
72	#120	—	103	2	no	—
73	C₄×D₄	—	103	4	no	—
74	#117	—	103	1	no	—
75	C₂⁴	—	103	1	no	—
76	#41	—	2	4	no	—
77	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic, complemented
78	C₂×D₄	—	103	4	no	—
79	C₂×D₄	C₂×D₁₀₃	1	2	yes	—
80	C₂×D₄	C₂×D₁₀₃	1	1	yes	characteristic, complemented
81	C₂²×C₄	—	103	1	no	—
82	C₂²×C₄	—	103	2	no	—
83	C₂²×C₄	—	103	2	no	—
84	C₂²×C₄	—	206	4	no	—
85	C₂²×C₄	C₂×D₁₀₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₁₀₃	1	2	yes	complemented
87	C₄⋊C₄	—	103	4	no	—
88	C₄⋊C₄	—	103	2	no	—
89	C₂²⋊C₄	—	103	2	no	—
90	C₂²⋊C₄	—	103	4	no	—
91	C₂²⋊C₄	—	103	4	no	—
92	C₄²	—	103	2	no	—
93	C₂³	—	103	2	no	—
94	C₂³	—	206	4	no	—
95	C₂³	—	206	2	no	—
96	C₂³	C₂²×D₁₀₃	1	2	yes	—
97	C₂³	C₂²×D₁₀₃	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	4	no	—
100	D₄	—	2	2	no	—
101	D₄	—	206	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	103	4	no	—
104	C₂×C₄	—	103	2	no	—
105	C₂×C₄	—	206	2	no	—
106	C₂×C₄	—	206	4	no	—
107	C₂×C₄	—	206	4	no	—
108	C₂×C₄	C₂²⋊D₁₀₃	1	4	yes	complemented
109	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
110	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	103	4	no	—
116	C₂²	—	206	8	no	—
117	C₂²	—	206	2	no	—
118	C₂²	—	206	2	no	—
119	C₂²	C₂×C₂²⋊D₁₀₃	1	2	yes	—
120	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	206	2	no	—
123	C₄	—	206	4	no	—
124	C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
125	C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	4	2	no	—
128	C₂	—	206	4	no	—
129	C₂	C₂²×C₂²⋊D₁₀₃	1	1	yes	characteristic
130	C₂	#41×D₁₀₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₃

C₂

yes

maximal, characteristic, complemented

#15693002

C₂

yes

maximal, complemented

#15692997

C₂

yes

maximal, complemented

#15692992

C₂

yes

maximal, complemented

#15692989

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₀₃

C₂

yes

maximal, complemented

#15692980

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₃

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₃

C₂²

yes

—

C₂²×Dic₁₀₃

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

complemented

Dic₁₀₃⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₁₀₃

C₂²

yes

—

C₂²⋊Dic₁₀₃

C₂²

yes

complemented

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

complemented

#41×C₁₀₃

—

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₀₃

C₂²

yes

—

C₂×D₄×C₁₀₃

C₂²

yes

characteristic, complemented

C₂²×C₄×C₁₀₃

C₂²

yes

characteristic

C₂²×C₄×C₁₀₃

C₂²

yes

complemented

C₂³×C₁₀₃

C₂³

yes

—

C₂³×C₁₀₃

C₂³

yes

characteristic

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂²×D₁₀₃

C₂³

yes

—

Q₈×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

D₄

yes

complemented

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×D₄

yes

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

#41

yes

complemented

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×D₄

yes

—

C₄×C₁₀₃

C₂×D₄

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂²×D₄

yes

characteristic

C₂×C₁₀₃

C₂×#41

yes

—

D₁₀₃

—

C₁₀₃

#525

yes

characteristic, complemented, Sylow

#525

—

103

maximal, Sylow

C₂×#41

D₁₀₃

yes

characteristic, complemented

C₂³×C₄

—

103

—

#123

—

103

—

C₂²⋊Q₈

—

103

—

#121

—

103

—

#120

—

103

—

C₄×D₄

—

103

—

#117

—

103

—

C₂⁴

—

103

—

#41

—

C₂×Q₈

C₂×D₁₀₃

yes

characteristic, complemented

C₂×D₄

—

103

—

C₂×D₄

C₂×D₁₀₃

yes

—

C₂×D₄

C₂×D₁₀₃

yes

characteristic, complemented

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂×D₁₀₃

yes

characteristic

C₂²×C₄

C₂×D₁₀₃

yes

complemented

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₄²

—

103

—

C₂³

—

103

—

C₂³

—

206

—

C₂³

—

206

—

C₂³

C₂²×D₁₀₃

yes

—

C₂³

C₂²×D₁₀₃

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

D₄

—

206

—

102

C₂×C₄

—

103

C₂×C₄

—

103

—

104

C₂×C₄

—

103

—

105

C₂×C₄

—

206

—

106

C₂×C₄

—

206

—

107

C₂×C₄

—

206

—

108

C₂×C₄

C₂²⋊D₁₀₃

yes

complemented

109

C₂×C₄

C₂²×D₁₀₃

yes

—

110

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

111

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

103

—

116

C₂²

—

206

—

117

C₂²

—

206

—

118

C₂²

—

206

—

119

C₂²

C₂×C₂²⋊D₁₀₃

yes

—

120

C₂²

C₂³×D₁₀₃

yes

characteristic

121

C₄

—

122

C₄

—

206

—

123

C₄

—

206

—

124

C₄

C₂×C₂²⋊D₁₀₃

yes

—

125

C₄

C₂×C₂²⋊D₁₀₃

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

206

—

129

C₂

C₂²×C₂²⋊D₁₀₃

yes

characteristic

130

C₂

#41×D₁₀₃

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Frattini	C₂²	C₂³×D₁₀₃
Fitting	C₂×#41×C₁₀₃	C₂
hypercenter	C₂×#41	D₁₀₃
hypocenter	C₁₀₃	#525
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂×C₄

C₂²×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂⁴

Frattini

C₂²

C₂³×D₁₀₃

Fitting

C₂×#41×C₁₀₃

C₂

hypercenter

C₂×#41

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×C₄×D₁₀₃	C₂	semidirect
2. #15693002	C₂	semidirect
3. #15692997	C₂	semidirect
4. #15692992	C₂	semidirect
5. #15692989	C₂	semidirect
6. C₄×C₂²⋊D₁₀₃	C₂	semidirect
7. #15692980	C₂	semidirect
8. C₂×#41×C₁₀₃	C₂	semidirect
15. C₄⋊Dic₁₀₃	C₂²	semidirect
16. Dic₁₀₃⋊C₄	C₂²	semidirect
18. C₂²⋊Dic₁₀₃	C₂²	semidirect
20. C₄×Dic₁₀₃	C₂²	semidirect
22. C₂×Q₈×C₁₀₃	C₂²	semidirect
24. C₂×D₄×C₁₀₃	C₂²	semidirect
26. C₂²×C₄×C₁₀₃	C₂²	semidirect
36. C₂×C₄×C₁₀₃	D₄	semidirect
54. C₂×D₁₀₃	#41	semidirect
65. C₁₀₃	#525	semidirect
67. C₂×#41	D₁₀₃	semidirect
77. C₂×Q₈	C₂×D₁₀₃	semidirect
80. C₂×D₄	C₂×D₁₀₃	semidirect
86. C₂²×C₄	C₂×D₁₀₃	semidirect
108. C₂×C₄	C₂²⋊D₁₀₃	semidirect

H ≅ G/N

type

1. C₂²×C₄×D₁₀₃

C₂

semidirect

2. #15693002

C₂

semidirect

3. #15692997

C₂

semidirect

4. #15692992

C₂

semidirect

5. #15692989

C₂

semidirect

6. C₄×C₂²⋊D₁₀₃

C₂

semidirect

7. #15692980

C₂

semidirect

8. C₂×#41×C₁₀₃

C₂

semidirect

15. C₄⋊Dic₁₀₃

C₂²

semidirect

16. Dic₁₀₃⋊C₄

C₂²

semidirect

18. C₂²⋊Dic₁₀₃

C₂²

semidirect

20. C₄×Dic₁₀₃

C₂²

semidirect

22. C₂×Q₈×C₁₀₃

C₂²

semidirect

24. C₂×D₄×C₁₀₃

C₂²

semidirect

26. C₂²×C₄×C₁₀₃

C₂²

semidirect

36. C₂×C₄×C₁₀₃

D₄

semidirect

54. C₂×D₁₀₃

#41

semidirect

65. C₁₀₃

#525

semidirect

67. C₂×#41

D₁₀₃

semidirect

77. C₂×Q₈

C₂×D₁₀₃

semidirect

80. C₂×D₄

C₂×D₁₀₃

semidirect

86. C₂²×C₄

C₂×D₁₀₃

semidirect

108. C₂×C₄

C₂²⋊D₁₀₃

semidirect

Subgroups — #20268298

Conjugacy classes of subgroups

Complements