Group #20268540 — subgroups

Subgroups — #20268540

Order 6592.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#15693019	C₂	1	1	yes	maximal, characteristic, complemented
2	#15693016	C₂	1	2	yes	maximal, complemented
3	#15693015	C₂	1	2	yes	maximal, complemented
4	#15693011	C₂	1	2	yes	maximal, complemented
5	#15693010	C₂	1	1	yes	maximal, characteristic
6	#15693006	C₂	1	2	yes	maximal
7	#15692999	C₂	1	2	yes	maximal, complemented
8	#15692997	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₀₆	C₂²	1	1	yes	characteristic
11	C₂×D₄₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
13	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
14	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
15	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
16	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
17	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
18	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
19	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₁₀₃)⋊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	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
30	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
31	C₂²×D₁₀₃	C₂³	1	2	yes	—
32	Q₈×C₁₀₃	—	4	1	no	—
33	Q₈×C₁₀₃	—	4	1	no	—
34	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
35	C₂×C₄×C₁₀₃	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×Dic₁₀₃	C₂³	1	2	yes	—
40	C₂×Dic₁₀₃	C₂³	1	2	yes	—
41	C₂×Dic₁₀₃	C₂³	1	2	yes	—
42	D₄₁₂	—	4	1	no	—
43	C₄×D₁₀₃	—	4	2	no	—
44	Dic₂₀₆	—	4	1	no	—
45	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
46	C₂×D₁₀₃	—	2	2	no	—
47	C₂×D₁₀₃	—	2	4	no	—
48	C₄×C₁₀₃	—	2	1	no	—
49	C₄×C₁₀₃	—	2	2	no	—
50	C₄×C₁₀₃	—	2	1	no	—
51	C₄×C₁₀₃	—	2	1	no	—
52	C₄×C₁₀₃	—	2	2	no	—
53	Dic₁₀₃	—	2	2	no	—
54	Dic₁₀₃	—	2	2	no	—
55	Dic₁₀₃	—	2	2	no	—
56	C₂×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₀₃	2₋¹⁺⁴	1	2	yes	—
58	D₁₀₃	—	4	2	no	—
59	C₁₀₃	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	103	1	no	maximal, Sylow
61	C₄⋊Q₈	—	103	1	no	—
62	C₄⋊Q₈	D₁₀₃	1	1	yes	characteristic, complemented
63	#126	—	103	2	no	—
64	#126	—	103	2	no	—
65	#125	—	103	2	no	—
66	#124	—	103	1	no	—
67	#123	—	103	2	no	—
68	C₂²⋊Q₈	—	103	2	no	—
69	C₂²⋊Q₈	—	103	2	no	—
70	C₂×Q₈	—	103	1	no	—
71	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
73	C₂×D₄	—	103	1	no	—
74	C₂²×C₄	—	103	2	no	—
75	C₄⋊C₄	—	103	2	no	—
76	C₄⋊C₄	—	103	2	no	—
77	C₄⋊C₄	—	103	2	no	—
78	C₄⋊C₄	—	103	2	no	—
79	C₄⋊C₄	—	103	2	no	—
80	C₄⋊C₄	—	103	2	no	—
81	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
82	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
83	C₂²⋊C₄	—	103	2	no	—
84	C₂²⋊C₄	—	103	2	no	—
85	C₂²⋊C₄	—	103	2	no	—
86	C₂²⋊C₄	—	103	2	no	—
87	C₂²⋊C₄	—	103	2	no	—
88	C₄²	—	103	1	no	—
89	C₄²	—	103	1	no	—
90	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
91	C₂³	—	103	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	412	1	no	—
95	D₄	—	412	1	no	—
96	C₂×C₄	—	103	2	no	—
97	C₂×C₄	—	103	2	no	—
98	C₂×C₄	—	103	2	no	—
99	C₂×C₄	—	412	2	no	—
100	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
103	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
105	C₂²	—	206	2	no	—
106	C₂²	—	206	4	no	—
107	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	206	2	no	—
114	C₄	—	206	2	no	—
115	C₄	—	206	2	no	—
116	C₂	—	412	2	no	—
117	C₂	#15693066	1	1	yes	characteristic
118	C₂	#15693069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15693019

C₂

yes

maximal, characteristic, complemented

#15693016

C₂

yes

maximal, complemented

#15693015

C₂

yes

maximal, complemented

#15693011

C₂

yes

maximal, complemented

#15693010

C₂

yes

maximal, characteristic

#15693006

C₂

yes

maximal

#15692999

C₂

yes

maximal, complemented

#15692997

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₆

C₂²

yes

characteristic

C₂×D₄₁₂

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

—

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

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

—

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

D₄₁₂

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₃

2₋¹⁺⁴

yes

—

D₁₀₃

—

C₁₀₃

#563

yes

characteristic, complemented, Sylow

#563

—

103

maximal, Sylow

C₄⋊Q₈

—

103

—

C₄⋊Q₈

D₁₀₃

yes

characteristic, complemented

#126

—

103

—

#126

—

103

—

#125

—

103

—

#124

—

103

—

#123

—

103

—

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×D₄

—

103

—

C₂²×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

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

C₂×D₁₀₃

yes

characteristic

C₂³

—

103

—

Q₈

—

Q₈

—

Q₈

—

412

—

D₄

—

412

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

412

—

100

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

101

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

102

C₂×C₄

C₂²×D₁₀₃

yes

—

103

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

104

C₂×C₄

C₂²×D₁₀₃

yes

—

105

C₂²

—

206

—

106

C₂²

—

206

—

107

C₂²

C₂³×D₁₀₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

206

—

114

C₄

—

206

—

115

C₄

—

206

—

116

C₂

—

412

—

117

C₂

#15693066

yes

characteristic

118

C₂

#15693069

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₃	C₂
hypercenter	C₄⋊Q₈	D₁₀₃
hypocenter	C₁₀₃	#563
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂⁴

Fitting

C₄⋊Q₈×C₁₀₃

C₂

hypercenter

C₄⋊Q₈

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #15693019	C₂	semidirect
2. #15693016	C₂	semidirect
3. #15693015	C₂	semidirect
4. #15693011	C₂	semidirect
7. #15692999	C₂	semidirect
8. #15692997	C₂	semidirect
9. C₄⋊Q₈×C₁₀₃	C₂	semidirect
59. C₁₀₃	#563	semidirect
62. C₄⋊Q₈	D₁₀₃	semidirect

H ≅ G/N

type

1. #15693019

C₂

semidirect

2. #15693016

C₂

semidirect

3. #15693015

C₂

semidirect

4. #15693011

C₂

semidirect

7. #15692999

C₂

semidirect

8. #15692997

C₂

semidirect

9. C₄⋊Q₈×C₁₀₃

C₂

semidirect

59. C₁₀₃

#563

semidirect

62. C₄⋊Q₈

D₁₀₃

semidirect

Subgroups — #20268540

Conjugacy classes of subgroups

Complements