Group #20268366 — subgroups

Subgroups — #20268366

Order 6592.

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

normal subgroups only

This group has 270 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	C₂×#13110106	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#15693019	C₂	1	1	yes	maximal, characteristic, complemented
4	#15693010	C₂	1	1	yes	maximal, characteristic, complemented
5	#15693009	C₂	1	1	yes	maximal, characteristic, complemented
6	#15693006	C₂	1	4	yes	maximal, complemented
7	#15692999	C₂	1	4	yes	maximal, complemented
8	#15692978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13110106	—	2	4	no	—
11	C₂×Dic₂₀₆	C₂²	1	2	yes	—
12	C₂×Dic₂₀₆	C₂²	1	1	yes	characteristic
13	Q₈×D₁₀₃	—	2	4	no	—
14	C₂×D₄₁₂	C₂²	1	2	yes	—
15	C₂×D₄₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
17	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
18	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
19	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
20	Dic₁₀₃⋊C₄	C₂²	1	4	yes	—
21	(C₂×D₁₀₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
23	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
24	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₃	C₂²	1	4	yes	—
28	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
29	C₂²×D₁₀₃	C₂³	1	2	yes	—
30	C₂²×D₁₀₃	C₂³	1	1	yes	characteristic
31	Q₈×C₁₀₃	—	2	2	no	—
32	Q₈×C₁₀₃	—	2	2	no	—
33	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
34	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
35	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
36	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₀₃	C₂³	1	2	yes	—
38	C₂×Dic₁₀₃	C₂³	1	2	yes	—
39	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
40	D₄₁₂	—	2	4	no	—
41	D₄₁₂	—	2	2	no	—
42	C₄×D₁₀₃	—	2	4	no	—
43	C₄×D₁₀₃	—	2	4	no	—
44	C₄×D₁₀₃	D₄	1	4	yes	—
45	Dic₂₀₆	—	2	2	no	—
46	Dic₂₀₆	—	2	4	no	—
47	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
48	C₂×D₁₀₃	—	2	2	no	—
49	C₂×D₁₀₃	—	2	2	no	—
50	C₂×D₁₀₃	—	2	4	no	—
51	C₂×D₁₀₃	C₂×D₄	1	2	yes	—
52	C₄×C₁₀₃	—	2	2	no	—
53	C₄×C₁₀₃	—	2	2	no	—
54	C₄×C₁₀₃	—	2	2	no	—
55	C₄×C₁₀₃	C₂×D₄	1	2	yes	—
56	Dic₁₀₃	—	2	2	no	—
57	Dic₁₀₃	—	2	2	no	—
58	Dic₁₀₃	C₂×D₄	1	2	yes	—
59	C₂×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₁₀₃	2₋¹⁺⁴	1	2	yes	—
61	D₁₀₃	—	2	2	no	—
62	D₁₀₃	—	4	2	no	—
63	C₁₀₃	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	103	1	no	maximal, Sylow
65	C₂×#41	—	103	1	no	—
66	C₂²×Q₈	—	103	1	no	—
67	C₄⋊Q₈	—	103	1	no	—
68	C₄⋊Q₈	D₁₀₃	1	1	yes	characteristic, complemented
69	#124	—	103	1	no	—
70	#124	—	103	1	no	—
71	#123	—	103	4	no	—
72	C₂²⋊Q₈	—	103	4	no	—
73	#117	—	103	1	no	—
74	#41	—	206	4	no	—
75	C₂×Q₈	—	103	2	no	—
76	C₂×Q₈	—	103	1	no	—
77	C₂×Q₈	—	206	4	no	—
78	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
80	C₂×D₄	—	103	2	no	—
81	C₂×D₄	—	103	1	no	—
82	C₂²×C₄	—	103	1	no	—
83	C₂²×C₄	—	103	2	no	—
84	C₂²×C₄	—	103	2	no	—
85	C₄⋊C₄	—	103	2	no	—
86	C₄⋊C₄	—	103	4	no	—
87	C₄⋊C₄	C₂×D₁₀₃	1	4	yes	—
88	C₂²⋊C₄	—	103	2	no	—
89	C₂²⋊C₄	—	103	4	no	—
90	C₂²⋊C₄	—	103	4	no	—
91	C₄²	—	103	1	no	—
92	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
93	C₂³	—	103	1	no	—
94	C₂³	—	103	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	206	4	no	—
98	Q₈	—	206	2	no	—
99	D₄	—	206	4	no	—
100	D₄	—	206	2	no	—
101	C₂×C₄	—	103	1	no	—
102	C₂×C₄	—	103	4	no	—
103	C₂×C₄	—	103	2	no	—
104	C₂×C₄	—	103	2	no	—
105	C₂×C₄	—	206	4	no	—
106	C₂×C₄	—	206	4	no	—
107	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
108	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
109	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
110	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
111	C₂²	—	103	2	no	—
112	C₂²	—	206	2	no	—
113	C₂²	—	206	4	no	—
114	C₂²	—	206	2	no	—
115	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	103	2	no	—
120	C₄	—	206	2	no	—
121	C₄	—	206	2	no	—
122	C₄	D₄×D₁₀₃	1	2	yes	—
123	C₂	—	412	2	no	—
124	C₂	—	206	2	no	—
125	C₂	C₂×D₄×D₁₀₃	1	1	yes	characteristic
126	C₂	#15693069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13110106

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₀₃

C₂

yes

maximal, characteristic, complemented

#15693019

C₂

yes

maximal, characteristic, complemented

#15693010

C₂

yes

maximal, characteristic, complemented

#15693009

C₂

yes

maximal, characteristic, complemented

#15693006

C₂

yes

maximal, complemented

#15692999

C₂

yes

maximal, complemented

#15692978

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#13110106

—

C₂×Dic₂₀₆

C₂²

yes

—

C₂×Dic₂₀₆

C₂²

yes

characteristic

Q₈×D₁₀₃

—

C₂×D₄₁₂

C₂²

yes

—

C₂×D₄₁₂

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

Dic₁₀₃⋊C₄

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic

C₂×Q₈×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

—

C₂²×D₁₀₃

C₂³

yes

characteristic

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

characteristic

D₄₁₂

—

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

C₄×D₁₀₃

D₄

yes

—

Dic₂₀₆

—

Dic₂₀₆

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

C₂×D₄

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×D₄

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

C₂×D₄

yes

—

C₂×C₁₀₃

C₂²×D₄

yes

characteristic

C₂×C₁₀₃

2₋¹⁺⁴

yes

—

D₁₀₃

—

D₁₀₃

—

C₁₀₃

#536

yes

characteristic, complemented, Sylow

#536

—

103

maximal, Sylow

C₂×#41

—

103

—

C₂²×Q₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

D₁₀₃

yes

characteristic, complemented

#124

—

103

—

#124

—

103

—

#123

—

103

—

C₂²⋊Q₈

—

103

—

#117

—

103

—

#41

—

206

—

C₂×Q₈

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₄²

—

103

—

C₄²

C₂×D₁₀₃

yes

characteristic

C₂³

—

103

—

C₂³

—

103

—

Q₈

—

Q₈

—

Q₈

—

206

—

Q₈

—

206

—

D₄

—

206

—

100

D₄

—

206

—

101

C₂×C₄

—

103

—

102

C₂×C₄

—

103

—

103

C₂×C₄

—

103

—

104

C₂×C₄

—

103

—

105

C₂×C₄

—

206

—

106

C₂×C₄

—

206

—

107

C₂×C₄

C₂²×D₁₀₃

yes

—

108

C₂×C₄

C₂²×D₁₀₃

yes

—

109

C₂×C₄

C₂²×D₁₀₃

yes

—

110

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

111

C₂²

—

103

—

112

C₂²

—

206

—

113

C₂²

—

206

—

114

C₂²

—

206

—

115

C₂²

C₂³×D₁₀₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

103

—

120

C₄

—

206

—

121

C₄

—

206

—

122

C₄

D₄×D₁₀₃

yes

—

123

C₂

—

412

—

124

C₂

—

206

—

125

C₂

C₂×D₄×D₁₀₃

yes

characteristic

126

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	#536
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
2. C₂×Q₈×D₁₀₃	C₂	semidirect
3. #15693019	C₂	semidirect
4. #15693010	C₂	semidirect
5. #15693009	C₂	semidirect
6. #15693006	C₂	semidirect
7. #15692999	C₂	semidirect
8. #15692978	C₂	semidirect
9. C₄⋊Q₈×C₁₀₃	C₂	semidirect
63. C₁₀₃	#536	semidirect
68. C₄⋊Q₈	D₁₀₃	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₀₃

C₂

semidirect

3. #15693019

C₂

semidirect

4. #15693010

C₂

semidirect

5. #15693009

C₂

semidirect

6. #15693006

C₂

semidirect

7. #15692999

C₂

semidirect

8. #15692978

C₂

semidirect

9. C₄⋊Q₈×C₁₀₃

C₂

semidirect

63. C₁₀₃

#536

semidirect

68. C₄⋊Q₈

D₁₀₃

semidirect

Subgroups — #20268366

Conjugacy classes of subgroups

Complements