Group #20268346 — subgroups

Subgroups — #20268346

Order 6592.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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	#15693012	C₂	1	2	yes	maximal, complemented
3	#15693001	C₂	1	2	yes	maximal
4	#15692997	C₂	1	2	yes	maximal
5	Q₈×Dic₁₀₃	C₂	1	2	yes	maximal, complemented
6	#15692986	C₂	1	2	yes	maximal, complemented
7	#15692979	C₂	1	2	yes	maximal
8	C₄²×D₁₀₃	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₀₆	C₂²	1	2	yes	—
11	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₀₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
15	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₃	C₂²	1	4	yes	—
17	Dic₁₀₃⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
20	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₀₃	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₀₃	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
26	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
27	C₂²×D₁₀₃	C₂³	1	1	yes	characteristic
28	Q₈×C₁₀₃	—	2	4	no	—
29	C₂×C₄×C₁₀₃	C₂³	1	4	yes	—
30	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₁₀₃	C₂³	1	4	yes	—
36	C₂×Dic₁₀₃	C₂³	1	1	yes	characteristic
37	C₄×D₁₀₃	—	2	2	no	—
38	C₄×D₁₀₃	—	2	2	no	—
39	C₄×D₁₀₃	Q₈	1	4	yes	complemented
40	Dic₂₀₆	—	2	4	no	—
41	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₀₃	—	2	1	no	—
43	C₂×D₁₀₃	—	2	1	no	—
44	C₂×D₁₀₃	C₂×Q₈	1	2	yes	—
45	C₄×C₁₀₃	—	2	4	no	—
46	C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
47	C₄×C₁₀₃	#41	1	2	yes	—
48	C₄×C₁₀₃	#41	1	2	yes	—
49	Dic₁₀₃	—	2	4	no	—
50	Dic₁₀₃	—	2	1	no	—
51	Dic₁₀₃	—	2	1	no	—
52	Dic₁₀₃	C₂×Q₈	1	2	yes	—
53	C₂×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
56	D₁₀₃	—	2	2	no	—
57	C₁₀₃	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	103	1	no	maximal, Sylow
59	C₄⋊Q₈	—	103	1	no	—
60	C₄⋊Q₈	D₁₀₃	1	1	yes	characteristic, complemented
61	#125	—	103	2	no	—
62	C₂²⋊Q₈	—	103	2	no	—
63	C₂²⋊Q₈	—	103	2	no	—
64	C₄×Q₈	—	103	2	no	—
65	C₄×Q₈	—	103	2	no	—
66	#117	—	103	2	no	—
67	C₂×C₄²	—	103	1	no	—
68	C₂×Q₈	—	103	2	no	—
69	C₂×Q₈	C₂×D₁₀₃	1	2	yes	—
70	C₂²×C₄	—	103	1	no	—
71	C₂²×C₄	—	103	1	no	—
72	C₂²×C₄	—	103	1	no	—
73	C₄⋊C₄	—	103	4	no	—
74	C₄⋊C₄	—	103	2	no	—
75	C₄⋊C₄	—	103	4	no	—
76	C₄⋊C₄	—	103	2	no	—
77	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
78	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
79	C₂²⋊C₄	—	103	4	no	—
80	C₄²	—	103	1	no	—
81	C₄²	—	103	1	no	—
82	C₄²	—	103	1	no	—
83	C₄²	—	103	4	no	—
84	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
85	C₂³	—	103	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	206	4	no	—
88	C₂×C₄	—	103	1	no	—
89	C₂×C₄	—	103	4	no	—
90	C₂×C₄	—	103	1	no	—
91	C₂×C₄	—	103	1	no	—
92	C₂×C₄	—	103	4	no	—
93	C₂×C₄	—	206	2	no	—
94	C₂×C₄	—	206	2	no	—
95	C₂×C₄	C₂²×D₁₀₃	1	4	yes	—
96	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
99	C₂²	—	103	2	no	—
100	C₂²	—	206	1	no	—
101	C₂²	—	206	1	no	—
102	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	103	2	no	—
105	C₄	—	206	1	no	—
106	C₄	—	206	1	no	—
107	C₄	—	206	4	no	—
108	C₄	Q₈×D₁₀₃	1	2	yes	—
109	C₄	#13110104	1	2	yes	—
110	C₄	#13110106	1	2	yes	—
111	C₂	—	206	2	no	—
112	C₂	C₂×Q₈×D₁₀₃	1	1	yes	characteristic
113	C₂	C₂×#13110104	1	1	yes	characteristic
114	C₂	C₂×#13110106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693021

C₂

yes

maximal, characteristic, complemented

#15693012

C₂

yes

maximal, complemented

#15693001

C₂

yes

maximal

#15692997

C₂

yes

maximal

Q₈×Dic₁₀₃

C₂

yes

maximal, complemented

#15692986

C₂

yes

maximal, complemented

#15692979

C₂

yes

maximal

C₄²×D₁₀₃

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₆

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₂×C₄×D₁₀₃

C₂²

yes

characteristic

C₄⋊Dic₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

characteristic

C₄×Dic₁₀₃

C₂²

yes

—

C₂×Q₈×C₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

characteristic

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×Dic₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

characteristic

C₄×D₁₀₃

—

C₄×D₁₀₃

—

C₄×D₁₀₃

Q₈

yes

complemented

Dic₂₀₆

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

C₂×Q₈

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×Q₈

yes

—

C₄×C₁₀₃

#41

yes

—

C₄×C₁₀₃

#41

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

C₂×Q₈

yes

—

C₂×C₁₀₃

C₂²×Q₈

yes

characteristic

C₂×C₁₀₃

C₂×#41

yes

characteristic

C₂×C₁₀₃

C₂×#41

yes

characteristic

D₁₀₃

—

C₁₀₃

#533

yes

characteristic, complemented, Sylow

#533

—

103

maximal, Sylow

C₄⋊Q₈

—

103

—

C₄⋊Q₈

D₁₀₃

yes

characteristic, complemented

#125

—

103

—

C₂²⋊Q₈

—

103

—

C₂²⋊Q₈

—

103

—

C₄×Q₈

—

103

—

C₄×Q₈

—

103

—

#117

—

103

—

C₂×C₄²

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

C₂×D₁₀₃

yes

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×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

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

C₂×D₁₀₃

yes

characteristic

C₂³

—

103

—

Q₈

—

Q₈

—

206

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

C₂²×D₁₀₃

yes

—

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

C₂²

—

103

—

100

C₂²

—

206

—

101

C₂²

—

206

—

102

C₂²

C₂³×D₁₀₃

yes

characteristic

103

C₄

—

104

C₄

—

103

—

105

C₄

—

206

—

106

C₄

—

206

—

107

C₄

—

206

—

108

C₄

Q₈×D₁₀₃

yes

—

109

C₄

#13110104

yes

—

110

C₄

#13110106

yes

—

111

C₂

—

206

—

112

C₂

C₂×Q₈×D₁₀₃

yes

characteristic

113

C₂

C₂×#13110104

yes

characteristic

114

C₂

C₂×#13110106

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₃	C₂
hypercenter	C₄⋊Q₈	D₁₀₃
hypocenter	C₁₀₃	#533
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. #15693021	C₂	semidirect
2. #15693012	C₂	semidirect
5. Q₈×Dic₁₀₃	C₂	semidirect
6. #15692986	C₂	semidirect
9. C₄⋊Q₈×C₁₀₃	C₂	semidirect
39. C₄×D₁₀₃	Q₈	semidirect
57. C₁₀₃	#533	semidirect
60. C₄⋊Q₈	D₁₀₃	semidirect

H ≅ G/N

type

1. #15693021

C₂

semidirect

2. #15693012

C₂

semidirect

5. Q₈×Dic₁₀₃

C₂

semidirect

6. #15692986

C₂

semidirect

9. C₄⋊Q₈×C₁₀₃

C₂

semidirect

39. C₄×D₁₀₃

Q₈

semidirect

57. C₁₀₃

#533

semidirect

60. C₄⋊Q₈

D₁₀₃

semidirect

Subgroups — #20268346

Conjugacy classes of subgroups

Complements