Group #20268526 — subgroups

Subgroups — #20268526

Order 6592.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#15693018	C₂	1	1	yes	maximal, characteristic
2	#15693010	C₂	1	2	yes	maximal, complemented
3	#15693009	C₂	1	2	yes	maximal, complemented
4	#15693006	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₂₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₁₂	C₂	1	2	yes	maximal, complemented
7	#15692983	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₀₆	C₂²	1	1	yes	characteristic
10	C₂×D₄₁₂	C₂²	1	2	yes	—
11	C₂×D₄₁₂	C₂²	1	2	yes	—
12	C₂×D₄₁₂	C₂²	1	2	yes	—
13	C₂×C₄×D₁₀₃	C₂²	1	4	yes	—
14	C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
15	Dic₁₀₃⋊C₄	C₂²	1	2	yes	—
16	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
17	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₀₃	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
23	C₄²×C₁₀₃	C₂²	1	2	yes	—
24	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
25	C₂²×D₁₀₃	C₂³	1	4	yes	—
26	Q₈×C₁₀₃	—	2	2	no	—
27	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
30	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₀₃	C₂³	1	2	yes	—
33	C₂×Dic₁₀₃	C₂³	1	2	yes	—
34	D₄₁₂	—	2	4	no	—
35	D₄₁₂	—	2	4	no	—
36	D₄₁₂	—	4	2	no	—
37	C₄×D₁₀₃	—	4	4	no	—
38	Dic₂₀₆	—	2	2	no	—
39	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
40	C₂×D₁₀₃	—	2	4	no	—
41	C₂×D₁₀₃	—	2	4	no	—
42	C₂×D₁₀₃	—	2	4	no	—
43	C₄×C₁₀₃	—	2	2	no	—
44	C₄×C₁₀₃	—	2	2	no	—
45	C₄×C₁₀₃	—	2	1	no	—
46	C₄×C₁₀₃	#41	1	2	yes	—
47	C₄×C₁₀₃	#41	1	2	yes	—
48	Dic₁₀₃	—	2	2	no	—
49	Dic₁₀₃	—	2	2	no	—
50	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₀₃	C₂×#41	1	1	yes	characteristic
52	C₂×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
53	D₁₀₃	—	4	4	no	—
54	C₁₀₃	#559	1	1	yes	characteristic, complemented, Sylow
55	#559	—	103	1	no	maximal, Sylow
56	#127	—	103	1	no	—
57	#124	—	103	2	no	—
58	#124	—	103	2	no	—
59	#123	—	103	4	no	—
60	C₄×Q₈	—	103	1	no	—
61	C₄×Q₈	D₁₀₃	1	1	yes	characteristic, complemented
62	C₄×D₄	—	103	2	no	—
63	C₄×D₄	—	103	2	no	—
64	C₂×Q₈	—	103	1	no	—
65	C₂×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
66	C₂×D₄	—	103	2	no	—
67	C₂×D₄	—	103	2	no	—
68	C₂×D₄	—	103	2	no	—
69	C₂²×C₄	—	103	4	no	—
70	C₄⋊C₄	—	103	2	no	—
71	C₄⋊C₄	—	103	1	no	—
72	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
73	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
74	C₂²⋊C₄	—	103	4	no	—
75	C₂²⋊C₄	—	103	4	no	—
76	C₂²⋊C₄	—	103	4	no	—
77	C₄²	—	103	2	no	—
78	C₄²	C₂×D₁₀₃	1	2	yes	—
79	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
80	C₂³	—	103	4	no	—
81	Q₈	—	2	2	no	—
82	Q₈	—	206	2	no	—
83	D₄	—	412	2	no	—
84	D₄	—	206	4	no	—
85	D₄	—	206	4	no	—
86	C₂×C₄	—	103	2	no	—
87	C₂×C₄	—	103	2	no	—
88	C₂×C₄	—	412	4	no	—
89	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
90	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
91	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
92	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
94	C₂²	—	206	4	no	—
95	C₂²	—	206	4	no	—
96	C₂²	—	206	4	no	—
97	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	206	2	no	—
102	C₄	—	206	2	no	—
103	C₄	#13110105	1	2	yes	—
104	C₄	#13110106	1	2	yes	—
105	C₂	—	412	4	no	—
106	C₂	C₂×#13110105	1	1	yes	characteristic
107	C₂	C₂×#13110106	1	1	yes	characteristic
108	C₂	#15693067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693018

C₂

yes

maximal, characteristic

#15693010

C₂

yes

maximal, complemented

#15693009

C₂

yes

maximal, complemented

#15693006

C₂

yes

maximal, complemented

C₄×Dic₂₀₆

C₂

yes

maximal, characteristic, complemented

C₄×D₄₁₂

C₂

yes

maximal, complemented

#15692983

C₂

yes

maximal, complemented

C₄×Q₈×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₆

C₂²

yes

characteristic

C₂×D₄₁₂

C₂²

yes

—

C₂×D₄₁₂

C₂²

yes

—

C₂×D₄₁₂

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

characteristic

Dic₁₀₃⋊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

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#41

yes

—

C₄×C₁₀₃

#41

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

C₂×#41

yes

characteristic

C₂×C₁₀₃

C₂×#41

yes

characteristic

C₂×C₁₀₃

H(5,𝔽₂)

yes

characteristic

D₁₀₃

—

C₁₀₃

#559

yes

characteristic, complemented, Sylow

#559

—

103

maximal, Sylow

#127

—

103

—

#124

—

103

—

#124

—

103

—

#123

—

103

—

C₄×Q₈

—

103

—

C₄×Q₈

D₁₀₃

yes

characteristic, complemented

C₄×D₄

—

103

—

C₄×D₄

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂²×C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₂²⋊C₄

—

103

—

C₄²

—

103

—

C₄²

C₂×D₁₀₃

yes

—

C₄²

C₂×D₁₀₃

yes

characteristic

C₂³

—

103

—

Q₈

—

Q₈

—

206

—

D₄

—

412

—

D₄

—

206

—

D₄

—

206

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

412

—

C₂×C₄

C₂²×D₁₀₃

yes

—

C₂×C₄

C₂²×D₁₀₃

yes

—

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

C₂²

—

206

—

C₂²

—

206

—

C₂²

—

206

—

C₂²

C₂³×D₁₀₃

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

206

—

102

C₄

—

206

—

103

C₄

#13110105

yes

—

104

C₄

#13110106

yes

—

105

C₂

—

412

—

106

C₂

C₂×#13110105

yes

characteristic

107

C₂

C₂×#13110106

yes

characteristic

108

C₂

#15693067

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	#559
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. #15693010	C₂	semidirect
3. #15693009	C₂	semidirect
4. #15693006	C₂	semidirect
5. C₄×Dic₂₀₆	C₂	semidirect
6. C₄×D₄₁₂	C₂	semidirect
7. #15692983	C₂	semidirect
8. C₄×Q₈×C₁₀₃	C₂	semidirect
54. C₁₀₃	#559	semidirect
61. C₄×Q₈	D₁₀₃	semidirect

H ≅ G/N

type

2. #15693010

C₂

semidirect

3. #15693009

C₂

semidirect

4. #15693006

C₂

semidirect

5. C₄×Dic₂₀₆

C₂

semidirect

6. C₄×D₄₁₂

C₂

semidirect

7. #15692983

C₂

semidirect

8. C₄×Q₈×C₁₀₃

C₂

semidirect

54. C₁₀₃

#559

semidirect

61. C₄×Q₈

D₁₀₃

semidirect

Subgroups — #20268526

Conjugacy classes of subgroups

Complements