Group #20265892 — subgroups

Subgroups — #20265892

Order 6464.

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	#15692255	C₂	1	1	yes	maximal, characteristic, complemented
2	#15692241	C₂	1	4	yes	maximal, complemented
3	#15692229	C₂	1	2	yes	maximal, complemented
4	#15692226	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692214	C₂	1	2	yes	maximal, complemented
7	#15692213	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₀₂	C₂²	1	2	yes	—
10	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
12	C₂×C₄×D₁₀₁	C₂²	1	2	yes	—
13	C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
14	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
17	#13108051	C₂²	1	4	yes	—
18	#13108051	C₂²	1	2	yes	—
19	C₄×Dic₁₀₁	C₂²	1	4	yes	—
20	C₂×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
24	C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂³×C₁₀₁	C₂³	1	2	yes	—
26	C₂²×D₁₀₁	C₂³	1	2	yes	—
27	D₄×C₁₀₁	—	2	2	no	—
28	C₂×C₄×C₁₀₁	—	2	4	no	—
29	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	C₂³	1	2	yes	—
31	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₀₁	—	4	4	no	—
34	C₂×Dic₁₀₁	C₂³	1	2	yes	—
35	C₂×Dic₁₀₁	C₂³	1	4	yes	—
36	D₄₀₄	—	2	2	no	—
37	C₄×D₁₀₁	—	2	4	no	—
38	Dic₂₀₂	—	4	2	no	—
39	C₂²×C₁₀₁	—	2	2	no	—
40	C₂²×C₁₀₁	—	2	2	no	—
41	C₂²×C₁₀₁	—	2	2	no	—
42	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₀₁	—	2	2	no	—
44	C₂×D₁₀₁	—	2	2	no	—
45	C₂×D₁₀₁	—	2	2	no	—
46	C₄×C₁₀₁	—	2	2	no	—
47	C₄×C₁₀₁	—	2	1	no	—
48	C₄×C₁₀₁	#41	1	2	yes	—
49	C₄×C₁₀₁	#41	1	2	yes	—
50	Dic₁₀₁	—	2	2	no	—
51	Dic₁₀₁	—	2	4	no	—
52	C₂×C₁₀₁	—	4	2	no	—
53	C₂×C₁₀₁	C₂×#41	1	1	yes	characteristic
54	C₂×C₁₀₁	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
56	D₁₀₁	—	4	2	no	—
57	C₁₀₁	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	101	1	no	maximal, Sylow
59	C₄⋊Q₈	—	101	1	no	—
60	#124	—	101	4	no	—
61	#121	—	101	2	no	—
62	#121	—	101	2	no	—
63	C₄×D₄	—	101	1	no	—
64	C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
65	#117	—	101	2	no	—
66	#117	—	101	2	no	—
67	C₂×Q₈	—	101	2	no	—
68	C₂×D₄	—	101	4	no	—
69	C₂×D₄	—	101	1	no	—
70	C₂×D₄	C₂×D₁₀₁	1	1	yes	characteristic
71	C₂²×C₄	—	101	2	no	—
72	C₂²×C₄	C₂×D₁₀₁	1	2	yes	—
73	C₄⋊C₄	—	101	1	no	—
74	C₄⋊C₄	—	101	2	no	—
75	C₄⋊C₄	—	101	2	no	—
76	C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
77	C₂²⋊C₄	—	101	4	no	—
78	C₂²⋊C₄	—	101	4	no	—
79	C₂²⋊C₄	—	101	2	no	—
80	C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
81	C₄²	—	101	4	no	—
82	C₄²	C₂×D₁₀₁	1	1	yes	characteristic
83	C₂³	—	101	2	no	—
84	C₂³	C₂²×D₁₀₁	1	2	yes	—
85	Q₈	—	404	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	404	4	no	—
88	D₄	—	202	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	101	2	no	—
91	C₂×C₄	—	101	4	no	—
92	C₂×C₄	—	202	4	no	—
93	C₂×C₄	C₂²×D₁₀₁	1	2	yes	—
94	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	202	2	no	—
101	C₂²	—	202	2	no	—
102	C₂²	—	202	2	no	—
103	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	202	2	no	—
107	C₄	—	202	4	no	—
108	C₄	#13108073	1	2	yes	—
109	C₄	#13108074	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	404	2	no	—
112	C₂	C₂×#13108073	1	1	yes	characteristic
113	C₂	C₂×#13108074	1	1	yes	characteristic
114	C₂	#15692301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15692255

C₂

yes

maximal, characteristic, complemented

#15692241

C₂

yes

maximal, complemented

#15692229

C₂

yes

maximal, complemented

#15692226

C₂

yes

maximal, complemented

C₄×D₄₀₄

C₂

yes

maximal, characteristic, complemented

#15692214

C₂

yes

maximal, complemented

#15692213

C₂

yes

maximal, complemented

C₄×D₄×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₂

C₂²

yes

—

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

—

C₄⋊Dic₁₀₁

C₂²

yes

—

C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₄⋊Dic₁₀₁

C₂²

yes

—

C₂²⋊Dic₁₀₁

C₂²

yes

—

#13108051

C₂²

yes

—

#13108051

C₂²

yes

—

C₄×Dic₁₀₁

C₂²

yes

—

C₂×D₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₄×C₁₀₁

C₂²

yes

—

C₄⋊C₄×C₁₀₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₁

C₂²

yes

—

C₄²×C₁₀₁

C₂²

yes

characteristic

C₂³×C₁₀₁

C₂³

yes

—

C₂²×D₁₀₁

C₂³

yes

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

D₄₀₄

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#41

yes

—

C₄×C₁₀₁

#41

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

H(5,𝔽₂)

yes

characteristic

D₁₀₁

—

C₁₀₁

#555

yes

characteristic, complemented, Sylow

#555

—

101

maximal, Sylow

C₄⋊Q₈

—

101

—

#124

—

101

—

#121

—

101

—

#121

—

101

—

C₄×D₄

—

101

—

C₄×D₄

D₁₀₁

yes

characteristic, complemented

#117

—

101

—

#117

—

101

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂×D₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₄

—

101

—

C₂²×C₄

C₂×D₁₀₁

yes

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

C₂×D₁₀₁

yes

characteristic

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

C₂×D₁₀₁

yes

—

C₄²

—

101

—

C₄²

C₂×D₁₀₁

yes

characteristic

C₂³

—

101

—

C₂³

C₂²×D₁₀₁

yes

—

Q₈

—

404

—

D₄

—

D₄

—

404

—

D₄

—

202

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

202

—

101

C₂²

—

202

—

102

C₂²

—

202

—

103

C₂²

C₂³×D₁₀₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

202

—

107

C₄

—

202

—

108

C₄

#13108073

yes

—

109

C₄

#13108074

yes

—

110

C₂

—

111

C₂

—

404

—

112

C₂

C₂×#13108073

yes

characteristic

113

C₂

C₂×#13108074

yes

characteristic

114

C₂

#15692301

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂⁴
Fitting	C₄×D₄×C₁₀₁	C₂
hypercenter	C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#555
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂⁴

Fitting

C₄×D₄×C₁₀₁

C₂

hypercenter

C₄×D₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #15692255	C₂	semidirect
2. #15692241	C₂	semidirect
3. #15692229	C₂	semidirect
4. #15692226	C₂	semidirect
5. C₄×D₄₀₄	C₂	semidirect
6. #15692214	C₂	semidirect
7. #15692213	C₂	semidirect
8. C₄×D₄×C₁₀₁	C₂	semidirect
57. C₁₀₁	#555	semidirect
64. C₄×D₄	D₁₀₁	semidirect

H ≅ G/N

type

1. #15692255

C₂

semidirect

2. #15692241

C₂

semidirect

3. #15692229

C₂

semidirect

4. #15692226

C₂

semidirect

5. C₄×D₄₀₄

C₂

semidirect

6. #15692214

C₂

semidirect

7. #15692213

C₂

semidirect

8. C₄×D₄×C₁₀₁

C₂

semidirect

57. C₁₀₁

#555

semidirect

64. C₄×D₄

D₁₀₁

semidirect

Subgroups — #20265892

Conjugacy classes of subgroups

Complements