Group #21445184 — subgroups

Subgroups — #21445184

Order 7808.

172 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840042	C₂	1	2	yes	maximal
3	#16839515	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839509	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	—	2	1	no	—
7	C₂×Dic₂₄₄	—	2	1	no	—
8	#14519777	C₂²	1	2	yes	—
9	#14519710	—	2	2	no	—
10	C₄⋊#92418	C₂²	1	2	yes	—
11	#14519698	—	2	2	no	—
12	C₂²×C₈×C₆₁	—	2	1	no	—
13	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	—	2	1	no	—
19	C₂×Dic₁₂₂	—	2	2	no	—
20	C₂×C₈×C₆₁	—	2	1	no	—
21	C₂×C₈×C₆₁	—	4	2	no	—
22	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂²⋊C₄×C₆₁	—	2	1	no	—
26	C₄²×C₆₁	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₆₁	—	2	2	no	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₂×#92418	—	2	2	no	—
30	Dic₂₄₄	—	4	2	no	—
31	Dic₂₄₄	—	4	2	no	—
32	C₂³×C₆₁	—	2	1	no	—
33	D₄×C₆₁	—	2	1	no	—
34	D₄×C₆₁	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₆₁	—	2	1	no	—
36	C₂×C₄×C₆₁	—	4	1	no	—
37	C₂×C₄×C₆₁	—	4	1	no	—
38	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
41	C₂×Dic₆₁	—	4	2	no	—
42	Dic₁₂₂	—	4	2	no	—
43	Dic₁₂₂	—	4	2	no	—
44	C₈×C₆₁	—	2	1	no	—
45	C₈×C₆₁	—	4	1	no	—
46	C₈×C₆₁	D₈	1	2	yes	—
47	#92418	—	4	2	no	—
48	C₂²×C₆₁	—	2	2	no	—
49	C₂²×C₆₁	—	4	1	no	—
50	C₂²×C₆₁	—	4	1	no	—
51	C₂²×C₆₁	#121	1	1	yes	characteristic
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	—	4	1	no	—
54	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
56	C₄×C₆₁	#135	1	1	yes	characteristic
57	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₆₁	—	8	2	no	—
59	#1669	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	—	4	2	no	—
61	C₂×C₆₁	#459	1	1	yes	characteristic
62	C₂×C₆₁	#467	1	1	yes	characteristic
63	C₂×C₆₁	#473	1	1	yes	characteristic
64	#494	—	61	1	no	—
65	#474	—	61	2	no	—
66	C₈×D₄	D₆₁	1	1	yes	characteristic
67	#335	—	61	1	no	—
68	#332	—	61	2	no	—
69	C₆₁	#1669	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₁₆	—	122	1	no	—
71	C₂×Q₁₆	—	122	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₄⋊Q₈	—	61	2	no	—
74	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
75	#107	—	122	2	no	—
76	C₄⋊C₈	—	61	2	no	—
77	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
78	#102	—	122	2	no	—
79	C₂²⋊C₈	—	2	1	no	—
80	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
81	C₂×Q₈	—	122	2	no	—
82	C₂×D₄	D₂₄₄	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	Q₁₆	—	244	2	no	—
85	Q₁₆	—	244	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	122	2	no	—
89	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
91	C₄⋊C₄	—	122	2	no	—
92	C₄⋊C₄	—	122	2	no	—
93	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
94	C₂²⋊C₄	—	2	1	no	—
95	C₄²	C₂²×D₆₁	1	1	yes	characteristic
96	C₂³	—	2	1	no	—
97	Q₈	—	244	2	no	—
98	Q₈	—	244	2	no	—
99	D₄	—	2	1	no	—
100	D₄	Dic₂₄₄	1	2	yes	complemented
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	244	2	no	—
105	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
106	C₂×C₄	#11758222	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	4	1	no	—
110	C₈	—	244	2	no	—
111	C₈	#11758199	1	2	yes	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	4	1	no	—
115	C₂²	#14519751	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	4	1	no	—
118	C₄	—	488	2	no	—
119	C₄	C₂×#11758199	1	1	yes	characteristic
120	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
121	C₄	#14519799	1	1	yes	characteristic
122	C₄	#14519802	1	1	yes	characteristic
123	C₂	—	4	2	no	—
124	C₂	#16839957	1	1	yes	characteristic
125	C₂	#16840003	1	1	yes	characteristic
126	C₂	#16840037	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840142

C₂

yes

maximal, characteristic, complemented

#16840042

C₂

yes

maximal

#16839515

C₂

yes

maximal, characteristic, complemented

#16839509

C₂

yes

maximal, complemented

C₈×D₄×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

C₂²

yes

—

#14519710

—

C₄⋊#92418

C₂²

yes

—

#14519698

—

C₂²×C₈×C₆₁

—

C₄×D₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

Q₁₆

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

D₈

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1669

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

C₈×D₄

D₆₁

yes

characteristic

#335

—

#332

—

C₆₁

#1669

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₆₁

yes

characteristic

#107

—

122

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#102

—

122

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×D₄

D₂₄₄

yes

characteristic

C₂²×C₄

—

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

D₂₄₄

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

—

Q₈

—

244

—

Q₈

—

244

—

D₄

—

100

D₄

Dic₂₄₄

yes

complemented

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

244

—

105

C₂×C₄

C₂×D₂₄₄

yes

characteristic

106

C₂×C₄

#11758222

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

244

—

111

C₈

#11758199

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#14519751

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

488

—

119

C₄

C₂×#11758199

yes

characteristic

120

C₄

C₂×Dic₂₄₄

yes

characteristic

121

C₄

#14519799

yes

characteristic

122

C₄

#14519802

yes

characteristic

123

C₂

—

124

C₂

#16839957

yes

characteristic

125

C₂

#16840003

yes

characteristic

126

C₂

#16840037

yes

characteristic

	subgroup	G/H
center	C₂²	#14519751
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈×D₄×C₆₁	C₂
socle	C₂²×C₆₁	#121
hypercenter	C₈×D₄	D₆₁
hypocenter	C₆₁	#1669
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#14519751

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈×D₄×C₆₁

C₂

socle

C₂²×C₆₁

#121

hypercenter

C₈×D₄

D₆₁

hypocenter

C₆₁

#1669

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #16840142	C₂	semidirect
3. #16839515	C₂	semidirect
4. #16839509	C₂	semidirect
34. D₄×C₆₁	Q₁₆	semidirect
69. C₆₁	#1669	semidirect
100. D₄	Dic₂₄₄	semidirect

H ≅ G/N

type

1. #16840142

C₂

semidirect

3. #16839515

C₂

semidirect

4. #16839509

C₂

semidirect

34. D₄×C₆₁

Q₁₆

semidirect

69. C₆₁

#1669

semidirect

100. D₄

Dic₂₄₄

semidirect

Subgroups — #21445184

Conjugacy classes of subgroups

Complements