Group #21447312 — subgroups

Subgroups — #21447312

Order 7808.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14519711	C₂	1	1	yes	maximal, characteristic
2	#16839660	C₂	1	2	yes	maximal
3	#16839525	C₂	1	2	yes	maximal
4	#16839525	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
8	C₂×#11758196	—	2	2	no	—
9	#14519711	—	2	2	no	—
10	#14519684	—	2	2	no	—
11	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	—	2	2	no	—
15	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	2	no	—
19	C₂²×Dic₆₁	—	2	2	no	—
20	C₂×C₈×C₆₁	—	2	2	no	—
21	C₂×C₈×C₆₁	—	2	1	no	—
22	C₂×C₈×C₆₁	—	2	2	no	—
23	C₂×C₈×C₆₁	D₄	1	2	yes	—
24	C₄²×C₆₁	—	2	1	no	—
25	C₄²×C₆₁	—	2	1	no	—
26	C₄⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	—	2	2	no	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₄⋊Dic₆₁	Q₈	1	4	yes	—
30	#11758196	—	4	4	no	—
31	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	—	2	1	no	—
33	C₂×C₄×C₆₁	—	2	1	no	—
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
37	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₆₁	#41	1	2	yes	—
40	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
41	C₂×Dic₆₁	—	2	4	no	—
42	C₂×Dic₆₁	—	4	2	no	—
43	C₂×Dic₆₁	—	4	2	no	—
44	C₂×Dic₆₁	—	4	2	no	—
45	C₂×Dic₆₁	—	4	2	no	—
46	C₂×Dic₆₁	—	4	2	no	—
47	C₈×C₆₁	—	2	2	no	—
48	C₈×C₆₁	—	4	1	no	—
49	C₂²×C₆₁	#121	1	1	yes	characteristic
50	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₆₁	#135	1	1	yes	characteristic
53	C₂²×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	—	4	1	no	—
56	C₄×C₆₁	#123	1	1	yes	characteristic
57	C₄×C₆₁	#125	1	2	yes	—
58	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₆₁	—	4	4	no	—
60	Dic₆₁	—	8	2	no	—
61	#2087	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	#398	1	1	yes	characteristic
63	C₂×C₆₁	#467	1	1	yes	characteristic
64	C₂×C₆₁	#469	1	1	yes	characteristic
65	C₂×C₆₁	#477	1	2	yes	—
66	C₂×C₆₁	#478	1	2	yes	—
67	C₂×#107	—	61	1	no	—
68	C₂×C₄⋊C₈	D₆₁	1	1	yes	characteristic
69	#389	—	61	2	no	—
70	#340	—	61	1	no	—
71	#340	—	61	2	no	—
72	C₆₁	#2087	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	61	2	no	—
76	C₂×C₄⋊C₄	—	61	2	no	—
77	C₂×C₄⋊C₄	—	122	2	no	—
78	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
79	#107	—	122	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	122	2	no	—
82	C₂²×C₄	—	122	2	no	—
83	C₂²×C₄	—	122	2	no	—
84	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
85	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
91	C₄⋊C₄	—	61	4	no	—
92	C₄⋊C₄	—	122	2	no	—
93	C₄⋊C₄	—	122	2	no	—
94	C₄⋊C₄	—	122	2	no	—
95	C₄⋊C₄	—	244	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	122	4	no	—
104	C₂×C₄	—	244	2	no	—
105	C₂×C₄	—	244	2	no	—
106	C₂×C₄	—	244	2	no	—
107	C₂×C₄	—	244	2	no	—
108	C₂×C₄	—	244	2	no	—
109	C₂×C₄	Dic₂₄₄	1	2	yes	—
110	C₂×C₄	#11758222	1	1	yes	characteristic
111	C₂×C₄	#11758224	1	2	yes	—
112	C₂×C₄	Q₈×D₆₁	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#14519751	1	1	yes	characteristic
117	C₂²	#14519757	1	2	yes	—
118	C₂²	C₂×Dic₂₄₄	1	1	yes	characteristic
119	C₂²	#14519799	1	1	yes	characteristic
120	C₂²	#14519805	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	488	2	no	—
124	C₄	—	244	4	no	—
125	C₄	#14519758	1	1	yes	characteristic
126	C₄	#14519767	1	2	yes	—
127	C₄	#14519775	1	1	yes	characteristic
128	C₂	#16839692	1	1	yes	characteristic
129	C₂	#16840003	1	1	yes	characteristic
130	C₂	#16840015	1	1	yes	characteristic
131	C₂	#16840058	1	2	yes	—
132	C₂	#16840063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519711

C₂

yes

maximal, characteristic

#16839660

C₂

yes

maximal

#16839525

C₂

yes

maximal

#16839525

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

#14519711

—

#14519684

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

Q₈

yes

—

#11758196

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#123

yes

characteristic

C₄×C₆₁

#125

yes

—

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#478

yes

—

C₂×#107

—

C₂×C₄⋊C₈

D₆₁

yes

—

—

—

C₆₁

#2087

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

122

—

C₄⋊C₈

—

#95

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄²

—

C₄²

—

C₂³

C₂×D₂₄₄

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

122

—

104

C₂×C₄

—

244

—

105

C₂×C₄

—

244

—

106

C₂×C₄

—

244

—

107

C₂×C₄

—

244

—

108

C₂×C₄

—

244

—

109

C₂×C₄

Dic₂₄₄

yes

—

110

C₂×C₄

#11758222

yes

characteristic

111

C₂×C₄

#11758224

yes

—

112

C₂×C₄

Q₈×D₆₁

yes

—

113

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#14519751

yes

characteristic

117

C₂²

#14519757

yes

—

118

C₂²

C₂×Dic₂₄₄

yes

characteristic

119

C₂²

#14519799

yes

characteristic

120

C₂²

#14519805

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

488

—

124

C₄

—

244

—

125

C₄

#14519758

yes

characteristic

126

C₄

#14519767

yes

—

127

C₄

#14519775

yes

characteristic

128

C₂

#16839692

yes

characteristic

129

C₂

#16840003

yes

characteristic

130

C₂

#16840015

yes

characteristic

131

C₂

#16840058

yes

—

132

C₂

#16840063

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄⋊C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#2087
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄⋊C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₆₁

hypocenter

C₆₁

#2087

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
72. C₆₁	#2087	semidirect

H ≅ G/N

type

72. C₆₁

#2087

semidirect

Subgroups — #21447312

Conjugacy classes of subgroups

Complements