Group #21447239 — subgroups

Subgroups — #21447239

Order 7808.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#14519702	C₂	1	2	yes	maximal
2	#16839668	C₂	1	1	yes	maximal, characteristic
3	#16839628	C₂	1	1	yes	maximal, characteristic
4	#16839525	C₂	1	2	yes	maximal
5	C₆₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
9	#14519702	—	2	4	no	—
10	#14519684	—	2	2	no	—
11	#14519684	—	2	2	no	—
12	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	D₄	1	2	yes	—
15	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
16	C₂²×Dic₆₁	—	2	1	no	—
17	C₂²×Dic₆₁	—	2	2	no	—
18	C₂²×Dic₆₁	—	2	1	no	—
19	C₂×Dic₁₂₂	—	2	2	no	—
20	C₂×Dic₁₂₂	—	2	2	no	—
21	C₂×Dic₁₂₂	—	2	1	no	—
22	C₂×Dic₁₂₂	D₄	1	2	yes	—
23	C₂×C₈×C₆₁	—	2	2	no	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₂×C₈×C₆₁	—	2	2	no	—
26	C₄²×C₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	—	2	2	no	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₄⋊Dic₆₁	—	2	2	no	—
30	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	—	2	2	no	—
34	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	#41	1	2	yes	—
36	C₂×C₄×C₆₁	#41	1	2	yes	—
37	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
38	C₂×Dic₆₁	—	2	2	no	—
39	C₂×Dic₆₁	—	4	1	no	—
40	C₂×Dic₆₁	—	4	1	no	—
41	C₂×Dic₆₁	—	4	2	no	—
42	C₂×Dic₆₁	—	4	1	no	—
43	C₂×Dic₆₁	—	4	2	no	—
44	C₂×Dic₆₁	—	4	2	no	—
45	C₂×Dic₆₁	—	4	2	no	—
46	Dic₁₂₂	—	2	4	no	—
47	Dic₁₂₂	—	4	2	no	—
48	C₈×C₆₁	—	4	2	no	—
49	C₂²×C₆₁	#121	1	1	yes	characteristic
50	C₂²×C₆₁	#123	1	1	yes	characteristic
51	C₂²×C₆₁	#124	1	1	yes	characteristic
52	C₂²×C₆₁	#135	1	2	yes	—
53	C₂²×C₆₁	#137	1	2	yes	—
54	C₄×C₆₁	—	4	2	no	—
55	C₄×C₆₁	#123	1	2	yes	—
56	C₄×C₆₁	#126	1	2	yes	—
57	Dic₆₁	—	4	2	no	—
58	Dic₆₁	—	8	2	no	—
59	Dic₆₁	—	8	1	no	—
60	#2074	—	61	1	no	maximal, Sylow
61	C₂×C₆₁	#397	1	1	yes	characteristic
62	C₂×C₆₁	#464	1	2	yes	—
63	C₂×C₆₁	#485	1	2	yes	—
64	C₂×C₆₁	#488	1	1	yes	characteristic
65	C₂×C₆₁	#491	1	1	yes	characteristic
66	C₂×#103	—	61	2	no	—
67	#391	—	61	1	no	—
68	#382	—	61	1	no	—
69	#340	—	61	2	no	—
70	#336	D₆₁	1	1	yes	characteristic
71	C₆₁	#2074	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	61	1	no	—
73	C₂²×C₈	C₂×D₆₁	1	2	yes	—
74	C₂×C₄⋊C₄	—	61	1	no	—
75	C₂×C₄⋊C₄	—	61	2	no	—
76	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
77	#103	—	122	4	no	—
78	#95	—	122	2	no	—
79	#95	—	122	2	no	—
80	C₂×Q₈	—	61	2	no	—
81	C₂×Q₈	—	122	1	no	—
82	C₂×Q₈	—	122	2	no	—
83	C₂×Q₈	—	122	2	no	—
84	C₂²×C₄	—	122	1	no	—
85	C₂²×C₄	—	122	2	no	—
86	C₂²×C₄	—	122	1	no	—
87	C₂²×C₄	D₂₄₄	1	2	yes	—
88	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	122	2	no	—
93	C₄⋊C₄	—	122	2	no	—
94	C₄⋊C₄	—	122	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
97	Q₈	—	122	4	no	—
98	Q₈	—	244	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	122	2	no	—
103	C₂×C₄	—	244	2	no	—
104	C₂×C₄	—	244	2	no	—
105	C₂×C₄	—	244	1	no	—
106	C₂×C₄	—	244	2	no	—
107	C₂×C₄	—	244	2	no	—
108	C₂×C₄	—	244	1	no	—
109	C₂×C₄	—	244	1	no	—
110	C₂×C₄	#11758222	1	2	yes	—
111	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
112	C₂×C₄	#11758224	1	2	yes	—
113	C₂×C₄	#11758226	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#14519752	1	1	yes	characteristic
116	C₂²	#14519761	1	1	yes	characteristic
117	C₂²	#14519765	1	1	yes	characteristic
118	C₂²	#14519799	1	2	yes	—
119	C₂²	#14519810	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	488	2	no	—
122	C₄	—	488	1	no	—
123	C₄	—	244	2	no	—
124	C₄	#14519759	1	2	yes	—
125	C₄	#14519772	1	2	yes	—
126	C₂	#16839689	1	1	yes	characteristic
127	C₂	#16839985	1	2	yes	—
128	C₂	#16840104	1	2	yes	—
129	C₂	#16840117	1	1	yes	characteristic
130	C₂	#16840131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519702

C₂

yes

maximal

#16839668

C₂

yes

maximal, characteristic

#16839628

C₂

yes

maximal, characteristic

#16839525

C₂

yes

maximal

C₆₁×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

—

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#137

yes

—

C₄×C₆₁

—

C₄×C₆₁

#123

yes

—

C₄×C₆₁

#126

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#485

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#2074

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

122

—

#95

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₂³

C₂×D₂₄₄

yes

characteristic

Q₈

—

122

—

Q₈

—

244

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

122

—

103

C₂×C₄

—

244

—

104

C₂×C₄

—

244

—

105

C₂×C₄

—

244

—

106

C₂×C₄

—

244

—

107

C₂×C₄

—

244

—

108

C₂×C₄

—

244

—

109

C₂×C₄

—

244

—

110

C₂×C₄

#11758222

yes

—

111

C₂×C₄

D₄×D₆₁

yes

characteristic

112

C₂×C₄

#11758224

yes

—

113

C₂×C₄

#11758226

yes

characteristic

114

C₈

—

115

C₂²

#14519752

yes

characteristic

116

C₂²

#14519761

yes

characteristic

117

C₂²

#14519765

yes

characteristic

118

C₂²

#14519799

yes

—

119

C₂²

#14519810

yes

—

120

C₄

—

121

C₄

—

488

—

122

C₄

—

488

—

123

C₄

—

244

—

124

C₄

#14519759

yes

—

125

C₄

#14519772

yes

—

126

C₂

#16839689

yes

characteristic

127

C₂

#16839985

yes

—

128

C₂

#16840104

yes

—

129

C₂

#16840117

yes

characteristic

130

C₂

#16840131

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#336	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	#336	D₆₁
hypocenter	C₆₁	#2074
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#336

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
71. C₆₁	#2074	semidirect

H ≅ G/N

type

71. C₆₁

#2074

semidirect

Subgroups — #21447239

Conjugacy classes of subgroups

Complements