Group #21445202 — subgroups

Subgroups — #21445202

Order 7808.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840160	C₂	1	1	yes	maximal, characteristic
2	#16840058	C₂	1	1	yes	maximal, characteristic
3	#16839973	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
5	#16839510	C₂	1	1	yes	maximal, characteristic
6	#16839509	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#334	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₄₄	—	2	1	no	—
9	#14519777	C₂²	1	1	yes	characteristic
10	#14519777	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
12	#14519711	—	2	1	no	—
13	#14519709	—	2	1	no	—
14	#14519709	—	2	1	no	—
15	C₄⋊#92418	C₂²	1	1	yes	characteristic
16	#14519706	—	2	1	no	—
17	#14519702	—	2	1	no	—
18	#14519702	—	2	1	no	—
19	C₈×Dic₆₁	—	2	1	no	—
20	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₂₂	—	2	1	no	—
24	C₂×Dic₁₂₂	—	2	1	no	—
25	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₆₁	—	2	1	no	—
32	C₄⋊Dic₆₁	—	2	1	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
36	#11758196	—	2	1	no	—
37	C₄×Dic₆₁	—	2	1	no	—
38	C₂×#92418	—	2	1	no	—
39	Dic₂₄₄	—	4	2	no	—
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂×Dic₆₁	—	2	1	no	—
44	C₂×Dic₆₁	—	2	1	no	—
45	C₂×Dic₆₁	—	4	1	no	—
46	C₂×Dic₆₁	—	4	1	no	—
47	Dic₁₂₂	—	2	1	no	—
48	Dic₁₂₂	—	4	1	no	—
49	Dic₁₂₂	—	4	1	no	—
50	Dic₁₂₂	—	8	1	no	—
51	Dic₁₂₂	SD₁₆	1	2	yes	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	—	4	1	no	—
54	C₈×C₆₁	—	4	1	no	—
55	C₈×C₆₁	Q₁₆	1	2	yes	complemented
56	#92418	—	4	1	no	—
57	C₂²×C₆₁	#121	1	1	yes	characteristic
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₆₁	#135	1	1	yes	characteristic
62	C₄×C₆₁	#137	1	1	yes	characteristic
63	Dic₆₁	—	2	2	no	—
64	Dic₆₁	—	4	1	no	—
65	Dic₆₁	—	8	1	no	—
66	Dic₆₁	—	8	1	no	—
67	#1672	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	#462	1	1	yes	characteristic
69	C₂×C₆₁	#465	1	1	yes	characteristic
70	C₂×C₆₁	#471	1	1	yes	characteristic
71	#498	—	61	1	no	—
72	#477	—	61	1	no	—
73	#462	—	61	1	no	—
74	C₈×Q₈	—	61	1	no	—
75	#334	D₆₁	1	1	yes	characteristic
76	#333	—	61	1	no	—
77	#332	—	61	1	no	—
78	C₆₁	#1672	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	122	1	no	—
80	C₄⋊Q₈	—	61	1	no	—
81	C₄⋊Q₈	—	61	1	no	—
82	C₄×Q₈	—	61	1	no	—
83	#107	—	122	1	no	—
84	#106	—	122	1	no	—
85	#106	—	122	1	no	—
86	C₄⋊C₈	—	61	1	no	—
87	C₄⋊C₈	—	122	1	no	—
88	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
90	#103	—	122	1	no	—
91	#103	—	122	1	no	—
92	C₄×C₈	—	122	1	no	—
93	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₂×Q₈	—	61	1	no	—
95	C₂×Q₈	—	122	1	no	—
96	C₂×Q₈	—	122	1	no	—
97	Q₁₆	—	244	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	122	1	no	—
101	C₂×C₈	D₂₄₄	1	1	yes	characteristic
102	C₂×C₈	D₂₄₄	1	1	yes	characteristic
103	C₄⋊C₄	—	61	1	no	—
104	C₄⋊C₄	—	122	1	no	—
105	C₄⋊C₄	—	122	1	no	—
106	C₄⋊C₄	—	122	1	no	—
107	C₄⋊C₄	—	122	1	no	—
108	C₄⋊C₄	—	122	1	no	—
109	C₄²	—	122	1	no	—
110	C₄²	C₂²×D₆₁	1	1	yes	characteristic
111	Q₈	—	61	2	no	—
112	Q₈	—	122	1	no	—
113	Q₈	—	488	1	no	—
114	Q₈	—	244	1	no	—
115	Q₈	—	244	1	no	—
116	C₂×C₄	—	122	1	no	—
117	C₂×C₄	—	122	1	no	—
118	C₂×C₄	—	244	1	no	—
119	C₂×C₄	—	244	1	no	—
120	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
121	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
122	C₂×C₄	#11758226	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	244	1	no	—
127	C₈	Dic₂₄₄	1	2	yes	complemented
128	C₂²	#14519752	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	122	2	no	—
131	C₄	—	488	1	no	—
132	C₄	—	488	1	no	—
133	C₄	—	244	1	no	—
134	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
135	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
136	C₄	#14519798	1	1	yes	characteristic
137	C₄	#14519810	1	1	yes	characteristic
138	C₂	#16839973	1	1	yes	characteristic
139	C₂	#16839992	1	1	yes	characteristic
140	C₂	#16840026	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840160

C₂

yes

maximal, characteristic

#16840058

C₂

yes

maximal, characteristic

#16839973

C₂

yes

maximal, characteristic

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic

#16839510

C₂

yes

maximal, characteristic

#16839509

C₂

yes

maximal, characteristic

C₆₁×#334

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

—

#14519777

C₂²

yes

characteristic

#14519777

C₂²

yes

characteristic

C₄×Dic₁₂₂

C₂²

yes

—

—

—

C₂²

yes

—

—

—

C₈×Dic₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

SD₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

Q₁₆

yes

complemented

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

C₈×Q₈

—

#334

D₆₁

yes

characteristic

#333

—

#332

—

C₆₁

#1672

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

122

—

#106

—

122

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

#103

—

122

—

C₄×C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

122

—

101

C₂×C₈

D₂₄₄

yes

characteristic

102

C₂×C₈

D₂₄₄

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

—

122

—

106

C₄⋊C₄

—

122

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄²

—

122

—

110

C₄²

C₂²×D₆₁

yes

characteristic

111

Q₈

—

112

Q₈

—

122

—

113

Q₈

—

488

—

114

Q₈

—

244

—

115

Q₈

—

244

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

122

—

118

C₂×C₄

—

244

—

119

C₂×C₄

—

244

—

120

C₂×C₄

C₂×D₂₄₄

yes

characteristic

121

C₂×C₄

D₄×D₆₁

yes

characteristic

122

C₂×C₄

#11758226

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

244

—

127

C₈

Dic₂₄₄

yes

complemented

128

C₂²

#14519752

yes

characteristic

129

C₄

—

130

C₄

—

122

—

131

C₄

—

488

—

132

C₄

—

488

—

133

C₄

—

244

—

134

C₄

SD₁₆×D₆₁

yes

characteristic

135

C₄

C₂×Dic₂₄₄

yes

characteristic

136

C₄

#14519798

yes

characteristic

137

C₄

#14519810

yes

characteristic

138

C₂

#16839973

yes

characteristic

139

C₂

#16839992

yes

characteristic

140

C₂

#16840026

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#334	C₂
socle	C₂²×C₆₁	#121
hypercenter	#334	D₆₁
hypocenter	C₆₁	#1672
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14519752

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#334

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
55. C₈×C₆₁	Q₁₆	semidirect
78. C₆₁	#1672	semidirect
127. C₈	Dic₂₄₄	semidirect

H ≅ G/N

type

55. C₈×C₆₁

Q₁₆

semidirect

78. C₆₁

#1672

semidirect

127. C₈

Dic₂₄₄

semidirect

Subgroups — #21445202

Conjugacy classes of subgroups

Complements