Group #21444249 — subgroups

Subgroups — #21444249

Order 7808.

248 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#16839786	C₂	1	1	yes	maximal, characteristic
3	#16839494	C₂	1	4	yes	maximal, complemented
4	C₆₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
6	C₂×#11758194	—	2	1	no	—
7	C₄×Dic₁₂₂	—	2	1	no	—
8	C₄×Dic₁₂₂	—	2	1	no	—
9	C₄×Dic₁₂₂	C₄	1	2	yes	—
10	C₄×Dic₁₂₂	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758196	—	2	1	no	—
13	C₂×C₄×Dic₆₁	—	2	1	no	—
14	C₄⋊#92418	—	2	1	no	—
15	C₄⋊#92418	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₆₁	—	2	1	no	—
18	C₂²⋊C₈×C₆₁	—	2	1	no	—
19	C₄×C₈×C₆₁	C₂²	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	—	2	1	no	—
24	C₂²×Dic₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	4	no	—
27	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	D₄	1	1	yes	characteristic
31	C₄²×C₆₁	D₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
35	#11758196	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	C₄×Dic₆₁	—	2	2	no	—
39	#11758194	—	4	2	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	4	1	no	—
44	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₆₁	—	2	4	no	—
53	C₂×Dic₆₁	—	2	2	no	—
54	C₂×Dic₆₁	—	2	2	no	—
55	C₂×Dic₆₁	—	4	2	no	—
56	Dic₁₂₂	—	2	4	no	—
57	Dic₁₂₂	—	2	2	no	—
58	Dic₁₂₂	M₁₆	1	4	yes	complemented
59	C₈×C₆₁	—	4	2	no	—
60	#92418	—	4	2	no	—
61	C₂²×C₆₁	—	2	1	no	—
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	#103	1	2	yes	—
64	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	#103	1	2	yes	—
69	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
70	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₆₁	—	2	4	no	—
73	Dic₆₁	—	4	2	no	—
74	#1487	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	—	2	2	no	—
76	C₂×C₆₁	#407	1	1	yes	characteristic
77	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
78	C₂×C₆₁	#421	1	1	yes	characteristic
79	C₂×C₄×Q₈	—	61	1	no	—
80	#431	D₆₁	1	1	yes	characteristic
81	#423	—	61	1	no	—
82	Q₈⋊C₈	—	61	4	no	—
83	C₆₁	#1487	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	61	1	no	—
85	C₂×M₁₆	—	122	1	no	—
86	C₄×Q₈	—	61	2	no	—
87	C₄×Q₈	—	61	2	no	—
88	C₄×Q₈	—	122	1	no	—
89	C₄×Q₈	—	122	1	no	—
90	C₂×C₄⋊C₄	—	61	1	no	—
91	C₂×C₄⋊C₄	—	122	1	no	—
92	C₂×C₄²	—	122	1	no	—
93	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	61	2	no	—
96	C₄⋊C₈	—	122	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	C₄×C₈	C₂×D₆₁	1	2	yes	—
99	C₂×Q₈	—	61	2	no	—
100	C₂×Q₈	—	122	4	no	—
101	C₂×Q₈	—	122	1	no	—
102	C₂²×C₄	—	122	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
105	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
107	M₁₆	—	244	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	122	2	no	—
110	C₄⋊C₄	—	61	2	no	—
111	C₄⋊C₄	—	122	1	no	—
112	C₄⋊C₄	—	122	2	no	—
113	C₄⋊C₄	—	122	2	no	—
114	C₄²	—	122	2	no	—
115	C₄²	—	122	2	no	—
116	C₄²	C₄×D₆₁	1	1	yes	characteristic
117	C₄²	D₂₄₄	1	1	yes	characteristic
118	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
119	C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	C₂³	#11758198	1	1	yes	characteristic
121	Q₈	—	61	4	no	—
122	Q₈	—	122	4	no	—
123	Q₈	—	122	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	122	4	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	122	2	no	—
129	C₂×C₄	—	244	2	no	—
130	C₂×C₄	#11758190	1	2	yes	—
131	C₂×C₄	Dic₂₄₄	1	2	yes	—
132	C₂×C₄	#11758198	1	1	yes	characteristic
133	C₂×C₄	#11758198	1	1	yes	characteristic
134	C₂×C₄	#11758198	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	244	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#14519702	1	2	yes	—
143	C₂²	C₂×#11758198	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	122	4	no	—
148	C₄	—	244	2	no	—
149	C₄	#14519702	1	2	yes	—
150	C₄	M₁₆×D₆₁	1	2	yes	—
151	C₄	C₂×#11758190	1	1	yes	characteristic
152	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	#16839725	1	1	yes	characteristic
155	C₂	C₂×#14519702	1	1	yes	characteristic
156	C₂	#16839780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

#16839786

C₂

yes

maximal, characteristic

#16839494

C₂

yes

maximal, complemented

C₆₁×#431

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₄

yes

characteristic

C₂×#11758194

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₄

yes

characteristic

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

M₁₆

yes

complemented

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#103

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#1487

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

—

#431

D₆₁

yes

characteristic

#423

—

Q₈⋊C₈

—

C₆₁

#1487

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

122

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

122

—

101

C₂×Q₈

—

122

—

102

C₂²×C₄

—

122

—

103

C₂²×C₄

—

122

—

104

C₂²×C₄

C₄×D₆₁

yes

characteristic

105

C₂²×C₄

D₂₄₄

yes

characteristic

106

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

107

M₁₆

—

244

—

108

C₂×C₈

—

109

C₂×C₈

—

122

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

122

—

113

C₄⋊C₄

—

122

—

114

C₄²

—

122

—

115

C₄²

—

122

—

116

C₄²

C₄×D₆₁

yes

characteristic

117

C₄²

D₂₄₄

yes

characteristic

118

C₄²

C₂²⋊D₆₁

yes

characteristic

119

C₄²

C₂²×D₆₁

yes

characteristic

120

C₂³

#11758198

yes

characteristic

121

Q₈

—

122

Q₈

—

122

—

123

Q₈

—

122

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

122

—

129

C₂×C₄

—

244

—

130

C₂×C₄

#11758190

yes

—

131

C₂×C₄

Dic₂₄₄

yes

—

132

C₂×C₄

#11758198

yes

characteristic

133

C₂×C₄

#11758198

yes

characteristic

134

C₂×C₄

#11758198

yes

characteristic

135

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

136

C₂×C₄

C₂×D₂₄₄

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

C₈

—

139

C₈

—

244

—

140

C₂²

—

141

C₂²

—

142

C₂²

#14519702

yes

—

143

C₂²

C₂×#11758198

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

122

—

148

C₄

—

244

—

149

C₄

#14519702

yes

—

150

C₄

M₁₆×D₆₁

yes

—

151

C₄

C₂×#11758190

yes

characteristic

152

C₄

C₂×Dic₂₄₄

yes

characteristic

153

C₂

—

154

C₂

#16839725

yes

characteristic

155

C₂

C₂×#14519702

yes

characteristic

156

C₂

#16839780

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#431	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#431	D₆₁
hypocenter	C₆₁	#1487
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#431

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16839494	C₂	semidirect
58. Dic₁₂₂	M₁₆	semidirect
83. C₆₁	#1487	semidirect

H ≅ G/N

type

3. #16839494

C₂

semidirect

58. Dic₁₂₂

M₁₆

semidirect

83. C₆₁

#1487

semidirect

Subgroups — #21444249

Conjugacy classes of subgroups

Complements