Group #21445237 — subgroups

Subgroups — #21445237

Order 7808.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16840110	C₂	1	1	yes	maximal, characteristic
2	#16839962	C₂	1	1	yes	maximal, characteristic
3	Q₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839511	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839505	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758201	—	2	1	no	—
9	#14519774	C₂²	1	1	yes	characteristic
10	C₄⋊#92418	—	2	1	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	#14519697	—	2	1	no	—
14	#14519697	—	2	1	no	—
15	#14519697	—	2	1	no	—
16	C₄×#92418	—	2	1	no	—
17	C₄×#92418	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	#107×C₆₁	—	2	1	no	—
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	#103×C₆₁	—	2	1	no	—
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×D₂₄₄	—	2	1	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₄²×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#11758201	—	4	2	no	—
36	C₂×#92418	—	2	1	no	—
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	D₄	1	1	yes	characteristic
39	C₂×#92418	D₄	1	1	yes	characteristic
40	C₂²×D₆₁	—	4	1	no	—
41	Q₈×C₆₁	—	2	1	no	—
42	Q₈×C₆₁	—	8	1	no	—
43	Q₈×C₆₁	SD₁₆	1	2	yes	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	—	4	1	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
50	D₂₄₄	—	4	1	no	—
51	D₂₄₄	—	4	1	no	—
52	D₂₄₄	—	4	1	no	—
53	D₂₄₄	—	4	1	no	—
54	D₂₄₄	—	8	1	no	—
55	C₈×C₆₁	—	4	1	no	—
56	#92418	—	2	1	no	—
57	#92418	—	2	1	no	—
58	#92418	—	4	1	no	—
59	#92418	—	4	1	no	—
60	C₂²×C₆₁	#121	1	1	yes	characteristic
61	C₂×D₆₁	—	8	1	no	—
62	C₂×D₆₁	—	8	1	no	—
63	C₂×D₆₁	—	8	1	no	—
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	—	8	1	no	—
68	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₆₁	#135	1	1	yes	characteristic
70	C₄×C₆₁	#135	1	1	yes	characteristic
71	C₄×C₆₁	#137	1	1	yes	characteristic
72	#1678	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	#464	1	1	yes	characteristic
74	C₂×C₆₁	#465	1	1	yes	characteristic
75	C₂×C₆₁	#472	1	1	yes	characteristic
76	D₆₁	—	16	1	no	—
77	#486	—	61	1	no	—
78	#477	D₆₁	1	1	yes	characteristic, complemented
79	#460	—	61	1	no	—
80	C₈×Q₈	—	61	1	no	—
81	#334	—	61	1	no	—
82	#333	—	61	1	no	—
83	#331	—	61	1	no	—
84	C₆₁	#1678	1	1	yes	characteristic, complemented, Sylow
85	C₂×SD₁₆	—	122	1	no	—
86	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
87	#127	—	61	1	no	—
88	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	C₄⋊C₈	—	61	1	no	—
91	C₄⋊C₈	—	61	1	no	—
92	C₄⋊C₈	—	122	1	no	—
93	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
94	#103	—	2	1	no	—
95	#102	—	122	1	no	—
96	#102	—	122	1	no	—
97	#102	—	122	1	no	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	—	122	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂×D₄	—	122	1	no	—
103	C₂×D₄	—	122	1	no	—
104	C₂×D₄	—	122	1	no	—
105	SD₁₆	—	244	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	122	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂²×D₆₁	1	1	yes	characteristic
117	C₂³	—	244	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	8	1	no	—
120	Q₈	#11758201	1	2	yes	—
121	D₄	—	488	1	no	—
122	D₄	—	244	1	no	—
123	D₄	—	244	1	no	—
124	D₄	—	244	1	no	—
125	D₄	—	244	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	#11758224	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	122	1	no	—
134	C₈	—	122	1	no	—
135	C₈	—	244	1	no	—
136	C₈	—	244	1	no	—
137	C₂²	—	488	1	no	—
138	C₂²	—	488	1	no	—
139	C₂²	—	488	1	no	—
140	C₂²	#14519749	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	8	1	no	—
145	C₄	C₂×#11758201	1	1	yes	characteristic
146	C₄	#14519797	1	1	yes	characteristic
147	C₄	#14519800	1	1	yes	characteristic
148	C₄	#14519809	1	1	yes	characteristic
149	C₂	—	976	1	no	—
150	C₂	#16839986	1	1	yes	characteristic
151	C₂	#16839993	1	1	yes	characteristic
152	C₂	#16840030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840110

C₂

yes

maximal, characteristic

#16839962

C₂

yes

maximal, characteristic

Q₈×#92418

C₂

yes

maximal, characteristic, complemented

#16839512

C₂

yes

maximal, characteristic, complemented

#16839511

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, characteristic

C₆₁×#477

C₂

yes

maximal, characteristic, complemented

C₂×#11758201

—

#14519774

C₂²

yes

characteristic

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

characteristic

#107×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

#103×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

—

—

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

SD₁₆

yes

—

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

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

—

—

—

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#486

—

#477

D₆₁

yes

characteristic, complemented

#460

—

C₈×Q₈

—

#334

—

#333

—

#331

—

C₆₁

#1678

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

#102

—

122

—

#102

—

122

—

#102

—

122

—

C₄×C₈

—

C₄×C₈

—

122

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

102

C₂×D₄

—

122

—

103

C₂×D₄

—

122

—

104

C₂×D₄

—

122

—

105

SD₁₆

—

244

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

122

—

110

C₂×C₈

—

122

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²⋊D₆₁

yes

characteristic

115

C₄²

—

116

C₄²

C₂²×D₆₁

yes

characteristic

117

C₂³

—

244

—

118

Q₈

—

119

Q₈

—

120

Q₈

#11758201

yes

—

121

D₄

—

488

—

122

D₄

—

244

—

123

D₄

—

244

—

124

D₄

—

244

—

125

D₄

—

244

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₆₁

yes

characteristic

130

C₂×C₄

#11758224

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

132

C₈

—

133

C₈

—

122

—

134

C₈

—

122

—

135

C₈

—

244

—

136

C₈

—

244

—

137

C₂²

—

488

—

138

C₂²

—

488

—

139

C₂²

—

488

—

140

C₂²

#14519749

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

C₂×#11758201

yes

characteristic

146

C₄

#14519797

yes

characteristic

147

C₄

#14519800

yes

characteristic

148

C₄

#14519809

yes

characteristic

149

C₂

—

976

—

150

C₂

#16839986

yes

characteristic

151

C₂

#16839993

yes

characteristic

152

C₂

#16840030

yes

characteristic

	subgroup	G/H
center	C₂²	#14519749
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#477	C₂
socle	C₂²×C₆₁	#121
hypercenter	#477	D₆₁
hypocenter	C₆₁	#1678
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14519749

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#477

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
3. Q₈×#92418	C₂	semidirect
4. #16839512	C₂	semidirect
5. #16839511	C₂	semidirect
7. C₆₁×#477	C₂	semidirect
78. #477	D₆₁	semidirect
84. C₆₁	#1678	semidirect

H ≅ G/N

type

3. Q₈×#92418

C₂

semidirect

4. #16839512

C₂

semidirect

5. #16839511

C₂

semidirect

7. C₆₁×#477

C₂

semidirect

78. #477

D₆₁

semidirect

84. C₆₁

#1678

semidirect

Subgroups — #21445237

Conjugacy classes of subgroups

Complements