Group #21454412 — subgroups

Subgroups — #21454412

Order 7808.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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×#14519784	C₂	1	1	yes	maximal, characteristic
2	#16840245	C₂	1	1	yes	maximal, characteristic
3	#16839805	C₂	1	3	yes	maximal, complemented
4	#16839802	C₂	1	3	yes	maximal, complemented
5	#16839798	C₂	1	3	yes	maximal, complemented
6	C₂×#14519708	C₂	1	3	yes	maximal, complemented
7	C₆₁×#520	C₂	1	1	yes	maximal, characteristic
8	#14519784	C₄	1	8	yes	complemented
9	C₂×#11758194	C₂²	1	3	yes	—
10	C₂²×#92418	C₂²	1	3	yes	—
11	Q₈×Dic₆₁	—	2	1	no	—
12	D₄×Dic₆₁	—	2	3	no	—
13	#14519736	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
15	#14519710	C₂²	1	3	yes	—
16	#14519710	C₂²	1	3	yes	—
17	#14519708	C₂²	1	3	yes	—
18	#14519708	C₂²	1	6	yes	—
19	#14519708	C₂²	1	1	yes	characteristic
20	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₆₁	—	2	1	no	—
22	C₄×D₄×C₆₁	—	2	3	no	—
23	#117×C₆₁	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
25	#41×C₆₁	C₂×C₄	1	4	yes	—
26	#41×C₆₁	Q₈	1	4	yes	—
27	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂×D₄×C₆₁	D₄	1	3	yes	—
29	C₂²×C₄×C₆₁	—	2	3	no	—
30	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
31	C₂²×Dic₆₁	—	2	3	no	—
32	C₄⋊C₄×C₆₁	—	2	3	no	—
33	C₄⋊C₄×C₆₁	C₂³	1	3	yes	—
34	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²⋊C₄×C₆₁	—	2	3	no	—
36	C₄²×C₆₁	—	2	3	no	—
37	C₂²⋊Dic₆₁	—	2	3	no	—
38	C₄⋊Dic₆₁	—	2	3	no	—
39	C₄⋊Dic₆₁	C₂³	1	3	yes	—
40	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
41	C₄×Dic₆₁	—	2	3	no	—
42	#11758194	C₂×C₄	1	12	yes	complemented
43	C₂×#92418	C₂×C₄	1	12	yes	complemented
44	C₂×#92418	C₂³	1	1	yes	characteristic
45	C₂×#92418	C₂³	1	3	yes	—
46	C₂³×C₆₁	C₂×D₄	1	3	yes	—
47	Q₈×C₆₁	C₄⋊C₄	1	4	yes	—
48	D₄×C₆₁	C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	3	no	—
51	C₂×C₄×C₆₁	—	8	3	no	—
52	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
53	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₆₁	C₂×Q₈	1	6	yes	—
55	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	1	no	—
57	C₂×Dic₆₁	—	2	3	no	—
58	C₂×Dic₆₁	—	8	3	no	—
59	#92418	C₂²×C₄	1	2	yes	—
60	#92418	C₂²×C₄	1	6	yes	—
61	C₂²×C₆₁	—	2	3	no	—
62	C₂²×C₆₁	—	2	3	no	—
63	C₂²×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₆₁	—	4	1	no	—
66	C₄×C₆₁	—	4	3	no	—
67	C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	—	4	3	no	—
72	#2912	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	—	2	6	no	—
74	C₂×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₆₁	#577	1	1	yes	characteristic
76	C₂×C₆₁	#577	1	1	yes	characteristic
77	C₂×#131	—	61	1	no	—
78	#520	—	61	1	no	—
79	#520	D₆₁	1	1	yes	characteristic
80	#428	—	61	3	no	—
81	#428	—	61	3	no	—
82	#427	—	61	3	no	—
83	C₂×#106	—	61	3	no	—
84	C₆₁	#2912	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
86	#131	—	61	8	no	—
87	C₂×M₁₆	—	61	3	no	—
88	C₂²×C₈	—	61	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	122	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	122	3	no	—
93	#117	—	61	3	no	—
94	#117	C₂×D₆₁	1	3	yes	—
95	C₂×C₄⋊C₄	—	61	3	no	—
96	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
97	#107	—	61	3	no	—
98	#107	—	61	3	no	—
99	#106	—	61	3	no	—
100	#106	—	61	6	no	—
101	#106	—	61	1	no	—
102	#41	Dic₁₂₂	1	4	yes	—
103	#41	C₄×D₆₁	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₆₁	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	122	3	no	—
108	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
109	M₁₆	—	61	12	no	—
110	C₂×C₈	—	61	12	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	61	3	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	61	3	no	—
115	C₄⋊C₄	—	61	1	no	—
116	C₄⋊C₄	—	122	3	no	—
117	C₄⋊C₄	C₂²×D₆₁	1	3	yes	—
118	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	122	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	122	3	no	—
123	C₂³	C₂×C₂²⋊D₆₁	1	3	yes	—
124	Q₈	#11758196	1	4	yes	—
125	D₄	#11758196	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	122	1	no	—
130	C₂×C₄	—	122	3	no	—
131	C₂×C₄	—	488	3	no	—
132	C₂×C₄	C₂×Dic₁₂₂	1	6	yes	—
133	C₂×C₄	C₂×C₄×D₆₁	1	6	yes	—
134	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
135	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
136	C₈	—	61	2	no	—
137	C₈	—	61	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×#11758196	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
142	C₄	—	4	1	no	—
143	C₄	—	4	3	no	—
144	C₄	—	244	3	no	—
145	C₄	—	244	1	no	—
146	C₄	C₂×#11758196	1	6	yes	—
147	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
148	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×#11758196	1	1	yes	characteristic
151	C₂	#16840570	1	1	yes	characteristic
152	C₂	#16840573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519784

C₂

yes

maximal, characteristic

#16840245

C₂

yes

maximal, characteristic

#16839805

C₂

yes

maximal, complemented

#16839802

C₂

yes

maximal, complemented

#16839798

C₂

yes

maximal, complemented

C₂×#14519708

C₂

yes

maximal, complemented

C₆₁×#520

C₂

yes

maximal, characteristic

#14519784

C₄

yes

complemented

C₂×#11758194

C₂²

yes

—

C₂²×#92418

C₂²

yes

—

Q₈×Dic₆₁

—

D₄×Dic₆₁

—

#14519736

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

#14519710

C₂²

yes

—

#14519710

C₂²

yes

—

#14519708

C₂²

yes

—

#14519708

C₂²

yes

—

#14519708

C₂²

yes

characteristic

C₂×#41×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

—

C₄×D₄×C₆₁

—

#117×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#41×C₆₁

C₂×C₄

yes

—

#41×C₆₁

Q₈

yes

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

characteristic

C₄×Dic₆₁

—

#11758194

C₂×C₄

yes

complemented

C₂×#92418

C₂×C₄

yes

complemented

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

—

Q₈×C₆₁

C₄⋊C₄

yes

—

D₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

C₂²×C₄

yes

—

#92418

C₂²×C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2912

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂²×C₄⋊C₄

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

D₆₁

yes

—

—

—

—

C₆₁

#2912

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×D₄

—

C₄×D₄

—

122

—

#117

—

#117

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

#107

—

#107

—

#106

—

100

#106

—

101

#106

—

102

#41

Dic₁₂₂

yes

—

103

#41

C₄×D₆₁

yes

—

104

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

105

C₂×D₄

C₂²⋊D₆₁

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

122

—

108

C₂²×C₄

C₂²×D₆₁

yes

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

C₂²×D₆₁

yes

—

118

C₄⋊C₄

C₂²×D₆₁

yes

characteristic

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

122

—

121

C₄²

—

122

C₄²

—

122

—

123

C₂³

C₂×C₂²⋊D₆₁

yes

—

124

Q₈

#11758196

yes

—

125

D₄

#11758196

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

488

—

132

C₂×C₄

C₂×Dic₁₂₂

yes

—

133

C₂×C₄

C₂×C₄×D₆₁

yes

—

134

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

135

C₂×C₄

C₂³×D₆₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×#11758196

yes

—

141

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

244

—

145

C₄

—

244

—

146

C₄

C₂×#11758196

yes

—

147

C₄

C₂²×C₄×D₆₁

yes

characteristic

148

C₄

C₂²×Dic₁₂₂

yes

characteristic

149

C₂

—

150

C₂

C₂²×#11758196

yes

characteristic

151

C₂

#16840570

yes

characteristic

152

C₂

#16840573

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#520	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#520	D₆₁
hypocenter	C₆₁	#2912
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₁

commutator

C₄×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#520

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #16839805	C₂	semidirect
4. #16839802	C₂	semidirect
5. #16839798	C₂	semidirect
6. C₂×#14519708	C₂	semidirect
8. #14519784	C₄	semidirect
42. #11758194	C₂×C₄	semidirect
43. C₂×#92418	C₂×C₄	semidirect
84. C₆₁	#2912	semidirect

H ≅ G/N

type

3. #16839805

C₂

semidirect

4. #16839802

C₂

semidirect

5. #16839798

C₂

semidirect

6. C₂×#14519708

C₂

semidirect

8. #14519784

C₄

semidirect

42. #11758194

C₂×C₄

semidirect

43. C₂×#92418

C₂×C₄

semidirect

84. C₆₁

#2912

semidirect

Subgroups — #21454412

Conjugacy classes of subgroups

Complements