Group #21456347 — subgroups

Subgroups — #21456347

Order 7808.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16840393	C₂	1	2	yes	maximal
2	#16840169	C₂	1	4	yes	maximal, complemented
3	#16840162	C₂	1	2	yes	maximal, complemented
4	#16840159	C₂	1	2	yes	maximal, complemented
5	#16839802	C₂	1	2	yes	maximal
6	#16839800	C₂	1	1	yes	maximal, characteristic
7	#16839717	C₂	1	1	yes	maximal, characteristic
8	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#11758194	C₂²	1	1	yes	characteristic
10	#14519776	—	2	2	no	—
11	#14519776	C₂²	1	4	yes	—
12	#14519768	—	2	2	no	—
13	#14519768	C₂²	1	4	yes	—
14	#14519756	—	2	4	no	—
15	#14519736	C₂²	1	2	yes	—
16	#14519736	C₂²	1	1	yes	characteristic
17	#14519711	C₂²	1	2	yes	—
18	#14519710	C₂²	1	4	yes	—
19	#14519709	C₂²	1	1	yes	characteristic
20	#14519709	C₂²	1	1	yes	characteristic
21	#14519708	C₂²	1	4	yes	—
22	#14519687	C₂²	1	2	yes	—
23	C₈×Dic₆₁	C₂²	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
26	#106×C₆₁	C₂²	1	2	yes	—
27	#106×C₆₁	C₂²	1	2	yes	—
28	C₂²×C₄×C₆₁	—	2	2	no	—
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₂₂	—	2	4	no	—
31	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
32	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
36	C₂²⋊Dic₆₁	—	2	2	no	—
37	C₂²⋊Dic₆₁	D₄	1	2	yes	—
38	C₄⋊Dic₆₁	—	2	4	no	—
39	C₄⋊Dic₆₁	D₄	1	2	yes	—
40	C₄⋊Dic₆₁	C₂³	1	2	yes	—
41	C₄⋊Dic₆₁	C₂³	1	2	yes	—
42	#11758196	—	2	4	no	—
43	#11758196	—	2	4	no	—
44	C₄×Dic₆₁	—	2	2	no	—
45	C₄×Dic₆₁	C₂³	1	2	yes	—
46	#11758194	Q₈	1	4	yes	complemented
47	C₂×#92418	C₂³	1	2	yes	—
48	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	4	no	—
50	C₂×C₄×C₆₁	—	8	2	no	—
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	2	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
58	Dic₁₂₂	—	8	4	no	—
59	C₈×C₆₁	C₂×Q₈	1	2	yes	—
60	C₈×C₆₁	C₂×Q₈	1	2	yes	—
61	#92418	C₂×Q₈	1	4	yes	—
62	C₂²×C₆₁	—	2	2	no	—
63	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₆₁	—	4	4	no	—
66	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
67	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
69	Dic₆₁	—	2	4	no	—
70	Dic₆₁	—	4	2	no	—
71	Dic₆₁	—	4	2	no	—
72	#3163	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	—	2	2	no	—
74	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₆₁	#577	1	2	yes	—
76	#544	—	61	2	no	—
77	#501	—	61	4	no	—
78	#499	—	61	2	no	—
79	#498	—	61	2	no	—
80	#428	—	61	2	no	—
81	#427	—	61	1	no	—
82	C₂×#106	D₆₁	1	1	yes	characteristic
83	#405	—	61	1	no	—
84	C₆₁	#3163	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	61	1	no	—
86	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₄⋊Q₈	—	61	4	no	—
88	C₄⋊Q₈	—	122	2	no	—
89	#125	—	61	4	no	—
90	#125	—	122	2	no	—
91	C₂²⋊Q₈	—	122	4	no	—
92	#117	—	61	1	no	—
93	#117	—	61	2	no	—
94	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
95	#107	—	61	2	no	—
96	#107	—	61	4	no	—
97	#106	—	61	1	no	—
98	#106	—	61	1	no	—
99	#106	—	61	4	no	—
100	#106	C₂×D₆₁	1	2	yes	—
101	#106	C₂×D₆₁	1	2	yes	—
102	#97	—	61	2	no	—
103	C₄×C₈	—	61	2	no	—
104	C₂×Q₈	—	122	4	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
107	M₁₆	—	61	4	no	—
108	C₂×C₈	—	61	2	no	—
109	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
110	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	61	2	no	—
114	C₄⋊C₄	—	61	2	no	—
115	C₄⋊C₄	—	61	2	no	—
116	C₄⋊C₄	—	122	4	no	—
117	C₄⋊C₄	—	122	4	no	—
118	C₄⋊C₄	—	122	4	no	—
119	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
120	C₂²⋊C₄	—	61	2	no	—
121	C₂²⋊C₄	—	122	2	no	—
122	C₄²	—	61	2	no	—
123	C₄²	—	122	2	no	—
124	C₂³	D₄×D₆₁	1	1	yes	characteristic
125	Q₈	—	488	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	61	4	no	—
129	C₂×C₄	—	122	2	no	—
130	C₂×C₄	—	122	2	no	—
131	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
132	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₆₁	1	2	yes	—
134	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
135	C₈	—	61	4	no	—
136	C₈	C₂×Dic₁₂₂	1	2	yes	—
137	C₈	C₂×Dic₁₂₂	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#14519776	1	2	yes	—
140	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	122	4	no	—
143	C₄	—	244	2	no	—
144	C₄	—	244	2	no	—
145	C₄	#14519776	1	2	yes	—
146	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
147	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#14519776	1	1	yes	characteristic
150	C₂	#16840568	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840393

C₂

yes

maximal

#16840169

C₂

yes

maximal, complemented

#16840162

C₂

yes

maximal, complemented

#16840159

C₂

yes

maximal, complemented

#16839802

C₂

yes

maximal

#16839800

C₂

yes

maximal, characteristic

#16839717

C₂

yes

maximal, characteristic

C₂×#106×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

characteristic

#14519776

—

#14519776

C₂²

yes

—

#14519768

—

#14519768

C₂²

yes

—

#14519756

—

#14519736

C₂²

yes

—

#14519736

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519710

C₂²

yes

—

#14519709

C₂²

yes

characteristic

#14519709

C₂²

yes

characteristic

#14519708

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

—

#11758194

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#3163

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

#577

yes

—

—

—

—

—

—

—

D₆₁

yes

characteristic

#405

—

C₆₁

#3163

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

#125

—

#125

—

122

—

C₂²⋊Q₈

—

122

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

—

—

—

—

—

100

#106

C₂×D₆₁

yes

—

101

#106

C₂×D₆₁

yes

—

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

122

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₆₁

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

Dic₁₂₂

yes

complemented

110

C₂×C₈

C₂²×D₆₁

yes

characteristic

111

C₂×C₈

C₂²×D₆₁

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

C₂²×D₆₁

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

—

122

C₄²

—

123

C₄²

—

122

—

124

C₂³

D₄×D₆₁

yes

characteristic

125

Q₈

—

488

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

122

—

131

C₂×C₄

C₂×Dic₁₂₂

yes

—

132

C₂×C₄

D₄×D₆₁

yes

characteristic

133

C₂×C₄

Q₈×D₆₁

yes

—

134

C₂×C₄

C₂³×D₆₁

yes

characteristic

135

C₈

—

136

C₈

C₂×Dic₁₂₂

yes

—

137

C₈

C₂×Dic₁₂₂

yes

—

138

C₂²

—

139

C₂²

#14519776

yes

—

140

C₂²

C₂×D₄×D₆₁

yes

characteristic

141

C₄

—

142

C₄

—

122

—

143

C₄

—

244

—

144

C₄

—

244

—

145

C₄

#14519776

yes

—

146

C₄

C₂×Q₈×D₆₁

yes

characteristic

147

C₄

C₂²×Dic₁₂₂

yes

characteristic

148

C₂

—

149

C₂

C₂×#14519776

yes

characteristic

150

C₂

#16840568

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#106×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂×#106	D₆₁
hypocenter	C₆₁	#3163
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#106×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #16840169	C₂	semidirect
3. #16840162	C₂	semidirect
4. #16840159	C₂	semidirect
31. C₂×C₈×C₆₁	Q₈	semidirect
46. #11758194	Q₈	semidirect
84. C₆₁	#3163	semidirect
109. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #16840169

C₂

semidirect

3. #16840162

C₂

semidirect

4. #16840159

C₂

semidirect

31. C₂×C₈×C₆₁

Q₈

semidirect

46. #11758194

Q₈

semidirect

84. C₆₁

#3163

semidirect

109. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21456347

Conjugacy classes of subgroups

Complements