Group #21445201 — subgroups

Subgroups — #21445201

Order 7808.

150 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16840057	C₂	1	1	yes	maximal, characteristic
3	#16839972	C₂	1	1	yes	maximal, characteristic
4	#16839513	C₂	1	1	yes	maximal, characteristic
5	#16839510	C₂	1	1	yes	maximal, characteristic
6	#16839507	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758202	—	2	1	no	—
9	#14519777	C₂²	1	1	yes	characteristic
10	#14519777	C₂²	1	1	yes	characteristic
11	#14519710	—	2	1	no	—
12	#14519709	—	2	1	no	—
13	#14519709	—	2	1	no	—
14	C₄⋊#92418	C₂²	1	1	yes	characteristic
15	C₄⋊#92418	C₂²	1	1	yes	characteristic
16	#14519701	—	2	1	no	—
17	#14519700	—	2	1	no	—
18	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	—	2	1	no	—
20	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₆₁	—	2	1	no	—
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₂	—	2	1	no	—
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
31	C₄²×C₆₁	—	2	1	no	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#11758202	—	4	2	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	1	no	—
37	C₄⋊Dic₆₁	—	2	1	no	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	Q₈×C₆₁	—	2	1	no	—
41	Q₈×C₆₁	SD₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
47	C₂×Dic₆₁	—	4	1	no	—
48	C₂×Dic₆₁	—	4	1	no	—
49	Dic₁₂₂	—	4	1	no	—
50	Dic₁₂₂	—	4	1	no	—
51	Dic₁₂₂	—	8	1	no	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	—	4	1	no	—
54	C₈×C₆₁	Q₁₆	1	2	yes	complemented
55	#92418	—	4	1	no	—
56	#92418	—	4	1	no	—
57	C₂²×C₆₁	#121	1	1	yes	characteristic
58	C₄×C₆₁	—	2	2	no	—
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	—	4	1	no	—
61	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₆₁	#135	1	1	yes	characteristic
64	C₄×C₆₁	#137	1	1	yes	characteristic
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	D₆₁	1	1	yes	characteristic
75	#334	—	61	1	no	—
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	C₂×D₆₁	1	1	yes	characteristic
83	#107	—	122	1	no	—
84	#106	—	122	1	no	—
85	#106	—	122	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	61	1	no	—
88	C₄⋊C₈	—	61	1	no	—
89	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
90	#103	—	122	1	no	—
91	#103	—	122	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₂×Q₈	—	122	1	no	—
95	C₂×Q₈	—	122	1	no	—
96	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
97	Q₁₆	—	244	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	122	1	no	—
100	C₂×C₈	—	122	1	no	—
101	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	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	D₂₄₄	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₆₁	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	488	1	no	—
113	Q₈	—	244	1	no	—
114	Q₈	—	244	1	no	—
115	Q₈	#11758190	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	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	#11758222	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	244	1	no	—
126	C₈	—	244	1	no	—
127	C₈	#11758202	1	2	yes	complemented
128	C₂²	#14519751	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	488	1	no	—
133	C₄	—	488	1	no	—
134	C₄	C₂×#11758190	1	1	yes	characteristic
135	C₄	C₂×#11758202	1	1	yes	characteristic
136	C₄	#14519799	1	1	yes	characteristic
137	C₄	#14519807	1	1	yes	characteristic
138	C₂	#16839972	1	1	yes	characteristic
139	C₂	#16839991	1	1	yes	characteristic
140	C₂	#16840025	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840160

C₂

yes

maximal, characteristic

#16840057

C₂

yes

maximal, characteristic

#16839972

C₂

yes

maximal, characteristic

#16839513

C₂

yes

maximal, characteristic

#16839510

C₂

yes

maximal, characteristic

#16839507

C₂

yes

maximal, characteristic

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758202

—

#14519777

C₂²

yes

characteristic

#14519777

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519701

—

#14519700

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

Q₈×C₆₁

—

Q₈×C₆₁

SD₁₆

yes

—

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

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

Q₁₆

yes

complemented

#92418

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

C₈×Q₈

D₆₁

yes

—

—

—

C₆₁

#1672

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

#107

—

122

—

#106

—

122

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

#103

—

122

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

D₂₄₄

yes

characteristic

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

100

C₂×C₈

—

122

—

101

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

—

122

—

106

C₄⋊C₄

—

122

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

D₂₄₄

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₆₁

yes

characteristic

111

Q₈

—

112

Q₈

—

488

—

113

Q₈

—

244

—

114

Q₈

—

244

—

115

Q₈

#11758190

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

244

—

119

C₂×C₄

—

244

—

120

C₂×C₄

C₂×D₂₄₄

yes

characteristic

121

C₂×C₄

#11758222

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

244

—

126

C₈

—

244

—

127

C₈

#11758202

yes

complemented

128

C₂²

#14519751

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

488

—

133

C₄

—

488

—

134

C₄

C₂×#11758190

yes

characteristic

135

C₄

C₂×#11758202

yes

characteristic

136

C₄

#14519799

yes

characteristic

137

C₄

#14519807

yes

characteristic

138

C₂

#16839972

yes

characteristic

139

C₂

#16839991

yes

characteristic

140

C₂

#16840025

yes

characteristic

	subgroup	G/H
center	C₂²	#14519751
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	#121
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#1672
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14519751

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

#121

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#1672

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
54. C₈×C₆₁	Q₁₆	semidirect
78. C₆₁	#1672	semidirect
127. C₈	#11758202	semidirect

H ≅ G/N

type

54. C₈×C₆₁

Q₁₆

semidirect

78. C₆₁

#1672

semidirect

127. C₈

#11758202

semidirect

Subgroups — #21445201

Conjugacy classes of subgroups

Complements