Group #21445174 — subgroups

Subgroups — #21445174

Order 7808.

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

normal subgroups only

This group has 204 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	#16840139	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839957	C₂	1	2	yes	maximal, complemented
3	#16839515	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839505	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₈₈	—	2	1	no	—
7	C₂×D₄₈₈	—	2	1	no	—
8	C₂×#11758199	—	2	2	no	—
9	#14519774	C₂²	1	2	yes	—
10	C₄⋊#92418	C₂²	1	2	yes	—
11	#14519697	—	2	2	no	—
12	C₂²×C₈×C₆₁	—	2	1	no	—
13	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	—	2	1	no	—
19	C₂×D₂₄₄	—	2	2	no	—
20	C₂×D₂₄₄	—	2	2	no	—
21	C₂×D₂₄₄	—	2	2	no	—
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₂×C₈×C₆₁	—	4	2	no	—
24	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
27	C₂²⋊C₄×C₆₁	—	2	1	no	—
28	C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	#11758199	—	4	4	no	—
30	C₂×#92418	—	2	2	no	—
31	D₄₈₈	—	4	2	no	—
32	D₄₈₈	—	4	2	no	—
33	C₂³×C₆₁	—	2	1	no	—
34	C₂²×D₆₁	—	4	2	no	—
35	D₄×C₆₁	—	2	1	no	—
36	D₄×C₆₁	D₈	1	2	yes	complemented
37	C₂×C₄×C₆₁	—	2	1	no	—
38	C₂×C₄×C₆₁	—	4	1	no	—
39	C₂×C₄×C₆₁	—	4	1	no	—
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
43	D₂₄₄	—	4	2	no	—
44	D₂₄₄	—	4	2	no	—
45	D₂₄₄	—	4	2	no	—
46	D₂₄₄	—	4	2	no	—
47	D₂₄₄	—	8	2	no	—
48	C₈×C₆₁	—	2	1	no	—
49	C₈×C₆₁	—	4	1	no	—
50	C₈×C₆₁	D₈	1	2	yes	complemented
51	#92418	—	4	2	no	—
52	C₂²×C₆₁	—	2	2	no	—
53	C₂²×C₆₁	—	4	1	no	—
54	C₂²×C₆₁	—	4	1	no	—
55	C₂²×C₆₁	#121	1	1	yes	characteristic
56	C₂×D₆₁	—	8	2	no	—
57	C₂×D₆₁	—	8	2	no	—
58	C₂×D₆₁	—	8	2	no	—
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	—	4	1	no	—
61	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
62	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₄×C₆₁	#135	1	1	yes	characteristic
64	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#1667	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	—	4	2	no	—
67	C₂×C₆₁	#459	1	1	yes	characteristic
68	C₂×C₆₁	#466	1	1	yes	characteristic
69	C₂×C₆₁	#472	1	1	yes	characteristic
70	D₆₁	—	16	2	no	—
71	#493	—	61	1	no	—
72	#459	—	61	2	no	—
73	C₈×D₄	D₆₁	1	1	yes	characteristic, complemented
74	#335	—	61	1	no	—
75	#331	—	61	2	no	—
76	C₆₁	#1667	1	1	yes	characteristic, complemented, Sylow
77	C₂×D₈	—	122	2	no	—
78	C₂×D₈	—	122	1	no	—
79	C₂×D₈	—	122	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	#127	—	61	2	no	—
82	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
83	C₄⋊C₈	—	61	2	no	—
84	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
85	#102	—	122	2	no	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×D₄	—	122	2	no	—
89	C₂×D₄	—	122	2	no	—
90	C₂×D₄	—	122	2	no	—
91	C₂×D₄	D₂₄₄	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	D₈	—	244	2	no	—
94	D₈	—	244	4	no	—
95	D₈	—	244	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	122	2	no	—
99	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
101	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
102	C₂²⋊C₄	—	2	1	no	—
103	C₄²	C₂²×D₆₁	1	1	yes	characteristic
104	C₂³	—	2	1	no	—
105	C₂³	—	244	2	no	—
106	D₄	—	2	1	no	—
107	D₄	—	488	2	no	—
108	D₄	—	244	2	no	—
109	D₄	—	244	2	no	—
110	D₄	—	244	2	no	—
111	D₄	—	244	2	no	—
112	D₄	D₄₈₈	1	2	yes	complemented
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
117	C₂×C₄	#11758222	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	244	2	no	—
122	C₈	#11758199	1	2	yes	complemented
123	C₂²	—	2	2	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	488	2	no	—
127	C₂²	—	488	2	no	—
128	C₂²	—	488	2	no	—
129	C₂²	#14519751	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	C₂×#11758199	1	1	yes	characteristic
133	C₄	C₂×D₄₈₈	1	1	yes	characteristic
134	C₄	#14519799	1	1	yes	characteristic
135	C₄	#14519802	1	1	yes	characteristic
136	C₂	—	4	2	no	—
137	C₂	—	976	2	no	—
138	C₂	#16839957	1	1	yes	characteristic
139	C₂	#16839998	1	1	yes	characteristic
140	C₂	#16840032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840139

C₂

yes

maximal, characteristic, complemented

#16839957

C₂

yes

maximal, complemented

#16839515

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, complemented

C₈×D₄×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×#11758199

—

#14519774

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519697

—

C₂²×C₈×C₆₁

—

C₄×D₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758199

—

C₂×#92418

—

D₄₈₈

—

D₄₈₈

—

C₂³×C₆₁

—

C₂²×D₆₁

—

D₄×C₆₁

—

D₄×C₆₁

D₈

yes

complemented

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

—

C₈×C₆₁

—

C₈×C₆₁

D₈

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#493

—

#459

—

C₈×D₄

D₆₁

yes

characteristic, complemented

#335

—

#331

—

C₆₁

#1667

yes

characteristic, complemented, Sylow

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂²×C₈

—

#127

—

C₄×D₄

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#102

—

122

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

D₂₄₄

yes

characteristic

C₂²×C₄

—

D₈

—

244

—

D₈

—

244

—

D₈

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

100

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

101

C₄⋊C₄

D₂₄₄

yes

characteristic

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₆₁

yes

characteristic

104

C₂³

—

105

C₂³

—

244

—

106

D₄

—

107

D₄

—

488

—

108

D₄

—

244

—

109

D₄

—

244

—

110

D₄

—

244

—

111

D₄

—

244

—

112

D₄

D₄₈₈

yes

complemented

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×D₂₄₄

yes

characteristic

117

C₂×C₄

#11758222

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

244

—

122

C₈

#11758199

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

488

—

127

C₂²

—

488

—

128

C₂²

—

488

—

129

C₂²

#14519751

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

C₂×#11758199

yes

characteristic

133

C₄

C₂×D₄₈₈

yes

characteristic

134

C₄

#14519799

yes

characteristic

135

C₄

#14519802

yes

characteristic

136

C₂

—

137

C₂

—

976

—

138

C₂

#16839957

yes

characteristic

139

C₂

#16839998

yes

characteristic

140

C₂

#16840032

yes

characteristic

	subgroup	G/H
center	C₂²	#14519751
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈×D₄×C₆₁	C₂
socle	C₂²×C₆₁	#121
hypercenter	C₈×D₄	D₆₁
hypocenter	C₆₁	#1667
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#14519751

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈×D₄×C₆₁

C₂

socle

C₂²×C₆₁

#121

hypercenter

C₈×D₄

D₆₁

hypocenter

C₆₁

#1667

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #16840139	C₂	semidirect
2. #16839957	C₂	semidirect
3. #16839515	C₂	semidirect
4. #16839505	C₂	semidirect
5. C₈×D₄×C₆₁	C₂	semidirect
36. D₄×C₆₁	D₈	semidirect
50. C₈×C₆₁	D₈	semidirect
73. C₈×D₄	D₆₁	semidirect
76. C₆₁	#1667	semidirect
112. D₄	D₄₈₈	semidirect
122. C₈	#11758199	semidirect

H ≅ G/N

type

1. #16840139

C₂

semidirect

2. #16839957

C₂

semidirect

3. #16839515

C₂

semidirect

4. #16839505

C₂

semidirect

5. C₈×D₄×C₆₁

C₂

semidirect

36. D₄×C₆₁

D₈

semidirect

50. C₈×C₆₁

D₈

semidirect

73. C₈×D₄

D₆₁

semidirect

76. C₆₁

#1667

semidirect

112. D₄

D₄₈₈

semidirect

122. C₈

#11758199

semidirect

Subgroups — #21445174

Conjugacy classes of subgroups

Complements