Group #21445175 — subgroups

Subgroups — #21445175

Order 7808.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16839958	C₂	1	2	yes	maximal, complemented
3	C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839505	C₂	1	2	yes	maximal, complemented
5	C₆₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₈₈	—	2	2	no	—
7	C₂×D₄₈₈	—	2	1	no	—
8	C₂×D₄₈₈	—	2	1	no	—
9	C₂×C₈×D₆₁	—	2	1	no	—
10	#14519774	C₂²	1	2	yes	—
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	#14519699	—	2	2	no	—
14	#14519689	—	2	1	no	—
15	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×D₂₄₄	—	2	2	no	—
18	C₂×D₂₄₄	—	2	2	no	—
19	C₂×D₂₄₄	—	2	2	no	—
20	C₂×D₂₄₄	D₄	1	1	yes	characteristic
21	C₂×C₄×D₆₁	—	2	1	no	—
22	C₂×C₈×C₆₁	—	2	2	no	—
23	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₄²×C₆₁	C₂³	1	1	yes	characteristic
26	#11758198	—	2	1	no	—
27	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
28	C₂×#92418	—	2	1	no	—
29	D₄₈₈	—	4	2	no	—
30	D₄₈₈	—	4	4	no	—
31	D₄₈₈	—	4	2	no	—
32	C₈×D₆₁	—	4	2	no	—
33	C₂²×D₆₁	—	2	1	no	—
34	C₂²×D₆₁	—	4	2	no	—
35	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
38	C₂×Dic₆₁	—	2	1	no	—
39	D₂₄₄	—	2	1	no	—
40	D₂₄₄	—	4	2	no	—
41	D₂₄₄	—	4	2	no	—
42	D₂₄₄	—	4	2	no	—
43	D₂₄₄	—	4	2	no	—
44	D₂₄₄	—	8	2	no	—
45	D₂₄₄	D₈	1	2	yes	complemented
46	C₄×D₆₁	—	4	1	no	—
47	C₄×D₆₁	—	4	1	no	—
48	C₈×C₆₁	—	2	1	no	—
49	C₈×C₆₁	—	4	2	no	—
50	C₈×C₆₁	D₈	1	2	yes	complemented
51	#92418	—	4	1	no	—
52	C₂²×C₆₁	#121	1	1	yes	characteristic
53	C₂×D₆₁	—	2	2	no	—
54	C₂×D₆₁	—	4	1	no	—
55	C₂×D₆₁	—	4	1	no	—
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	C₂×D₈	1	1	yes	characteristic
61	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
62	C₄×C₆₁	#135	1	1	yes	characteristic
63	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₆₁	—	4	1	no	—
65	#1667	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	#459	1	1	yes	characteristic
67	C₂×C₆₁	#466	1	1	yes	characteristic
68	C₂×C₆₁	#472	1	1	yes	characteristic
69	D₆₁	—	4	2	no	—
70	D₆₁	—	16	2	no	—
71	#493	—	61	1	no	—
72	#459	—	61	2	no	—
73	C₈×D₄	—	61	1	no	—
74	#335	D₆₁	1	1	yes	characteristic, complemented
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	—	122	1	no	—
81	#127	—	61	2	no	—
82	C₄×D₄	—	61	1	no	—
83	C₄⋊C₈	—	61	1	no	—
84	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
85	#102	—	122	2	no	—
86	C₂²⋊C₈	—	122	1	no	—
87	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×D₄	—	61	1	no	—
89	C₂×D₄	—	122	2	no	—
90	C₂×D₄	—	122	2	no	—
91	C₂×D₄	—	122	2	no	—
92	C₂²×C₄	—	122	1	no	—
93	D₈	—	244	2	no	—
94	D₈	—	244	4	no	—
95	D₈	—	244	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	122	1	no	—
98	C₂×C₈	—	244	2	no	—
99	C₂×C₈	D₂₄₄	1	1	yes	characteristic
100	C₂×C₈	D₂₄₄	1	1	yes	characteristic
101	C₄⋊C₄	—	61	1	no	—
102	C₂²⋊C₄	—	122	1	no	—
103	C₄²	C₂²×D₆₁	1	1	yes	characteristic
104	C₂³	—	122	1	no	—
105	C₂³	—	244	2	no	—
106	D₄	—	61	2	no	—
107	D₄	—	122	1	no	—
108	D₄	—	488	2	no	—
109	D₄	—	244	2	no	—
110	D₄	—	244	2	no	—
111	D₄	—	244	2	no	—
112	D₄	—	244	2	no	—
113	C₂×C₄	—	122	1	no	—
114	C₂×C₄	—	244	1	no	—
115	C₂×C₄	—	244	1	no	—
116	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
117	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
118	C₂×C₄	#11758226	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	2	no	—
121	C₈	—	244	1	no	—
122	C₈	D₄₈₈	1	2	yes	complemented
123	C₂²	—	122	2	no	—
124	C₂²	—	488	2	no	—
125	C₂²	—	488	2	no	—
126	C₂²	—	488	2	no	—
127	C₂²	—	244	1	no	—
128	C₂²	—	244	1	no	—
129	C₂²	#14519752	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	244	1	no	—
132	C₄	D₈×D₆₁	1	1	yes	characteristic
133	C₄	C₂×D₄₈₈	1	1	yes	characteristic
134	C₄	#14519800	1	1	yes	characteristic
135	C₄	#14519805	1	1	yes	characteristic
136	C₂	—	976	2	no	—
137	C₂	—	244	2	no	—
138	C₂	#16839958	1	1	yes	characteristic
139	C₂	#16839999	1	1	yes	characteristic
140	C₂	#16840033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840139

C₂

yes

maximal, characteristic, complemented

#16839958

C₂

yes

maximal, complemented

C₈×D₂₄₄

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, complemented

C₆₁×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×C₈×D₆₁

—

#14519774

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519699

—

#14519689

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

D₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₂×#92418

—

D₄₈₈

—

D₄₈₈

—

D₄₈₈

—

C₈×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

D₈

yes

complemented

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

D₈

yes

complemented

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

#493

—

#459

—

C₈×D₄

—

#335

D₆₁

yes

characteristic, complemented

#331

—

C₆₁

#1667

yes

characteristic, complemented, Sylow

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂²×C₈

—

122

—

#127

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

—

#102

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂²×C₄

—

122

—

D₈

—

244

—

D₈

—

244

—

D₈

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

D₂₄₄

yes

characteristic

100

C₂×C₈

D₂₄₄

yes

characteristic

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

122

—

103

C₄²

C₂²×D₆₁

yes

characteristic

104

C₂³

—

122

—

105

C₂³

—

244

—

106

D₄

—

107

D₄

—

122

—

108

D₄

—

488

—

109

D₄

—

244

—

110

D₄

—

244

—

111

D₄

—

244

—

112

D₄

—

244

—

113

C₂×C₄

—

122

—

114

C₂×C₄

—

244

—

115

C₂×C₄

—

244

—

116

C₂×C₄

C₂×D₂₄₄

yes

characteristic

117

C₂×C₄

D₄×D₆₁

yes

characteristic

118

C₂×C₄

#11758226

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

244

—

122

C₈

D₄₈₈

yes

complemented

123

C₂²

—

122

—

124

C₂²

—

488

—

125

C₂²

—

488

—

126

C₂²

—

488

—

127

C₂²

—

244

—

128

C₂²

—

244

—

129

C₂²

#14519752

yes

characteristic

130

C₄

—

131

C₄

—

244

—

132

C₄

D₈×D₆₁

yes

characteristic

133

C₄

C₂×D₄₈₈

yes

characteristic

134

C₄

#14519800

yes

characteristic

135

C₄

#14519805

yes

characteristic

136

C₂

—

976

—

137

C₂

—

244

—

138

C₂

#16839958

yes

characteristic

139

C₂

#16839999

yes

characteristic

140

C₂

#16840033

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#335	C₂
socle	C₂²×C₆₁	#121
hypercenter	#335	D₆₁
hypocenter	C₆₁	#1667
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14519752

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#335

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #16840139	C₂	semidirect
2. #16839958	C₂	semidirect
3. C₈×D₂₄₄	C₂	semidirect
4. #16839505	C₂	semidirect
5. C₆₁×#335	C₂	semidirect
45. D₂₄₄	D₈	semidirect
50. C₈×C₆₁	D₈	semidirect
74. #335	D₆₁	semidirect
76. C₆₁	#1667	semidirect
122. C₈	D₄₈₈	semidirect

H ≅ G/N

type

1. #16840139

C₂

semidirect

2. #16839958

C₂

semidirect

3. C₈×D₂₄₄

C₂

semidirect

4. #16839505

C₂

semidirect

5. C₆₁×#335

C₂

semidirect

45. D₂₄₄

D₈

semidirect

50. C₈×C₆₁

D₈

semidirect

74. #335

D₆₁

semidirect

76. C₆₁

#1667

semidirect

122. C₈

D₄₈₈

semidirect

Subgroups — #21445175

Conjugacy classes of subgroups

Complements