Group #21447738 — subgroups

Subgroups — #21447738

Order 7808.

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

normal subgroups only

This group has 226 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	C₂²×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14519722	C₂	1	2	yes	maximal, complemented
3	#16839998	C₂	1	1	yes	maximal, characteristic
4	#16839567	C₂	1	2	yes	maximal, complemented
5	C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₁₂₂	—	2	1	no	—
7	C₂×Dic₂₄₄	—	2	1	no	—
8	C₂×Dic₂₄₄	—	2	4	no	—
9	C₂×Dic₂₄₄	C₂²	1	2	yes	—
10	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
11	#14519751	—	2	1	no	—
12	#14519722	—	2	4	no	—
13	#14519711	C₂²	1	1	yes	characteristic
14	#14519699	—	2	1	no	—
15	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₂²⋊D₆₁	—	4	1	no	—
19	C₂²×Dic₆₁	—	4	1	no	—
20	C₂×D₂₄₄	—	2	1	no	—
21	C₂×Dic₁₂₂	—	2	4	no	—
22	C₂×Dic₁₂₂	—	2	2	no	—
23	C₂×Dic₁₂₂	—	4	2	no	—
24	C₂×Dic₁₂₂	—	4	1	no	—
25	C₂×C₈×C₆₁	—	2	2	no	—
26	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	#11758198	—	4	1	no	—
29	C₄⋊Dic₆₁	—	2	1	no	—
30	Dic₂₄₄	—	2	4	no	—
31	Dic₂₄₄	—	2	2	no	—
32	Dic₂₄₄	D₄	1	4	yes	complemented
33	D₄₈₈	—	2	2	no	—
34	C₁₆×C₆₁	—	2	2	no	—
35	C₂³×C₆₁	D₈	1	1	yes	characteristic
36	C₂²×D₆₁	—	4	1	no	—
37	C₂×C₄×C₆₁	—	2	1	no	—
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²⋊D₆₁	—	8	2	no	—
42	C₂×Dic₆₁	—	4	4	no	—
43	C₂×Dic₆₁	—	4	2	no	—
44	C₂×Dic₆₁	—	4	1	no	—
45	D₂₄₄	—	4	1	no	—
46	D₂₄₄	—	4	1	no	—
47	Dic₁₂₂	—	2	4	no	—
48	Dic₁₂₂	—	4	4	no	—
49	Dic₁₂₂	—	4	2	no	—
50	C₈×C₆₁	—	2	1	no	—
51	C₈×C₆₁	C₂×D₄	1	2	yes	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	1	no	—
54	C₂²×C₆₁	SD₃₂	1	2	yes	complemented
55	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
56	C₂×D₆₁	—	8	1	no	—
57	C₂×D₆₁	—	8	1	no	—
58	C₂×D₆₁	—	8	1	no	—
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	#120	1	1	yes	characteristic
61	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₆₁	—	4	4	no	—
63	Dic₆₁	—	8	1	no	—
64	#2185	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	#447	1	1	yes	characteristic
67	C₂×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
68	C₂×C₆₁	#510	1	1	yes	characteristic
69	D₆₁	—	16	1	no	—
70	C₂²×Q₁₆	—	61	1	no	—
71	C₂×SD₃₂	—	61	2	no	—
72	#466	—	61	1	no	—
73	#358	—	61	2	no	—
74	C₂²⋊C₁₆	D₆₁	1	1	yes	characteristic, complemented
75	C₆₁	#2185	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	122	1	no	—
77	C₂×Q₁₆	—	61	2	no	—
78	C₂×Q₁₆	—	122	4	no	—
79	C₂×Q₁₆	—	122	1	no	—
80	C₂×D₈	—	61	1	no	—
81	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
82	#121	—	122	1	no	—
83	SD₃₂	—	122	4	no	—
84	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
85	#107	—	61	1	no	—
86	#102	—	122	1	no	—
87	C₂×Q₈	—	122	4	no	—
88	C₂×Q₈	—	122	2	no	—
89	C₂×Q₈	—	244	2	no	—
90	C₂×Q₈	—	244	1	no	—
91	C₂×D₄	—	122	1	no	—
92	C₂×D₄	—	244	1	no	—
93	C₂²×C₄	—	244	1	no	—
94	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
95	Q₁₆	—	61	4	no	—
96	Q₁₆	—	122	4	no	—
97	Q₁₆	—	122	2	no	—
98	D₈	—	122	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	D₂₄₄	1	1	yes	characteristic
101	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
102	C₄⋊C₄	—	122	1	no	—
103	C₂²⋊C₄	—	244	1	no	—
104	C₁₆	—	2	2	no	—
105	C₂³	—	244	1	no	—
106	C₂³	D₄₈₈	1	1	yes	characteristic
107	Q₈	—	122	4	no	—
108	Q₈	—	244	4	no	—
109	Q₈	—	244	2	no	—
110	D₄	—	488	2	no	—
111	D₄	—	244	1	no	—
112	D₄	—	244	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	244	1	no	—
116	C₂×C₄	—	244	4	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	D₄₈₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	D₄×D₆₁	1	2	yes	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	488	1	no	—
125	C₂²	—	488	1	no	—
126	C₂²	—	488	1	no	—
127	C₂²	#14519722	1	2	yes	complemented
128	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	488	1	no	—
131	C₄	—	244	4	no	—
132	C₄	#14519747	1	1	yes	characteristic
133	C₄	#14519805	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	—	976	1	no	—
136	C₂	#16839898	1	1	yes	characteristic
137	C₂	C₂×#14519722	1	1	yes	characteristic
138	C₂	#16840205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₄₄

C₂

yes

maximal, characteristic, complemented

C₂×#14519722

C₂

yes

maximal, complemented

#16839998

C₂

yes

maximal, characteristic

#16839567

C₂

yes

maximal, complemented

C₆₁×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

—

—

C₂²

yes

characteristic

#14519699

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

D₄

yes

complemented

D₄₈₈

—

C₁₆×C₆₁

—

C₂³×C₆₁

D₈

yes

characteristic

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

SD₃₂

yes

complemented

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

#2185

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#447

yes

characteristic

C₂×C₆₁

C₂×SD₃₂

yes

characteristic

C₂×C₆₁

#510

yes

characteristic

D₆₁

—

C₂²×Q₁₆

—

C₂×SD₃₂

—

#466

—

#358

—

C₂²⋊C₁₆

D₆₁

yes

characteristic, complemented

C₆₁

#2185

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂×D₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

#121

—

122

—

SD₃₂

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

—

#107

—

#102

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂²×C₄

—

244

—

C₂²×C₄

D₂₄₄

yes

characteristic

Q₁₆

—

Q₁₆

—

122

—

Q₁₆

—

122

—

D₈

—

122

—

C₂×C₈

—

100

C₂×C₈

D₂₄₄

yes

characteristic

101

C₂×C₈

C₂²×D₆₁

yes

characteristic

102

C₄⋊C₄

—

122

—

103

C₂²⋊C₄

—

244

—

104

C₁₆

—

105

C₂³

—

244

—

106

C₂³

D₄₈₈

yes

characteristic

107

Q₈

—

122

—

108

Q₈

—

244

—

109

Q₈

—

244

—

110

D₄

—

488

—

111

D₄

—

244

—

112

D₄

—

244

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

244

—

116

C₂×C₄

—

244

—

117

C₂×C₄

—

244

—

118

C₂×C₄

D₄₈₈

yes

characteristic

119

C₂×C₄

C₂×D₂₄₄

yes

characteristic

120

C₈

—

121

C₈

D₄×D₆₁

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

488

—

125

C₂²

—

488

—

126

C₂²

—

488

—

127

C₂²

#14519722

yes

complemented

128

C₂²

C₂×D₄₈₈

yes

characteristic

129

C₄

—

130

C₄

—

488

—

131

C₄

—

244

—

132

C₄

#14519747

yes

characteristic

133

C₄

#14519805

yes

characteristic

134

C₂

—

135

C₂

—

976

—

136

C₂

#16839898

yes

characteristic

137

C₂

C₂×#14519722

yes

characteristic

138

C₂

#16840205

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×C₂²⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂²⋊C₁₆	D₆₁
hypocenter	C₆₁	#2185
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₈₈

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×C₂²⋊C₁₆

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₂²⋊C₁₆

D₆₁

hypocenter

C₆₁

#2185

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
1. C₂²×Dic₂₄₄	C₂	semidirect
2. C₂×#14519722	C₂	semidirect
4. #16839567	C₂	semidirect
5. C₆₁×C₂²⋊C₁₆	C₂	semidirect
32. Dic₂₄₄	D₄	semidirect
54. C₂²×C₆₁	SD₃₂	semidirect
74. C₂²⋊C₁₆	D₆₁	semidirect
75. C₆₁	#2185	semidirect
127. C₂²	#14519722	semidirect

H ≅ G/N

type

1. C₂²×Dic₂₄₄

C₂

semidirect

2. C₂×#14519722

C₂

semidirect

4. #16839567

C₂

semidirect

5. C₆₁×C₂²⋊C₁₆

C₂

semidirect

32. Dic₂₄₄

D₄

semidirect

54. C₂²×C₆₁

SD₃₂

semidirect

74. C₂²⋊C₁₆

D₆₁

semidirect

75. C₆₁

#2185

semidirect

127. C₂²

#14519722

semidirect

Subgroups — #21447738

Conjugacy classes of subgroups

Complements