Group #21456564 — subgroups

Subgroups — #21456564

Order 7808.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16840319	C₂	1	2	yes	maximal
2	#16840160	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840113	C₂	1	2	yes	maximal, complemented
5	#16840020	C₂	1	4	yes	maximal
6	#16839870	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₄₄	C₂²	1	4	yes	—
9	#14519777	C₂²	1	2	yes	—
10	#14519777	C₂²	1	2	yes	—
11	#14519769	—	2	2	no	—
12	#14519756	—	2	2	no	—
13	#14519756	C₂²	1	4	yes	—
14	C₄×Dic₁₂₂	—	2	2	no	—
15	C₄×Dic₁₂₂	C₂²	1	4	yes	—
16	#14519736	—	2	2	no	—
17	#14519709	C₂²	1	4	yes	—
18	#14519702	C₂²	1	8	yes	—
19	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	#97×C₆₁	C₂²	1	2	yes	—
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	D₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	C₂³	1	4	yes	—
28	M₁₆×C₆₁	—	2	4	no	—
29	C₂×C₈×C₆₁	C₂³	1	2	yes	—
30	C₂×C₈×C₆₁	C₂³	1	2	yes	—
31	C₄²×C₆₁	D₄	1	2	yes	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₆₁	—	2	4	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	—	2	2	no	—
37	C₄⋊Dic₆₁	—	2	2	no	—
38	C₄⋊Dic₆₁	C₂³	1	4	yes	—
39	#11758196	—	2	4	no	—
40	#11758196	—	2	4	no	—
41	C₄×Dic₆₁	—	2	4	no	—
42	Dic₂₄₄	—	2	8	no	—
43	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	2	4	no	—
52	Dic₁₂₂	—	2	8	no	—
53	Dic₁₂₂	—	4	4	no	—
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	#41	1	4	yes	—
56	C₂²×C₆₁	—	2	1	no	—
57	C₂²×C₆₁	—	2	2	no	—
58	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	—	2	2	no	—
61	C₄×C₆₁	#137	1	4	yes	—
62	C₄×C₆₁	C₂×#41	1	2	yes	—
63	Dic₆₁	—	4	4	no	—
64	Dic₆₁	—	4	4	no	—
65	#3185	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	—	4	1	no	—
67	C₂×C₆₁	#532	1	1	yes	characteristic
68	C₂×C₆₁	C₂×#137	1	2	yes	—
69	#533	—	61	2	no	—
70	#498	—	61	1	no	—
71	#494	—	61	1	no	—
72	#487	—	61	2	no	—
73	#470	—	61	4	no	—
74	#441	—	61	4	no	—
75	C₄×M₁₆	D₆₁	1	1	yes	characteristic
76	C₆₁	#3185	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	61	4	no	—
78	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
79	C₄⋊Q₈	—	61	2	no	—
80	C₄⋊Q₈	—	61	2	no	—
81	#125	—	122	2	no	—
82	C₂²⋊Q₈	—	61	4	no	—
83	C₂²⋊Q₈	—	122	2	no	—
84	C₄×Q₈	—	61	4	no	—
85	C₄×Q₈	—	122	2	no	—
86	#117	—	122	2	no	—
87	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
88	#106	—	61	4	no	—
89	#103	—	61	8	no	—
90	#97	C₂×D₆₁	1	2	yes	—
91	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
93	C₂×Q₈	—	61	4	no	—
94	C₂×Q₈	—	122	2	no	—
95	C₂²×C₄	D₂₄₄	1	2	yes	—
96	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
97	Q₁₆	—	122	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂²×D₆₁	1	2	yes	—
100	C₂×C₈	C₂²×D₆₁	1	2	yes	—
101	C₄⋊C₄	—	61	4	no	—
102	C₄⋊C₄	—	122	4	no	—
103	C₄⋊C₄	—	122	4	no	—
104	C₄⋊C₄	—	122	2	no	—
105	C₄⋊C₄	—	122	2	no	—
106	C₄⋊C₄	—	122	2	no	—
107	C₂²⋊C₄	—	122	4	no	—
108	C₄²	—	122	4	no	—
109	C₄²	D₂₄₄	1	2	yes	—
110	C₄²	C₂²×D₆₁	1	1	yes	characteristic
111	C₄²	C₂²×D₆₁	1	1	yes	characteristic
112	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
113	Q₈	—	122	8	no	—
114	Q₈	—	244	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	122	4	no	—
118	C₂×C₄	—	122	4	no	—
119	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
120	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
122	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#11758222	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	244	4	no	—
131	C₄	—	244	4	no	—
132	C₄	#14519810	1	4	yes	—
133	C₄	C₂×#11758222	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#16840312	1	1	yes	characteristic
136	C₂	C₂×#14519810	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal

#16840160

C₂

yes

maximal, characteristic, complemented

#16840142

C₂

yes

maximal, characteristic, complemented

#16840113

C₂

yes

maximal, complemented

#16840020

C₂

yes

maximal

#16839870

C₂

yes

maximal, complemented

C₄×M₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

—

#14519777

C₂²

yes

—

#14519777

C₂²

yes

—

#14519769

—

#14519756

—

#14519756

C₂²

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

—

#14519709

C₂²

yes

—

#14519702

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#137

yes

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

Dic₆₁

—

#3185

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#532

yes

characteristic

C₂×C₆₁

C₂×#137

yes

—

—

—

—

—

—

—

C₄×M₁₆

D₆₁

yes

characteristic

C₆₁

#3185

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄²

C₂×D₆₁

yes

—

—

C₂×D₆₁

yes

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

Q₁₆

—

122

—

M₁₆

—

C₂×C₈

C₂²×D₆₁

yes

—

100

C₂×C₈

C₂²×D₆₁

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

122

—

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

—

122

—

106

C₄⋊C₄

—

122

—

107

C₂²⋊C₄

—

122

—

108

C₄²

—

122

—

109

C₄²

D₂₄₄

yes

—

110

C₄²

C₂²×D₆₁

yes

characteristic

111

C₄²

C₂²×D₆₁

yes

characteristic

112

C₂³

C₂×D₂₄₄

yes

characteristic

113

Q₈

—

122

—

114

Q₈

—

244

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

122

—

118

C₂×C₄

—

122

—

119

C₂×C₄

C₂×D₂₄₄

yes

—

120

C₂×C₄

C₂×D₂₄₄

yes

characteristic

121

C₂×C₄

C₂×D₂₄₄

yes

—

122

C₂×C₄

C₂³×D₆₁

yes

characteristic

123

C₈

—

124

C₈

#11758222

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₂₄₄

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

244

—

131

C₄

—

244

—

132

C₄

#14519810

yes

—

133

C₄

C₂×#11758222

yes

—

134

C₂

—

135

C₂

#16840312

yes

characteristic

136

C₂

C₂×#14519810

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₄×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₄×M₁₆	D₆₁
hypocenter	C₆₁	#3185
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₄×M₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₄×M₁₆

D₆₁

hypocenter

C₆₁

#3185

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #16840160	C₂	semidirect
3. #16840142	C₂	semidirect
4. #16840113	C₂	semidirect
6. #16839870	C₂	semidirect
76. C₆₁	#3185	semidirect

H ≅ G/N

type

2. #16840160

C₂

semidirect

3. #16840142

C₂

semidirect

4. #16840113

C₂

semidirect

6. #16839870

C₂

semidirect

76. C₆₁

#3185

semidirect

Subgroups — #21456564

Conjugacy classes of subgroups

Complements