Group #21461370 — subgroups

Subgroups — #21461370

Order 7808.

180 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 92 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	#16840778	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
3	#16840625	C₂	1	2	yes	maximal, complemented
4	#16840623	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92420	—	2	1	no	—
7	C₄⋊#92420	—	2	1	no	—
8	C₄⋊#92420	C₂²	1	2	yes	—
9	C₄⋊#92420	C₂²	1	1	yes	characteristic
10	C₄⋊#92420	C₂²	1	1	yes	characteristic
11	C₂²⋊#92420	—	2	1	no	—
12	C₄×#92420	—	2	1	no	—
13	C₂×#11758194	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
15	#14519687	C₄	1	2	yes	—
16	C₈×Dic₆₁	C₂²	1	2	yes	—
17	C₂×M₁₆×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂²×Dic₆₁	Q₈	1	1	yes	characteristic
21	C₂×#92420	—	2	1	no	—
22	C₂×#92420	—	2	1	no	—
23	C₂×#92420	—	2	2	no	—
24	C₂×#92420	—	4	2	no	—
25	M₁₆×C₆₁	C₈	1	4	yes	complemented
26	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₆₁	D₄	1	1	yes	characteristic
29	C₄×Dic₆₁	Q₈	1	1	yes	characteristic
30	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	#11758194	—	2	2	no	—
32	C₂×#92418	C₂×C₄	1	2	yes	—
33	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
37	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₆₁	—	2	1	no	—
39	C₂×Dic₆₁	—	2	1	no	—
40	C₂×Dic₆₁	—	4	1	no	—
41	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
45	#92420	—	4	1	no	—
46	#92420	—	4	2	no	—
47	#92420	—	4	1	no	—
48	C₈×C₆₁	C₂×C₈	1	4	yes	complemented
49	#92418	—	2	2	no	—
50	C₂²×C₆₁	—	2	1	no	—
51	C₂²×C₆₁	—	2	1	no	—
52	C₂²×C₆₁	C₄⋊C₈	1	2	yes	—
53	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₆₁	C₄⋊C₈	1	2	yes	—
55	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
57	Dic₆₁	—	2	1	no	—
58	Dic₆₁	—	2	1	no	—
59	Dic₆₁	—	2	1	no	—
60	Dic₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₆₁	#137	1	1	yes	characteristic
62	#1565	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	—	2	2	no	—
64	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₆₁	#428	1	1	yes	characteristic
66	C₂×C₆₁	#430	1	1	yes	characteristic
67	#431	—	61	1	no	—
68	C₂×C₄⋊C₈	—	61	1	no	—
69	C₄×M₁₆	—	61	1	no	—
70	#335	—	61	2	no	—
71	#334	—	61	2	no	—
72	C₆₁	#1565	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	61	1	no	—
74	C₂×M₁₆	#6923	1	1	yes	characteristic
75	C₂²×C₈	—	122	1	no	—
76	C₂×C₄²	—	61	1	no	—
77	C₄⋊C₈	—	61	1	no	—
78	C₄⋊C₈	—	61	2	no	—
79	C₄⋊C₈	—	61	1	no	—
80	C₄⋊C₈	—	122	1	no	—
81	C₂²⋊C₈	—	122	1	no	—
82	#97	—	61	2	no	—
83	C₄×C₈	—	61	2	no	—
84	C₄×C₈	—	122	1	no	—
85	C₂²×C₄	—	61	1	no	—
86	C₂²×C₄	—	61	1	no	—
87	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
88	M₁₆	—	122	2	no	—
89	M₁₆	#92420	1	4	yes	complemented
90	C₂×C₈	—	61	2	no	—
91	C₂×C₈	—	122	1	no	—
92	C₂×C₈	—	122	1	no	—
93	C₂×C₈	—	122	2	no	—
94	C₂×C₈	—	244	2	no	—
95	C₂×C₈	C₂×#6923	1	2	yes	—
96	C₄²	—	61	1	no	—
97	C₄²	—	61	1	no	—
98	C₄²	—	61	1	no	—
99	C₄²	—	61	1	no	—
100	C₂³	C₄⋊#6923	1	1	yes	characteristic
101	C₂×C₄	—	61	1	no	—
102	C₂×C₄	—	61	1	no	—
103	C₂×C₄	—	61	1	no	—
104	C₂×C₄	—	61	1	no	—
105	C₂×C₄	—	122	1	no	—
106	C₂×C₄	—	122	1	no	—
107	C₂×C₄	—	244	1	no	—
108	C₂×C₄	#11758213	1	2	yes	—
109	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
110	C₂×C₄	C₂×#92420	1	2	yes	—
111	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
112	C₈	—	122	2	no	—
113	C₈	—	244	1	no	—
114	C₈	—	244	1	no	—
115	C₈	—	244	2	no	—
116	C₈	C₂×#92420	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊#92420	1	2	yes	—
120	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
121	C₄	—	61	1	no	—
122	C₄	—	61	1	no	—
123	C₄	—	122	1	no	—
124	C₄	—	122	1	no	—
125	C₄	—	122	1	no	—
126	C₄	C₄⋊#92420	1	2	yes	—
127	C₄	C₂²×#92420	1	1	yes	characteristic
128	C₄	C₂×#11758213	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊#92420	1	1	yes	characteristic
131	C₂	#16840773	1	1	yes	characteristic
132	C₂	#16840777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840778

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

C₂

yes

maximal, characteristic

#16840625

C₂

yes

maximal, complemented

#16840623

C₂

yes

maximal, complemented

M₁₆×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

—

C₄⋊#92420

—

C₄⋊#92420

C₂²

yes

—

C₄⋊#92420

C₂²

yes

characteristic

C₄⋊#92420

C₂²

yes

—

—

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519687

C₄

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×M₁₆×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

Q₈

yes

—

—

—

—

M₁₆×C₆₁

C₈

yes

complemented

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

Q₈

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

#11758194

—

C₂×#92418

C₂×C₄

yes

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×Q₈

yes

—

—

—

C₈×C₆₁

C₂×C₈

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊C₈

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₄⋊C₈

yes

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊C₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#335

—

#334

—

C₆₁

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

#6923

yes

characteristic

C₂²×C₈

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

—

C₄×C₈

—

C₄×C₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

characteristic

M₁₆

—

122

—

M₁₆

#92420

yes

complemented

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₂×#6923

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₂³

C₄⋊#6923

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

122

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

244

—

108

C₂×C₄

#11758213

yes

—

109

C₂×C₄

C₄⋊#6923

yes

characteristic

110

C₂×C₄

C₂×#92420

yes

—

111

C₂×C₄

C₂²×#6923

yes

characteristic

112

C₈

—

122

—

113

C₈

—

244

—

114

C₈

—

244

—

115

C₈

—

244

—

116

C₈

C₂×#92420

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊#92420

yes

—

120

C₂²

C₂×C₄⋊#6923

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

122

—

124

C₄

—

122

—

125

C₄

—

122

—

126

C₄

C₄⋊#92420

yes

—

127

C₄

C₂²×#92420

yes

characteristic

128

C₄

C₂×#11758213

yes

characteristic

129

C₂

—

130

C₂

C₂×C₄⋊#92420

yes

characteristic

131

C₂

#16840773

yes

characteristic

132

C₂

#16840777

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×M₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₂×M₁₆	#6923
hypocenter	C₆₁	#1565
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂²

C₂×C₄⋊#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₂×M₁₆×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₄⋊C₄

C₂×M₁₆

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #16840625	C₂	semidirect
4. #16840623	C₂	semidirect
25. M₁₆×C₆₁	C₈	semidirect
48. C₈×C₆₁	C₂×C₈	semidirect
72. C₆₁	#1565	semidirect
89. M₁₆	#92420	semidirect
116. C₈	C₂×#92420	semidirect

H ≅ G/N

type

3. #16840625

C₂

semidirect

4. #16840623

C₂

semidirect

25. M₁₆×C₆₁

C₈

semidirect

48. C₈×C₆₁

C₂×C₈

semidirect

72. C₆₁

#1565

semidirect

89. M₁₆

#92420

semidirect

116. C₈

C₂×#92420

semidirect

Subgroups — #21461370

Conjugacy classes of subgroups

Complements