Group #21444571 — subgroups

Subgroups — #21444571

Order 7808.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16839815	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#16839515	C₂	1	2	yes	maximal, complemented
4	#16839513	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	—	2	1	no	—
7	C₄⋊#92418	—	2	1	no	—
8	C₄⋊#92418	C₂²	1	1	yes	characteristic
9	C₄⋊#92418	C₂²	1	2	yes	—
10	C₄⋊#92418	C₂²	1	1	yes	characteristic
11	C₂²⋊#92418	—	2	1	no	—
12	C₄×#92418	—	2	1	no	—
13	C₂×M₁₆×C₆₁	C₄	1	1	yes	characteristic
14	C₂×M₁₆×C₆₁	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	#97×C₆₁	C₄	1	2	yes	—
17	C₄×C₈×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
21	M₁₆×C₆₁	—	2	2	no	—
22	M₁₆×C₆₁	C₈	1	4	yes	complemented
23	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₆₁	D₄	1	1	yes	characteristic
27	C₄²×C₆₁	Q₈	1	1	yes	characteristic
28	C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×#92418	—	2	1	no	—
30	C₂×#92418	—	2	1	no	—
31	C₂×#92418	—	2	2	no	—
32	C₂×#92418	—	4	2	no	—
33	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	—	2	1	no	—
36	C₂×C₄×C₆₁	—	4	1	no	—
37	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₆₁	—	2	2	no	—
46	C₈×C₆₁	C₂×C₈	1	4	yes	complemented
47	#92418	—	4	1	no	—
48	#92418	—	4	2	no	—
49	#92418	—	4	1	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	—	2	1	no	—
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	2	1	no	—
57	C₄×C₆₁	C₄⋊C₈	1	2	yes	—
58	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₆₁	#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	D₆₁	1	1	yes	characteristic
70	#335	—	61	2	no	—
71	#334	—	61	2	no	—
72	C₆₁	#1565	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	Dic₆₁	1	1	yes	characteristic
74	C₂×M₁₆	Dic₆₁	1	1	yes	characteristic
75	C₂²×C₈	—	122	1	no	—
76	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
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	Dic₆₁	1	2	yes	—
83	C₄×C₈	—	122	1	no	—
84	C₄×C₈	C₂×D₆₁	1	2	yes	—
85	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
86	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	#92418	1	4	yes	complemented
90	C₂×C₈	—	122	1	no	—
91	C₂×C₈	—	122	1	no	—
92	C₂×C₈	—	122	2	no	—
93	C₂×C₈	—	244	2	no	—
94	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
95	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
96	C₄²	Dic₁₂₂	1	1	yes	characteristic
97	C₄²	D₂₄₄	1	1	yes	characteristic
98	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
99	C₄²	C₂²×D₆₁	1	1	yes	characteristic
100	C₂³	C₄⋊Dic₆₁	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×#92418	1	2	yes	—
105	C₂×C₄	#11758194	1	2	yes	—
106	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
107	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
108	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
111	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	244	1	no	—
114	C₈	—	244	1	no	—
115	C₈	—	244	2	no	—
116	C₈	C₂×#92418	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊#92418	1	2	yes	—
120	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	C₄⋊#92418	1	2	yes	—
125	C₄	C₂²×#92418	1	1	yes	characteristic
126	C₄	C₂×#11758194	1	1	yes	characteristic
127	C₄	#14519805	1	1	yes	characteristic
128	C₄	#14519810	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊#92418	1	1	yes	characteristic
131	C₂	#16839804	1	1	yes	characteristic
132	C₂	#16839811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839815

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

#16839515

C₂

yes

maximal, complemented

#16839513

C₂

yes

maximal, complemented

C₄×M₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

C₄⋊#92418

C₂²

yes

characteristic

C₂²⋊#92418

—

C₄×#92418

—

C₂×M₁₆×C₆₁

C₄

yes

characteristic

C₂×M₁₆×C₆₁

C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₄

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

C₈

yes

complemented

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

—

—

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×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×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

C₂×C₈

yes

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊C₈

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₈

yes

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊C₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₆₁

yes

characteristic

#335

—

#334

—

C₆₁

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

Dic₆₁

yes

characteristic

C₂×M₁₆

Dic₆₁

yes

characteristic

C₂²×C₈

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

Dic₆₁

yes

—

C₄×C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

—

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

M₁₆

—

M₁₆

#92418

yes

complemented

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₄²

Dic₁₂₂

yes

characteristic

C₄²

D₂₄₄

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

100

C₂³

C₄⋊Dic₆₁

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×#92418

yes

—

105

C₂×C₄

#11758194

yes

—

106

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

107

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

108

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

109

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

110

C₂×C₄

C₂×D₂₄₄

yes

characteristic

111

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

112

C₈

—

113

C₈

—

244

—

114

C₈

—

244

—

115

C₈

—

244

—

116

C₈

C₂×#92418

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊#92418

yes

—

120

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₄⋊#92418

yes

—

125

C₄

C₂²×#92418

yes

characteristic

126

C₄

C₂×#11758194

yes

characteristic

127

C₄

#14519805

yes

characteristic

128

C₄

#14519810

yes

characteristic

129

C₂

—

130

C₂

C₂×C₄⋊#92418

yes

characteristic

131

C₂

#16839804

yes

characteristic

132

C₂

#16839811

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₄²	C₂²×D₆₁
Fitting	C₄×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₆₁
hypocenter	C₆₁	#1565
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₄²

C₂²×D₆₁

Fitting

C₄×M₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₆₁

hypocenter

C₆₁

#1565

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #16839515	C₂	semidirect
4. #16839513	C₂	semidirect
22. M₁₆×C₆₁	C₈	semidirect
46. C₈×C₆₁	C₂×C₈	semidirect
72. C₆₁	#1565	semidirect
89. M₁₆	#92418	semidirect
116. C₈	C₂×#92418	semidirect

H ≅ G/N

type

3. #16839515

C₂

semidirect

4. #16839513

C₂

semidirect

22. M₁₆×C₆₁

C₈

semidirect

46. C₈×C₆₁

C₂×C₈

semidirect

72. C₆₁

#1565

semidirect

89. M₁₆

#92418

semidirect

116. C₈

C₂×#92418

semidirect

Subgroups — #21444571

Conjugacy classes of subgroups

Complements