Group #21447514 — subgroups

Subgroups — #21447514

Order 7808.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16839746	C₂	1	2	yes	maximal
2	#16839561	C₂	1	4	yes	maximal, complemented
3	C₆₁×#536	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758194	—	2	2	no	—
5	#14519696	C₂²	1	4	yes	—
6	#14519694	—	2	2	no	—
7	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₆₁	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
10	#124×C₆₁	C₄	1	2	yes	—
11	#123×C₆₁	—	2	2	no	—
12	C₂²⋊Q₈×C₆₁	—	2	2	no	—
13	#117×C₆₁	C₄	1	1	yes	characteristic
14	#41×C₆₁	—	4	2	no	—
15	C₂×Q₈×C₆₁	—	2	1	no	—
16	C₂×Q₈×C₆₁	—	4	2	no	—
17	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
19	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₆₁	D₄	1	2	yes	—
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	D₄	1	2	yes	—
24	C₄⋊C₄×C₆₁	—	2	1	no	—
25	C₄⋊C₄×C₆₁	—	2	4	no	—
26	C₂²⋊C₄×C₆₁	—	2	1	no	—
27	C₂²⋊C₄×C₆₁	—	2	4	no	—
28	C₄²×C₆₁	C₂×C₄	1	2	yes	—
29	#11758194	—	2	4	no	—
30	#11758194	—	4	2	no	—
31	C₂×#92418	—	4	2	no	—
32	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	Q₈×C₆₁	—	4	1	no	—
35	Q₈×C₆₁	—	4	2	no	—
36	Q₈×C₆₁	—	4	2	no	—
37	D₄×C₆₁	—	4	1	no	—
38	D₄×C₆₁	—	4	2	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
48	#92418	—	4	4	no	—
49	C₂²×C₆₁	—	2	1	no	—
50	C₂²×C₆₁	—	2	2	no	—
51	C₂²×C₆₁	—	4	2	no	—
52	C₂²×C₆₁	—	4	1	no	—
53	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	4	2	no	—
57	C₄×C₆₁	—	4	2	no	—
58	C₄×C₆₁	#99	1	2	yes	—
59	#2135	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	—	2	1	no	—
61	C₂×C₆₁	—	4	1	no	—
62	C₂×C₆₁	—	4	2	no	—
63	C₂×C₆₁	C₂×#99	1	1	yes	characteristic
64	#536	D₆₁	1	1	yes	characteristic
65	#413	—	61	2	no	—
66	#356	—	61	4	no	—
67	C₆₁	#2135	1	1	yes	characteristic, complemented, Sylow
68	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
69	C₂²×Q₈	Dic₆₁	1	1	yes	characteristic
70	C₂×M₁₆	—	122	2	no	—
71	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
72	#124	Dic₆₁	1	2	yes	—
73	#123	—	2	2	no	—
74	C₂²⋊Q₈	—	2	2	no	—
75	#117	Dic₆₁	1	1	yes	characteristic
76	#101	—	61	4	no	—
77	#100	—	122	2	no	—
78	#41	—	4	2	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
82	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
83	C₂×D₄	C₂×Dic₆₁	1	1	yes	characteristic
84	C₂×D₄	C₂²⋊D₆₁	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
88	M₁₆	—	122	4	no	—
89	M₁₆	—	244	2	no	—
90	C₂×C₈	—	244	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₂²⋊C₄	—	2	1	no	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₄²	C₂×Dic₆₁	1	2	yes	—
96	C₂³	C₂²⋊Dic₆₁	1	2	yes	—
97	C₂³	C₂²⋊Dic₆₁	1	1	yes	characteristic
98	Q₈	—	4	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	1	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
112	C₈	—	244	4	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	#14519692	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	4	1	no	—
125	C₂	—	4	2	no	—
126	C₂	C₂×#14519692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839746

C₂

yes

maximal

#16839561

C₂

yes

maximal, complemented

C₆₁×#536

C₂

yes

maximal, characteristic

C₂×#11758194

—

#14519696

C₂²

yes

—

#14519694

—

C₂×#41×C₆₁

C₂²

yes

characteristic

C₂²×Q₈×C₆₁

C₄

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

—

#124×C₆₁

C₄

yes

—

#123×C₆₁

—

C₂²⋊Q₈×C₆₁

—

#117×C₆₁

C₄

yes

characteristic

#41×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

C₂×C₄

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#99

yes

—

#2135

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#99

yes

characteristic

#536

D₆₁

yes

characteristic

#413

—

#356

—

C₆₁

#2135

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

C₂²×Q₈

Dic₆₁

yes

characteristic

C₂×M₁₆

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#124

Dic₆₁

yes

—

#123

—

C₂²⋊Q₈

—

#117

Dic₆₁

yes

characteristic

#101

—

#100

—

122

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂×Q₈

C₂²×D₆₁

yes

characteristic

C₂×D₄

C₂×Dic₆₁

yes

characteristic

C₂×D₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

—

M₁₆

—

122

—

M₁₆

—

244

—

C₂×C₈

—

244

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₆₁

yes

—

C₂³

C₂²⋊Dic₆₁

yes

—

C₂³

C₂²⋊Dic₆₁

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₆₁

yes

characteristic

110

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

112

C₈

—

244

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#14519692

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂×#14519692

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14519692
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₆₁
Fitting	C₆₁×#536	C₂
socle	C₂×C₆₁	C₂×#99
hypercenter	#536	D₆₁
hypocenter	C₆₁	#2135
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

subgroup

G/H

center

C₂

C₂×#14519692

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₆₁

Fitting

C₆₁×#536

C₂

socle

C₂×C₆₁

C₂×#99

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

N	H ≅ G/N	type
2. #16839561	C₂	semidirect
67. C₆₁	#2135	semidirect

H ≅ G/N

type

2. #16839561

C₂

semidirect

67. C₆₁

#2135

semidirect

Subgroups — #21447514

Conjugacy classes of subgroups

Complements