Group #21454759 — subgroups

Subgroups — #21454759

Order 7808.

348 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 189 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	1	yes	maximal, characteristic
2	#16839890	C₂	1	8	yes	maximal, complemented
3	#16839788	C₂	1	2	yes	maximal
4	#16839786	C₂	1	1	yes	maximal, characteristic
5	#16839717	C₂	1	2	yes	maximal
6	C₂×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#11758194	C₂²	1	2	yes	—
8	#14519777	C₄	1	2	yes	—
9	#14519769	C₄	1	2	yes	—
10	#14519756	C₄	1	4	yes	—
11	C₄×Dic₁₂₂	C₂²	1	4	yes	—
12	#14519736	C₂²	1	2	yes	—
13	C₄⋊#92418	C₂²	1	4	yes	—
14	#14519706	C₂²	1	8	yes	—
15	#14519687	C₂²	1	4	yes	—
16	C₈×Dic₆₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	#97×C₆₁	C₂²	1	4	yes	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
26	C₂×C₈×C₆₁	C₂³	1	2	yes	—
27	C₂×C₈×C₆₁	C₂³	1	2	yes	—
28	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₂³	1	2	yes	—
31	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
34	#11758196	C₂×C₄	1	8	yes	—
35	C₄×Dic₆₁	C₂³	1	4	yes	—
36	#11758194	—	2	4	no	—
37	C₂×#92418	C₂³	1	4	yes	—
38	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	4	2	no	—
40	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₁	#41	1	2	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	C₂²×C₄	1	8	yes	—
47	Dic₁₂₂	—	4	4	no	—
48	C₈×C₆₁	—	2	2	no	—
49	C₈×C₆₁	C₂×Q₈	1	4	yes	—
50	#92418	—	2	4	no	—
51	C₂²×C₆₁	—	2	2	no	—
52	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
53	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	C₄×Q₈	1	2	yes	—
57	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
59	Dic₆₁	—	2	8	no	—
60	#2961	—	61	1	no	maximal, Sylow
61	C₂×C₆₁	—	2	2	no	—
62	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₆₁	#568	1	2	yes	—
64	#533	—	61	1	no	—
65	#446	—	61	8	no	—
66	#424	—	61	2	no	—
67	#423	—	61	1	no	—
68	#405	—	61	2	no	—
69	C₂×#97	D₆₁	1	1	yes	characteristic
70	C₆₁	#2961	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	61	2	no	—
72	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
74	C₄⋊Q₈	—	61	2	no	—
75	#125	—	61	2	no	—
76	C₂²⋊Q₈	—	61	4	no	—
77	C₄×Q₈	—	61	4	no	—
78	#117	—	61	2	no	—
79	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
80	C₄⋊C₈	—	61	8	no	—
81	C₄⋊C₈	—	61	4	no	—
82	#97	—	61	4	no	—
83	#97	C₂×D₆₁	1	4	yes	—
84	C₄×C₈	—	61	4	no	—
85	C₂×Q₈	—	61	4	no	—
86	C₂²×C₄	C₄×D₆₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
88	M₁₆	—	122	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	61	4	no	—
91	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
92	C₂×C₈	C₂²×D₆₁	1	2	yes	—
93	C₂×C₈	C₂²×D₆₁	1	2	yes	—
94	C₄⋊C₄	—	61	8	no	—
95	C₄⋊C₄	—	61	4	no	—
96	C₄⋊C₄	—	61	4	no	—
97	C₂²⋊C₄	—	61	4	no	—
98	C₄²	—	61	4	no	—
99	C₄²	C₄×D₆₁	1	1	yes	characteristic
100	C₄²	C₄×D₆₁	1	1	yes	characteristic
101	C₄²	C₂²×D₆₁	1	2	yes	—
102	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
103	Q₈	—	244	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	61	8	no	—
106	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
110	C₂×C₄	#11758222	1	2	yes	—
111	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	122	4	no	—
114	C₈	C₂×Dic₁₂₂	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₁₂₂	1	2	yes	—
117	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	122	8	no	—
121	C₄	C₄×Dic₁₂₂	1	2	yes	—
122	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
123	C₄	C₂×#11758222	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
126	C₂	#16840528	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16839890

C₂

yes

maximal, complemented

#16839788

C₂

yes

maximal

#16839786

C₂

yes

maximal, characteristic

#16839717

C₂

yes

maximal

C₂×#97×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

#14519777

C₄

yes

—

#14519769

C₄

yes

—

#14519756

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

C₂²⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

#2961

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

#568

yes

—

—

—

—

—

—

C₂×#97

D₆₁

yes

characteristic

C₆₁

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

C₂×D₆₁

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

complemented

C₂×C₈

C₂²×D₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₆₁

yes

characteristic

100

C₄²

C₄×D₆₁

yes

characteristic

101

C₄²

C₂²×D₆₁

yes

—

102

C₂³

C₂×C₄×D₆₁

yes

characteristic

103

Q₈

—

244

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₁₂₂

yes

—

107

C₂×C₄

C₂×C₄×D₆₁

yes

—

108

C₂×C₄

C₂×C₄×D₆₁

yes

—

109

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

110

C₂×C₄

#11758222

yes

—

111

C₂×C₄

C₂³×D₆₁

yes

characteristic

112

C₈

—

113

C₈

—

122

—

114

C₈

C₂×Dic₁₂₂

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₁₂₂

yes

—

117

C₂²

C₂²×C₄×D₆₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

122

—

121

C₄

C₄×Dic₁₂₂

yes

—

122

C₄

C₂²×Dic₁₂₂

yes

characteristic

123

C₄

C₂×#11758222

yes

characteristic

124

C₂

—

125

C₂

C₂×C₄×Dic₁₂₂

yes

characteristic

126

C₂

#16840528

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×#97×C₆₁	C₂
hypercenter	C₂×#97	D₆₁
hypocenter	C₆₁	#2961
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₂×#97×C₆₁

C₂

hypercenter

C₂×#97

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #16839890	C₂	semidirect
25. C₂×C₈×C₆₁	Q₈	semidirect
70. C₆₁	#2961	semidirect
91. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #16839890

C₂

semidirect

25. C₂×C₈×C₆₁

Q₈

semidirect

70. C₆₁

#2961

semidirect

91. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21454759

Conjugacy classes of subgroups

Complements