Group #21454195 — subgroups

Subgroups — #21454195

Order 7808.

504 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#14519776	C₂	1	2	yes	maximal
2	#16839666	C₂	1	8	yes	maximal
3	#16839628	C₂	1	1	yes	maximal, characteristic
4	#16839627	C₂	1	2	yes	maximal
5	C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#382	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	C₂²	1	4	yes	—
8	#14519776	—	2	8	no	—
9	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	C₂×#11758196	C₂²	1	4	yes	—
11	C₂×C₄×Dic₆₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519684	C₂²	1	8	yes	—
15	#14519684	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	#95×C₆₁	C₂²	1	4	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	C₂³	1	4	yes	—
23	C₂²×Dic₆₁	C₂³	1	2	yes	—
24	C₂²×Dic₆₁	C₂³	1	2	yes	—
25	C₂×Dic₁₂₂	—	2	8	no	—
26	C₂×Dic₁₂₂	—	2	16	no	—
27	C₄⋊C₄×C₆₁	—	2	4	no	—
28	C₄²×C₆₁	Q₈	1	4	yes	complemented
29	C₄⋊Dic₆₁	—	2	4	no	—
30	#11758196	—	2	8	no	—
31	C₄×Dic₆₁	—	2	8	no	—
32	C₄×Dic₆₁	D₄	1	4	yes	—
33	C₄×Dic₆₁	Q₈	1	4	yes	complemented
34	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₆₁	—	2	4	no	—
36	C₂×C₄×C₆₁	—	2	4	no	—
37	C₂×C₄×C₆₁	—	2	8	no	—
38	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
40	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
42	C₂×Dic₆₁	—	2	4	no	—
43	C₂×Dic₆₁	—	2	4	no	—
44	C₂×Dic₆₁	—	2	8	no	—
45	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
46	C₂×Dic₆₁	C₂×Q₈	1	4	yes	—
47	C₂×Dic₆₁	#41	1	8	yes	—
48	Dic₁₂₂	—	4	16	no	—
49	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₆₁	C₂×#41	1	4	yes	—
53	C₄×C₆₁	—	2	4	no	—
54	C₄×C₆₁	—	4	4	no	—
55	C₄×C₆₁	C₄⋊Q₈	1	4	yes	—
56	Dic₆₁	—	2	4	no	—
57	Dic₆₁	—	4	4	no	—
58	Dic₆₁	#124	1	8	yes	—
59	#2862	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	C₂×#124	1	2	yes	—
61	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₆₁	#533	1	2	yes	—
63	C₂×C₆₁	#533	1	2	yes	—
64	C₂×C₄⋊Q₈	—	61	2	no	—
65	#391	—	61	8	no	—
66	#382	—	61	2	no	—
67	#382	—	61	1	no	—
68	#382	D₆₁	1	1	yes	characteristic
69	C₄³	—	61	1	no	—
70	C₆₁	#2862	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₈	—	61	4	no	—
72	C₄⋊Q₈	—	122	8	no	—
73	C₂×C₄⋊C₄	—	61	2	no	—
74	C₂×C₄⋊C₄	—	61	4	no	—
75	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
76	C₂×C₄²	—	61	1	no	—
77	C₂×C₄²	—	61	1	no	—
78	C₂×C₄²	—	61	4	no	—
79	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
80	#95	—	61	8	no	—
81	#95	—	61	4	no	—
82	#95	C₂×D₆₁	1	4	yes	—
83	C₂×Q₈	—	122	8	no	—
84	C₂×Q₈	—	122	16	no	—
85	C₂²×C₄	—	61	2	no	—
86	C₂²×C₄	—	61	2	no	—
87	C₂²×C₄	—	61	4	no	—
88	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
89	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	122	8	no	—
93	C₄⋊C₄	—	122	4	no	—
94	C₄²	—	61	4	no	—
95	C₄²	—	61	4	no	—
96	C₄²	—	122	8	no	—
97	C₄²	Dic₁₂₂	1	4	yes	complemented
98	C₂³	C₂³×D₆₁	1	1	yes	characteristic
99	Q₈	—	244	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	61	4	no	—
104	C₂×C₄	—	61	8	no	—
105	C₂×C₄	—	61	4	no	—
106	C₂×C₄	—	122	4	no	—
107	C₂×C₄	—	122	4	no	—
108	C₂×C₄	—	122	8	no	—
109	C₂×C₄	C₂×Dic₁₂₂	1	4	yes	—
110	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
111	C₂×C₄	D₄×D₆₁	1	2	yes	—
112	C₂×C₄	Q₈×D₆₁	1	2	yes	—
113	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
114	C₂²	C₂×Q₈×D₆₁	1	1	yes	characteristic
115	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
116	C₂²	#41×D₆₁	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	61	8	no	—
120	C₄	—	122	4	no	—
121	C₄	—	244	4	no	—
122	C₄	#14519776	1	4	yes	—
123	C₂	#124×D₆₁	1	2	yes	—
124	C₂	C₂×#14519776	1	1	yes	characteristic
125	C₂	#16840315	1	2	yes	—
126	C₂	#16840318	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519776

C₂

yes

maximal

#16839666

C₂

yes

maximal

#16839628

C₂

yes

maximal, characteristic

#16839627

C₂

yes

maximal

C₄²×Dic₆₁

C₂

yes

maximal, characteristic

C₆₁×#382

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

#14519776

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#95×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

Q₈

yes

complemented

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

Q₈

yes

complemented

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

#41

yes

—

Dic₁₂₂

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂×#41

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#124

yes

—

#2862

—

maximal, Sylow

C₂×C₆₁

C₂×#124

yes

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

#533

yes

—

C₂×C₆₁

#533

yes

—

C₂×C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic

C₄³

—

C₆₁

#2862

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

—

—

C₂×D₆₁

yes

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₄²

Dic₁₂₂

yes

complemented

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

244

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

122

—

108

C₂×C₄

—

122

—

109

C₂×C₄

C₂×Dic₁₂₂

yes

—

110

C₂×C₄

C₂×Dic₁₂₂

yes

—

111

C₂×C₄

D₄×D₆₁

yes

—

112

C₂×C₄

Q₈×D₆₁

yes

—

113

C₂²

C₂×D₄×D₆₁

yes

characteristic

114

C₂²

C₂×Q₈×D₆₁

yes

characteristic

115

C₂²

C₂²×Dic₁₂₂

yes

characteristic

116

C₂²

#41×D₆₁

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

122

—

121

C₄

—

244

—

122

C₄

#14519776

yes

—

123

C₂

#124×D₆₁

yes

—

124

C₂

C₂×#14519776

yes

characteristic

125

C₂

#16840315

yes

—

126

C₂

#16840318

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₆₁×#382	C₂
hypercenter	#382	D₆₁
hypocenter	C₆₁	#2862
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₆₁×#382

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
28. C₄²×C₆₁	Q₈	semidirect
33. C₄×Dic₆₁	Q₈	semidirect
70. C₆₁	#2862	semidirect
97. C₄²	Dic₁₂₂	semidirect

H ≅ G/N

type

28. C₄²×C₆₁

Q₈

semidirect

33. C₄×Dic₆₁

Q₈

semidirect

70. C₆₁

#2862

semidirect

97. C₄²

Dic₁₂₂

semidirect

Subgroups — #21454195

Conjugacy classes of subgroups

Complements