Group #16840489 — subgroups

Subgroups — #16840489

Order 3904.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14519776	C₂	1	2	yes	maximal
2	#14519776	C₂	1	1	yes	maximal, characteristic
3	#14519768	C₂	1	1	yes	maximal, characteristic
4	#14519767	C₂	1	2	yes	maximal
5	#14519767	C₂	1	2	yes	maximal
6	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
8	#14519742	C₂	1	1	yes	maximal, characteristic
9	#14519742	C₂	1	2	yes	maximal
10	#14519742	C₂	1	1	yes	maximal, characteristic
11	#125×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₂₂	C₂²	1	2	yes	—
13	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₆₁	C₂²	1	2	yes	—
23	#11758196	C₂²	1	1	yes	characteristic
24	#11758196	C₂²	1	2	yes	—
25	#11758196	C₂²	1	2	yes	—
26	#11758196	C₂²	1	2	yes	—
27	#11758196	C₂²	1	1	yes	characteristic
28	#11758196	C₂²	1	2	yes	—
29	#11758196	C₂²	1	2	yes	—
30	C₄×Dic₆₁	C₂²	1	2	yes	—
31	C₄×Dic₆₁	C₂²	1	2	yes	—
32	C₄×Dic₆₁	C₂²	1	2	yes	—
33	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂³	1	2	yes	—
37	C₂×C₄×C₆₁	C₂³	1	2	yes	—
38	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
40	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₆₁	C₂³	1	2	yes	—
43	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₆₁	C₂³	1	2	yes	—
45	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
46	Dic₁₂₂	—	2	4	no	—
47	Dic₁₂₂	—	2	2	no	—
48	Dic₁₂₂	Q₈	1	4	yes	complemented
49	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
50	C₄×C₆₁	—	2	2	no	—
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	—	2	2	no	—
54	C₄×C₆₁	C₂×Q₈	1	2	yes	—
55	Dic₆₁	—	2	2	no	—
56	Dic₆₁	—	2	1	no	—
57	Dic₆₁	—	2	1	no	—
58	Dic₆₁	—	2	1	no	—
59	Dic₆₁	C₂×Q₈	1	4	yes	—
60	Dic₆₁	#41	1	2	yes	—
61	C₂×C₆₁	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
63	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
64	#557	—	61	1	no	maximal, Sylow
65	C₆₁	#557	1	1	yes	characteristic, complemented, Sylow
66	C₄⋊Q₈	—	61	2	no	—
67	C₄⋊Q₈	—	61	1	no	—
68	#125	—	61	2	no	—
69	#125	—	61	2	no	—
70	#125	—	61	1	no	—
71	#125	D₆₁	1	1	yes	characteristic
72	C₄×Q₈	—	61	1	no	—
73	C₄×Q₈	—	61	2	no	—
74	C₄×Q₈	—	61	1	no	—
75	C₄×Q₈	—	61	1	no	—
76	C₄×Q₈	—	61	1	no	—
77	C₂×Q₈	—	61	1	no	—
78	C₂×Q₈	—	61	2	no	—
79	C₂×Q₈	—	61	1	no	—
80	C₄⋊C₄	—	61	1	no	—
81	C₄⋊C₄	—	61	2	no	—
82	C₄⋊C₄	—	61	2	no	—
83	C₄⋊C₄	—	61	1	no	—
84	C₄⋊C₄	—	61	2	no	—
85	C₄⋊C₄	—	61	2	no	—
86	C₄⋊C₄	—	61	2	no	—
87	C₄⋊C₄	—	61	1	no	—
88	C₄⋊C₄	—	61	1	no	—
89	C₄⋊C₄	—	61	2	no	—
90	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
91	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
94	C₄²	—	61	2	no	—
95	C₄²	—	61	2	no	—
96	C₄²	—	61	1	no	—
97	C₄²	—	61	2	no	—
98	C₄²	—	61	1	no	—
99	C₄²	C₂×D₆₁	1	1	yes	characteristic
100	Q₈	—	61	4	no	—
101	Q₈	—	122	4	no	—
102	Q₈	—	122	2	no	—
103	C₂×C₄	—	61	2	no	—
104	C₂×C₄	—	61	1	no	—
105	C₂×C₄	—	61	1	no	—
106	C₂×C₄	—	61	1	no	—
107	C₂×C₄	—	61	1	no	—
108	C₂×C₄	—	61	2	no	—
109	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₆₁	1	2	yes	—
111	C₂×C₄	C₂²×D₆₁	1	2	yes	—
112	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
114	C₂²	C₂³×D₆₁	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	61	4	no	—
120	C₄	—	61	2	no	—
121	C₄	—	122	1	no	—
122	C₄	—	122	2	no	—
123	C₄	—	122	1	no	—
124	C₄	—	122	1	no	—
125	C₄	Q₈×D₆₁	1	2	yes	—
126	C₂	C₂×Q₈×D₆₁	1	1	yes	characteristic
127	C₂	#41×D₆₁	1	1	yes	characteristic
128	C₂	#14519824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14519776

C₂

yes

maximal

#14519776

C₂

yes

maximal, characteristic

#14519768

C₂

yes

maximal, characteristic

#14519767

C₂

yes

maximal

#14519767

C₂

yes

maximal

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

#14519742

C₂

yes

maximal, characteristic

#14519742

C₂

yes

maximal

#14519742

C₂

yes

maximal, characteristic

#125×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₁₂₂

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

—

C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

—

#11758196

C₂²

yes

characteristic

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

characteristic

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

Q₈

yes

complemented

C₂²×C₆₁

C₂⁴

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×Q₈

yes

—

Dic₆₁

#41

yes

—

C₂×C₆₁

C₂²×Q₈

yes

characteristic

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

—

C₆₁

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₁

yes

—

C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₄⋊C₄

C₂×D₆₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₆₁

yes

characteristic

100

Q₈

—

101

Q₈

—

122

—

102

Q₈

—

122

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×D₆₁

yes

characteristic

110

C₂×C₄

C₂²×D₆₁

yes

—

111

C₂×C₄

C₂²×D₆₁

yes

—

112

C₂×C₄

C₂²×D₆₁

yes

characteristic

113

C₂×C₄

C₂²×D₆₁

yes

characteristic

114

C₂²

C₂³×D₆₁

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

—

122

C₄

—

122

—

123

C₄

—

122

—

124

C₄

—

122

—

125

C₄

Q₈×D₆₁

yes

—

126

C₂

C₂×Q₈×D₆₁

yes

characteristic

127

C₂

#41×D₆₁

yes

characteristic

128

C₂

#14519824

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, prime radical, cosocle	C₂²×C₆₁	C₂⁴
Fitting	#125×C₆₁	C₂
hypercenter	#125	D₆₁
hypocenter	C₆₁	#557
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, prime radical, cosocle

C₂²×C₆₁

C₂⁴

Fitting

#125×C₆₁

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
48. Dic₁₂₂	Q₈	semidirect
65. C₆₁	#557	semidirect

H ≅ G/N

type

48. Dic₁₂₂

Q₈

semidirect

65. C₆₁

#557

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14519776	2
#14519768	2
#14519767	2
C₄×Dic₁₂₂	2
#14519742	2
#125×C₆₁	2
#557	61

subgroup

index

C₄×Dic₁₂₂

#125×C₆₁

Subgroups — #16840489

Conjugacy classes of subgroups

Complements

Maximal subgroups