Group #16840370 — subgroups

Subgroups — #16840370

Order 3904.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14519775	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519772	C₂	1	2	yes	maximal
3	#14519771	C₂	1	2	yes	maximal
4	#14519769	C₂	1	2	yes	maximal, complemented
5	#14519768	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519767	C₂	1	2	yes	maximal, complemented
7	#14519757	C₂	1	1	yes	maximal, characteristic
8	#14519753	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#14519734	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₆₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
17	C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₄²×C₆₁	C₂²	1	2	yes	—
19	#11758198	C₂²	1	1	yes	characteristic
20	#11758198	C₂²	1	1	yes	characteristic
21	#11758198	C₂²	1	2	yes	—
22	#11758198	C₂²	1	2	yes	—
23	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₆₁	C₂²	1	2	yes	—
25	C₄⋊Dic₆₁	C₂²	1	2	yes	—
26	#11758196	C₂²	1	1	yes	characteristic
27	#11758196	C₂²	1	1	yes	characteristic
28	#11758196	C₂²	1	2	yes	—
29	#11758196	C₂²	1	2	yes	—
30	#11758196	C₂²	1	2	yes	—
31	#11758196	C₂²	1	2	yes	—
32	#11758196	C₂²	1	2	yes	—
33	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₆₁	C₂²	1	2	yes	—
35	C₂²×D₆₁	C₂³	1	1	yes	characteristic
36	Q₈×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄×C₆₁	C₂³	1	2	yes	—
41	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₆₁	C₂³	1	2	yes	—
45	C₂×Dic₆₁	C₂³	1	2	yes	—
46	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
47	C₄×D₆₁	—	2	2	no	—
48	Dic₁₂₂	—	2	2	no	—
49	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₆₁	—	2	1	no	—
51	C₂×D₆₁	—	2	1	no	—
52	C₂×D₆₁	—	2	1	no	—
53	C₄×C₆₁	—	2	2	no	—
54	C₄×C₆₁	—	2	1	no	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	—	2	1	no	—
57	C₄×C₆₁	#41	1	2	yes	—
58	Dic₆₁	—	2	2	no	—
59	Dic₆₁	—	2	1	no	—
60	Dic₆₁	—	2	1	no	—
61	Dic₆₁	—	2	1	no	—
62	Dic₆₁	—	2	2	no	—
63	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
64	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
65	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₆₁	—	4	1	no	—
67	#541	—	61	1	no	maximal, Sylow
68	C₆₁	#541	1	1	yes	characteristic, complemented, Sylow
69	C₄⋊Q₈	—	61	1	no	—
70	#126	—	61	2	no	—
71	#126	—	61	2	no	—
72	#125	—	61	2	no	—
73	#125	—	61	1	no	—
74	#125	—	61	2	no	—
75	C₂²⋊Q₈	—	61	1	no	—
76	C₂²⋊Q₈	—	61	1	no	—
77	C₄×Q₈	—	61	1	no	—
78	C₄×Q₈	D₆₁	1	1	yes	characteristic, complemented
79	#117	—	61	1	no	—
80	C₂×Q₈	—	61	1	no	—
81	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
82	C₂²×C₄	—	61	1	no	—
83	C₄⋊C₄	—	61	1	no	—
84	C₄⋊C₄	—	61	1	no	—
85	C₄⋊C₄	—	61	2	no	—
86	C₄⋊C₄	—	61	2	no	—
87	C₄⋊C₄	—	61	2	no	—
88	C₄⋊C₄	—	61	2	no	—
89	C₄⋊C₄	—	61	2	no	—
90	C₄⋊C₄	—	61	1	no	—
91	C₄⋊C₄	—	61	2	no	—
92	C₄⋊C₄	—	61	2	no	—
93	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
95	C₂²⋊C₄	—	61	1	no	—
96	C₂²⋊C₄	—	61	1	no	—
97	C₂²⋊C₄	—	61	2	no	—
98	C₂²⋊C₄	—	61	2	no	—
99	C₄²	—	61	1	no	—
100	C₄²	—	61	2	no	—
101	C₄²	C₂×D₆₁	1	1	yes	characteristic
102	C₄²	C₂×D₆₁	1	2	yes	—
103	C₂³	—	61	1	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	122	2	no	—
106	C₂×C₄	—	61	1	no	—
107	C₂×C₄	—	61	1	no	—
108	C₂×C₄	—	61	2	no	—
109	C₂×C₄	—	61	2	no	—
110	C₂×C₄	—	61	1	no	—
111	C₂×C₄	—	122	2	no	—
112	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₆₁	1	2	yes	—
115	C₂×C₄	C₂²×D₆₁	1	2	yes	—
116	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
117	C₂²	—	122	1	no	—
118	C₂²	—	122	1	no	—
119	C₂²	—	122	1	no	—
120	C₂²	C₂³×D₆₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	122	1	no	—
126	C₄	—	122	2	no	—
127	C₄	—	122	1	no	—
128	C₄	—	122	1	no	—
129	C₄	—	122	2	no	—
130	C₄	#11758222	1	2	yes	—
131	C₂	—	244	1	no	—
132	C₂	C₂×#11758222	1	1	yes	characteristic
133	C₂	#14519824	1	1	yes	characteristic
134	C₂	#14519825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14519775

C₂

yes

maximal, characteristic, complemented

#14519772

C₂

yes

maximal

#14519771

C₂

yes

maximal

#14519769

C₂

yes

maximal, complemented

#14519768

C₂

yes

maximal, characteristic, complemented

#14519767

C₂

yes

maximal, complemented

#14519757

C₂

yes

maximal, characteristic

#14519753

C₂

yes

maximal, characteristic

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

#14519734

C₂

yes

maximal, characteristic

C₄×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₆₁

C₂²

yes

characteristic

C₂×C₄×D₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

characteristic

C₄²×C₆₁

C₂²

yes

—

#11758198

C₂²

yes

characteristic

#11758198

C₂²

yes

characteristic

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

—

#11758196

C₂²

yes

characteristic

#11758196

C₂²

yes

characteristic

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₆₁

C₂²

yes

—

C₂²×D₆₁

C₂³

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₂³

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×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₄×D₆₁

—

Dic₁₂₂

—

C₂²×C₆₁

C₂⁴

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#41

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

characteristic

D₆₁

—

#541

—

maximal, Sylow

C₆₁

#541

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₆₁

yes

characteristic, complemented

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₆₁

yes

characteristic

C₂²×C₄

—

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

characteristic

C₄⋊C₄

C₂×D₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×D₆₁

yes

characteristic

102

C₄²

C₂×D₆₁

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

122

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

C₂²×D₆₁

yes

characteristic

113

C₂×C₄

C₂²×D₆₁

yes

characteristic

114

C₂×C₄

C₂²×D₆₁

yes

—

115

C₂×C₄

C₂²×D₆₁

yes

—

116

C₂×C₄

C₂²×D₆₁

yes

characteristic

117

C₂²

—

122

—

118

C₂²

—

122

—

119

C₂²

—

122

—

120

C₂²

C₂³×D₆₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

122

—

126

C₄

—

122

—

127

C₄

—

122

—

128

C₄

—

122

—

129

C₄

—

122

—

130

C₄

#11758222

yes

—

131

C₂

—

244

—

132

C₂

C₂×#11758222

yes

characteristic

133

C₂

#14519824

yes

characteristic

134

C₂

#14519825

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₄×Q₈×C₆₁	C₂
prime radical	C₂²×D₆₁	C₂³
hypercenter	C₄×Q₈	D₆₁
hypocenter	C₆₁	#541
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, cosocle

C₂²×C₆₁

C₂⁴

Fitting

C₄×Q₈×C₆₁

C₂

prime radical

C₂²×D₆₁

C₂³

hypercenter

C₄×Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #14519775	C₂	semidirect
4. #14519769	C₂	semidirect
5. #14519768	C₂	semidirect
6. #14519767	C₂	semidirect
9. C₄×Dic₁₂₂	C₂	semidirect
11. C₄×Q₈×C₆₁	C₂	semidirect
68. C₆₁	#541	semidirect
78. C₄×Q₈	D₆₁	semidirect

H ≅ G/N

type

1. #14519775

C₂

semidirect

4. #14519769

C₂

semidirect

5. #14519768

C₂

semidirect

6. #14519767

C₂

semidirect

9. C₄×Dic₁₂₂

C₂

semidirect

11. C₄×Q₈×C₆₁

C₂

semidirect

68. C₆₁

#541

semidirect

78. C₄×Q₈

D₆₁

semidirect

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

subgroup	index
#14519775	2
#14519772	2
#14519771	2
#14519769	2
#14519768	2
#14519767	2
#14519757	2
#14519753	2
C₄×Dic₁₂₂	2
#14519734	2
C₄×Q₈×C₆₁	2
#541	61

subgroup

index

C₄×Dic₁₂₂

C₄×Q₈×C₆₁

Subgroups — #16840370

Conjugacy classes of subgroups

Complements

Maximal subgroups