Group #16840337 — subgroups

Subgroups — #16840337

Order 3904.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#11758226	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#14519775	C₂	1	1	yes	maximal, characteristic, complemented
4	#14519766	C₂	1	1	yes	maximal, characteristic, complemented
5	#14519765	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519762	C₂	1	4	yes	maximal, complemented
7	#14519755	C₂	1	4	yes	maximal, complemented
8	#14519734	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
12	#11758226	—	2	4	no	—
13	Q₈×D₆₁	—	2	4	no	—
14	C₂×D₂₄₄	C₂²	1	2	yes	—
15	C₂×D₂₄₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₆₁	C₂²	1	2	yes	—
17	C₂×C₄×D₆₁	C₂²	1	2	yes	—
18	C₂×C₄×D₆₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₂₂	C₂²	1	2	yes	—
20	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
22	C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	#11758198	C₂²	1	2	yes	—
24	#11758198	C₂²	1	4	yes	—
25	#11758198	C₂²	1	4	yes	—
26	#11758196	C₂²	1	2	yes	—
27	#11758196	C₂²	1	4	yes	—
28	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
29	C₂²×D₆₁	C₂³	1	2	yes	—
30	C₂²×D₆₁	C₂³	1	1	yes	characteristic
31	Q₈×C₆₁	—	2	2	no	—
32	Q₈×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	C₂³	1	2	yes	—
34	C₂×C₄×C₆₁	C₂³	1	2	yes	—
35	C₂×C₄×C₆₁	C₂³	1	2	yes	—
36	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×Dic₆₁	C₂³	1	2	yes	—
38	C₂×Dic₆₁	C₂³	1	2	yes	—
39	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
40	D₂₄₄	—	2	4	no	—
41	D₂₄₄	—	2	2	no	—
42	C₄×D₆₁	—	2	4	no	—
43	C₄×D₆₁	—	2	4	no	—
44	C₄×D₆₁	D₄	1	4	yes	—
45	Dic₁₂₂	—	2	2	no	—
46	Dic₁₂₂	—	2	4	no	—
47	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
48	C₂×D₆₁	—	2	2	no	—
49	C₂×D₆₁	—	2	2	no	—
50	C₂×D₆₁	—	2	4	no	—
51	C₂×D₆₁	C₂×D₄	1	2	yes	—
52	C₄×C₆₁	—	2	2	no	—
53	C₄×C₆₁	—	2	2	no	—
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	C₂×D₄	1	2	yes	—
56	Dic₆₁	—	2	2	no	—
57	Dic₆₁	—	2	2	no	—
58	Dic₆₁	C₂×D₄	1	2	yes	—
59	C₂×C₆₁	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₆₁	2₋¹⁺⁴	1	2	yes	—
61	D₆₁	—	2	2	no	—
62	D₆₁	—	4	2	no	—
63	#536	—	61	1	no	maximal, Sylow
64	C₆₁	#536	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	61	1	no	—
66	C₂²×Q₈	—	61	1	no	—
67	C₄⋊Q₈	—	61	1	no	—
68	C₄⋊Q₈	D₆₁	1	1	yes	characteristic, complemented
69	#124	—	61	1	no	—
70	#124	—	61	1	no	—
71	#123	—	61	4	no	—
72	C₂²⋊Q₈	—	61	4	no	—
73	#117	—	61	1	no	—
74	#41	—	122	4	no	—
75	C₂×Q₈	—	61	2	no	—
76	C₂×Q₈	—	61	1	no	—
77	C₂×Q₈	—	122	4	no	—
78	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
80	C₂×D₄	—	61	2	no	—
81	C₂×D₄	—	61	1	no	—
82	C₂²×C₄	—	61	1	no	—
83	C₂²×C₄	—	61	2	no	—
84	C₂²×C₄	—	61	2	no	—
85	C₄⋊C₄	—	61	2	no	—
86	C₄⋊C₄	—	61	4	no	—
87	C₄⋊C₄	C₂×D₆₁	1	4	yes	—
88	C₂²⋊C₄	—	61	2	no	—
89	C₂²⋊C₄	—	61	4	no	—
90	C₂²⋊C₄	—	61	4	no	—
91	C₄²	—	61	1	no	—
92	C₄²	C₂×D₆₁	1	1	yes	characteristic
93	C₂³	—	61	1	no	—
94	C₂³	—	61	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	122	4	no	—
98	Q₈	—	122	2	no	—
99	D₄	—	122	4	no	—
100	D₄	—	122	2	no	—
101	C₂×C₄	—	61	1	no	—
102	C₂×C₄	—	61	4	no	—
103	C₂×C₄	—	61	2	no	—
104	C₂×C₄	—	61	2	no	—
105	C₂×C₄	—	122	4	no	—
106	C₂×C₄	—	122	4	no	—
107	C₂×C₄	C₂²×D₆₁	1	2	yes	—
108	C₂×C₄	C₂²×D₆₁	1	2	yes	—
109	C₂×C₄	C₂²×D₆₁	1	2	yes	—
110	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
111	C₂²	—	61	2	no	—
112	C₂²	—	122	2	no	—
113	C₂²	—	122	2	no	—
114	C₂²	—	122	4	no	—
115	C₂²	C₂³×D₆₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	61	2	no	—
120	C₄	—	122	2	no	—
121	C₄	—	122	2	no	—
122	C₄	D₄×D₆₁	1	2	yes	—
123	C₂	—	122	2	no	—
124	C₂	—	244	2	no	—
125	C₂	C₂×D₄×D₆₁	1	1	yes	characteristic
126	C₂	#14519825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758226

C₂

yes

maximal, characteristic

C₂×Q₈×D₆₁

C₂

yes

maximal, characteristic, complemented

#14519775

C₂

yes

maximal, characteristic, complemented

#14519766

C₂

yes

maximal, characteristic, complemented

#14519765

C₂

yes

maximal, characteristic, complemented

#14519762

C₂

yes

maximal, complemented

#14519755

C₂

yes

maximal, complemented

#14519734

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂²

yes

characteristic

#11758226

—

Q₈×D₆₁

—

C₂×D₂₄₄

C₂²

yes

—

C₂×D₂₄₄

C₂²

yes

characteristic

C₂×C₄×D₆₁

C₂²

yes

—

C₂×C₄×D₆₁

C₂²

yes

—

C₂×C₄×D₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

characteristic

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

characteristic

C₂²×D₆₁

C₂³

yes

—

C₂²×D₆₁

C₂³

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₂²×C₆₁

C₂⁴

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×D₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₄

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×D₄

yes

—

C₂×C₆₁

C₂²×D₄

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

—

D₆₁

—

D₆₁

—

#536

—

maximal, Sylow

C₆₁

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₆₁

yes

characteristic, complemented

#124

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₆₁

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

122

—

Q₈

—

122

—

D₄

—

122

—

100

D₄

—

122

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

122

—

106

C₂×C₄

—

122

—

107

C₂×C₄

C₂²×D₆₁

yes

—

108

C₂×C₄

C₂²×D₆₁

yes

—

109

C₂×C₄

C₂²×D₆₁

yes

—

110

C₂×C₄

C₂²×D₆₁

yes

characteristic

111

C₂²

—

112

C₂²

—

122

—

113

C₂²

—

122

—

114

C₂²

—

122

—

115

C₂²

C₂³×D₆₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

122

—

121

C₄

—

122

—

122

C₄

D₄×D₆₁

yes

—

123

C₂

—

122

—

124

C₂

—

244

—

125

C₂

C₂×D₄×D₆₁

yes

characteristic

126

C₂

#14519825

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₄⋊Q₈×C₆₁	C₂
prime radical	C₂×#11758226	C₂
hypercenter	C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#536
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

C₂×#11758226

C₂

hypercenter

C₄⋊Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₆₁	C₂	semidirect
3. #14519775	C₂	semidirect
4. #14519766	C₂	semidirect
5. #14519765	C₂	semidirect
6. #14519762	C₂	semidirect
7. #14519755	C₂	semidirect
8. #14519734	C₂	semidirect
9. C₄⋊Q₈×C₆₁	C₂	semidirect
64. C₆₁	#536	semidirect
68. C₄⋊Q₈	D₆₁	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₆₁

C₂

semidirect

3. #14519775

C₂

semidirect

4. #14519766

C₂

semidirect

5. #14519765

C₂

semidirect

6. #14519762

C₂

semidirect

7. #14519755

C₂

semidirect

8. #14519734

C₂

semidirect

9. C₄⋊Q₈×C₆₁

C₂

semidirect

64. C₆₁

#536

semidirect

68. C₄⋊Q₈

D₆₁

semidirect

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

subgroup	index
C₂×#11758226	2
C₂×Q₈×D₆₁	2
#14519775	2
#14519766	2
#14519765	2
#14519762	2
#14519755	2
#14519734	2
C₄⋊Q₈×C₆₁	2
#536	61

subgroup

index

C₂×Q₈×D₆₁

C₄⋊Q₈×C₆₁

Subgroups — #16840337

Conjugacy classes of subgroups

Complements

Maximal subgroups