Group #21454839 — subgroups

Subgroups — #21454839

Order 7808.

402 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×#92418	C₂	1	2	yes	maximal, complemented
2	#16839840	C₂	1	6	yes	maximal, complemented
3	#16839819	C₂	1	3	yes	maximal, complemented
4	C₄×#11758194	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	6	yes	—
7	C₄⋊#92418	C₂²	1	6	yes	—
8	C₂²⋊#92418	C₂²	1	6	yes	—
9	#14519688	C₂²	1	1	yes	characteristic
10	#14519688	C₂²	1	3	yes	—
11	C₄×#92418	C₂²	1	6	yes	—
12	C₂×#41×C₆₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₆₁	C₄	1	3	yes	—
16	C₄×D₄×C₆₁	C₂²	1	3	yes	—
17	#117×C₆₁	C₄	1	3	yes	—
18	C₂×C₄²×C₆₁	C₂²	1	3	yes	—
19	#41×C₆₁	—	2	4	no	—
20	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₆₁	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
25	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
26	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
27	C₂²⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
28	C₄²×C₆₁	C₂×C₄	1	3	yes	—
29	C₄²×C₆₁	C₂×C₄	1	3	yes	—
30	C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄²×C₆₁	C₂³	1	3	yes	—
32	#11758194	—	2	12	no	—
33	C₂×#92418	C₂³	1	6	yes	—
34	C₂×#92418	C₂³	1	2	yes	—
35	C₂³×C₆₁	C₂²×C₄	1	3	yes	—
36	Q₈×C₆₁	M₁₆	1	4	yes	complemented
37	D₄×C₆₁	—	2	6	no	—
38	C₂×C₄×C₆₁	—	2	6	no	—
39	C₂×C₄×C₆₁	—	2	6	no	—
40	C₂×C₄×C₆₁	—	4	3	no	—
41	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
47	#92418	—	2	6	no	—
48	#92418	#41	1	4	yes	complemented
49	C₂²×C₆₁	—	2	3	no	—
50	C₂²×C₆₁	—	2	3	no	—
51	C₂²×C₆₁	—	2	3	no	—
52	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₆₁	—	2	3	no	—
54	C₄×C₆₁	—	2	1	no	—
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	2	3	no	—
57	C₄×C₆₁	C₂×M₁₆	1	6	yes	—
58	C₄×C₆₁	C₂×#41	1	2	yes	—
59	#2971	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	—	4	3	no	—
61	C₂×C₆₁	C₄×#41	1	1	yes	characteristic
62	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
63	C₂×C₆₁	#568	1	1	yes	characteristic
64	C₄×#41	D₆₁	1	1	yes	characteristic
65	C₈×Q₈	—	61	2	no	—
66	#436	—	61	6	no	—
67	#432	—	61	3	no	—
68	C₄×M₁₆	—	61	3	no	—
69	C₆₁	#2971	1	1	yes	characteristic, complemented, Sylow
70	C₂×#41	Dic₆₁	1	1	yes	characteristic
71	C₂×M₁₆	—	61	6	no	—
72	C₄×Q₈	Dic₆₁	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
74	C₄×D₄	Dic₆₁	1	3	yes	—
75	C₄×D₄	C₂×D₆₁	1	3	yes	—
76	#117	Dic₆₁	1	3	yes	—
77	C₂×C₄²	C₂×D₆₁	1	3	yes	—
78	C₄⋊C₈	—	61	6	no	—
79	C₂²⋊C₈	—	61	6	no	—
80	#97	—	61	1	no	—
81	#97	—	61	3	no	—
82	C₄×C₈	—	61	6	no	—
83	#41	—	2	4	no	—
84	C₂×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
85	C₂×D₄	C₂×Dic₆₁	1	3	yes	—
86	C₂²×C₄	C₂×Dic₆₁	1	3	yes	—
87	C₂²×C₄	C₂×Dic₆₁	1	3	yes	—
88	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
89	M₁₆	—	122	12	no	—
90	C₂×C₈	—	61	2	no	—
91	C₂×C₈	—	61	6	no	—
92	C₄⋊C₄	C₂×Dic₆₁	1	3	yes	—
93	C₄⋊C₄	C₂×Dic₆₁	1	3	yes	—
94	C₂²⋊C₄	C₂×Dic₆₁	1	6	yes	—
95	C₄²	C₂×Dic₆₁	1	3	yes	—
96	C₄²	C₂×Dic₆₁	1	3	yes	—
97	C₄²	C₂²×D₆₁	1	3	yes	—
98	C₄²	C₂²×D₆₁	1	1	yes	characteristic
99	C₂³	C₂²×Dic₆₁	1	3	yes	—
100	Q₈	#11758194	1	4	yes	complemented
101	D₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
108	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
109	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
110	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
111	C₈	—	61	4	no	—
112	C₈	—	122	6	no	—
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	—	2	3	no	—
116	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
117	C₄	—	2	3	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	3	no	—
121	C₄	C₂×#11758194	1	6	yes	—
122	C₄	#41×D₆₁	1	2	yes	—
123	C₂	—	4	3	no	—
124	C₂	#41×Dic₆₁	1	1	yes	characteristic
125	C₂	C₂²×#11758194	1	1	yes	characteristic
126	C₂	#16840529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#92418

C₂

yes

maximal, complemented

#16839840

C₂

yes

maximal, complemented

#16839819

C₂

yes

maximal, complemented

C₄×#11758194

C₂

yes

maximal, complemented

C₄×#41×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

characteristic

#14519688

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₂×#41×C₆₁

C₄

yes

characteristic

C₄×Q₈×C₆₁

C₄

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄×D₄×C₆₁

C₄

yes

—

C₄×D₄×C₆₁

C₂²

yes

—

#117×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

—

Q₈×C₆₁

M₁₆

yes

complemented

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

yes

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂×#41

yes

—

#2971

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₄×#41

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

#568

yes

characteristic

C₄×#41

D₆₁

yes

characteristic

C₈×Q₈

—

#436

—

#432

—

C₄×M₁₆

—

C₆₁

#2971

yes

characteristic, complemented, Sylow

C₂×#41

Dic₆₁

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₆₁

yes

characteristic

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

Dic₆₁

yes

—

C₄×D₄

C₂×D₆₁

yes

—

#117

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×Dic₆₁

yes

characteristic

C₂×D₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₂²⋊C₄

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

C₂²×Dic₆₁

yes

—

100

Q₈

#11758194

yes

complemented

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

106

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

107

C₂×C₄

C₂²×Dic₆₁

yes

—

108

C₂×C₄

C₂²×Dic₆₁

yes

—

109

C₂×C₄

C₂²×Dic₆₁

yes

—

110

C₂×C₄

C₂³×D₆₁

yes

characteristic

111

C₈

—

112

C₈

—

122

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×Dic₆₁

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#11758194

yes

—

122

C₄

#41×D₆₁

yes

—

123

C₂

—

124

C₂

#41×Dic₆₁

yes

characteristic

125

C₂

C₂²×#11758194

yes

characteristic

126

C₂

#16840529

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×#41×C₆₁	C₂
hypercenter	C₄×#41	D₆₁
hypocenter	C₆₁	#2971
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₄×#41×C₆₁

C₂

hypercenter

C₄×#41

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×#92418	C₂	semidirect
2. #16839840	C₂	semidirect
3. #16839819	C₂	semidirect
4. C₄×#11758194	C₂	semidirect
36. Q₈×C₆₁	M₁₆	semidirect
48. #92418	#41	semidirect
69. C₆₁	#2971	semidirect
100. Q₈	#11758194	semidirect

H ≅ G/N

type

1. Q₈×#92418

C₂

semidirect

2. #16839840

C₂

semidirect

3. #16839819

C₂

semidirect

4. C₄×#11758194

C₂

semidirect

36. Q₈×C₆₁

M₁₆

semidirect

48. #92418

#41

semidirect

69. C₆₁

#2971

semidirect

100. Q₈

#11758194

semidirect

Subgroups — #21454839

Conjugacy classes of subgroups

Complements