Group #21454951 — subgroups

Subgroups — #21454951

Order 7808.

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

normal subgroups only

This group has 394 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	#16839815	C₂	1	3	yes	maximal, complemented
4	#16839786	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	3	yes	—
8	C₄⋊#92418	C₂²	1	6	yes	—
9	C₄⋊#92418	C₂²	1	1	yes	characteristic
10	C₂²⋊#92418	C₂²	1	6	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	—	4	6	no	—
33	C₂×#92418	C₂³	1	2	yes	—
34	C₂×#92418	C₂³	1	6	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	—	2	3	no	—
41	C₂×C₄×C₆₁	—	2	3	no	—
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	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	#92418	—	2	6	no	—
49	#92418	—	2	2	no	—
50	C₂²×C₆₁	—	2	3	no	—
51	C₂²×C₆₁	—	2	3	no	—
52	C₂²×C₆₁	—	2	3	no	—
53	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₆₁	—	2	3	no	—
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	2	3	no	—
57	C₄×C₆₁	C₂×M₁₆	1	6	yes	—
58	C₄×C₆₁	#131	1	2	yes	—
59	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
61	#2991	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	4	3	no	—
63	C₂×C₆₁	#520	1	1	yes	characteristic
64	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
66	C₄×#41	D₆₁	1	1	yes	characteristic
67	C₈×Q₈	—	61	2	no	—
68	#436	—	61	6	no	—
69	#431	—	61	3	no	—
70	#423	—	61	3	no	—
71	C₆₁	#2991	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	Dic₆₁	1	1	yes	characteristic
73	C₂×M₁₆	—	61	6	no	—
74	C₄×Q₈	Dic₆₁	1	1	yes	characteristic
75	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
76	C₄×D₄	Dic₆₁	1	3	yes	—
77	C₄×D₄	C₂×D₆₁	1	3	yes	—
78	#117	Dic₆₁	1	3	yes	—
79	C₂×C₄²	C₂×D₆₁	1	3	yes	—
80	C₄⋊C₈	—	61	3	no	—
81	C₄⋊C₈	—	61	6	no	—
82	C₄⋊C₈	—	61	1	no	—
83	C₂²⋊C₈	—	61	6	no	—
84	C₄×C₈	—	61	6	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₆₁	1	3	yes	—
88	C₂²×C₄	C₂×Dic₆₁	1	3	yes	—
89	C₂²×C₄	C₂×Dic₆₁	1	3	yes	—
90	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
91	M₁₆	—	244	6	no	—
92	C₂×C₈	—	61	2	no	—
93	C₂×C₈	—	61	6	no	—
94	C₄⋊C₄	C₂×Dic₆₁	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₆₁	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₆₁	1	6	yes	—
97	C₄²	C₂×Dic₆₁	1	3	yes	—
98	C₄²	C₂×Dic₆₁	1	3	yes	—
99	C₄²	C₂²×D₆₁	1	3	yes	—
100	C₄²	C₂²×D₆₁	1	1	yes	characteristic
101	C₂³	C₂²×Dic₆₁	1	3	yes	—
102	Q₈	#11758194	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	6	no	—
108	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
110	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
111	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
112	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
113	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
114	C₈	—	122	6	no	—
115	C₈	—	122	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#11758194	1	6	yes	—
124	C₄	#14519784	1	2	yes	—
125	C₄	#14519823	1	1	yes	characteristic
126	C₄	#14519824	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#16840245	1	1	yes	characteristic
129	C₂	C₂²×#11758194	1	1	yes	characteristic
130	C₂	C₂×#14519784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#92418

C₂

yes

maximal, complemented

#16839840

C₂

yes

maximal, complemented

#16839815

C₂

yes

maximal, complemented

#16839786

C₂

yes

maximal, complemented

C₄×#41×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

C₄⋊#92418

C₂²

yes

characteristic

C₂²⋊#92418

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

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

#2991

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#520

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

C₄×#41

D₆₁

yes

characteristic

C₈×Q₈

—

#436

—

#431

—

#423

—

C₆₁

#2991

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

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

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

—

244

—

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

—

100

C₄²

C₂²×D₆₁

yes

characteristic

101

C₂³

C₂²×Dic₆₁

yes

—

102

Q₈

#11758194

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

109

C₂×C₄

C₂²×Dic₆₁

yes

—

110

C₂×C₄

C₂²×Dic₆₁

yes

—

111

C₂×C₄

C₂²×Dic₆₁

yes

—

112

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

113

C₂×C₄

C₂³×D₆₁

yes

characteristic

114

C₈

—

122

—

115

C₈

—

122

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×Dic₆₁

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#11758194

yes

—

124

C₄

#14519784

yes

—

125

C₄

#14519823

yes

characteristic

126

C₄

#14519824

yes

characteristic

127

C₂

—

128

C₂

#16840245

yes

characteristic

129

C₂

C₂²×#11758194

yes

characteristic

130

C₂

C₂×#14519784

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	#2991
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. #16839815	C₂	semidirect
4. #16839786	C₂	semidirect
36. Q₈×C₆₁	M₁₆	semidirect
71. C₆₁	#2991	semidirect
102. Q₈	#11758194	semidirect

H ≅ G/N

type

1. Q₈×#92418

C₂

semidirect

2. #16839840

C₂

semidirect

3. #16839815

C₂

semidirect

4. #16839786

C₂

semidirect

36. Q₈×C₆₁

M₁₆

semidirect

71. C₆₁

#2991

semidirect

102. Q₈

#11758194

semidirect

Subgroups — #21454951

Conjugacy classes of subgroups

Complements