Group #21444574 — subgroups

Subgroups — #21444574

Order 7808.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16839786	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
3	#16839512	C₂	1	4	yes	maximal, complemented
4	C₆₁×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	—	2	1	no	—
6	C₂²×#92418	C₄	1	2	yes	—
7	C₄⋊#92418	—	2	1	no	—
8	C₄⋊#92418	C₂²	1	2	yes	—
9	C₄×#92418	C₄	1	1	yes	characteristic
10	C₄×#92418	C₄	1	1	yes	characteristic
11	C₄×#92418	C₂²	1	2	yes	—
12	C₂×M₁₆×C₆₁	—	2	1	no	—
13	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	—	2	1	no	—
15	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
19	M₁₆×C₆₁	—	4	2	no	—
20	C₂×C₈×C₆₁	—	2	2	no	—
21	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₆₁	D₄	1	1	yes	characteristic
23	C₄²×C₆₁	Q₈	1	1	yes	characteristic
24	C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	#11758194	—	4	2	no	—
26	C₂×#92418	—	2	4	no	—
27	C₂×#92418	—	2	2	no	—
28	C₂×#92418	C₂×C₄	1	2	yes	—
29	C₂×#92418	C₂×C₄	1	2	yes	—
30	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
37	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₆₁	—	4	2	no	—
42	#92418	—	2	2	no	—
43	#92418	—	4	2	no	—
44	#92418	M₁₆	1	4	yes	complemented
45	C₂²×C₆₁	—	2	1	no	—
46	C₂²×C₆₁	—	2	1	no	—
47	C₂²×C₆₁	#106	1	2	yes	—
48	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	#106	1	2	yes	—
52	C₄×C₆₁	#108	1	2	yes	—
53	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
54	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
56	#1566	—	61	1	no	maximal, Sylow
57	C₂×C₆₁	—	2	2	no	—
58	C₂×C₆₁	#423	1	1	yes	characteristic
59	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
60	C₂×C₆₁	C₂×#108	1	1	yes	characteristic
61	#423	—	61	1	no	—
62	#423	D₆₁	1	1	yes	characteristic
63	C₂×C₄×C₈	—	61	1	no	—
64	#334	—	61	4	no	—
65	C₆₁	#1566	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	122	1	no	—
68	C₂²×C₈	—	61	2	no	—
69	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	61	2	no	—
72	C₄⋊C₈	—	122	1	no	—
73	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
74	C₄×C₈	—	61	1	no	—
75	C₄×C₈	—	61	1	no	—
76	C₄×C₈	—	61	2	no	—
77	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	244	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	61	2	no	—
84	C₂×C₈	—	61	2	no	—
85	C₂×C₈	—	122	4	no	—
86	C₂×C₈	—	122	2	no	—
87	C₄²	Dic₁₂₂	1	1	yes	characteristic
88	C₄²	C₄×D₆₁	1	1	yes	characteristic
89	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
90	C₄²	C₂²×D₆₁	1	1	yes	characteristic
91	C₂³	#11758196	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	#11758196	1	1	yes	characteristic
95	C₂×C₄	#11758196	1	1	yes	characteristic
96	C₂×C₄	#11758196	1	1	yes	characteristic
97	C₂×C₄	#11758200	1	2	yes	—
98	C₂×C₄	#11758201	1	2	yes	—
99	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	61	4	no	—
104	C₈	—	122	2	no	—
105	C₈	—	244	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#14519708	1	2	yes	—
109	C₂²	C₂×#11758196	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#14519708	1	2	yes	—
113	C₄	#14519712	1	2	yes	—
114	C₄	M₁₆×D₆₁	1	2	yes	—
115	C₄	C₂×#11758201	1	1	yes	characteristic
116	C₄	C₂×#11758200	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#16839785	1	1	yes	characteristic
119	C₂	C₂×#14519708	1	1	yes	characteristic
120	C₂	C₂×#14519712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839786

C₂

yes

maximal, characteristic

C₂×C₄×#92418

C₂

yes

maximal, characteristic

#16839512

C₂

yes

maximal, complemented

C₆₁×#423

C₂

yes

maximal, characteristic

C₂×#11758194

—

C₂²×#92418

C₄

yes

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

—

C₄×#92418

C₄

yes

characteristic

C₄×#92418

C₄

yes

characteristic

C₄×#92418

C₂²

yes

—

C₂×M₁₆×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₂×#92418

C₂×C₄

yes

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

—

#92418

—

#92418

—

#92418

M₁₆

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#106

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#106

yes

—

C₄×C₆₁

#108

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

#1566

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

#334

—

C₆₁

#1566

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

Dic₁₂₂

yes

characteristic

C₄²

C₄×D₆₁

yes

characteristic

C₄²

C₂²⋊D₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

#11758196

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#11758201

yes

—

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

100

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

122

—

105

C₈

—

244

—

106

C₂²

—

107

C₂²

—

108

C₂²

#14519708

yes

—

109

C₂²

C₂×#11758196

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#14519708

yes

—

113

C₄

#14519712

yes

—

114

C₄

M₁₆×D₆₁

yes

—

115

C₄

C₂×#11758201

yes

characteristic

116

C₄

C₂×#11758200

yes

characteristic

117

C₂

—

118

C₂

#16839785

yes

characteristic

119

C₂

C₂×#14519708

yes

characteristic

120

C₂

C₂×#14519712

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#423	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	#423	D₆₁
hypocenter	C₆₁	#1566
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#423

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16839512	C₂	semidirect
44. #92418	M₁₆	semidirect
65. C₆₁	#1566	semidirect

H ≅ G/N

type

3. #16839512

C₂

semidirect

44. #92418

M₁₆

semidirect

65. C₆₁

#1566

semidirect

Subgroups — #21444574

Conjugacy classes of subgroups

Complements