Group #21445209 — subgroups

Subgroups — #21445209

Order 7808.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840167	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839968	C₂	1	2	yes	maximal
3	C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic
4	#16839509	C₂	1	2	yes	maximal, complemented
5	C₆₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758190	—	2	2	no	—
7	C₂×C₈×D₆₁	—	2	1	no	—
8	#14519777	C₂²	1	2	yes	—
9	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
10	#14519711	—	2	1	no	—
11	#14519711	—	2	1	no	—
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	#14519702	—	2	2	no	—
14	#14519689	—	2	1	no	—
15	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×D₂₄₄	D₄	1	1	yes	characteristic
18	C₂×C₄×D₆₁	—	2	1	no	—
19	C₂×Dic₁₂₂	—	2	2	no	—
20	C₂×C₈×C₆₁	—	2	2	no	—
21	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	#11758198	—	2	1	no	—
25	C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
28	C₂×#92418	—	2	1	no	—
29	#11758190	—	4	4	no	—
30	C₈×D₆₁	—	4	2	no	—
31	C₂²×D₆₁	—	2	1	no	—
32	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
35	C₂×Dic₆₁	—	2	1	no	—
36	C₂×Dic₆₁	—	4	2	no	—
37	D₂₄₄	—	2	1	no	—
38	D₂₄₄	D₈	1	2	yes	—
39	C₄×D₆₁	—	4	1	no	—
40	C₄×D₆₁	—	4	1	no	—
41	Dic₁₂₂	—	4	2	no	—
42	Dic₁₂₂	—	4	2	no	—
43	C₈×C₆₁	—	2	1	no	—
44	C₈×C₆₁	—	4	2	no	—
45	C₈×C₆₁	SD₁₆	1	2	yes	complemented
46	#92418	—	4	1	no	—
47	C₂²×C₆₁	#121	1	1	yes	characteristic
48	C₂×D₆₁	—	2	2	no	—
49	C₂×D₆₁	—	4	1	no	—
50	C₂×D₆₁	—	4	1	no	—
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
53	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₆₁	#135	1	1	yes	characteristic
55	C₄×C₆₁	#137	1	1	yes	characteristic
56	Dic₆₁	—	4	1	no	—
57	Dic₆₁	—	8	2	no	—
58	#1673	—	61	1	no	maximal, Sylow
59	C₂×C₆₁	#461	1	1	yes	characteristic
60	C₂×C₆₁	#466	1	1	yes	characteristic
61	C₂×C₆₁	#473	1	1	yes	characteristic
62	D₆₁	—	4	2	no	—
63	#500	—	61	1	no	—
64	#461	—	61	2	no	—
65	C₈×D₄	—	61	1	no	—
66	#335	D₆₁	1	1	yes	characteristic, complemented
67	#332	—	61	2	no	—
68	C₆₁	#1673	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₁₆	—	122	2	no	—
70	C₂²×C₈	—	122	1	no	—
71	C₄⋊Q₈	—	61	2	no	—
72	C₄×D₄	—	61	1	no	—
73	#107	—	122	1	no	—
74	#107	—	122	1	no	—
75	C₄⋊C₈	—	61	1	no	—
76	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
77	#103	—	122	2	no	—
78	C₂²⋊C₈	—	122	1	no	—
79	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
80	C₂×Q₈	—	122	2	no	—
81	C₂×D₄	—	61	1	no	—
82	C₂²×C₄	—	122	1	no	—
83	SD₁₆	—	244	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	122	1	no	—
86	C₂×C₈	—	244	2	no	—
87	C₂×C₈	D₂₄₄	1	1	yes	characteristic
88	C₂×C₈	D₂₄₄	1	1	yes	characteristic
89	C₄⋊C₄	—	61	1	no	—
90	C₄⋊C₄	—	122	2	no	—
91	C₄⋊C₄	—	122	2	no	—
92	C₂²⋊C₄	—	122	1	no	—
93	C₄²	C₂²×D₆₁	1	1	yes	characteristic
94	C₂³	—	122	1	no	—
95	Q₈	—	244	2	no	—
96	Q₈	—	244	2	no	—
97	D₄	—	61	2	no	—
98	D₄	—	122	1	no	—
99	C₂×C₄	—	122	1	no	—
100	C₂×C₄	—	244	2	no	—
101	C₂×C₄	—	244	1	no	—
102	C₂×C₄	—	244	1	no	—
103	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
104	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
105	C₂×C₄	#11758226	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	4	2	no	—
108	C₈	—	244	1	no	—
109	C₈	#11758190	1	2	yes	complemented
110	C₂²	—	122	2	no	—
111	C₂²	—	244	1	no	—
112	C₂²	—	244	1	no	—
113	C₂²	#14519752	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	488	2	no	—
116	C₄	—	244	1	no	—
117	C₄	D₈×D₆₁	1	1	yes	characteristic
118	C₄	C₂×#11758190	1	1	yes	characteristic
119	C₄	#14519800	1	1	yes	characteristic
120	C₄	#14519810	1	1	yes	characteristic
121	C₂	—	244	2	no	—
122	C₂	#16839968	1	1	yes	characteristic
123	C₂	#16839999	1	1	yes	characteristic
124	C₂	#16840038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840167

C₂

yes

maximal, characteristic, complemented

#16839968

C₂

yes

maximal

C₈×D₂₄₄

C₂

yes

maximal, characteristic

#16839509

C₂

yes

maximal, complemented

C₆₁×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11758190

—

C₂×C₈×D₆₁

—

#14519777

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

—

C₂²

yes

characteristic

#14519702

—

#14519689

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

D₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₂×#92418

—

#11758190

—

C₈×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

D₈

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

SD₁₆

yes

complemented

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#500

—

#461

—

C₈×D₄

—

#335

D₆₁

yes

characteristic, complemented

#332

—

C₆₁

#1673

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

122

—

#107

—

122

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

—

#103

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×D₄

—

C₂²×C₄

—

122

—

SD₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

122

—

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

—

122

—

Q₈

—

244

—

Q₈

—

244

—

D₄

—

D₄

—

122

—

C₂×C₄

—

122

—

100

C₂×C₄

—

244

—

101

C₂×C₄

—

244

—

102

C₂×C₄

—

244

—

103

C₂×C₄

C₂×D₂₄₄

yes

characteristic

104

C₂×C₄

D₄×D₆₁

yes

characteristic

105

C₂×C₄

#11758226

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

244

—

109

C₈

#11758190

yes

complemented

110

C₂²

—

122

—

111

C₂²

—

244

—

112

C₂²

—

244

—

113

C₂²

#14519752

yes

characteristic

114

C₄

—

115

C₄

—

488

—

116

C₄

—

244

—

117

C₄

D₈×D₆₁

yes

characteristic

118

C₄

C₂×#11758190

yes

characteristic

119

C₄

#14519800

yes

characteristic

120

C₄

#14519810

yes

characteristic

121

C₂

—

244

—

122

C₂

#16839968

yes

characteristic

123

C₂

#16839999

yes

characteristic

124

C₂

#16840038

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#335	C₂
socle	C₂²×C₆₁	#121
hypercenter	#335	D₆₁
hypocenter	C₆₁	#1673
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14519752

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#335

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #16840167	C₂	semidirect
4. #16839509	C₂	semidirect
5. C₆₁×#335	C₂	semidirect
45. C₈×C₆₁	SD₁₆	semidirect
66. #335	D₆₁	semidirect
68. C₆₁	#1673	semidirect
109. C₈	#11758190	semidirect

H ≅ G/N

type

1. #16840167

C₂

semidirect

4. #16839509

C₂

semidirect

5. C₆₁×#335

C₂

semidirect

45. C₈×C₆₁

SD₁₆

semidirect

66. #335

D₆₁

semidirect

68. C₆₁

#1673

semidirect

109. C₈

#11758190

semidirect

Subgroups — #21445209

Conjugacy classes of subgroups

Complements