Group #21445180 — subgroups

Subgroups — #21445180

Order 7808.

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

normal subgroups only

This group has 188 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	#16840139	C₂	1	1	yes	maximal, characteristic
2	#16839963	C₂	1	2	yes	maximal, complemented
3	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839505	C₂	1	2	yes	maximal, complemented
5	C₆₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758190	—	2	2	no	—
7	C₂×D₄₈₈	—	2	1	no	—
8	C₂×D₄₈₈	—	2	1	no	—
9	#14519774	C₂²	1	2	yes	—
10	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	#14519706	—	2	1	no	—
13	#14519699	—	2	2	no	—
14	C₈×Dic₆₁	—	2	1	no	—
15	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×D₂₄₄	—	2	2	no	—
18	C₂×D₂₄₄	—	2	2	no	—
19	C₂×D₂₄₄	—	2	2	no	—
20	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	—	2	2	no	—
22	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
26	#11758196	—	2	1	no	—
27	C₄×Dic₆₁	—	2	1	no	—
28	C₂×#92418	—	2	1	no	—
29	D₄₈₈	—	4	2	no	—
30	D₄₈₈	—	4	2	no	—
31	#11758190	—	4	4	no	—
32	C₂²×D₆₁	—	4	2	no	—
33	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
36	C₂×Dic₆₁	—	2	1	no	—
37	C₂×Dic₆₁	—	2	1	no	—
38	D₂₄₄	—	4	2	no	—
39	D₂₄₄	—	4	2	no	—
40	D₂₄₄	—	4	2	no	—
41	D₂₄₄	—	4	2	no	—
42	D₂₄₄	—	8	2	no	—
43	Dic₁₂₂	—	2	1	no	—
44	Dic₁₂₂	D₈	1	2	yes	complemented
45	C₈×C₆₁	—	2	1	no	—
46	C₈×C₆₁	—	4	2	no	—
47	C₈×C₆₁	SD₁₆	1	2	yes	complemented
48	#92418	—	4	1	no	—
49	C₂²×C₆₁	#121	1	1	yes	characteristic
50	C₂×D₆₁	—	8	2	no	—
51	C₂×D₆₁	—	8	2	no	—
52	C₂×D₆₁	—	8	2	no	—
53	C₄×C₆₁	—	2	1	no	—
54	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₆₁	#135	1	1	yes	characteristic
57	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₆₁	—	2	2	no	—
59	Dic₆₁	—	4	1	no	—
60	#1668	—	61	1	no	maximal, Sylow
61	C₂×C₆₁	#460	1	1	yes	characteristic
62	C₂×C₆₁	#466	1	1	yes	characteristic
63	C₂×C₆₁	#472	1	1	yes	characteristic
64	D₆₁	—	16	2	no	—
65	#493	—	61	1	no	—
66	#460	—	61	2	no	—
67	C₈×Q₈	—	61	1	no	—
68	#335	D₆₁	1	1	yes	characteristic, complemented
69	#331	—	61	2	no	—
70	C₆₁	#1668	1	1	yes	characteristic, complemented, Sylow
71	C₂×SD₁₆	—	122	2	no	—
72	C₂×D₈	—	122	1	no	—
73	C₂×D₈	—	122	1	no	—
74	#127	—	61	2	no	—
75	C₄×Q₈	—	61	1	no	—
76	C₄⋊C₈	—	61	1	no	—
77	C₄⋊C₈	—	122	1	no	—
78	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
79	#102	—	122	2	no	—
80	C₄×C₈	—	122	1	no	—
81	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
82	C₂×Q₈	—	61	1	no	—
83	C₂×D₄	—	122	2	no	—
84	C₂×D₄	—	122	2	no	—
85	C₂×D₄	—	122	2	no	—
86	SD₁₆	—	244	4	no	—
87	D₈	—	244	2	no	—
88	D₈	—	244	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	122	1	no	—
91	C₂×C₈	D₂₄₄	1	1	yes	characteristic
92	C₂×C₈	D₂₄₄	1	1	yes	characteristic
93	C₄⋊C₄	—	61	1	no	—
94	C₄⋊C₄	—	122	1	no	—
95	C₄²	—	122	1	no	—
96	C₄²	C₂²×D₆₁	1	1	yes	characteristic
97	C₂³	—	244	2	no	—
98	Q₈	—	61	2	no	—
99	Q₈	—	122	1	no	—
100	D₄	—	488	2	no	—
101	D₄	—	244	2	no	—
102	D₄	—	244	2	no	—
103	D₄	—	244	2	no	—
104	D₄	—	244	2	no	—
105	C₂×C₄	—	122	1	no	—
106	C₂×C₄	—	122	1	no	—
107	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
108	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
109	C₂×C₄	#11758226	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	244	1	no	—
113	C₈	#11758190	1	2	yes	complemented
114	C₂²	—	488	2	no	—
115	C₂²	—	488	2	no	—
116	C₂²	—	488	2	no	—
117	C₂²	#14519752	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	122	2	no	—
120	C₄	—	244	1	no	—
121	C₄	D₈×D₆₁	1	1	yes	characteristic
122	C₄	C₂×#11758190	1	1	yes	characteristic
123	C₄	#14519800	1	1	yes	characteristic
124	C₄	#14519805	1	1	yes	characteristic
125	C₂	—	976	2	no	—
126	C₂	#16839963	1	1	yes	characteristic
127	C₂	#16839999	1	1	yes	characteristic
128	C₂	#16840033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840139

C₂

yes

maximal, characteristic

#16839963

C₂

yes

maximal, complemented

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, complemented

C₆₁×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11758190

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519774

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519706

—

#14519699

—

C₈×Dic₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

D₄₈₈

—

D₄₈₈

—

#11758190

—

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

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

D₈

yes

complemented

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

—

#493

—

#460

—

C₈×Q₈

—

#335

D₆₁

yes

characteristic, complemented

#331

—

C₆₁

#1668

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

—

#102

—

122

—

C₄×C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

SD₁₆

—

244

—

D₈

—

244

—

D₈

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄²

—

122

—

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

—

244

—

Q₈

—

Q₈

—

122

—

100

D₄

—

488

—

101

D₄

—

244

—

102

D₄

—

244

—

103

D₄

—

244

—

104

D₄

—

244

—

105

C₂×C₄

—

122

—

106

C₂×C₄

—

122

—

107

C₂×C₄

C₂×D₂₄₄

yes

characteristic

108

C₂×C₄

D₄×D₆₁

yes

characteristic

109

C₂×C₄

#11758226

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

244

—

113

C₈

#11758190

yes

complemented

114

C₂²

—

488

—

115

C₂²

—

488

—

116

C₂²

—

488

—

117

C₂²

#14519752

yes

characteristic

118

C₄

—

119

C₄

—

122

—

120

C₄

—

244

—

121

C₄

D₈×D₆₁

yes

characteristic

122

C₄

C₂×#11758190

yes

characteristic

123

C₄

#14519800

yes

characteristic

124

C₄

#14519805

yes

characteristic

125

C₂

—

976

—

126

C₂

#16839963

yes

characteristic

127

C₂

#16839999

yes

characteristic

128

C₂

#16840033

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	#1668
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
2. #16839963	C₂	semidirect
3. C₈×Dic₁₂₂	C₂	semidirect
4. #16839505	C₂	semidirect
5. C₆₁×#335	C₂	semidirect
44. Dic₁₂₂	D₈	semidirect
47. C₈×C₆₁	SD₁₆	semidirect
68. #335	D₆₁	semidirect
70. C₆₁	#1668	semidirect
113. C₈	#11758190	semidirect

H ≅ G/N

type

2. #16839963

C₂

semidirect

3. C₈×Dic₁₂₂

C₂

semidirect

4. #16839505

C₂

semidirect

5. C₆₁×#335

C₂

semidirect

44. Dic₁₂₂

D₈

semidirect

47. C₈×C₆₁

SD₁₆

semidirect

68. #335

D₆₁

semidirect

70. C₆₁

#1668

semidirect

113. C₈

#11758190

semidirect

Subgroups — #21445180

Conjugacy classes of subgroups

Complements