Group #21445946 — subgroups

Subgroups — #21445946

Order 7808.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
2	#16839632	C₂	1	1	yes	maximal, characteristic
3	#16839524	C₂	1	4	yes	maximal
4	C₆₁×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92418	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₆₁	—	2	1	no	—
7	C₂×C₄⋊Dic₆₁	—	2	1	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
9	C₄×#92418	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	—	2	2	no	—
15	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	2	no	—
19	C₄⋊C₄×C₆₁	—	2	4	no	—
20	C₄⋊C₄×C₆₁	—	4	2	no	—
21	C₄⋊C₄×C₆₁	—	4	2	no	—
22	C₄²×C₆₁	C₂×C₄	1	2	yes	—
23	C₄²×C₆₁	Q₈	1	2	yes	—
24	C₄⋊Dic₆₁	—	2	4	no	—
25	C₄⋊Dic₆₁	—	4	2	no	—
26	C₄⋊Dic₆₁	—	4	2	no	—
27	C₂×#92418	—	2	2	no	—
28	C₂×#92418	—	2	2	no	—
29	C₂×#92418	C₂×C₄	1	4	yes	—
30	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	4	2	no	—
32	C₂×C₄×C₆₁	—	4	2	no	—
33	C₂×C₄×C₆₁	—	4	2	no	—
34	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₆₁	D₈	1	2	yes	—
37	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
39	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
40	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#41	1	2	yes	—
43	C₂×C₄×C₆₁	#41	1	2	yes	—
44	C₂×Dic₆₁	—	4	2	no	—
45	C₂×Dic₆₁	—	4	2	no	—
46	C₂×Dic₆₁	—	4	2	no	—
47	#92418	—	2	4	no	—
48	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₆₁	#124	1	1	yes	characteristic
50	C₂²×C₆₁	#124	1	1	yes	characteristic
51	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₆₁	—	8	2	no	—
56	C₄×C₆₁	#106	1	4	yes	—
57	C₄×C₆₁	#107	1	4	yes	—
58	C₄×C₆₁	#117	1	2	yes	—
59	C₄×C₆₁	#125	1	2	yes	—
60	Dic₆₁	—	8	2	no	—
61	#1834	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	#382	1	1	yes	characteristic
63	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
64	C₂×C₆₁	C₂×#107	1	1	yes	characteristic
65	C₂×C₆₁	#486	1	1	yes	characteristic
66	C₂×C₆₁	#486	1	1	yes	characteristic
67	C₂×C₆₁	#487	1	1	yes	characteristic
68	C₂×C₆₁	#487	1	1	yes	characteristic
69	C₂×C₄×C₈	—	61	1	no	—
70	#384	—	61	1	no	—
71	#384	D₆₁	1	1	yes	characteristic
72	#340	—	61	4	no	—
73	C₆₁	#1834	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	61	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	61	2	no	—
78	C₂×C₄⋊C₄	—	122	1	no	—
79	C₂×C₄⋊C₄	—	122	1	no	—
80	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
81	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
82	C₄×C₈	—	61	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	122	2	no	—
85	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
88	C₂×C₈	—	61	4	no	—
89	C₂×C₈	—	122	2	no	—
90	C₂×C₈	—	122	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	122	4	no	—
95	C₄⋊C₄	—	244	2	no	—
96	C₄⋊C₄	—	244	2	no	—
97	C₄²	Dic₁₂₂	1	2	yes	—
98	C₄²	C₄×D₆₁	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	244	2	no	—
104	C₂×C₄	—	244	2	no	—
105	C₂×C₄	—	244	2	no	—
106	C₂×C₄	#11758196	1	2	yes	—
107	C₂×C₄	#11758196	1	2	yes	—
108	C₂×C₄	#11758199	1	2	yes	—
109	C₂×C₄	#11758200	1	2	yes	—
110	C₂×C₄	#11758201	1	2	yes	—
111	C₂×C₄	#11758202	1	2	yes	—
112	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
114	C₂×C₄	#11758224	1	2	yes	—
115	C₂×C₄	#11758226	1	2	yes	—
116	C₈	—	122	4	no	—
117	C₂²	C₂×#11758196	1	1	yes	characteristic
118	C₂²	#14519764	1	1	yes	characteristic
119	C₂²	#14519766	1	1	yes	characteristic
120	C₂²	C₂×#11758199	1	1	yes	characteristic
121	C₂²	C₂×#11758201	1	1	yes	characteristic
122	C₂²	C₂×#11758200	1	1	yes	characteristic
123	C₂²	C₂×#11758202	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	488	2	no	—
126	C₄	#14519708	1	4	yes	—
127	C₄	#14519710	1	4	yes	—
128	C₄	#14519735	1	2	yes	—
129	C₄	#14519768	1	2	yes	—
130	C₂	#16839627	1	1	yes	characteristic
131	C₂	C₂×#14519708	1	1	yes	characteristic
132	C₂	C₂×#14519710	1	1	yes	characteristic
133	C₂	#16840108	1	1	yes	characteristic
134	C₂	#16840110	1	1	yes	characteristic
135	C₂	#16840112	1	1	yes	characteristic
136	C₂	#16840114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#92418

C₂

yes

maximal, characteristic

#16839632

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

C₆₁×#384

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₄×#92418

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

Q₈

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#106

yes

—

C₄×C₆₁

#107

yes

—

C₄×C₆₁

#117

yes

—

C₄×C₆₁

#125

yes

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×C₈

—

#384

—

#384

D₆₁

yes

characteristic

#340

—

C₆₁

#1834

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

Dic₁₂₂

yes

—

C₄²

C₄×D₆₁

yes

—

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

244

—

104

C₂×C₄

—

244

—

105

C₂×C₄

—

244

—

106

C₂×C₄

#11758196

yes

—

107

C₂×C₄

#11758196

yes

—

108

C₂×C₄

#11758199

yes

—

109

C₂×C₄

#11758200

yes

—

110

C₂×C₄

#11758201

yes

—

111

C₂×C₄

#11758202

yes

—

112

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

113

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

114

C₂×C₄

#11758224

yes

—

115

C₂×C₄

#11758226

yes

—

116

C₈

—

122

—

117

C₂²

C₂×#11758196

yes

characteristic

118

C₂²

#14519764

yes

characteristic

119

C₂²

#14519766

yes

characteristic

120

C₂²

C₂×#11758199

yes

characteristic

121

C₂²

C₂×#11758201

yes

characteristic

122

C₂²

C₂×#11758200

yes

characteristic

123

C₂²

C₂×#11758202

yes

characteristic

124

C₄

—

125

C₄

—

488

—

126

C₄

#14519708

yes

—

127

C₄

#14519710

yes

—

128

C₄

#14519735

yes

—

129

C₄

#14519768

yes

—

130

C₂

#16839627

yes

characteristic

131

C₂

C₂×#14519708

yes

characteristic

132

C₂

C₂×#14519710

yes

characteristic

133

C₂

#16840108

yes

characteristic

134

C₂

#16840110

yes

characteristic

135

C₂

#16840112

yes

characteristic

136

C₂

#16840114

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#384	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	#384	D₆₁
hypocenter	C₆₁	#1834
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#384

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#92418	C₄	semidirect
73. C₆₁	#1834	semidirect

H ≅ G/N

type

9. C₄×#92418

C₄

semidirect

73. C₆₁

#1834

semidirect

Subgroups — #21445946

Conjugacy classes of subgroups

Complements