Group #21447161 — subgroups

Subgroups — #21447161

Order 7808.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16839805	C₂	1	2	yes	maximal
2	#16839631	C₂	1	1	yes	maximal, characteristic
3	#16839531	C₂	1	1	yes	maximal, characteristic
4	#16839521	C₂	1	2	yes	maximal
5	C₆₁×#544	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
8	C₂×#11758196	—	2	2	no	—
9	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	#14519710	—	2	2	no	—
11	#14519708	—	2	2	no	—
12	C₄⋊Q₈×C₆₁	—	2	2	no	—
13	#125×C₆₁	—	2	2	no	—
14	C₂²⋊Q₈×C₆₁	—	2	2	no	—
15	#117×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	2	no	—
18	C₂²×C₄×C₆₁	D₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₆₁	—	2	1	no	—
21	C₂²×Dic₆₁	—	2	1	no	—
22	C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₄⋊C₄×C₆₁	—	2	4	no	—
24	C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
26	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
27	C₂²⋊C₄×C₆₁	—	2	2	no	—
28	C₄²×C₆₁	—	2	2	no	—
29	C₄⋊Dic₆₁	—	2	2	no	—
30	#11758196	—	4	4	no	—
31	C₄×Dic₆₁	—	2	1	no	—
32	C₄×Dic₆₁	Q₈	1	2	yes	complemented
33	#11758194	—	2	2	no	—
34	#11758194	Q₈	1	4	yes	complemented
35	C₂×#92418	—	2	2	no	—
36	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₆₁	—	8	2	no	—
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	2	no	—
47	C₂×Dic₆₁	—	2	2	no	—
48	C₂×Dic₆₁	—	2	2	no	—
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	4	1	no	—
51	#92418	—	2	4	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	1	no	—
54	C₂²×C₆₁	#120	1	1	yes	characteristic
55	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₆₁	—	4	2	no	—
58	C₄×C₆₁	—	4	2	no	—
59	C₄×C₆₁	—	4	2	no	—
60	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
62	Dic₆₁	—	2	4	no	—
63	Dic₆₁	—	8	1	no	—
64	#2060	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	1	no	—
66	C₂×C₆₁	—	2	1	no	—
67	C₂×C₆₁	#395	1	1	yes	characteristic
68	C₂×C₆₁	#453	1	1	yes	characteristic
69	C₂×C₆₁	#456	1	1	yes	characteristic
70	#544	D₆₁	1	1	yes	characteristic
71	#428	—	61	2	no	—
72	#384	—	61	1	no	—
73	#343	—	61	1	no	—
74	#339	—	61	2	no	—
75	C₆₁	#2060	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	61	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	#125	—	2	2	no	—
79	C₂²⋊Q₈	—	2	2	no	—
80	#117	C₂×D₆₁	1	2	yes	—
81	C₂×C₄⋊C₄	—	61	1	no	—
82	C₂×C₄⋊C₄	—	122	2	no	—
83	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
84	C₂×C₄²	—	61	1	no	—
85	#107	—	122	2	no	—
86	#106	—	122	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂²×C₄	—	122	1	no	—
89	C₂²×C₄	—	122	1	no	—
90	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
92	M₁₆	—	61	4	no	—
93	M₁₆	—	122	2	no	—
94	C₂×C₈	—	122	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	122	2	no	—
99	C₄⋊C₄	—	244	4	no	—
100	C₄⋊C₄	C₂²⋊D₆₁	1	2	yes	—
101	C₄⋊C₄	C₂²⋊D₆₁	1	2	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	61	2	no	—
105	C₄²	—	122	1	no	—
106	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	122	2	no	—
113	C₂×C₄	—	122	2	no	—
114	C₂×C₄	—	122	2	no	—
115	C₂×C₄	—	244	2	no	—
116	C₂×C₄	—	244	1	no	—
117	C₂×C₄	Q₈×D₆₁	1	2	yes	—
118	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
119	C₂×C₄	#11758226	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
121	C₈	—	122	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#14519746	1	1	yes	characteristic
125	C₂²	#14519753	1	1	yes	characteristic
126	C₂²	#14519775	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	122	4	no	—
131	C₄	—	488	1	no	—
132	C₄	#14519753	1	2	yes	—
133	C₄	#14519775	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#16839681	1	1	yes	characteristic
137	C₂	#16839930	1	1	yes	characteristic
138	C₂	#16839945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839805

C₂

yes

maximal

#16839631

C₂

yes

maximal, characteristic

#16839531

C₂

yes

maximal, characteristic

#16839521

C₂

yes

maximal

C₆₁×#544

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519710

—

#14519708

—

C₄⋊Q₈×C₆₁

—

#125×C₆₁

—

C₂²⋊Q₈×C₆₁

—

#117×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

Q₈

yes

complemented

#11758194

—

#11758194

Q₈

yes

complemented

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#120

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

#2060

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

—

—

—

C₆₁

#2060

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂×C₄²

—

#107

—

122

—

#106

—

122

—

C₂×Q₈

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

100

C₄⋊C₄

C₂²⋊D₆₁

yes

—

101

C₄⋊C₄

C₂²⋊D₆₁

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

122

—

106

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

122

—

113

C₂×C₄

—

122

—

114

C₂×C₄

—

122

—

115

C₂×C₄

—

244

—

116

C₂×C₄

—

244

—

117

C₂×C₄

Q₈×D₆₁

yes

—

118

C₂×C₄

Q₈×D₆₁

yes

characteristic

119

C₂×C₄

#11758226

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

121

C₈

—

122

—

122

C₂²

—

123

C₂²

—

124

C₂²

#14519746

yes

characteristic

125

C₂²

#14519753

yes

characteristic

126

C₂²

#14519775

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

122

—

131

C₄

—

488

—

132

C₄

#14519753

yes

—

133

C₄

#14519775

yes

—

134

C₂

—

135

C₂

—

136

C₂

#16839681

yes

characteristic

137

C₂

#16839930

yes

characteristic

138

C₂

#16839945

yes

characteristic

	subgroup	G/H
center	C₂²	#14519746
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#544	C₂
socle	C₂²×C₆₁	#120
hypercenter	#544	D₆₁
hypocenter	C₆₁	#2060
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#14519746

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#544

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
32. C₄×Dic₆₁	Q₈	semidirect
34. #11758194	Q₈	semidirect
75. C₆₁	#2060	semidirect

H ≅ G/N

type

32. C₄×Dic₆₁

Q₈

semidirect

34. #11758194

Q₈

semidirect

75. C₆₁

#2060

semidirect

Subgroups — #21447161

Conjugacy classes of subgroups

Complements