Group #21461950 — subgroups

Subgroups — #21461950

Order 7808.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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×#14519840	C₂	1	4	yes	maximal
2	#16840721	C₂	1	1	yes	maximal, characteristic
3	#16840706	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₆₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#6923	C₂²	1	2	yes	—
7	#14519840	—	2	8	no	—
8	#14519832	—	2	2	no	—
9	(C₂×#6923)⋊C₄	—	2	2	no	—
10	C₂×Q₈×D₆₁	C₂²	1	2	yes	—
11	#14519777	C₄	1	1	yes	characteristic, complemented
12	#14519776	—	2	2	no	—
13	#14519775	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₆₁	—	2	2	no	—
15	C₄²×D₆₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
17	C₂²×#6923	—	2	2	no	—
18	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
19	Q₈×D₆₁	—	2	4	no	—
20	Q₈×D₆₁	—	2	4	no	—
21	C₂×C₄×D₆₁	—	2	2	no	—
22	C₂×C₄×D₆₁	D₄	1	2	yes	—
23	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₂₂	—	2	2	no	—
25	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
26	C₂×#92420	—	2	2	no	—
27	C₄⋊#6923	—	2	4	no	—
28	D₆₁⋊C₈	—	2	2	no	—
29	D₆₁⋊C₈	D₄	1	4	yes	—
30	C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₆₁	—	2	2	no	—
33	#11758196	—	2	4	no	—
34	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₆₁	D₄	1	2	yes	—
36	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×#6923	—	4	2	no	—
38	C₂×#6923	—	4	4	no	—
39	Q₈×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
46	C₄×D₆₁	—	2	4	no	—
47	C₄×D₆₁	—	4	2	no	—
48	C₄×D₆₁	—	4	2	no	—
49	C₄×D₆₁	C₂×D₄	1	2	yes	—
50	C₄×D₆₁	#41	1	2	yes	—
51	Dic₁₂₂	—	2	4	no	—
52	Dic₁₂₂	—	4	4	no	—
53	#92420	—	2	4	no	—
54	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₆₁	#124	1	1	yes	characteristic
56	C₂×D₆₁	#127	1	1	yes	characteristic
57	C₂×D₆₁	#137	1	4	yes	—
58	#6923	—	8	2	no	—
59	C₄×C₆₁	—	2	2	no	—
60	C₄×C₆₁	—	4	2	no	—
61	C₄×C₆₁	C₄×D₄	1	2	yes	—
62	Dic₆₁	—	2	2	no	—
63	Dic₆₁	—	4	2	no	—
64	Dic₆₁	#121	1	2	yes	—
65	#1960	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	#390	1	1	yes	characteristic
67	C₂×C₆₁	#419	1	2	yes	—
68	D₆₁	#491	1	2	yes	—
69	D₆₁	#497	1	2	yes	—
70	C₂×C₄⋊Q₈	—	61	1	no	—
71	C₂×#103	—	61	4	no	—
72	C₂×#97	—	61	1	no	—
73	#382	—	61	1	no	—
74	C₆₁	#1960	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	61	2	no	—
76	C₂²×C₈	—	61	2	no	—
77	C₄⋊Q₈	—	61	2	no	—
78	C₄⋊Q₈	—	61	1	no	—
79	C₄⋊Q₈	—	122	2	no	—
80	C₄⋊Q₈	#6923	1	1	yes	characteristic, complemented
81	C₂×C₄⋊C₄	—	61	2	no	—
82	C₂×C₄⋊C₄	—	122	2	no	—
83	C₂×C₄²	—	61	1	no	—
84	#103	—	122	8	no	—
85	#97	—	122	2	no	—
86	#95	—	122	2	no	—
87	C₂×Q₈	—	61	2	no	—
88	C₂×Q₈	—	122	2	no	—
89	C₂×Q₈	—	122	4	no	—
90	C₂×Q₈	—	122	4	no	—
91	C₂×Q₈	C₂×#6923	1	2	yes	—
92	C₂²×C₄	—	61	1	no	—
93	C₂²×C₄	—	61	2	no	—
94	C₂²×C₄	—	122	2	no	—
95	C₂²×C₄	—	122	2	no	—
96	C₂×C₈	—	61	4	no	—
97	C₂×C₈	—	122	2	no	—
98	C₂×C₈	—	122	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	122	4	no	—
101	C₄⋊C₄	—	122	4	no	—
102	C₄⋊C₄	—	122	2	no	—
103	C₄²	—	61	2	no	—
104	C₄²	—	61	1	no	—
105	C₄²	C₂×#6923	1	1	yes	characteristic
106	C₂³	—	61	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	122	4	no	—
109	Q₈	—	244	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	61	1	no	—
112	C₂×C₄	—	61	2	no	—
113	C₂×C₄	—	61	2	no	—
114	C₂×C₄	—	61	2	no	—
115	C₂×C₄	—	122	4	no	—
116	C₂×C₄	—	122	2	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	—	244	2	no	—
119	C₂×C₄	—	244	2	no	—
120	C₂×C₄	—	244	4	no	—
121	C₂×C₄	C₂²⋊#6923	1	2	yes	—
122	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
123	C₈	—	122	4	no	—
124	C₂²	—	61	1	no	—
125	C₂²	—	61	1	no	—
126	C₂²	—	61	4	no	—
127	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	61	2	no	—
131	C₄	—	122	2	no	—
132	C₄	—	488	2	no	—
133	C₄	—	244	2	no	—
134	C₄	D₄×#6923	1	2	yes	—
135	C₂	—	61	2	no	—
136	C₂	—	61	2	no	—
137	C₂	#16840714	1	1	yes	characteristic
138	C₂	#16840755	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519840

C₂

yes

maximal

#16840721

C₂

yes

maximal, characteristic

#16840706

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

C₂²

yes

—

C₂×C₄⋊#6923

C₂²

yes

—

#14519840

—

#14519832

—

(C₂×#6923)⋊C₄

—

C₂×Q₈×D₆₁

C₂²

yes

—

#14519777

C₄

yes

characteristic, complemented

#14519776

—

#14519775

C₄

yes

complemented

C₄⋊C₄×D₆₁

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

Q₈×D₆₁

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

C₄⋊#6923

—

D₆₁⋊C₈

—

D₆₁⋊C₈

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

C₂×#6923

—

C₂×#6923

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₄

yes

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

C₂×D₆₁

yes

C₂×D₆₁

yes

C₂×D₆₁

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#121

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

#491

yes

—

D₆₁

#497

yes

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

—

C₆₁

#1960

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

#6923

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

#103

—

122

—

#97

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

100

C₄⋊C₄

—

122

—

101

C₄⋊C₄

—

122

—

102

C₄⋊C₄

—

122

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×#6923

yes

characteristic

106

C₂³

—

107

Q₈

—

108

Q₈

—

122

—

109

Q₈

—

244

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

122

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

244

—

118

C₂×C₄

—

244

—

119

C₂×C₄

—

244

—

120

C₂×C₄

—

244

—

121

C₂×C₄

C₂²⋊#6923

yes

—

122

C₂×C₄

C₂²×#6923

yes

characteristic

123

C₈

—

122

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×C₂²⋊#6923

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

122

—

132

C₄

—

488

—

133

C₄

—

244

—

134

C₄

D₄×#6923

yes

—

135

C₂

—

136

C₂

—

137

C₂

#16840714

yes

characteristic

138

C₂

#16840755

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1960
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #14519777	C₄	semidirect
13. #14519775	C₄	semidirect
16. C₄⋊Q₈×C₆₁	C₄	semidirect
74. C₆₁	#1960	semidirect
80. C₄⋊Q₈	#6923	semidirect

H ≅ G/N

type

11. #14519777

C₄

semidirect

13. #14519775

C₄

semidirect

16. C₄⋊Q₈×C₆₁

C₄

semidirect

74. C₆₁

#1960

semidirect

80. C₄⋊Q₈

#6923

semidirect

Subgroups — #21461950

Conjugacy classes of subgroups

Complements