Group #21456341 — subgroups

Subgroups — #21456341

Order 7808.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16840319	C₂	1	1	yes	maximal, characteristic
2	#16840172	C₂	1	4	yes	maximal, complemented
3	#16840168	C₂	1	4	yes	maximal, complemented
4	#16839798	C₂	1	4	yes	maximal
5	C₂×#14519688	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#533	C₂	1	1	yes	maximal, characteristic
7	C₂²×#92418	C₂²	1	2	yes	—
8	#14519777	C₂²	1	2	yes	—
9	#14519769	C₂²	1	2	yes	—
10	#14519756	—	2	2	no	—
11	C₄×Dic₁₂₂	—	2	2	no	—
12	#14519736	C₂²	1	2	yes	—
13	#14519710	C₂²	1	8	yes	—
14	#14519708	C₂²	1	8	yes	—
15	#14519688	C₂²	1	2	yes	—
16	#14519688	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
18	#125×C₆₁	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₆₁	—	2	2	no	—
20	C₄×Q₈×C₆₁	—	2	2	no	—
21	#117×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	D₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₄⋊C₄×C₆₁	—	2	4	no	—
28	C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
30	C₂²⋊C₄×C₆₁	—	2	2	no	—
31	C₄²×C₆₁	—	2	2	no	—
32	C₄²×C₆₁	D₄	1	2	yes	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	—	2	2	no	—
37	C₄⋊Dic₆₁	C₂³	1	4	yes	—
38	#11758196	—	2	4	no	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₂×#92418	Q₈	1	8	yes	complemented
41	C₂×#92418	C₂³	1	4	yes	—
42	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₆₁	—	4	4	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	4	no	—
52	Dic₁₂₂	—	4	4	no	—
53	#92418	C₂×Q₈	1	8	yes	—
54	C₂²×C₆₁	—	2	2	no	—
55	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₆₁	—	2	2	no	—
58	C₄×C₆₁	—	2	2	no	—
59	C₄×C₆₁	—	4	4	no	—
60	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
61	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
62	Dic₆₁	—	4	4	no	—
63	#3162	—	61	1	no	maximal, Sylow
64	C₂×C₆₁	—	2	2	no	—
65	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₆₁	#575	1	2	yes	—
67	#533	—	61	1	no	—
68	#533	D₆₁	1	1	yes	characteristic
69	#501	—	61	4	no	—
70	#501	—	61	4	no	—
71	#427	—	61	4	no	—
72	C₂×#97	—	61	1	no	—
73	C₆₁	#3162	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	61	2	no	—
75	C₄⋊Q₈	—	61	2	no	—
76	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
77	#125	—	61	2	no	—
78	#125	C₂×D₆₁	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₂²⋊Q₈	—	122	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	122	2	no	—
83	#117	—	61	2	no	—
84	#117	C₂×D₆₁	1	2	yes	—
85	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
86	#107	—	61	8	no	—
87	#106	—	61	8	no	—
88	#97	—	61	2	no	—
89	#97	—	61	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	122	2	no	—
92	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
94	C₂×C₈	—	61	8	no	—
95	C₂×C₈	—	61	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	61	4	no	—
99	C₄⋊C₄	—	122	4	no	—
100	C₄⋊C₄	—	122	2	no	—
101	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	122	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	122	2	no	—
106	C₄²	C₂²⋊D₆₁	1	2	yes	—
107	C₄²	C₂²×D₆₁	1	1	yes	characteristic
108	C₄²	C₂²×D₆₁	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	244	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	122	4	no	—
115	C₂×C₄	Q₈×D₆₁	1	4	yes	—
116	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
120	C₈	—	61	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#14519775	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	244	4	no	—
128	C₄	#14519775	1	2	yes	—
129	C₄	C₂×Q₈×D₆₁	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#14519775	1	1	yes	characteristic
132	C₂	#16840560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16840172

C₂

yes

maximal, complemented

#16840168

C₂

yes

maximal, complemented

#16839798

C₂

yes

maximal

C₂×#14519688

C₂

yes

maximal, characteristic

C₆₁×#533

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

#14519777

C₂²

yes

—

#14519769

C₂²

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂²

yes

—

#14519710

C₂²

yes

—

#14519708

C₂²

yes

—

#14519688

C₂²

yes

—

#14519688

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

#125×C₆₁

C₂²

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

#3162

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

#575

yes

—

#533

—

#533

D₆₁

yes

—

—

—

C₂×#97

—

C₆₁

#3162

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#125

—

#125

C₂×D₆₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

#117

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

100

C₄⋊C₄

—

122

—

101

C₄⋊C₄

C₂²×D₆₁

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

122

—

104

C₄²

—

105

C₄²

—

122

—

106

C₄²

C₂²⋊D₆₁

yes

—

107

C₄²

C₂²×D₆₁

yes

characteristic

108

C₄²

C₂²×D₆₁

yes

characteristic

109

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

110

Q₈

—

111

Q₈

—

244

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

122

—

115

C₂×C₄

Q₈×D₆₁

yes

—

116

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

118

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

119

C₂×C₄

C₂³×D₆₁

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

#14519775

yes

—

123

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

244

—

128

C₄

#14519775

yes

—

129

C₄

C₂×Q₈×D₆₁

yes

—

130

C₂

—

131

C₂

C₂×#14519775

yes

characteristic

132

C₂

#16840560

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#533	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#533	D₆₁
hypocenter	C₆₁	#3162
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#533

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16840172	C₂	semidirect
3. #16840168	C₂	semidirect
40. C₂×#92418	Q₈	semidirect
73. C₆₁	#3162	semidirect

H ≅ G/N

type

2. #16840172

C₂

semidirect

3. #16840168

C₂

semidirect

40. C₂×#92418

Q₈

semidirect

73. C₆₁

#3162

semidirect

Subgroups — #21456341

Conjugacy classes of subgroups

Complements