Group #21446659 — subgroups

Subgroups — #21446659

Order 7808.

396 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14519774	C₂	1	1	yes	maximal, characteristic
2	#16839786	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14519693	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₄₄	C₄	1	4	yes	—
6	C₂²×D₂₄₄	C₂²	1	2	yes	—
7	C₂×#11758194	C₂²	1	2	yes	—
8	#14519774	—	2	2	no	—
9	#14519774	—	2	1	no	—
10	#14519774	—	2	1	no	—
11	C₄⋊#92418	—	2	2	no	—
12	#14519693	—	2	8	no	—
13	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	#97×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	—	2	1	no	—
17	C₂³×D₆₁	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂×D₂₄₄	—	2	4	no	—
21	C₂×D₂₄₄	—	2	4	no	—
22	C₂×D₂₄₄	—	2	4	no	—
23	C₂×D₂₄₄	—	2	2	no	—
24	C₂×D₂₄₄	—	2	2	no	—
25	C₂×D₂₄₄	—	2	4	no	—
26	C₂×D₂₄₄	—	4	8	no	—
27	M₁₆×C₆₁	—	2	2	no	—
28	M₁₆×C₆₁	D₄	1	4	yes	complemented
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₄²×C₆₁	D₄	1	2	yes	—
31	C₄²×C₆₁	D₄	1	1	yes	characteristic
32	C₄²×C₆₁	D₄	1	1	yes	characteristic
33	#11758194	—	2	4	no	—
34	C₂×#92418	—	2	2	no	—
35	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₆₁	—	2	4	no	—
37	C₂²×D₆₁	—	2	8	no	—
38	C₂²×D₆₁	—	4	8	no	—
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#41	1	2	yes	—
46	D₂₄₄	—	4	8	no	—
47	D₂₄₄	—	4	8	no	—
48	D₂₄₄	—	4	4	no	—
49	D₂₄₄	—	4	2	no	—
50	D₂₄₄	—	4	2	no	—
51	C₈×C₆₁	—	2	4	no	—
52	#92418	—	4	2	no	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₆₁	C₄×D₄	1	2	yes	—
56	C₂×D₆₁	—	4	4	no	—
57	C₂×D₆₁	—	4	8	no	—
58	C₂×D₆₁	—	8	8	no	—
59	C₄×C₆₁	—	2	2	no	—
60	C₄×C₆₁	#100	1	4	yes	complemented
61	C₄×C₆₁	#121	1	2	yes	—
62	C₄×C₆₁	#124	1	1	yes	characteristic
63	C₄×C₆₁	#127	1	1	yes	characteristic
64	#1965	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	#390	1	1	yes	characteristic
67	C₂×C₆₁	C₂×#100	1	2	yes	—
68	D₆₁	—	8	4	no	—
69	C₂×#127	—	61	1	no	—
70	#423	—	61	1	no	—
71	C₂×#100	—	61	4	no	—
72	C₄×M₁₆	D₆₁	1	1	yes	characteristic, complemented
73	C₆₁	#1965	1	1	yes	characteristic, complemented, Sylow
74	C₂²×D₄	—	61	4	no	—
75	C₂²×D₄	—	61	2	no	—
76	C₂×M₁₆	—	61	2	no	—
77	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
78	#127	—	122	2	no	—
79	#127	—	122	1	no	—
80	#127	—	122	1	no	—
81	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
82	C₄⋊C₈	—	122	2	no	—
83	#100	—	122	8	no	—
84	#97	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₂⁴	—	61	4	no	—
87	C₂×D₄	—	122	4	no	—
88	C₂×D₄	—	122	4	no	—
89	C₂×D₄	—	122	4	no	—
90	C₂×D₄	—	122	2	no	—
91	C₂×D₄	—	122	4	no	—
92	C₂×D₄	—	122	2	no	—
93	C₂×D₄	—	244	8	no	—
94	C₂²×C₄	C₄×D₆₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	122	4	no	—
98	M₁₆	D₂₄₄	1	4	yes	complemented
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	122	2	no	—
101	C₄²	D₂₄₄	1	2	yes	—
102	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
103	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂³	—	122	8	no	—
105	C₂³	—	122	4	no	—
106	C₂³	—	244	8	no	—
107	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
108	D₄	—	244	8	no	—
109	D₄	—	244	8	no	—
110	D₄	—	244	4	no	—
111	D₄	—	244	2	no	—
112	D₄	—	244	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	#11758198	1	2	yes	—
115	C₂×C₄	#11758198	1	2	yes	—
116	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
117	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
118	C₂×C₄	#11758222	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	244	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	488	8	no	—
124	C₂²	—	244	8	no	—
125	C₂²	—	244	4	no	—
126	C₂²	C₂×#11758198	1	1	yes	characteristic
127	C₂²	C₄×D₂₄₄	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#14519693	1	4	yes	complemented
130	C₄	#14519751	1	2	yes	—
131	C₄	#14519765	1	1	yes	characteristic
132	C₄	#14519774	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	488	4	no	—
135	C₂	#16839664	1	1	yes	characteristic
136	C₂	C₂×#14519693	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519774

C₂

yes

maximal, characteristic

#16839786

C₂

yes

maximal, characteristic, complemented

C₂×#14519693

C₂

yes

maximal, complemented

C₄×M₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

C₄

yes

—

C₂²×D₂₄₄

C₂²

yes

—

C₂×#11758194

C₂²

yes

—

—

—

—

—

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₂³×D₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

M₁₆×C₆₁

—

M₁₆×C₆₁

D₄

yes

complemented

C₂×C₈×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#100

yes

complemented

C₄×C₆₁

#121

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

C₂×#100

yes

—

D₆₁

—

C₂×#127

—

#423

—

C₂×#100

—

C₄×M₁₆

D₆₁

yes

characteristic, complemented

C₆₁

#1965

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

#127

—

122

—

#127

—

122

—

#127

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

#100

—

122

—

#97

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

122

—

M₁₆

D₂₄₄

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

122

—

101

C₄²

D₂₄₄

yes

—

102

C₄²

C₂²⋊D₆₁

yes

characteristic

103

C₄²

C₂²⋊D₆₁

yes

characteristic

104

C₂³

—

122

—

105

C₂³

—

122

—

106

C₂³

—

244

—

107

C₂³

C₂×C₄×D₆₁

yes

characteristic

108

D₄

—

244

—

109

D₄

—

244

—

110

D₄

—

244

—

111

D₄

—

244

—

112

D₄

—

244

—

113

C₂×C₄

—

114

C₂×C₄

#11758198

yes

—

115

C₂×C₄

#11758198

yes

—

116

C₂×C₄

C₂×D₂₄₄

yes

—

117

C₂×C₄

C₂×D₂₄₄

yes

characteristic

118

C₂×C₄

#11758222

yes

—

119

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

120

C₈

—

121

C₈

—

244

—

122

C₂²

—

123

C₂²

—

488

—

124

C₂²

—

244

—

125

C₂²

—

244

—

126

C₂²

C₂×#11758198

yes

characteristic

127

C₂²

C₄×D₂₄₄

yes

—

128

C₄

—

129

C₄

#14519693

yes

complemented

130

C₄

#14519751

yes

—

131

C₄

#14519765

yes

characteristic

132

C₄

#14519774

yes

characteristic

133

C₂

—

134

C₂

—

488

—

135

C₂

#16839664

yes

characteristic

136

C₂

C₂×#14519693

yes

—

	subgroup	G/H
center	C₂²	C₂×#11758198
commutator	C₂³×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₄×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₆₁
hypocenter	C₆₁	#1965
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#11758198

commutator

C₂³×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₄×M₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₆₁

hypocenter

C₆₁

#1965

LCS

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #16839786	C₂	semidirect
3. C₂×#14519693	C₂	semidirect
4. C₄×M₁₆×C₆₁	C₂	semidirect
28. M₁₆×C₆₁	D₄	semidirect
60. C₄×C₆₁	#100	semidirect
72. C₄×M₁₆	D₆₁	semidirect
73. C₆₁	#1965	semidirect
98. M₁₆	D₂₄₄	semidirect
129. C₄	#14519693	semidirect

H ≅ G/N

type

2. #16839786

C₂

semidirect

3. C₂×#14519693

C₂

semidirect

4. C₄×M₁₆×C₆₁

C₂

semidirect

28. M₁₆×C₆₁

D₄

semidirect

60. C₄×C₆₁

#100

semidirect

72. C₄×M₁₆

D₆₁

semidirect

73. C₆₁

#1965

semidirect

98. M₁₆

D₂₄₄

semidirect

129. C₄

#14519693

semidirect

Subgroups — #21446659

Conjugacy classes of subgroups

Complements