Group #22601437 — subgroups

Subgroups — #504×D₇₁ #22601437

Order 9088.

368 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 211 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	#131×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#17986417	C₂	1	3	yes	maximal, complemented
3	#17986416	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986415	C₂	1	3	yes	maximal, complemented
5	M₃₂×D₇₁	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₆×D₇₁	C₂	1	3	yes	maximal, complemented
7	#504×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×D₇₁	C₄	1	1	yes	characteristic
9	#14556136	C₄	1	3	yes	—
10	#14556135	C₄	1	1	yes	characteristic
11	#14556134	C₂²	1	3	yes	—
12	M₁₆×D₇₁	C₄	1	3	yes	—
13	C₂×C₈×D₇₁	C₂²	1	3	yes	—
14	#14556068	C₂²	1	3	yes	complemented
15	#14556067	C₂²	1	6	yes	complemented
16	C₂×C₇₁⋊C₁₆	C₂²	1	3	yes	complemented
17	C₁₆×D₇₁	C₂²	1	1	yes	characteristic
18	C₁₆×D₇₁	C₂²	1	3	yes	—
19	C₁₆×D₇₁	C₂²	1	3	yes	complemented
20	C₁₆×D₇₁	C₂²	1	3	yes	complemented
21	#131×C₇₁	C₂²	1	1	yes	characteristic
22	M₃₂×C₇₁	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₇₁	C₂²	1	3	yes	complemented
24	#11765866	C₈	1	1	yes	characteristic
25	#11765865	C₂×C₄	1	3	yes	—
26	#11765864	C₈	1	3	yes	—
27	Q₈×D₇₁	C₈	1	1	yes	characteristic
28	D₄×D₇₁	C₈	1	3	yes	—
29	C₂×C₄×D₇₁	C₂×C₄	1	3	yes	—
30	#11765849	C₂×C₄	1	3	yes	—
31	#11765848	C₂×C₄	1	6	yes	—
32	C₂×C₇₁⋊C₈	C₂×C₄	1	3	yes	—
33	C₈×D₇₁	C₂×C₄	1	3	yes	—
34	C₈×D₇₁	C₂×C₄	1	3	yes	—
35	C₈×D₇₁	C₂³	1	1	yes	characteristic
36	C₈×D₇₁	C₂³	1	3	yes	—
37	C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
38	C₇₁⋊C₁₆	C₂³	1	3	yes	—
39	#41×C₇₁	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₇₁	C₂×C₄	1	3	yes	—
41	C₂×C₈×C₇₁	C₂³	1	3	yes	—
42	C₁₆×C₇₁	C₂³	1	3	yes	—
43	C₁₆×C₇₁	C₂³	1	1	yes	characteristic
44	C₂²×D₇₁	C₂×C₈	1	3	yes	—
45	Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
46	D₄×C₇₁	C₂×C₈	1	3	yes	—
47	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
48	C₂²⋊D₇₁	C₂×C₈	1	6	yes	—
49	C₂×Dic₇₁	C₂×C₈	1	3	yes	—
50	D₂₈₄	C₂×C₈	1	3	yes	—
51	C₄×D₇₁	C₂×C₈	1	3	yes	—
52	C₄×D₇₁	C₂×C₈	1	3	yes	—
53	C₄×D₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₇₁	C₂²×C₄	1	3	yes	—
55	Dic₁₄₂	C₂×C₈	1	3	yes	—
56	C₈×C₇₁	C₂²×C₄	1	3	yes	—
57	C₈×C₇₁	C₂⁴	1	1	yes	characteristic
58	C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
59	C₇₁⋊C₈	C₂²×C₄	1	3	yes	—
60	C₂²×C₇₁	C₂²×C₈	1	3	yes	—
61	C₂×D₇₁	—	2	6	no	—
62	C₂×D₇₁	C₂²×C₈	1	3	yes	—
63	C₂×D₇₁	C₂²×C₈	1	1	yes	characteristic
64	C₄×C₇₁	C₂²×C₈	1	3	yes	—
65	C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
66	Dic₇₁	C₂²×C₈	1	3	yes	—
67	Dic₇₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₇₁	—	2	3	no	—
69	C₂×C₇₁	C₂³×C₈	1	1	yes	characteristic
70	D₇₁	—	2	3	no	—
71	D₇₁	#504	1	2	yes	direct product factor
72	C₂×#504	—	71	1	no	maximal, Sylow
73	C₇₁	C₂×#504	1	1	yes	characteristic, complemented, Sylow
74	C₂×#131	—	71	1	no	—
75	#504	—	71	1	no	—
76	#504	—	71	3	no	—
77	#504	—	71	3	no	—
78	#504	D₇₁	1	1	yes	characteristic, direct product factor
79	C₂×M₃₂	—	71	3	no	—
80	C₂²×C₁₆	—	71	3	no	—
81	C₂×#41	—	71	1	no	—
82	#131	—	71	1	no	—
83	#131	—	71	3	no	—
84	#131	—	71	3	no	—
85	#131	C₂×D₇₁	1	1	yes	characteristic
86	C₂×M₁₆	—	71	3	no	—
87	C₂²×C₈	—	71	3	no	—
88	M₃₂	—	71	3	no	—
89	M₃₂	—	71	6	no	—
90	M₃₂	C₂×D₇₁	1	3	yes	complemented
91	C₂×C₁₆	—	71	3	no	—
92	C₂×C₁₆	—	71	3	no	—
93	C₂×C₁₆	—	71	3	no	—
94	C₂×C₁₆	—	71	1	no	—
95	C₂×C₁₆	—	71	3	no	—
96	C₂×C₁₆	C₂×D₇₁	1	3	yes	complemented
97	#41	—	71	1	no	—
98	#41	—	71	3	no	—
99	#41	—	71	3	no	—
100	#41	C₄×D₇₁	1	1	yes	characteristic
101	C₂×Q₈	—	71	1	no	—
102	C₂×D₄	—	71	3	no	—
103	C₂²×C₄	—	71	3	no	—
104	M₁₆	—	71	3	no	—
105	M₁₆	—	71	6	no	—
106	M₁₆	C₄×D₇₁	1	3	yes	—
107	C₂×C₈	—	71	3	no	—
108	C₂×C₈	—	71	3	no	—
109	C₂×C₈	—	71	3	no	—
110	C₂×C₈	—	71	1	no	—
111	C₂×C₈	—	71	3	no	—
112	C₂×C₈	C₂²×D₇₁	1	3	yes	—
113	C₁₆	—	71	1	no	—
114	C₁₆	—	71	3	no	—
115	C₁₆	C₂²×D₇₁	1	3	yes	—
116	C₁₆	C₂²×D₇₁	1	1	yes	characteristic
117	C₂³	—	71	3	no	—
118	Q₈	—	71	3	no	—
119	Q₈	C₈×D₇₁	1	1	yes	characteristic
120	D₄	—	71	6	no	—
121	D₄	—	71	3	no	—
122	D₄	C₈×D₇₁	1	3	yes	—
123	C₂×C₄	—	71	3	no	—
124	C₂×C₄	—	71	1	no	—
125	C₂×C₄	—	71	3	no	—
126	C₂×C₄	—	71	3	no	—
127	C₂×C₄	—	71	3	no	—
128	C₂×C₄	C₂×C₄×D₇₁	1	3	yes	—
129	C₈	—	71	1	no	—
130	C₈	—	71	3	no	—
131	C₈	C₂×C₄×D₇₁	1	3	yes	—
132	C₈	C₂³×D₇₁	1	1	yes	characteristic
133	C₂²	—	71	1	no	—
134	C₂²	—	71	3	no	—
135	C₂²	—	142	6	no	—
136	C₂²	C₂×C₈×D₇₁	1	3	yes	—
137	C₄	—	71	1	no	—
138	C₄	—	71	3	no	—
139	C₄	C₂×C₈×D₇₁	1	3	yes	—
140	C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
141	C₂	—	2	3	no	—
142	C₂	—	71	2	no	—
143	C₂	—	142	3	no	—
144	C₂	C₂²×C₈×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#131×D₇₁

C₂

yes

maximal, characteristic

#17986417

C₂

yes

maximal, complemented

#17986416

C₂

yes

maximal, characteristic, complemented

#17986415

C₂

yes

maximal, complemented

M₃₂×D₇₁

C₂

yes

maximal, complemented

C₂×C₁₆×D₇₁

C₂

yes

maximal, complemented

#504×C₇₁

C₂

yes

maximal, characteristic, complemented

#41×D₇₁

C₄

yes

characteristic

#14556136

C₄

yes

—

#14556135

C₄

yes

characteristic

#14556134

C₂²

yes

—

M₁₆×D₇₁

C₄

yes

—

C₂×C₈×D₇₁

C₂²

yes

—

#14556068

C₂²

yes

complemented

#14556067

C₂²

yes

complemented

C₂×C₇₁⋊C₁₆

C₂²

yes

complemented

C₁₆×D₇₁

C₂²

yes

characteristic

C₁₆×D₇₁

C₂²

yes

—

C₁₆×D₇₁

C₂²

yes

complemented

C₁₆×D₇₁

C₂²

yes

complemented

#131×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

C₂²

yes

complemented

C₂×C₁₆×C₇₁

C₂²

yes

complemented

#11765866

C₈

yes

characteristic

#11765865

C₂×C₄

yes

—

#11765864

C₈

yes

—

Q₈×D₇₁

C₈

yes

characteristic

D₄×D₇₁

C₈

yes

—

C₂×C₄×D₇₁

C₂×C₄

yes

—

#11765849

C₂×C₄

yes

—

#11765848

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₈×D₇₁

C₂×C₄

yes

—

C₈×D₇₁

C₂×C₄

yes

—

C₈×D₇₁

C₂³

yes

characteristic

C₈×D₇₁

C₂³

yes

—

C₇₁⋊C₁₆

C₂³

yes

characteristic

C₇₁⋊C₁₆

C₂³

yes

—

#41×C₇₁

C₂×C₄

yes

characteristic

M₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂³

yes

—

C₁₆×C₇₁

C₂³

yes

—

C₁₆×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₂×C₈

yes

—

Q₈×C₇₁

C₂×C₈

yes

characteristic

D₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂²⋊D₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

C₂×C₈

yes

—

D₂₈₄

C₂×C₈

yes

—

C₄×D₇₁

C₂×C₈

yes

—

C₄×D₇₁

C₂×C₈

yes

—

C₄×D₇₁

C₂²×C₄

yes

characteristic

C₄×D₇₁

C₂²×C₄

yes

—

Dic₁₄₂

C₂×C₈

yes

—

C₈×C₇₁

C₂²×C₄

yes

—

C₈×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

C₂²×C₄

yes

characteristic

C₇₁⋊C₈

C₂²×C₄

yes

—

C₂²×C₇₁

C₂²×C₈

yes

—

C₂×D₇₁

—

C₂×D₇₁

C₂²×C₈

yes

—

C₂×D₇₁

C₂²×C₈

yes

characteristic

C₄×C₇₁

C₂²×C₈

yes

—

C₄×C₇₁

C₂³×C₄

yes

characteristic

Dic₇₁

C₂²×C₈

yes

—

Dic₇₁

C₂²×C₈

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

C₂³×C₈

yes

characteristic

D₇₁

—

D₇₁

#504

yes

direct product factor

C₂×#504

—

maximal, Sylow

C₇₁

C₂×#504

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic, direct product factor

C₂×M₃₂

—

C₂²×C₁₆

—

C₂×#41

—

—

—

—

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₇₁

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

—

—

—

100

#41

C₄×D₇₁

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

C₄×D₇₁

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₇₁

yes

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

C₂²×D₇₁

yes

—

116

C₁₆

C₂²×D₇₁

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

C₈×D₇₁

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₈×D₇₁

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₄×D₇₁

yes

—

129

C₈

—

130

C₈

—

131

C₈

C₂×C₄×D₇₁

yes

—

132

C₈

C₂³×D₇₁

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

142

—

136

C₂²

C₂×C₈×D₇₁

yes

—

137

C₄

—

138

C₄

—

139

C₄

C₂×C₈×D₇₁

yes

—

140

C₄

C₂²×C₄×D₇₁

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

142

—

144

C₂

C₂²×C₈×D₇₁

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂²×D₇₁
commutator, socle	C₂×C₇₁	C₂³×C₈
Frattini	C₈	C₂³×D₇₁
Fitting	#504×C₇₁	C₂
hypercenter	#504	D₇₁
hypocenter	C₇₁	C₂×#504
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₁₆ ◁ #504

subgroup

G/H

center

C₁₆

C₂²×D₇₁

commutator, socle

C₂×C₇₁

C₂³×C₈

Frattini

C₈

C₂³×D₇₁

Fitting

#504×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₁₆ ◁ #504

N	H ≅ G/N	type
2. #17986417	C₂	semidirect
3. #17986416	C₂	semidirect
4. #17986415	C₂	semidirect
5. M₃₂×D₇₁	C₂	semidirect
6. C₂×C₁₆×D₇₁	C₂	semidirect
7. #504×C₇₁	C₂	semidirect
14. #14556068	C₂²	semidirect
15. #14556067	C₂²	semidirect
16. C₂×C₇₁⋊C₁₆	C₂²	semidirect
19. C₁₆×D₇₁	C₂²	semidirect
20. C₁₆×D₇₁	C₂²	semidirect
22. M₃₂×C₇₁	C₂²	semidirect
23. C₂×C₁₆×C₇₁	C₂²	semidirect
71. D₇₁	#504	direct (N × H)
73. C₇₁	C₂×#504	semidirect
78. #504	D₇₁	direct (N × H)
90. M₃₂	C₂×D₇₁	semidirect
96. C₂×C₁₆	C₂×D₇₁	semidirect

H ≅ G/N

type

2. #17986417

C₂

semidirect

3. #17986416

C₂

semidirect

4. #17986415

C₂

semidirect

5. M₃₂×D₇₁

C₂

semidirect

6. C₂×C₁₆×D₇₁

C₂

semidirect

7. #504×C₇₁

C₂

semidirect

14. #14556068

C₂²

semidirect

15. #14556067

C₂²

semidirect

16. C₂×C₇₁⋊C₁₆

C₂²

semidirect

19. C₁₆×D₇₁

C₂²

semidirect

20. C₁₆×D₇₁

C₂²

semidirect

22. M₃₂×C₇₁

C₂²

semidirect

23. C₂×C₁₆×C₇₁

C₂²

semidirect

71. D₇₁

#504

direct (N × H)

73. C₇₁

C₂×#504

semidirect

78. #504

D₇₁

direct (N × H)

90. M₃₂

C₂×D₇₁

semidirect

96. C₂×C₁₆

C₂×D₇₁

semidirect

Subgroups — #504×D₇₁ #22601437

Conjugacy classes of subgroups

Complements

Subgroups — #504×D71 #22601437

Conjugacy classes of subgroups

Complements

Subgroups — #504×D₇₁ #22601437