Group #22590091 — subgroups

Subgroups — C₄⋊C₁₆×D₇₁ #22590091

Order 9088.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₁₆×D₇₁	C₂	1	2	yes	maximal
2	C₄×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₇₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
7	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
8	C₄²×D₇₁	C₄	1	1	yes	characteristic
9	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	C₁₆×D₇₁	—	2	4	no	—
11	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
12	C₈×Dic₇₁	C₄	1	1	yes	characteristic
13	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	C₂²	1	2	yes	—
15	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₇₁	C₈	1	1	yes	characteristic
17	C₂×C₄×D₇₁	C₈	1	1	yes	characteristic
18	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×D₇₁	—	2	2	no	—
22	C₈×D₇₁	D₄	1	2	yes	—
23	C₈×D₇₁	Q₈	1	2	yes	—
24	C₄×Dic₇₁	C₈	1	1	yes	characteristic
25	C₄×Dic₇₁	C₈	1	1	yes	characteristic
26	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	C₇₁⋊C₁₆	—	2	2	no	—
28	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₁₆×C₇₁	—	2	2	no	—
32	C₂²×D₇₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
39	C₄×D₇₁	—	2	2	no	—
40	C₄×D₇₁	C₁₆	1	4	yes	complemented
41	C₄×D₇₁	C₄⋊C₄	1	2	yes	—
42	C₄×D₇₁	M₁₆	1	2	yes	—
43	C₈×C₇₁	—	2	1	no	—
44	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₇₁⋊C₈	—	2	1	no	—
47	C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
49	C₂×D₇₁	C₄⋊C₈	1	2	yes	—
50	C₂×D₇₁	C₂×C₁₆	1	2	yes	—
51	C₂×D₇₁	M₃₂	1	2	yes	—
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	C₂×C₁₆	1	2	yes	complemented
54	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
56	Dic₇₁	—	2	1	no	—
57	Dic₇₁	C₄⋊C₈	1	2	yes	—
58	Dic₇₁	C₂×C₁₆	1	2	yes	complemented
59	C₂×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
61	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
62	D₇₁	C₄⋊C₁₆	1	4	yes	direct product factor
63	C₂×C₄⋊C₁₆	—	71	1	no	maximal, Sylow
64	C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₁₆	—	71	2	no	—
66	C₂×C₄×C₈	—	71	1	no	—
67	C₄⋊C₁₆	—	71	2	no	—
68	C₄⋊C₁₆	—	71	1	no	—
69	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic, direct product factor
70	C₂²×C₈	—	71	1	no	—
71	C₂²×C₈	—	71	1	no	—
72	C₂×C₄²	—	71	1	no	—
73	C₂×C₁₆	—	71	2	no	—
74	C₂×C₁₆	—	142	4	no	—
75	C₂×C₁₆	C₂×D₇₁	1	2	yes	—
76	C₄×C₈	—	71	1	no	—
77	C₄×C₈	—	71	1	no	—
78	C₄×C₈	—	71	1	no	—
79	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
80	C₂²×C₄	—	71	1	no	—
81	C₂²×C₄	—	71	1	no	—
82	C₂²×C₄	—	71	1	no	—
83	C₂×C₈	—	71	1	no	—
84	C₂×C₈	—	71	2	no	—
85	C₂×C₈	—	71	2	no	—
86	C₂×C₈	—	71	1	no	—
87	C₂×C₈	—	142	2	no	—
88	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
90	C₄²	—	71	1	no	—
91	C₄²	—	71	1	no	—
92	C₄²	—	71	1	no	—
93	C₄²	C₄×D₇₁	1	1	yes	characteristic
94	C₁₆	—	2	2	no	—
95	C₁₆	—	142	2	no	—
96	C₂³	—	71	1	no	—
97	C₂×C₄	—	71	1	no	—
98	C₂×C₄	—	71	1	no	—
99	C₂×C₄	—	71	4	no	—
100	C₂×C₄	—	71	2	no	—
101	C₂×C₄	—	71	2	no	—
102	C₂×C₄	—	71	1	no	—
103	C₂×C₄	—	142	2	no	—
104	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
105	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	71	2	no	—
109	C₈	—	142	1	no	—
110	C₈	D₄×D₇₁	1	1	yes	characteristic
111	C₈	Q₈×D₇₁	1	1	yes	characteristic
112	C₂²	—	71	2	no	—
113	C₂²	—	71	2	no	—
114	C₂²	—	71	2	no	—
115	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	71	2	no	—
118	C₄	—	71	2	no	—
119	C₄	—	142	1	no	—
120	C₄	C₁₆×D₇₁	1	2	yes	complemented
121	C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
122	C₄	M₁₆×D₇₁	1	1	yes	characteristic
123	C₂	—	71	4	no	—
124	C₂	C₄⋊C₈×D₇₁	1	1	yes	characteristic
125	C₂	C₂×C₁₆×D₇₁	1	1	yes	characteristic
126	C₂	M₃₂×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₇₁

C₂

yes

maximal

C₄×C₈×D₇₁

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

Dic₇₁⋊C₁₆

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₁

C₄

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

C₄²×D₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂²

yes

—

C₁₆×D₇₁

—

C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₈×Dic₇₁

C₄

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄×D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

D₄

yes

—

C₈×D₇₁

Q₈

yes

—

C₄×Dic₇₁

C₈

yes

characteristic

C₄×Dic₇₁

C₈

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

C₁₆

yes

complemented

C₄×D₇₁

C₄⋊C₄

yes

—

C₄×D₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

C₄⋊C₈

yes

—

C₂×D₇₁

C₂×C₁₆

yes

—

C₂×D₇₁

M₃₂

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₁₆

yes

complemented

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₂×M₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

C₄⋊C₈

yes

—

Dic₇₁

C₂×C₁₆

yes

complemented

C₂×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₇₁

C₂²×C₁₆

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

characteristic

D₇₁

C₄⋊C₁₆

yes

direct product factor

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₇₁

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₇₁

yes

characteristic, direct product factor

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×D₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₇₁

yes

characteristic

C₁₆

—

C₁₆

—

142

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

142

—

104

C₂×C₄

C₈×D₇₁

yes

characteristic

105

C₂×C₄

C₈×D₇₁

yes

characteristic

106

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

142

—

110

C₈

D₄×D₇₁

yes

characteristic

111

C₈

Q₈×D₇₁

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₈×D₇₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

142

—

120

C₄

C₁₆×D₇₁

yes

complemented

121

C₄

C₄⋊C₄×D₇₁

yes

characteristic

122

C₄

M₁₆×D₇₁

yes

characteristic

123

C₂

—

124

C₂

C₄⋊C₈×D₇₁

yes

characteristic

125

C₂

C₂×C₁₆×D₇₁

yes

characteristic

126

C₂

M₃₂×D₇₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator	C₂×C₇₁	C₂²×C₁₆
Fitting	C₄⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₈
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₁

commutator

C₂×C₇₁

C₂²×C₁₆

Fitting

C₄⋊C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×C₈

hypercenter

C₄⋊C₁₆

D₇₁

hypocenter

C₇₁

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
3. C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. Dic₇₁⋊C₁₆	C₂	semidirect
5. C₄⋊C₁₆×C₇₁	C₂	semidirect
40. C₄×D₇₁	C₁₆	semidirect
53. C₄×C₇₁	C₂×C₁₆	semidirect
58. Dic₇₁	C₂×C₁₆	semidirect
62. D₇₁	C₄⋊C₁₆	direct (N × H)
64. C₇₁	C₂×C₄⋊C₁₆	semidirect
69. C₄⋊C₁₆	D₇₁	direct (N × H)
120. C₄	C₁₆×D₇₁	semidirect

H ≅ G/N

type

3. C₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

4. Dic₇₁⋊C₁₆

C₂

semidirect

5. C₄⋊C₁₆×C₇₁

C₂

semidirect

40. C₄×D₇₁

C₁₆

semidirect

53. C₄×C₇₁

C₂×C₁₆

semidirect

58. Dic₇₁

C₂×C₁₆

semidirect

62. D₇₁

C₄⋊C₁₆

direct (N × H)

64. C₇₁

C₂×C₄⋊C₁₆

semidirect

69. C₄⋊C₁₆

D₇₁

direct (N × H)

120. C₄

C₁₆×D₇₁

semidirect

Subgroups — C₄⋊C₁₆×D₇₁ #22590091

Conjugacy classes of subgroups

Complements

Subgroups — C4⋊C16×D71 #22590091

Conjugacy classes of subgroups

Complements

Subgroups — C₄⋊C₁₆×D₇₁ #22590091