Group #20239376 — subgroups

Subgroups — C₄×C₄⋊C₈×C₄₇ #20239376

Order 6016.

316 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 240 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×C₄⋊C₈×C₄₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₄₇	C₂	1	2	yes	maximal
3	C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#336	C₂	1	2	yes	maximal
5	C₂²×C₈×C₄₇	C₂²	1	4	yes	—
6	C₂×C₄²×C₄₇	C₄	1	2	yes	—
7	C₂×C₄²×C₄₇	C₄	1	2	yes	—
8	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₇	C₄	1	8	yes	direct product factor
12	C₄×C₈×C₄₇	—	2	4	no	—
13	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₄₇	—	2	4	no	—
19	C₂×C₈×C₄₇	—	2	4	no	—
20	C₂×C₈×C₄₇	C₂×C₄	1	8	yes	—
21	C₄²×C₄₇	—	2	4	no	—
22	C₄²×C₄₇	—	2	2	no	—
23	C₄²×C₄₇	C₈	1	8	yes	complemented
24	C₄²×C₄₇	C₂×C₄	1	4	yes	—
25	C₄²×C₄₇	D₄	1	2	yes	—
26	C₄²×C₄₇	Q₈	1	2	yes	—
27	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₄₇	—	2	2	no	—
29	C₂×C₄×C₄₇	—	2	2	no	—
30	C₂×C₄×C₄₇	—	2	2	no	—
31	C₂×C₄×C₄₇	C₄²	1	2	yes	—
32	C₂×C₄×C₄₇	C₄²	1	2	yes	—
33	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₄₇	M₁₆	1	4	yes	—
38	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₄₇	#41	1	2	yes	—
42	C₈×C₄₇	—	2	8	no	—
43	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄₇	#117	1	1	yes	characteristic
46	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
47	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
48	C₄×C₄₇	—	2	2	no	—
49	C₄×C₄₇	—	2	4	no	—
50	C₄×C₄₇	C₄×C₈	1	4	yes	complemented
51	C₄×C₄₇	C₄⋊C₈	1	8	yes	direct product factor
52	C₄×C₄₇	C₄×D₄	1	2	yes	—
53	C₄×C₄₇	C₄×Q₈	1	2	yes	—
54	C₄×C₄⋊C₈	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
57	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
59	C₂×C₄₇	#431	1	2	yes	—
60	C₂×C₄⋊C₈	C₂×C₄₇	1	2	yes	—
61	C₂×C₄×C₈	C₂×C₄₇	1	2	yes	—
62	C₄³	C₂×C₄₇	1	1	yes	characteristic
63	#336	C₂×C₄₇	1	2	yes	—
64	C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₈	C₂²×C₄₇	1	4	yes	—
66	C₂×C₄²	C₄×C₄₇	1	2	yes	—
67	C₂×C₄²	C₄×C₄₇	1	2	yes	—
68	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
69	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
70	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
71	C₄⋊C₈	C₄×C₄₇	1	8	yes	direct product factor
72	C₄×C₈	—	2	4	no	—
73	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
74	C₂²×C₄	C₂×C₄×C₄₇	1	2	yes	—
75	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄×C₄₇	1	2	yes	—
77	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	C₂×C₄×C₄₇	1	8	yes	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	2	no	—
83	C₄²	C₈×C₄₇	1	8	yes	complemented
84	C₄²	C₂×C₄×C₄₇	1	4	yes	—
85	C₄²	D₄×C₄₇	1	2	yes	—
86	C₄²	Q₈×C₄₇	1	2	yes	—
87	C₂³	C₂²×C₄×C₄₇	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₄²×C₄₇	1	2	yes	—
92	C₂×C₄	C₄²×C₄₇	1	2	yes	—
93	C₂×C₄	C₄⋊C₄×C₄₇	1	4	yes	—
94	C₂×C₄	C₂×C₈×C₄₇	1	4	yes	—
95	C₂×C₄	C₂×C₈×C₄₇	1	4	yes	—
96	C₂×C₄	C₂×C₈×C₄₇	1	4	yes	—
97	C₂×C₄	M₁₆×C₄₇	1	4	yes	—
98	C₂×C₄	C₂²×C₄×C₄₇	1	2	yes	—
99	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
100	C₂×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
101	C₂×C₄	#41×C₄₇	1	2	yes	—
102	C₈	—	2	8	no	—
103	C₂²	C₂×C₄²×C₄₇	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
105	C₂²	#117×C₄₇	1	1	yes	characteristic
106	C₂²	C₂²×C₈×C₄₇	1	2	yes	—
107	C₂²	C₂×M₁₆×C₄₇	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	C₄×C₈×C₄₇	1	4	yes	complemented
111	C₄	C₄⋊C₈×C₄₇	1	8	yes	direct product factor
112	C₄	C₄×D₄×C₄₇	1	2	yes	—
113	C₄	C₄×Q₈×C₄₇	1	2	yes	—
114	C₂	C₄×C₄⋊C₄×C₄₇	1	1	yes	characteristic
115	C₂	C₂×C₄×C₈×C₄₇	1	1	yes	characteristic
116	C₂	C₄×M₁₆×C₄₇	1	1	yes	characteristic
117	C₂	C₂×C₄⋊C₈×C₄₇	1	2	yes	—
118	C₂	C₄₇×#431	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal

C₂×C₄×C₈×C₄₇

C₂

yes

maximal

C₄³×C₄₇

C₂

yes

maximal, characteristic

C₄₇×#336

C₂

yes

maximal

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₄

yes

direct product factor

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₈

yes

complemented

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

Q₈

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₈×C₄₇

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#117

yes

characteristic

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₈

yes

complemented

C₄×C₄₇

C₄⋊C₈

yes

direct product factor

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₄×Q₈

yes

—

C₄×C₄⋊C₈

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

characteristic

C₂×C₄₇

C₂×C₄⋊C₈

yes

—

C₂×C₄₇

#431

yes

—

C₂×C₄⋊C₈

C₂×C₄₇

yes

—

C₂×C₄×C₈

C₂×C₄₇

yes

—

C₄³

C₂×C₄₇

yes

characteristic

#336

C₂×C₄₇

yes

—

C₄₇

C₄×C₄⋊C₈

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂²×C₄₇

yes

—

C₂×C₄²

C₄×C₄₇

yes

—

C₂×C₄²

C₄×C₄₇

yes

—

C₂×C₄²

C₂²×C₄₇

yes

characteristic

C₂×C₄²

C₂²×C₄₇

yes

characteristic

C₂×C₄²

C₂²×C₄₇

yes

characteristic

C₄⋊C₈

C₄×C₄₇

yes

direct product factor

C₄×C₈

—

C₂²×C₄

C₂×C₄×C₄₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₄₇

yes

—

C₂²×C₄

C₂×C₄×C₄₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₄₇

yes

—

C₂²×C₄

C₂³×C₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₄₇

yes

—

C₄²

—

C₄²

—

C₄²

C₈×C₄₇

yes

complemented

C₄²

C₂×C₄×C₄₇

yes

—

C₄²

D₄×C₄₇

yes

—

C₄²

Q₈×C₄₇

yes

—

C₂³

C₂²×C₄×C₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₄₇

yes

—

C₂×C₄

C₄²×C₄₇

yes

—

C₂×C₄

C₄⋊C₄×C₄₇

yes

—

C₂×C₄

C₂×C₈×C₄₇

yes

—

C₂×C₄

C₂×C₈×C₄₇

yes

—

C₂×C₄

C₂×C₈×C₄₇

yes

—

C₂×C₄

M₁₆×C₄₇

yes

—

C₂×C₄

C₂²×C₄×C₄₇

yes

—

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

100

C₂×C₄

C₂×Q₈×C₄₇

yes

characteristic

101

C₂×C₄

#41×C₄₇

yes

—

102

C₈

—

103

C₂²

C₂×C₄²×C₄₇

yes

characteristic

104

C₂²

C₂×C₄⋊C₄×C₄₇

yes

characteristic

105

C₂²

#117×C₄₇

yes

characteristic

106

C₂²

C₂²×C₈×C₄₇

yes

—

107

C₂²

C₂×M₁₆×C₄₇

yes

—

108

C₄

—

109

C₄

—

110

C₄

C₄×C₈×C₄₇

yes

complemented

111

C₄

C₄⋊C₈×C₄₇

yes

direct product factor

112

C₄

C₄×D₄×C₄₇

yes

—

113

C₄

C₄×Q₈×C₄₇

yes

—

114

C₂

C₄×C₄⋊C₄×C₄₇

yes

characteristic

115

C₂

C₂×C₄×C₈×C₄₇

yes

characteristic

116

C₂

C₄×M₁₆×C₄₇

yes

characteristic

117

C₂

C₂×C₄⋊C₈×C₄₇

yes

—

118

C₂

C₄₇×#431

yes

—

	subgroup	G/H
center	C₂×C₄²×C₄₇	C₂²
commutator	C₂	C₂×C₄×C₈×C₄₇
Frattini, cosocle	C₂²×C₄	C₂³×C₄₇
socle, prime radical	C₂³×C₄₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center

C₂×C₄²×C₄₇

C₂²

commutator

C₂

C₂×C₄×C₈×C₄₇

Frattini, cosocle

C₂²×C₄

C₂³×C₄₇

socle, prime radical

C₂³×C₄₇

C₂²×C₄

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
11. C₄⋊C₈×C₄₇	C₄	direct (N × H)
23. C₄²×C₄₇	C₈	semidirect
50. C₄×C₄₇	C₄×C₈	semidirect
51. C₄×C₄₇	C₄⋊C₈	direct (N × H)
54. C₄×C₄⋊C₈	C₄₇	direct (N × H)
64. C₄₇	C₄×C₄⋊C₈	direct (N × H)
71. C₄⋊C₈	C₄×C₄₇	direct (N × H)
83. C₄²	C₈×C₄₇	semidirect
110. C₄	C₄×C₈×C₄₇	semidirect
111. C₄	C₄⋊C₈×C₄₇	direct (N × H)

H ≅ G/N

type

11. C₄⋊C₈×C₄₇

C₄

direct (N × H)

23. C₄²×C₄₇

C₈

semidirect

50. C₄×C₄₇

C₄×C₈

semidirect

51. C₄×C₄₇

C₄⋊C₈

direct (N × H)

54. C₄×C₄⋊C₈

C₄₇

direct (N × H)

64. C₄₇

C₄×C₄⋊C₈

direct (N × H)

71. C₄⋊C₈

C₄×C₄₇

direct (N × H)

83. C₄²

C₈×C₄₇

semidirect

110. C₄

C₄×C₈×C₄₇

semidirect

111. C₄

C₄⋊C₈×C₄₇

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊C₈×C₄₇	2
C₂×C₄×C₈×C₄₇	2
C₄³×C₄₇	2
C₄₇×#336	2
C₄×C₄⋊C₈	47

subgroup

index

C₂×C₄⋊C₈×C₄₇

C₂×C₄×C₈×C₄₇

C₄³×C₄₇

C₄₇×#336

C₄×C₄⋊C₈

Subgroups — C₄×C₄⋊C₈×C₄₇ #20239376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×C4⋊C8×C47 #20239376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊C₈×C₄₇ #20239376