Group #11764691 — subgroups

Subgroups — C₈×D₅×D₇ #11764691

Order 1120.

152 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₄×D₅×D₇	C₂	1	1	yes	maximal, characteristic
2	C₈×D₃₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#10587837	C₂	1	1	yes	maximal, characteristic, complemented
6	D₅×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₇×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅×D₇	C₄	1	1	yes	characteristic
9	C₄×D₃₅	C₂²	1	1	yes	characteristic
10	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
11	C₄×C₅×D₇	C₂²	1	1	yes	characteristic
12	#63304	C₄	1	1	yes	characteristic
13	D₅×Dic₇	C₄	1	1	yes	characteristic
14	D₇×Dic₅	C₄	1	1	yes	characteristic
15	C₅×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
16	C₇⋊#168	C₂²	1	1	yes	characteristic, complemented
17	C₅×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
18	C₇×#168	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₈×D₇	—	5	1	no	maximal, Hall
20	C₂×C₈×D₅	—	7	1	no	maximal, Hall
21	C₂×D₃₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×D₇	C₂×C₄	1	1	yes	characteristic
24	D₅×D₇	C₈	1	4	yes	direct product factor
25	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
26	Dic₃₅	C₂×C₄	1	1	yes	characteristic
27	C₅×Dic₇	C₂×C₄	1	1	yes	characteristic
28	C₇×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₇	—	5	1	no	—
30	C₂×C₇×C₈	—	5	1	no	—
31	C₂×C₇⋊C₈	—	5	1	no	—
32	C₈×D₇	—	5	1	no	—
33	C₈×D₇	—	5	1	no	—
34	C₈×D₇	—	5	1	no	—
35	C₈×D₇	D₅	1	1	yes	characteristic, direct product factor
36	C₂×C₄×D₅	—	7	1	no	—
37	C₂×C₅×C₈	—	7	1	no	—
38	C₂×#168	—	7	1	no	—
39	C₈×D₅	—	7	1	no	—
40	C₈×D₅	—	7	1	no	—
41	C₈×D₅	—	7	1	no	—
42	C₈×D₅	D₇	1	1	yes	characteristic, direct product factor
43	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
44	D₃₅	C₂×C₈	1	2	yes	complemented
45	C₅×D₇	C₂×C₈	1	2	yes	complemented
46	C₇×D₅	C₂×C₈	1	2	yes	complemented
47	C₂²×D₇	—	5	1	no	—
48	C₂×C₄×C₇	—	5	1	no	—
49	C₂×Dic₇	—	5	1	no	—
50	C₄×D₇	—	5	1	no	—
51	C₄×D₇	—	5	1	no	—
52	C₄×D₇	—	5	1	no	—
53	C₄×D₇	C₂×D₅	1	1	yes	characteristic
54	C₇×C₈	—	5	1	no	—
55	C₇×C₈	C₂×D₅	1	1	yes	characteristic, complemented
56	C₇⋊C₈	—	5	1	no	—
57	C₇⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
58	C₂²×D₅	—	7	1	no	—
59	C₂×C₄×C₅	—	7	1	no	—
60	C₂×Dic₅	—	7	1	no	—
61	C₄×D₅	—	7	1	no	—
62	C₄×D₅	—	7	1	no	—
63	C₄×D₅	—	7	1	no	—
64	C₄×D₅	C₂×D₇	1	1	yes	characteristic
65	C₅×C₈	—	7	1	no	—
66	C₅×C₈	C₂×D₇	1	1	yes	characteristic, complemented
67	#168	—	7	1	no	—
68	#168	C₂×D₇	1	1	yes	characteristic, complemented
69	C₅×C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
70	C₂²×C₈	—	35	1	no	Sylow
71	C₂²×C₇	—	5	1	no	—
72	C₂×D₇	—	5	1	no	—
73	C₂×D₇	—	5	4	no	—
74	C₂×D₇	C₄×D₅	1	1	yes	characteristic
75	C₄×C₇	—	5	1	no	—
76	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
77	Dic₇	—	5	1	no	—
78	Dic₇	C₄×D₅	1	1	yes	characteristic
79	C₂²×C₅	—	7	1	no	—
80	C₂×D₅	—	7	1	no	—
81	C₂×D₅	—	7	4	no	—
82	C₂×D₅	C₄×D₇	1	1	yes	characteristic
83	C₄×C₅	—	7	1	no	—
84	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
85	Dic₅	—	7	1	no	—
86	Dic₅	C₄×D₇	1	1	yes	characteristic
87	C₂²×C₄	—	35	1	no	—
88	C₂×C₈	—	5	1	no	—
89	C₂×C₈	—	7	1	no	—
90	C₂×C₈	—	35	1	no	—
91	C₂×C₈	—	35	1	no	—
92	C₂×C₈	—	35	1	no	—
93	C₂×C₈	—	35	1	no	—
94	C₂×C₇	—	5	2	no	—
95	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
96	D₇	—	5	2	no	—
97	D₇	C₈×D₅	1	2	yes	direct product factor
98	C₂×C₅	—	7	2	no	—
99	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
100	D₅	—	7	2	no	—
101	D₅	C₈×D₇	1	2	yes	direct product factor
102	C₂³	—	35	1	no	—
103	C₂×C₄	—	5	1	no	—
104	C₂×C₄	—	7	1	no	—
105	C₂×C₄	—	35	1	no	—
106	C₂×C₄	—	35	1	no	—
107	C₂×C₄	—	35	1	no	—
108	C₂×C₄	—	35	1	no	—
109	C₈	—	5	1	no	—
110	C₈	—	7	1	no	—
111	C₈	—	35	1	no	—
112	C₈	D₅×D₇	1	1	yes	characteristic, direct product factor
113	C₇	C₂×C₈×D₅	1	1	yes	characteristic, complemented, Sylow
114	C₅	C₂×C₈×D₇	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	5	1	no	—
116	C₂²	—	7	1	no	—
117	C₂²	—	35	4	no	—
118	C₂²	—	35	1	no	—
119	C₄	—	5	1	no	—
120	C₄	—	7	1	no	—
121	C₄	—	35	1	no	—
122	C₄	C₂×D₅×D₇	1	1	yes	characteristic
123	C₂	—	5	2	no	—
124	C₂	—	7	2	no	—
125	C₂	—	35	2	no	—
126	C₂	C₄×D₅×D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅×D₇

C₂

yes

maximal, characteristic

C₈×D₃₅

C₂

yes

maximal, characteristic, complemented

C₇×C₈×D₅

C₂

yes

maximal, characteristic, complemented

C₅×C₈×D₇

C₂

yes

maximal, characteristic, complemented

#10587837

C₂

yes

maximal, characteristic, complemented

D₅×C₇⋊C₈

C₂

yes

maximal, characteristic, complemented

D₇×#168

C₂

yes

maximal, characteristic, complemented

C₂×D₅×D₇

C₄

yes

characteristic

C₄×D₃₅

C₂²

yes

characteristic

C₄×C₇×D₅

C₂²

yes

characteristic

C₄×C₅×D₇

C₂²

yes

characteristic

#63304

C₄

yes

characteristic

D₅×Dic₇

C₄

yes

characteristic

D₇×Dic₅

C₄

yes

characteristic

C₅×C₇×C₈

C₂²

yes

characteristic, complemented

C₇⋊#168

C₂²

yes

characteristic, complemented

C₅×C₇⋊C₈

C₂²

yes

characteristic, complemented

C₇×#168

C₂²

yes

characteristic, complemented

C₂×C₈×D₇

—

maximal, Hall

C₂×C₈×D₅

—

maximal, Hall

C₂×D₃₅

C₂×C₄

yes

characteristic

C₂×C₇×D₅

C₂×C₄

yes

characteristic

C₂×C₅×D₇

C₂×C₄

yes

characteristic

D₅×D₇

C₈

yes

direct product factor

C₄×C₅×C₇

C₂³

yes

characteristic

Dic₃₅

C₂×C₄

yes

characteristic

C₅×Dic₇

C₂×C₄

yes

characteristic

C₇×Dic₅

C₂×C₄

yes

characteristic

C₂×C₄×D₇

—

C₂×C₇×C₈

—

C₂×C₇⋊C₈

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

D₅

yes

characteristic, direct product factor

C₂×C₄×D₅

—

C₂×C₅×C₈

—

C₂×#168

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

D₇

yes

characteristic, direct product factor

C₂×C₅×C₇

C₂²×C₄

yes

characteristic

D₃₅

C₂×C₈

yes

complemented

C₅×D₇

C₂×C₈

yes

complemented

C₇×D₅

C₂×C₈

yes

complemented

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

C₂×D₅

yes

characteristic

C₇×C₈

—

C₇×C₈

C₂×D₅

yes

characteristic, complemented

C₇⋊C₈

—

C₇⋊C₈

C₂×D₅

yes

characteristic, complemented

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×D₇

yes

characteristic

C₅×C₈

—

C₅×C₈

C₂×D₇

yes

characteristic, complemented

#168

—

#168

C₂×D₇

yes

characteristic, complemented

C₅×C₇

C₂²×C₈

yes

characteristic, complemented, Hall

C₂²×C₈

—

Sylow

C₂²×C₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₄×D₅

yes

characteristic

C₄×C₇

—

C₄×C₇

C₂²×D₅

yes

characteristic

Dic₇

—

Dic₇

C₄×D₅

yes

characteristic

C₂²×C₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

C₄×D₇

yes

characteristic

C₄×C₅

—

C₄×C₅

C₂²×D₇

yes

characteristic

Dic₅

—

Dic₅

C₄×D₇

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₇

—

C₂×C₇

C₂×C₄×D₅

yes

characteristic

D₇

—

D₇

C₈×D₅

yes

direct product factor

C₂×C₅

—

C₂×C₅

C₂×C₄×D₇

yes

characteristic

100

D₅

—

101

D₅

C₈×D₇

yes

direct product factor

102

C₂³

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

D₅×D₇

yes

characteristic, direct product factor

113

C₇

C₂×C₈×D₅

yes

characteristic, complemented, Sylow

114

C₅

C₂×C₈×D₇

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂×D₅×D₇

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₄×D₅×D₇

yes

characteristic

	subgroup	G/H
center, hypercenter	C₈	D₅×D₇
commutator, hypocenter	C₅×C₇	C₂²×C₈
Frattini	C₄	C₂×D₅×D₇
Fitting	C₅×C₇×C₈	C₂²
socle	C₂×C₅×C₇	C₂²×C₄
prime radical	C₂×D₅×D₇	C₄
cosocle	C₄×C₅×C₇	C₂³
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₈

subgroup

G/H

center, hypercenter

C₈

D₅×D₇

commutator, hypocenter

C₅×C₇

C₂²×C₈

Frattini

C₄

C₂×D₅×D₇

Fitting

C₅×C₇×C₈

C₂²

socle

C₂×C₅×C₇

C₂²×C₄

prime radical

C₂×D₅×D₇

C₄

cosocle

C₄×C₅×C₇

C₂³

LCS

G ▷ C₅×C₇

UCS

C₁ ◁ C₈

N	H ≅ G/N	type
2. C₈×D₃₅	C₂	semidirect
3. C₇×C₈×D₅	C₂	semidirect
4. C₅×C₈×D₇	C₂	semidirect
5. #10587837	C₂	semidirect
6. D₅×C₇⋊C₈	C₂	semidirect
7. D₇×#168	C₂	semidirect
15. C₅×C₇×C₈	C₂²	semidirect
16. C₇⋊#168	C₂²	semidirect
17. C₅×C₇⋊C₈	C₂²	semidirect
18. C₇×#168	C₂²	semidirect
24. D₅×D₇	C₈	direct (N × H)
35. C₈×D₇	D₅	direct (N × H)
42. C₈×D₅	D₇	direct (N × H)
44. D₃₅	C₂×C₈	semidirect
45. C₅×D₇	C₂×C₈	semidirect
46. C₇×D₅	C₂×C₈	semidirect
55. C₇×C₈	C₂×D₅	semidirect
57. C₇⋊C₈	C₂×D₅	semidirect
66. C₅×C₈	C₂×D₇	semidirect
68. #168	C₂×D₇	semidirect
69. C₅×C₇	C₂²×C₈	semidirect
97. D₇	C₈×D₅	direct (N × H)
101. D₅	C₈×D₇	direct (N × H)
112. C₈	D₅×D₇	direct (N × H)
113. C₇	C₂×C₈×D₅	semidirect
114. C₅	C₂×C₈×D₇	semidirect

H ≅ G/N

type

2. C₈×D₃₅

C₂

semidirect

3. C₇×C₈×D₅

C₂

semidirect

4. C₅×C₈×D₇

C₂

semidirect

5. #10587837

C₂

semidirect

6. D₅×C₇⋊C₈

C₂

semidirect

7. D₇×#168

C₂

semidirect

15. C₅×C₇×C₈

C₂²

semidirect

16. C₇⋊#168

C₂²

semidirect

17. C₅×C₇⋊C₈

C₂²

semidirect

18. C₇×#168

C₂²

semidirect

24. D₅×D₇

C₈

direct (N × H)

35. C₈×D₇

D₅

direct (N × H)

42. C₈×D₅

D₇

direct (N × H)

44. D₃₅

C₂×C₈

semidirect

45. C₅×D₇

C₂×C₈

semidirect

46. C₇×D₅

C₂×C₈

semidirect

55. C₇×C₈

C₂×D₅

semidirect

57. C₇⋊C₈

C₂×D₅

semidirect

66. C₅×C₈

C₂×D₇

semidirect

68. #168

C₂×D₇

semidirect

69. C₅×C₇

C₂²×C₈

semidirect

97. D₇

C₈×D₅

direct (N × H)

101. D₅

C₈×D₇

direct (N × H)

112. C₈

D₅×D₇

direct (N × H)

113. C₇

C₂×C₈×D₅

semidirect

114. C₅

C₂×C₈×D₇

semidirect

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

subgroup	index
C₄×D₅×D₇	2
C₈×D₃₅	2
C₇×C₈×D₅	2
C₅×C₈×D₇	2
#10587837	2
D₅×C₇⋊C₈	2
D₇×#168	2
C₂×C₈×D₇	5
C₂×C₈×D₅	7

subgroup

index

C₄×D₅×D₇

C₈×D₃₅

C₇×C₈×D₅

C₅×C₈×D₇

#10587837

D₅×C₇⋊C₈

D₇×#168

C₂×C₈×D₇

C₂×C₈×D₅

Subgroups — C₈×D₅×D₇ #11764691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×D5×D7 #11764691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₅×D₇ #11764691