Group #42124574 — subgroups

Subgroups — D₄₅⋊D₁₅₅ #42124574

Order 27900.

144 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 34 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	C₅×D₁₃₉₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₃₁×D₄₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₅⋊D₁₅₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₅⋊D₁₅₅	—	3	1	no	maximal
5	C₅²×C₉×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₄₅⋊D₃₁	—	5	1	no	maximal
7	D₉×D₁₅₅	—	5	1	no	maximal
8	C₅×D₄₆₅	—	3	1	no	—
9	C₅×D₁₅×C₃₁	—	3	1	no	—
10	C₃×C₅⋊D₁₅₅	D₃	1	1	yes	characteristic
11	D₅×D₁₅₅	—	9	1	no	Hall
12	D₁₃₉₅	D₅	1	1	yes	characteristic, complemented
13	C₃₁×D₄₅	D₅	1	1	yes	characteristic, complemented
14	C₅×D₂₇₉	—	5	1	no	—
15	C₅×D₉×C₃₁	—	5	1	no	—
16	C₉×D₁₅₅	—	5	1	no	—
17	C₉×D₁₅₅	—	5	1	no	—
18	C₉×D₁₅₅	—	10	2	no	—
19	C₃×C₅²×C₃₁	C₂×D₃	1	1	yes	characteristic
20	D₁₅⋊D₃₁	—	15	1	no	—
21	D₃×D₁₅₅	—	15	1	no	—
22	C₅⋊D₁₅₅	D₉	1	1	yes	characteristic, complemented
23	C₅×D₁₅₅	—	9	1	no	—
24	C₅×D₅×C₃₁	—	9	1	no	—
25	C₅×C₉×C₃₁	—	2	2	no	—
26	C₅×C₉×C₃₁	C₂×D₅	1	1	yes	characteristic, complemented
27	C₅×C₉×C₃₁	C₂×D₅	1	1	yes	characteristic, complemented
28	D₉×D₃₁	—	25	1	no	Hall
29	D₄₆₅	—	3	1	no	—
30	D₁₅×C₃₁	—	3	1	no	—
31	C₅×D₉₃	—	15	1	no	—
32	C₅×D₃×C₃₁	—	15	1	no	—
33	C₃×D₁₅₅	—	5	1	no	—
34	C₃×D₁₅₅	—	5	1	no	—
35	C₃×D₁₅₅	—	10	2	no	—
36	D₄₅⋊D₅	—	31	1	no	maximal, Hall
37	C₅²×C₃₁	C₂×D₉	1	1	yes	characteristic, complemented, Hall
38	C₂×D₁₅₅	—	45	1	no	—
39	D₅×D₃₁	—	45	1	no	—
40	D₂₇₉	—	5	1	no	—
41	C₉×D₃₁	—	25	1	no	—
42	D₉×C₃₁	—	5	1	no	—
43	C₃×C₅×C₃₁	—	2	2	no	—
44	C₃×C₅×C₃₁	D₃×D₅	1	1	yes	characteristic
45	C₃×C₅×C₃₁	D₃×D₅	1	1	yes	characteristic
46	C₅×D₄₅	—	31	1	no	—
47	C₅×D₄₅	D₃₁	1	1	yes	characteristic, complemented
48	C₉×C₅⋊D₅	—	31	1	no	—
49	D₃×D₃₁	—	75	1	no	—
50	C₂×C₅×C₃₁	—	45	1	no	—
51	D₁₅₅	—	5	1	no	—
52	D₁₅₅	—	5	1	no	—
53	D₁₅₅	—	9	1	no	—
54	D₁₅₅	—	10	2	no	—
55	C₅×D₃₁	—	45	1	no	—
56	D₅×C₃₁	—	9	1	no	—
57	D₁₅⋊D₅	—	93	1	no	—
58	C₉×C₃₁	D₅²	1	1	yes	characteristic, complemented, Hall
59	C₅²×C₉	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
60	D₉₃	—	15	1	no	—
61	C₃×D₃₁	—	25	1	no	—
62	D₃×C₃₁	—	15	1	no	—
63	D₅×D₉	—	155	1	no	—
64	D₅×D₉	—	155	1	no	—
65	C₅×C₃₁	—	2	2	no	—
66	C₅×C₃₁	D₅×D₉	1	1	yes	characteristic, complemented
67	C₅×C₃₁	D₅×D₉	1	1	yes	characteristic, complemented
68	C₅×D₁₅	—	3	1	no	—
69	C₅×D₁₅	—	93	1	no	—
70	C₃×C₅⋊D₅	—	31	1	no	—
71	C₂×D₃₁	—	225	1	no	Hall
72	D₅²	—	279	1	no	Hall
73	C₃×C₃₁	D₁₅⋊D₅	1	1	yes	characteristic
74	D₄₅	—	31	1	no	—
75	D₄₅	D₁₅₅	1	1	yes	characteristic, complemented
76	C₉×D₅	—	310	2	no	—
77	C₉×D₅	—	155	1	no	—
78	C₉×D₅	—	155	1	no	—
79	C₅×D₉	—	5	1	no	—
80	C₅×D₉	—	155	1	no	—
81	C₃×C₅²	D₃×D₃₁	1	1	yes	characteristic
82	C₂×C₃₁	—	45	1	no	—
83	D₃₁	—	25	1	no	—
84	D₃₁	—	45	1	no	—
85	D₃×D₅	—	465	1	no	—
86	D₃×D₅	—	465	1	no	—
87	C₅⋊D₅	—	31	1	no	—
88	C₅×D₅	—	9	1	no	—
89	C₅×D₅	—	279	1	no	—
90	C₅×C₉	—	2	2	no	—
91	C₅×C₉	D₅×D₃₁	1	1	yes	characteristic, complemented
92	C₅×C₉	C₂×D₁₅₅	1	1	yes	characteristic, complemented
93	C₂×D₉	—	775	1	no	Hall
94	C₃₁	D₄₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
95	D₁₅	—	3	1	no	—
96	D₁₅	—	93	1	no	—
97	C₃×D₅	—	310	2	no	—
98	C₃×D₅	—	155	1	no	—
99	C₃×D₅	—	155	1	no	—
100	C₅×D₃	—	15	1	no	—
101	C₅×D₃	—	465	1	no	—
102	C₅²	D₉×D₃₁	1	1	yes	characteristic, complemented, Sylow
103	C₂×D₅	—	1395	1	no	—
104	C₂×D₅	—	1395	1	no	—
105	C₂×C₉	—	775	1	no	—
106	D₉	—	5	1	no	—
107	D₉	—	155	1	no	—
108	C₃×C₅	—	2	2	no	—
109	C₃×C₅	D₃×D₁₅₅	1	1	yes	characteristic
110	C₃×C₅	D₁₅⋊D₃₁	1	1	yes	characteristic
111	C₂×D₃	—	2325	1	no	—
112	C₂×C₅	—	45	1	no	—
113	C₂×C₅	—	1395	1	no	—
114	D₅	—	9	1	no	—
115	D₅	—	279	1	no	—
116	D₅	—	310	2	no	—
117	D₅	—	155	1	no	—
118	D₅	—	155	1	no	—
119	C₉	D₅×D₁₅₅	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₃	—	775	1	no	—
121	D₃	—	15	1	no	—
122	D₃	—	465	1	no	—
123	C₅	—	2	2	no	—
124	C₅	D₉×D₁₅₅	1	1	yes	characteristic, complemented
125	C₅	D₄₅⋊D₃₁	1	1	yes	characteristic, complemented
126	C₂²	—	6975	1	no	Sylow
127	C₃	D₁₅⋊D₁₅₅	1	1	yes	characteristic
128	C₂	—	775	1	no	—
129	C₂	—	45	1	no	—
130	C₂	—	1395	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×D₁₃₉₅

C₂

yes

maximal, characteristic, complemented

C₅×C₃₁×D₄₅

C₂

yes

maximal, characteristic, complemented

C₉×C₅⋊D₁₅₅

C₂

yes

maximal, characteristic, complemented

D₁₅⋊D₁₅₅

—

maximal

C₅²×C₉×C₃₁

C₂²

yes

characteristic, complemented, Hall

D₄₅⋊D₃₁

—

maximal

D₉×D₁₅₅

—

maximal

C₅×D₄₆₅

—

C₅×D₁₅×C₃₁

—

C₃×C₅⋊D₁₅₅

D₃

yes

characteristic

D₅×D₁₅₅

—

Hall

D₁₃₉₅

D₅

yes

characteristic, complemented

C₃₁×D₄₅

D₅

yes

characteristic, complemented

C₅×D₂₇₉

—

C₅×D₉×C₃₁

—

C₉×D₁₅₅

—

C₉×D₁₅₅

—

C₉×D₁₅₅

—

C₃×C₅²×C₃₁

C₂×D₃

yes

characteristic

D₁₅⋊D₃₁

—

D₃×D₁₅₅

—

C₅⋊D₁₅₅

D₉

yes

characteristic, complemented

C₅×D₁₅₅

—

C₅×D₅×C₃₁

—

C₅×C₉×C₃₁

—

C₅×C₉×C₃₁

C₂×D₅

yes

characteristic, complemented

C₅×C₉×C₃₁

C₂×D₅

yes

characteristic, complemented

D₉×D₃₁

—

Hall

D₄₆₅

—

D₁₅×C₃₁

—

C₅×D₉₃

—

C₅×D₃×C₃₁

—

C₃×D₁₅₅

—

C₃×D₁₅₅

—

C₃×D₁₅₅

—

D₄₅⋊D₅

—

maximal, Hall

C₅²×C₃₁

C₂×D₉

yes

characteristic, complemented, Hall

C₂×D₁₅₅

—

D₅×D₃₁

—

D₂₇₉

—

C₉×D₃₁

—

D₉×C₃₁

—

C₃×C₅×C₃₁

—

C₃×C₅×C₃₁

D₃×D₅

yes

characteristic

C₃×C₅×C₃₁

D₃×D₅

yes

characteristic

C₅×D₄₅

—

C₅×D₄₅

D₃₁

yes

characteristic, complemented

C₉×C₅⋊D₅

—

D₃×D₃₁

—

C₂×C₅×C₃₁

—

D₁₅₅

—

D₁₅₅

—

D₁₅₅

—

D₁₅₅

—

C₅×D₃₁

—

D₅×C₃₁

—

D₁₅⋊D₅

—

C₉×C₃₁

D₅²

yes

characteristic, complemented, Hall

C₅²×C₉

C₂×D₃₁

yes

characteristic, complemented, Hall

D₉₃

—

C₃×D₃₁

—

D₃×C₃₁

—

D₅×D₉

—

155

—

D₅×D₉

—

155

—

C₅×C₃₁

—

C₅×C₃₁

D₅×D₉

yes

characteristic, complemented

C₅×C₃₁

D₅×D₉

yes

characteristic, complemented

C₅×D₁₅

—

C₅×D₁₅

—

C₃×C₅⋊D₅

—

C₂×D₃₁

—

225

Hall

D₅²

—

279

Hall

C₃×C₃₁

D₁₅⋊D₅

yes

characteristic

D₄₅

—

D₄₅

D₁₅₅

yes

characteristic, complemented

C₉×D₅

—

310

—

C₉×D₅

—

155

—

C₉×D₅

—

155

—

C₅×D₉

—

C₅×D₉

—

155

—

C₃×C₅²

D₃×D₃₁

yes

characteristic

C₂×C₃₁

—

D₃₁

—

D₃₁

—

D₃×D₅

—

465

—

D₃×D₅

—

465

—

C₅⋊D₅

—

C₅×D₅

—

C₅×D₅

—

279

—

C₅×C₉

—

C₅×C₉

D₅×D₃₁

yes

characteristic, complemented

C₅×C₉

C₂×D₁₅₅

yes

characteristic, complemented

C₂×D₉

—

775

Hall

C₃₁

D₄₅⋊D₅

yes

characteristic, complemented, Sylow

D₁₅

—

D₁₅

—

C₃×D₅

—

310

—

C₃×D₅

—

155

—

C₃×D₅

—

155

—

100

C₅×D₃

—

101

C₅×D₃

—

465

—

102

C₅²

D₉×D₃₁

yes

characteristic, complemented, Sylow

103

C₂×D₅

—

1395

—

104

C₂×D₅

—

1395

—

105

C₂×C₉

—

775

—

106

D₉

—

107

D₉

—

155

—

108

C₃×C₅

—

109

C₃×C₅

D₃×D₁₅₅

yes

characteristic

110

C₃×C₅

D₁₅⋊D₃₁

yes

characteristic

111

C₂×D₃

—

2325

—

112

C₂×C₅

—

113

C₂×C₅

—

1395

—

114

D₅

—

115

D₅

—

279

—

116

D₅

—

310

—

117

D₅

—

155

—

118

D₅

—

155

—

119

C₉

D₅×D₁₅₅

yes

characteristic, complemented, Sylow

120

C₂×C₃

—

775

—

121

D₃

—

122

D₃

—

465

—

123

C₅

—

124

C₅

D₉×D₁₅₅

yes

characteristic, complemented

125

C₅

D₄₅⋊D₃₁

yes

characteristic, complemented

126

C₂²

—

6975

Sylow

127

C₃

D₁₅⋊D₁₅₅

yes

characteristic

128

C₂

—

775

—

129

C₂

—

130

C₂

—

1395

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅²×C₉×C₃₁	C₂²
Frattini	C₃	D₁₅⋊D₁₅₅
socle	C₃×C₅²×C₃₁	C₂×D₃
LCS	G ▷ C₅²×C₉×C₃₁
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, Fitting, cosocle, hypocenter

C₅²×C₉×C₃₁

C₂²

Frattini

C₃

D₁₅⋊D₁₅₅

socle

C₃×C₅²×C₃₁

C₂×D₃

LCS

G ▷ C₅²×C₉×C₃₁

UCS

C₁

N	H ≅ G/N	type
1. C₅×D₁₃₉₅	C₂	semidirect
2. C₅×C₃₁×D₄₅	C₂	semidirect
3. C₉×C₅⋊D₁₅₅	C₂	semidirect
5. C₅²×C₉×C₃₁	C₂²	semidirect
12. D₁₃₉₅	D₅	semidirect
13. C₃₁×D₄₅	D₅	semidirect
22. C₅⋊D₁₅₅	D₉	semidirect
26. C₅×C₉×C₃₁	C₂×D₅	semidirect
27. C₅×C₉×C₃₁	C₂×D₅	semidirect
37. C₅²×C₃₁	C₂×D₉	semidirect
47. C₅×D₄₅	D₃₁	semidirect
58. C₉×C₃₁	D₅²	semidirect
59. C₅²×C₉	C₂×D₃₁	semidirect
66. C₅×C₃₁	D₅×D₉	semidirect
67. C₅×C₃₁	D₅×D₉	semidirect
75. D₄₅	D₁₅₅	semidirect
91. C₅×C₉	D₅×D₃₁	semidirect
92. C₅×C₉	C₂×D₁₅₅	semidirect
94. C₃₁	D₄₅⋊D₅	semidirect
102. C₅²	D₉×D₃₁	semidirect
119. C₉	D₅×D₁₅₅	semidirect
124. C₅	D₉×D₁₅₅	semidirect
125. C₅	D₄₅⋊D₃₁	semidirect

H ≅ G/N

type

1. C₅×D₁₃₉₅

C₂

semidirect

2. C₅×C₃₁×D₄₅

C₂

semidirect

3. C₉×C₅⋊D₁₅₅

C₂

semidirect

5. C₅²×C₉×C₃₁

C₂²

semidirect

12. D₁₃₉₅

D₅

semidirect

13. C₃₁×D₄₅

D₅

semidirect

22. C₅⋊D₁₅₅

D₉

semidirect

26. C₅×C₉×C₃₁

C₂×D₅

semidirect

27. C₅×C₉×C₃₁

C₂×D₅

semidirect

37. C₅²×C₃₁

C₂×D₉

semidirect

47. C₅×D₄₅

D₃₁

semidirect

58. C₉×C₃₁

D₅²

semidirect

59. C₅²×C₉

C₂×D₃₁

semidirect

66. C₅×C₃₁

D₅×D₉

semidirect

67. C₅×C₃₁

D₅×D₉

semidirect

75. D₄₅

D₁₅₅

semidirect

91. C₅×C₉

D₅×D₃₁

semidirect

92. C₅×C₉

C₂×D₁₅₅

semidirect

94. C₃₁

D₄₅⋊D₅

semidirect

102. C₅²

D₉×D₃₁

semidirect

119. C₉

D₅×D₁₅₅

semidirect

124. C₅

D₉×D₁₅₅

semidirect

125. C₅

D₄₅⋊D₃₁

semidirect

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

subgroup	index
C₅×D₁₃₉₅	2
C₅×C₃₁×D₄₅	2
C₉×C₅⋊D₁₅₅	2
D₁₅⋊D₁₅₅	3
D₄₅⋊D₃₁	5
D₉×D₁₅₅	5
D₄₅⋊D₅	31

subgroup

index

C₅×D₁₃₉₅

C₅×C₃₁×D₄₅

C₉×C₅⋊D₁₅₅

D₁₅⋊D₁₅₅

D₄₅⋊D₃₁

D₉×D₁₅₅

D₄₅⋊D₅

Subgroups — D₄₅⋊D₁₅₅ #42124574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D45⋊D155 #42124574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₅⋊D₁₅₅ #42124574