Group #28413539 — subgroups

Subgroups — D₂₁×C₅⋊D₃₅ #28413539

Order 14700.

416 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₃⋊(C₅⋊(C₅⋊(C₇⋊D₇)))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₇×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×C₅⋊D₃₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×C₅⋊D₃₅	—	3	1	no	maximal, Hall
5	C₃×C₅²×C₇²	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₂₁×D₃₅	—	5	6	no	maximal
7	C₅⋊(C₅⋊(C₇⋊D₇))	—	3	1	no	—
8	C₇×C₅⋊D₃₅	D₃	1	1	yes	characteristic, complemented
9	C₅²×C₇×D₇	—	3	1	no	—
10	D₃×C₅⋊D₃₅	—	7	1	no	maximal
11	D₂₁×C₅⋊D₅	—	7	1	no	maximal
12	C₃⋊(C₅⋊(C₇⋊D₇))	—	5	6	no	—
13	C₅×C₇×D₂₁	D₅	1	6	yes	complemented
14	C₃×C₇×D₃₅	—	5	6	no	—
15	C₅²×C₇²	C₂×D₃	1	1	yes	characteristic, complemented, Hall
16	C₃⋊(C₅⋊D₃₅)	—	7	1	no	—
17	C₃⋊(C₅⋊D₃₅)	—	7	1	no	—
18	C₃⋊(C₅⋊D₃₅)	—	14	3	no	—
19	C₅²×D₂₁	D₇	1	1	yes	characteristic, complemented
20	C₅²×D₃×C₇	—	7	1	no	—
21	C₃×C₅⋊D₃₅	D₇	1	1	yes	characteristic, complemented
22	C₃×C₇×C₅⋊D₅	—	7	1	no	—
23	D₇×D₃₅	—	15	6	no	—
24	C₃×C₅×C₇²	C₂×D₅	1	6	yes	complemented
25	C₂×C₅⋊D₃₅	—	21	1	no	—
26	D₇×C₅⋊D₅	—	21	1	no	—
27	D₇×D₂₁	—	25	1	no	Hall
28	C₃×C₅²×C₇	—	2	3	no	—
29	C₃×C₅²×C₇	C₂×D₇	1	1	yes	characteristic, complemented
30	C₃×C₅²×C₇	C₂×D₇	1	1	yes	characteristic, complemented
31	C₅⋊(C₇⋊D₇)	—	15	6	no	—
32	C₇×D₃₅	—	5	6	no	—
33	C₅×C₇×D₇	—	3	6	no	—
34	D₃×D₃₅	—	35	6	no	—
35	D₅×D₂₁	—	35	6	no	—
36	C₂×C₅²×C₇	—	21	1	no	—
37	C₅⋊D₃₅	—	21	1	no	—
38	C₅⋊D₃₅	—	21	1	no	—
39	C₅⋊D₃₅	—	42	3	no	—
40	C₅⋊D₃₅	D₂₁	1	1	yes	characteristic, direct product factor
41	C₇×C₅⋊D₅	—	7	1	no	—
42	C₅²×D₇	—	3	1	no	—
43	D₃×C₅⋊D₅	—	49	1	no	Hall
44	C₃⋊(C₇⋊D₇)	—	25	1	no	—
45	C₇×D₂₁	C₅⋊D₅	1	1	yes	characteristic, complemented
46	C₃×C₇×D₇	—	25	1	no	—
47	C₅×C₇²	D₃×D₅	1	6	yes	complemented
48	D₁₀₅	—	35	6	no	—
49	D₁₀₅	—	35	6	no	—
50	D₁₀₅	—	70	18	no	—
51	C₅×D₂₁	D₃₅	1	6	yes	complemented
52	C₅×D₃×C₇	—	7	6	no	—
53	C₃×D₃₅	—	5	6	no	—
54	C₃×C₇×D₅	—	35	6	no	—
55	D₇²	—	75	1	no	Hall
56	C₅²×C₇	—	2	3	no	—
57	C₅²×C₇	D₃×D₇	1	1	yes	characteristic, complemented
58	C₅²×C₇	C₂×D₂₁	1	1	yes	characteristic, complemented
59	C₃⋊(C₅⋊D₅)	—	49	1	no	—
60	C₅²×D₃	—	7	1	no	—
61	C₃×C₅⋊D₅	—	7	1	no	—
62	C₃×C₇²	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
63	C₂×D₃₅	—	105	6	no	—
64	D₅×D₇	—	105	6	no	—
65	C₃×C₅×C₇	—	2	18	no	—
66	C₃×C₅×C₇	D₅×D₇	1	6	yes	complemented
67	C₃×C₅×C₇	C₂×D₃₅	1	6	yes	complemented
68	C₂×C₅⋊D₅	—	147	1	no	Hall
69	C₇⋊D₇	—	75	1	no	—
70	C₇×D₇	—	3	1	no	—
71	C₇×D₇	—	25	1	no	—
72	C₂×D₂₁	—	175	1	no	—
73	D₃×D₇	—	175	1	no	—
74	C₃×C₅²	D₇²	1	1	yes	characteristic, complemented, Hall
75	C₂×C₅×C₇	—	21	6	no	—
76	D₃₅	—	5	6	no	—
77	D₃₅	—	105	6	no	—
78	D₃₅	—	105	6	no	—
79	D₃₅	—	210	18	no	—
80	C₅×D₇	—	3	6	no	—
81	C₇×D₅	—	35	6	no	—
82	D₃×D₅	—	245	6	no	—
83	C₂×C₅²	—	21	1	no	—
84	C₅⋊D₅	—	7	1	no	—
85	C₅⋊D₅	—	147	1	no	—
86	C₇²	D₃×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₃×C₇	—	175	1	no	—
88	D₂₁	—	350	3	no	—
89	D₂₁	—	175	1	no	—
90	D₂₁	—	175	1	no	—
91	D₂₁	C₅⋊D₃₅	1	1	yes	characteristic, direct product factor
92	C₃×D₇	—	25	1	no	—
93	D₃×C₇	—	7	1	no	—
94	C₅×C₇	—	2	18	no	—
95	C₅×C₇	D₅×D₂₁	1	6	yes	complemented
96	C₅×C₇	D₃×D₃₅	1	6	yes	complemented
97	D₁₅	—	245	6	no	—
98	C₃×D₅	—	35	6	no	—
99	C₅×D₃	—	7	6	no	—
100	C₂×D₇	—	525	1	no	—
101	C₂×D₇	—	525	1	no	—
102	C₅²	D₇×D₂₁	1	1	yes	characteristic, complemented, Sylow
103	C₃×C₇	—	2	3	no	—
104	C₃×C₇	D₇×C₅⋊D₅	1	1	yes	characteristic, complemented
105	C₃×C₇	C₂×C₅⋊D₃₅	1	1	yes	characteristic, complemented
106	C₂×D₅	—	735	6	no	—
107	C₃×C₅	D₇×D₃₅	1	6	yes	complemented
108	C₂×C₇	—	21	1	no	—
109	C₂×C₇	—	175	1	no	—
110	D₇	—	3	1	no	—
111	D₇	—	525	1	no	—
112	D₇	—	525	1	no	—
113	D₇	—	25	1	no	—
114	D₇	—	1050	3	no	—
115	C₂×D₃	—	1225	1	no	Hall
116	C₂×C₅	—	21	6	no	—
117	D₅	—	35	6	no	—
118	D₅	—	735	6	no	—
119	C₇	—	2	3	no	—
120	C₇	D₂₁×C₅⋊D₅	1	1	yes	characteristic, complemented
121	C₇	D₃×C₅⋊D₃₅	1	1	yes	characteristic, complemented
122	C₂×C₃	—	175	1	no	—
123	D₃	—	7	1	no	—
124	D₃	—	1225	1	no	—
125	C₅	D₂₁×D₃₅	1	6	yes	complemented
126	C₂²	—	3675	1	no	Sylow
127	C₃	D₇×C₅⋊D₃₅	1	1	yes	characteristic, complemented, Sylow
128	C₂	—	21	1	no	—
129	C₂	—	3675	1	no	—
130	C₂	—	175	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⋊(C₅⋊(C₅⋊(C₇⋊D₇)))

C₂

yes

maximal, characteristic, complemented

C₅²×C₇×D₂₁

C₂

yes

maximal, characteristic, complemented

C₃×C₇×C₅⋊D₃₅

C₂

yes

maximal, characteristic, complemented

D₇×C₅⋊D₃₅

—

maximal, Hall

C₃×C₅²×C₇²

C₂²

yes

characteristic, complemented, Hall

D₂₁×D₃₅

—

maximal

C₅⋊(C₅⋊(C₇⋊D₇))

—

C₇×C₅⋊D₃₅

D₃

yes

characteristic, complemented

C₅²×C₇×D₇

—

D₃×C₅⋊D₃₅

—

maximal

D₂₁×C₅⋊D₅

—

maximal

C₃⋊(C₅⋊(C₇⋊D₇))

—

C₅×C₇×D₂₁

D₅

yes

complemented

C₃×C₇×D₃₅

—

C₅²×C₇²

C₂×D₃

yes

characteristic, complemented, Hall

C₃⋊(C₅⋊D₃₅)

—

C₃⋊(C₅⋊D₃₅)

—

C₃⋊(C₅⋊D₃₅)

—

C₅²×D₂₁

D₇

yes

characteristic, complemented

C₅²×D₃×C₇

—

C₃×C₅⋊D₃₅

D₇

yes

characteristic, complemented

C₃×C₇×C₅⋊D₅

—

D₇×D₃₅

—

C₃×C₅×C₇²

C₂×D₅

yes

complemented

C₂×C₅⋊D₃₅

—

D₇×C₅⋊D₅

—

D₇×D₂₁

—

Hall

C₃×C₅²×C₇

—

C₃×C₅²×C₇

C₂×D₇

yes

characteristic, complemented

C₃×C₅²×C₇

C₂×D₇

yes

characteristic, complemented

C₅⋊(C₇⋊D₇)

—

C₇×D₃₅

—

C₅×C₇×D₇

—

D₃×D₃₅

—

D₅×D₂₁

—

C₂×C₅²×C₇

—

C₅⋊D₃₅

—

C₅⋊D₃₅

—

C₅⋊D₃₅

—

C₅⋊D₃₅

D₂₁

yes

characteristic, direct product factor

C₇×C₅⋊D₅

—

C₅²×D₇

—

D₃×C₅⋊D₅

—

Hall

C₃⋊(C₇⋊D₇)

—

C₇×D₂₁

C₅⋊D₅

yes

characteristic, complemented

C₃×C₇×D₇

—

C₅×C₇²

D₃×D₅

yes

complemented

D₁₀₅

—

D₁₀₅

—

D₁₀₅

—

C₅×D₂₁

D₃₅

yes

complemented

C₅×D₃×C₇

—

C₃×D₃₅

—

C₃×C₇×D₅

—

D₇²

—

Hall

C₅²×C₇

—

C₅²×C₇

D₃×D₇

yes

characteristic, complemented

C₅²×C₇

C₂×D₂₁

yes

characteristic, complemented

C₃⋊(C₅⋊D₅)

—

C₅²×D₃

—

C₃×C₅⋊D₅

—

C₃×C₇²

C₂×C₅⋊D₅

yes

characteristic, complemented, Hall

C₂×D₃₅

—

105

—

D₅×D₇

—

105

—

C₃×C₅×C₇

—

C₃×C₅×C₇

D₅×D₇

yes

complemented

C₃×C₅×C₇

C₂×D₃₅

yes

complemented

C₂×C₅⋊D₅

—

147

Hall

C₇⋊D₇

—

C₇×D₇

—

C₇×D₇

—

C₂×D₂₁

—

175

—

D₃×D₇

—

175

—

C₃×C₅²

D₇²

yes

characteristic, complemented, Hall

C₂×C₅×C₇

—

D₃₅

—

D₃₅

—

105

—

D₃₅

—

105

—

D₃₅

—

210

—

C₅×D₇

—

C₇×D₅

—

D₃×D₅

—

245

—

C₂×C₅²

—

C₅⋊D₅

—

C₅⋊D₅

—

147

—

C₇²

D₃×C₅⋊D₅

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

175

—

D₂₁

—

350

—

D₂₁

—

175

—

D₂₁

—

175

—

D₂₁

C₅⋊D₃₅

yes

characteristic, direct product factor

C₃×D₇

—

D₃×C₇

—

C₅×C₇

—

C₅×C₇

D₅×D₂₁

yes

complemented

C₅×C₇

D₃×D₃₅

yes

complemented

D₁₅

—

245

—

C₃×D₅

—

C₅×D₃

—

100

C₂×D₇

—

525

—

101

C₂×D₇

—

525

—

102

C₅²

D₇×D₂₁

yes

characteristic, complemented, Sylow

103

C₃×C₇

—

104

C₃×C₇

D₇×C₅⋊D₅

yes

characteristic, complemented

105

C₃×C₇

C₂×C₅⋊D₃₅

yes

characteristic, complemented

106

C₂×D₅

—

735

—

107

C₃×C₅

D₇×D₃₅

yes

complemented

108

C₂×C₇

—

109

C₂×C₇

—

175

—

110

D₇

—

111

D₇

—

525

—

112

D₇

—

525

—

113

D₇

—

114

D₇

—

1050

—

115

C₂×D₃

—

1225

Hall

116

C₂×C₅

—

117

D₅

—

118

D₅

—

735

—

119

C₇

—

120

C₇

D₂₁×C₅⋊D₅

yes

characteristic, complemented

121

C₇

D₃×C₅⋊D₃₅

yes

characteristic, complemented

122

C₂×C₃

—

175

—

123

D₃

—

124

D₃

—

1225

—

125

C₅

D₂₁×D₃₅

yes

complemented

126

C₂²

—

3675

Sylow

127

C₃

D₇×C₅⋊D₃₅

yes

characteristic, complemented, Sylow

128

C₂

—

129

C₂

—

3675

—

130

C₂

—

175

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₇²	C₂²
LCS	G ▷ C₃×C₅²×C₇²
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃×C₅²×C₇²

C₂²

LCS

G ▷ C₃×C₅²×C₇²

UCS

C₁

N	H ≅ G/N	type
1. C₃⋊(C₅⋊(C₅⋊(C₇⋊D₇)))	C₂	semidirect
2. C₅²×C₇×D₂₁	C₂	semidirect
3. C₃×C₇×C₅⋊D₃₅	C₂	semidirect
5. C₃×C₅²×C₇²	C₂²	semidirect
8. C₇×C₅⋊D₃₅	D₃	semidirect
13. C₅×C₇×D₂₁	D₅	semidirect
15. C₅²×C₇²	C₂×D₃	semidirect
19. C₅²×D₂₁	D₇	semidirect
21. C₃×C₅⋊D₃₅	D₇	semidirect
24. C₃×C₅×C₇²	C₂×D₅	semidirect
29. C₃×C₅²×C₇	C₂×D₇	semidirect
30. C₃×C₅²×C₇	C₂×D₇	semidirect
40. C₅⋊D₃₅	D₂₁	direct (N × H)
45. C₇×D₂₁	C₅⋊D₅	semidirect
47. C₅×C₇²	D₃×D₅	semidirect
51. C₅×D₂₁	D₃₅	semidirect
57. C₅²×C₇	D₃×D₇	semidirect
58. C₅²×C₇	C₂×D₂₁	semidirect
62. C₃×C₇²	C₂×C₅⋊D₅	semidirect
66. C₃×C₅×C₇	D₅×D₇	semidirect
67. C₃×C₅×C₇	C₂×D₃₅	semidirect
74. C₃×C₅²	D₇²	semidirect
86. C₇²	D₃×C₅⋊D₅	semidirect
91. D₂₁	C₅⋊D₃₅	direct (N × H)
95. C₅×C₇	D₅×D₂₁	semidirect
96. C₅×C₇	D₃×D₃₅	semidirect
102. C₅²	D₇×D₂₁	semidirect
104. C₃×C₇	D₇×C₅⋊D₅	semidirect
105. C₃×C₇	C₂×C₅⋊D₃₅	semidirect
107. C₃×C₅	D₇×D₃₅	semidirect
120. C₇	D₂₁×C₅⋊D₅	semidirect
121. C₇	D₃×C₅⋊D₃₅	semidirect
125. C₅	D₂₁×D₃₅	semidirect
127. C₃	D₇×C₅⋊D₃₅	semidirect

H ≅ G/N

type

1. C₃⋊(C₅⋊(C₅⋊(C₇⋊D₇)))

C₂

semidirect

2. C₅²×C₇×D₂₁

C₂

semidirect

3. C₃×C₇×C₅⋊D₃₅

C₂

semidirect

5. C₃×C₅²×C₇²

C₂²

semidirect

8. C₇×C₅⋊D₃₅

D₃

semidirect

13. C₅×C₇×D₂₁

D₅

semidirect

15. C₅²×C₇²

C₂×D₃

semidirect

19. C₅²×D₂₁

D₇

semidirect

21. C₃×C₅⋊D₃₅

D₇

semidirect

24. C₃×C₅×C₇²

C₂×D₅

semidirect

29. C₃×C₅²×C₇

C₂×D₇

semidirect

30. C₃×C₅²×C₇

C₂×D₇

semidirect

40. C₅⋊D₃₅

D₂₁

direct (N × H)

45. C₇×D₂₁

C₅⋊D₅

semidirect

47. C₅×C₇²

D₃×D₅

semidirect

51. C₅×D₂₁

D₃₅

semidirect

57. C₅²×C₇

D₃×D₇

semidirect

58. C₅²×C₇

C₂×D₂₁

semidirect

62. C₃×C₇²

C₂×C₅⋊D₅

semidirect

66. C₃×C₅×C₇

D₅×D₇

semidirect

67. C₃×C₅×C₇

C₂×D₃₅

semidirect

74. C₃×C₅²

D₇²

semidirect

86. C₇²

D₃×C₅⋊D₅

semidirect

91. D₂₁

C₅⋊D₃₅

direct (N × H)

95. C₅×C₇

D₅×D₂₁

semidirect

96. C₅×C₇

D₃×D₃₅

semidirect

102. C₅²

D₇×D₂₁

semidirect

104. C₃×C₇

D₇×C₅⋊D₅

semidirect

105. C₃×C₇

C₂×C₅⋊D₃₅

semidirect

107. C₃×C₅

D₇×D₃₅

semidirect

120. C₇

D₂₁×C₅⋊D₅

semidirect

121. C₇

D₃×C₅⋊D₃₅

semidirect

125. C₅

D₂₁×D₃₅

semidirect

127. C₃

D₇×C₅⋊D₃₅

semidirect

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

subgroup	index
C₃⋊(C₅⋊(C₅⋊(C₇⋊D₇)))	2
C₅²×C₇×D₂₁	2
C₃×C₇×C₅⋊D₃₅	2
D₇×C₅⋊D₃₅	3
D₂₁×D₃₅	5
D₃×C₅⋊D₃₅	7
D₂₁×C₅⋊D₅	7

subgroup

index

C₃⋊(C₅⋊(C₅⋊(C₇⋊D₇)))

C₅²×C₇×D₂₁

C₃×C₇×C₅⋊D₃₅

D₇×C₅⋊D₃₅

D₂₁×D₃₅

D₃×C₅⋊D₃₅

D₂₁×C₅⋊D₅

Subgroups — D₂₁×C₅⋊D₃₅ #28413539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D21×C5⋊D35 #28413539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₁×C₅⋊D₃₅ #28413539