Group #44472964 — subgroups

Subgroups — C₁₇×D₂₁² #44472964

Order 29988.

248 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 52 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×C₇×C₁₇×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₁₇×C₃⋊(C₃⋊(C₇⋊D₇))	C₂	1	1	yes	maximal, characteristic, complemented
3	D₇×C₁₇×D₂₁	—	3	2	no	maximal
4	C₃²×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₇×C₃⋊(C₇⋊D₇)	—	3	2	no	—
6	C₁₇×C₃⋊(C₇⋊D₇)	—	6	1	no	—
7	C₇×C₁₇×D₂₁	D₃	1	2	yes	complemented
8	C₃×C₇×D₇×C₁₇	—	3	2	no	—
9	D₃×C₁₇×D₂₁	—	7	2	no	maximal
10	D₇²×C₁₇	—	9	1	no	Hall
11	C₃×C₇²×C₁₇	—	2	1	no	—
12	C₃×C₇²×C₁₇	C₂×D₃	1	2	yes	complemented
13	C₁₇×C₃⋊D₂₁	—	7	2	no	—
14	C₁₇×C₃⋊D₂₁	—	14	3	no	—
15	C₃×C₁₇×D₂₁	D₇	1	2	yes	complemented
16	C₃×D₃×C₇×C₁₇	—	7	2	no	—
17	D₂₁²	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₁₇×C₇⋊D₇	—	9	1	no	—
19	C₇×D₇×C₁₇	—	3	2	no	—
20	C₂×C₁₇×D₂₁	—	21	2	no	—
21	D₃×D₇×C₁₇	—	21	2	no	—
22	C₃²×C₇×C₁₇	—	2	3	no	—
23	C₃²×C₇×C₁₇	C₂×D₇	1	2	yes	complemented
24	C₃⋊(C₃⋊(C₇⋊D₇))	C₂×C₁₇	1	1	yes	characteristic, complemented
25	C₃×C₇×D₂₁	C₂×C₁₇	1	2	yes	complemented
26	C₇²×C₁₇	D₃²	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₇×C₁₇	—	21	2	no	—
28	C₁₇×D₂₁	—	21	2	no	—
29	C₁₇×D₂₁	—	21	2	no	—
30	C₁₇×D₂₁	—	42	2	no	—
31	C₁₇×D₂₁	—	42	6	no	—
32	C₁₇×D₂₁	—	42	6	no	—
33	C₁₇×D₂₁	D₂₁	1	2	yes	direct product factor
34	D₃×C₇×C₁₇	—	7	2	no	—
35	C₃×D₇×C₁₇	—	3	2	no	—
36	D₃²×C₁₇	—	49	1	no	Hall
37	D₇×D₂₁	—	3	2	no	—
38	C₂×D₇×C₁₇	—	63	2	no	—
39	C₃²×C₇²	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
40	C₃×C₇×C₁₇	—	2	6	no	—
41	C₃×C₇×C₁₇	—	2	2	no	—
42	C₃×C₇×C₁₇	—	2	6	no	—
43	C₃×C₇×C₁₇	D₃×D₇	1	2	yes	complemented
44	C₃×C₇×C₁₇	C₂×D₂₁	1	2	yes	complemented
45	C₁₇×C₃⋊D₃	—	49	1	no	—
46	C₃×D₃×C₁₇	—	7	2	no	—
47	C₃⋊(C₇⋊D₇)	—	3	2	no	—
48	C₃⋊(C₇⋊D₇)	—	6	1	no	—
49	C₇×D₂₁	D₃×C₁₇	1	2	yes	complemented
50	C₃×C₇×D₇	—	3	2	no	—
51	D₃×D₂₁	—	7	2	no	—
52	C₂×C₇×C₁₇	—	21	2	no	—
53	D₇×C₁₇	—	3	2	no	—
54	D₇×C₁₇	—	63	2	no	—
55	D₇×C₁₇	—	126	3	no	—
56	C₂×D₃×C₁₇	—	147	2	no	—
57	D₇²	—	9	1	no	Hall
58	C₃²×C₁₇	D₇²	1	1	yes	characteristic, complemented, Hall
59	C₃×C₇²	—	2	1	no	—
60	C₃×C₇²	C₂×D₃×C₁₇	1	2	yes	complemented
61	C₃⋊D₂₁	—	7	2	no	—
62	C₃⋊D₂₁	—	14	3	no	—
63	C₃×D₂₁	D₇×C₁₇	1	2	yes	complemented
64	C₃×D₃×C₇	—	7	2	no	—
65	C₇×C₁₇	—	2	3	no	—
66	C₇×C₁₇	D₃×D₂₁	1	2	yes	complemented
67	C₂×C₃×C₁₇	—	21	2	no	—
68	D₃×C₁₇	—	7	2	no	—
69	D₃×C₁₇	—	294	1	no	—
70	D₃×C₁₇	—	147	2	no	—
71	C₇⋊D₇	—	9	1	no	—
72	C₇×D₇	—	3	2	no	—
73	C₂×D₂₁	—	21	2	no	—
74	D₃×D₇	—	21	2	no	—
75	C₂²×C₁₇	—	441	1	no	Hall
76	C₃²×C₇	—	2	3	no	—
77	C₃²×C₇	C₂×D₇×C₁₇	1	2	yes	complemented
78	C₃×C₁₇	—	2	1	no	—
79	C₃×C₁₇	D₇×D₂₁	1	2	yes	complemented
80	C₇²	D₃²×C₁₇	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃×C₇	—	21	2	no	—
82	D₂₁	—	21	2	no	—
83	D₂₁	—	21	2	no	—
84	D₂₁	—	42	2	no	—
85	D₂₁	—	42	6	no	—
86	D₂₁	—	42	6	no	—
87	D₂₁	C₁₇×D₂₁	1	2	yes	direct product factor
88	C₃×D₇	—	3	2	no	—
89	D₃×C₇	—	7	2	no	—
90	D₃²	—	49	1	no	Hall
91	C₂×C₁₇	—	21	2	no	—
92	C₂×C₁₇	—	441	1	no	—
93	C₂×D₇	—	63	2	no	—
94	C₃×C₇	—	2	6	no	—
95	C₃×C₇	—	2	2	no	—
96	C₃×C₇	—	2	6	no	—
97	C₃×C₇	D₃×D₇×C₁₇	1	2	yes	complemented
98	C₃×C₇	C₂×C₁₇×D₂₁	1	2	yes	complemented
99	C₃⋊D₃	—	49	1	no	—
100	C₃×D₃	—	7	2	no	—
101	C₁₇	D₂₁²	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×C₇	—	21	2	no	—
103	D₇	—	3	2	no	—
104	D₇	—	63	2	no	—
105	D₇	—	126	3	no	—
106	C₂×D₃	—	147	2	no	—
107	C₃²	D₇²×C₁₇	1	1	yes	characteristic, complemented, Sylow
108	C₇	—	2	3	no	—
109	C₇	D₃×C₁₇×D₂₁	1	2	yes	complemented
110	C₂×C₃	—	21	2	no	—
111	D₃	—	7	2	no	—
112	D₃	—	294	1	no	—
113	D₃	—	147	2	no	—
114	C₂²	—	441	1	no	Sylow
115	C₃	—	2	1	no	—
116	C₃	D₇×C₁₇×D₂₁	1	2	yes	complemented
117	C₂	—	21	2	no	—
118	C₂	—	441	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₇×C₁₇×D₂₁

C₂

yes

maximal, complemented

C₁₇×C₃⋊(C₃⋊(C₇⋊D₇))

C₂

yes

maximal, characteristic, complemented

D₇×C₁₇×D₂₁

—

maximal

C₃²×C₇²×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₁₇×C₃⋊(C₇⋊D₇)

—

C₁₇×C₃⋊(C₇⋊D₇)

—

C₇×C₁₇×D₂₁

D₃

yes

complemented

C₃×C₇×D₇×C₁₇

—

D₃×C₁₇×D₂₁

—

maximal

D₇²×C₁₇

—

Hall

C₃×C₇²×C₁₇

—

C₃×C₇²×C₁₇

C₂×D₃

yes

complemented

C₁₇×C₃⋊D₂₁

—

C₁₇×C₃⋊D₂₁

—

C₃×C₁₇×D₂₁

D₇

yes

complemented

C₃×D₃×C₇×C₁₇

—

D₂₁²

C₁₇

yes

maximal, characteristic, direct product factor, Hall

C₁₇×C₇⋊D₇

—

C₇×D₇×C₁₇

—

C₂×C₁₇×D₂₁

—

D₃×D₇×C₁₇

—

C₃²×C₇×C₁₇

—

C₃²×C₇×C₁₇

C₂×D₇

yes

complemented

C₃⋊(C₃⋊(C₇⋊D₇))

C₂×C₁₇

yes

characteristic, complemented

C₃×C₇×D₂₁

C₂×C₁₇

yes

complemented

C₇²×C₁₇

D₃²

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

C₁₇×D₂₁

D₂₁

yes

direct product factor

D₃×C₇×C₁₇

—

C₃×D₇×C₁₇

—

D₃²×C₁₇

—

Hall

D₇×D₂₁

—

C₂×D₇×C₁₇

—

C₃²×C₇²

C₂²×C₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

D₃×D₇

yes

complemented

C₃×C₇×C₁₇

C₂×D₂₁

yes

complemented

C₁₇×C₃⋊D₃

—

C₃×D₃×C₁₇

—

C₃⋊(C₇⋊D₇)

—

C₃⋊(C₇⋊D₇)

—

C₇×D₂₁

D₃×C₁₇

yes

complemented

C₃×C₇×D₇

—

D₃×D₂₁

—

C₂×C₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

126

—

C₂×D₃×C₁₇

—

147

—

D₇²

—

Hall

C₃²×C₁₇

D₇²

yes

characteristic, complemented, Hall

C₃×C₇²

—

C₃×C₇²

C₂×D₃×C₁₇

yes

complemented

C₃⋊D₂₁

—

C₃⋊D₂₁

—

C₃×D₂₁

D₇×C₁₇

yes

complemented

C₃×D₃×C₇

—

C₇×C₁₇

—

C₇×C₁₇

D₃×D₂₁

yes

complemented

C₂×C₃×C₁₇

—

D₃×C₁₇

—

D₃×C₁₇

—

294

—

D₃×C₁₇

—

147

—

C₇⋊D₇

—

C₇×D₇

—

C₂×D₂₁

—

D₃×D₇

—

C₂²×C₁₇

—

441

Hall

C₃²×C₇

—

C₃²×C₇

C₂×D₇×C₁₇

yes

complemented

C₃×C₁₇

—

C₃×C₁₇

D₇×D₂₁

yes

complemented

C₇²

D₃²×C₁₇

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

D₂₁

—

D₂₁

—

D₂₁

—

D₂₁

—

D₂₁

—

D₂₁

C₁₇×D₂₁

yes

direct product factor

C₃×D₇

—

D₃×C₇

—

D₃²

—

Hall

C₂×C₁₇

—

C₂×C₁₇

—

441

—

C₂×D₇

—

C₃×C₇

—

C₃×C₇

—

C₃×C₇

—

C₃×C₇

D₃×D₇×C₁₇

yes

complemented

C₃×C₇

C₂×C₁₇×D₂₁

yes

complemented

C₃⋊D₃

—

100

C₃×D₃

—

101

C₁₇

D₂₁²

yes

characteristic, direct product factor, Sylow

102

C₂×C₇

—

103

D₇

—

104

D₇

—

105

D₇

—

126

—

106

C₂×D₃

—

147

—

107

C₃²

D₇²×C₁₇

yes

characteristic, complemented, Sylow

108

C₇

—

109

C₇

D₃×C₁₇×D₂₁

yes

complemented

110

C₂×C₃

—

111

D₃

—

112

D₃

—

294

—

113

D₃

—

147

—

114

C₂²

—

441

Sylow

115

C₃

—

116

C₃

D₇×C₁₇×D₂₁

yes

complemented

117

C₂

—

118

C₂

—

441

—

	subgroup	G/H
center, hypercenter	C₁₇	D₂₁²
commutator, cosocle, hypocenter	C₃²×C₇²	C₂²×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂²
LCS	G ▷ C₃²×C₇²
UCS	C₁ ◁ C₁₇

subgroup

G/H

center, hypercenter

C₁₇

D₂₁²

commutator, cosocle, hypocenter

C₃²×C₇²

C₂²×C₁₇

Frattini

C₁

Fitting, socle

C₃²×C₇²×C₁₇

C₂²

LCS

G ▷ C₃²×C₇²

UCS

C₁ ◁ C₁₇

N	H ≅ G/N	type
1. C₃×C₇×C₁₇×D₂₁	C₂	semidirect
2. C₁₇×C₃⋊(C₃⋊(C₇⋊D₇))	C₂	semidirect
4. C₃²×C₇²×C₁₇	C₂²	semidirect
7. C₇×C₁₇×D₂₁	D₃	semidirect
12. C₃×C₇²×C₁₇	C₂×D₃	semidirect
15. C₃×C₁₇×D₂₁	D₇	semidirect
17. D₂₁²	C₁₇	direct (N × H)
23. C₃²×C₇×C₁₇	C₂×D₇	semidirect
24. C₃⋊(C₃⋊(C₇⋊D₇))	C₂×C₁₇	semidirect
25. C₃×C₇×D₂₁	C₂×C₁₇	semidirect
26. C₇²×C₁₇	D₃²	semidirect
33. C₁₇×D₂₁	D₂₁	direct (N × H)
39. C₃²×C₇²	C₂²×C₁₇	semidirect
43. C₃×C₇×C₁₇	D₃×D₇	semidirect
44. C₃×C₇×C₁₇	C₂×D₂₁	semidirect
49. C₇×D₂₁	D₃×C₁₇	semidirect
58. C₃²×C₁₇	D₇²	semidirect
60. C₃×C₇²	C₂×D₃×C₁₇	semidirect
63. C₃×D₂₁	D₇×C₁₇	semidirect
66. C₇×C₁₇	D₃×D₂₁	semidirect
77. C₃²×C₇	C₂×D₇×C₁₇	semidirect
79. C₃×C₁₇	D₇×D₂₁	semidirect
80. C₇²	D₃²×C₁₇	semidirect
87. D₂₁	C₁₇×D₂₁	direct (N × H)
97. C₃×C₇	D₃×D₇×C₁₇	semidirect
98. C₃×C₇	C₂×C₁₇×D₂₁	semidirect
101. C₁₇	D₂₁²	direct (N × H)
107. C₃²	D₇²×C₁₇	semidirect
109. C₇	D₃×C₁₇×D₂₁	semidirect
116. C₃	D₇×C₁₇×D₂₁	semidirect

H ≅ G/N

type

1. C₃×C₇×C₁₇×D₂₁

C₂

semidirect

2. C₁₇×C₃⋊(C₃⋊(C₇⋊D₇))

C₂

semidirect

4. C₃²×C₇²×C₁₇

C₂²

semidirect

7. C₇×C₁₇×D₂₁

D₃

semidirect

12. C₃×C₇²×C₁₇

C₂×D₃

semidirect

15. C₃×C₁₇×D₂₁

D₇

semidirect

17. D₂₁²

C₁₇

direct (N × H)

23. C₃²×C₇×C₁₇

C₂×D₇

semidirect

24. C₃⋊(C₃⋊(C₇⋊D₇))

C₂×C₁₇

semidirect

25. C₃×C₇×D₂₁

C₂×C₁₇

semidirect

26. C₇²×C₁₇

D₃²

semidirect

33. C₁₇×D₂₁

D₂₁

direct (N × H)

39. C₃²×C₇²

C₂²×C₁₇

semidirect

43. C₃×C₇×C₁₇

D₃×D₇

semidirect

44. C₃×C₇×C₁₇

C₂×D₂₁

semidirect

49. C₇×D₂₁

D₃×C₁₇

semidirect

58. C₃²×C₁₇

D₇²

semidirect

60. C₃×C₇²

C₂×D₃×C₁₇

semidirect

63. C₃×D₂₁

D₇×C₁₇

semidirect

66. C₇×C₁₇

D₃×D₂₁

semidirect

77. C₃²×C₇

C₂×D₇×C₁₇

semidirect

79. C₃×C₁₇

D₇×D₂₁

semidirect

80. C₇²

D₃²×C₁₇

semidirect

87. D₂₁

C₁₇×D₂₁

direct (N × H)

97. C₃×C₇

D₃×D₇×C₁₇

semidirect

98. C₃×C₇

C₂×C₁₇×D₂₁

semidirect

101. C₁₇

D₂₁²

direct (N × H)

107. C₃²

D₇²×C₁₇

semidirect

109. C₇

D₃×C₁₇×D₂₁

semidirect

116. C₃

D₇×C₁₇×D₂₁

semidirect

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

subgroup	index
C₃×C₇×C₁₇×D₂₁	2
C₁₇×C₃⋊(C₃⋊(C₇⋊D₇))	2
D₇×C₁₇×D₂₁	3
D₃×C₁₇×D₂₁	7
D₂₁²	17

subgroup

index

C₃×C₇×C₁₇×D₂₁

C₁₇×C₃⋊(C₃⋊(C₇⋊D₇))

D₇×C₁₇×D₂₁

D₃×C₁₇×D₂₁

D₂₁²

Subgroups — C₁₇×D₂₁² #44472964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×D212 #44472964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×D₂₁² #44472964