Group #17953831 — subgroups

Subgroups — #17953831

Order 4352.

422 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 148 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	#14544720	C₂	1	2	yes	maximal
2	#14544705	C₂	1	4	yes	maximal
3	C₈²×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587534	—	2	4	no	—
5	#11763755	C₂²	1	4	yes	—
6	#11763683	—	2	8	no	—
7	C₄×C₈×D₁₇	C₂²	1	2	yes	—
8	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₈×#3622	C₄	1	1	yes	characteristic, complemented
10	C₈×#3622	C₄	1	2	yes	complemented
11	C₈²×C₁₇	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄⋊#595	—	2	4	no	—
13	C₂×C₄⋊#595	—	2	8	no	—
14	C₂×C₈×D₁₇	D₄	1	4	yes	—
15	C₂×C₈×D₁₇	D₄	1	2	yes	—
16	C₄²×D₁₇	C₂³	1	1	yes	characteristic
17	#10587534	—	4	8	no	—
18	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₁₇	Q₈	1	4	yes	—
21	C₈×Dic₁₇	Q₈	1	2	yes	—
22	C₄×#3622	C₂×C₄	1	2	yes	—
23	C₄×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²×#595	—	4	4	no	—
25	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
27	C₄⋊#595	—	4	8	no	—
28	C₄⋊#595	—	4	16	no	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
31	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
33	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
34	C₂×#3622	C₄⋊C₄	1	4	yes	—
35	C₂×#3622	C₄⋊C₄	1	2	yes	—
36	C₈×D₁₇	—	2	8	no	—
37	C₈×D₁₇	D₈	1	4	yes	—
38	C₈×D₁₇	Q₁₆	1	4	yes	—
39	#8816	—	17	1	no	maximal, Sylow
40	C₂²×D₁₇	#127	1	1	yes	characteristic
41	C₂×#595	—	8	4	no	—
42	C₂×#595	—	8	8	no	—
43	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
45	C₂×Dic₁₇	C₄⋊Q₈	1	2	yes	—
46	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×D₁₇	C₂×D₈	1	2	yes	—
48	C₄×D₁₇	C₂×Q₁₆	1	2	yes	—
49	C₄×D₁₇	#135	1	8	yes	—
50	C₈×C₁₇	—	2	4	no	—
51	C₈×C₁₇	#107	1	4	yes	—
52	#3622	—	2	4	no	—
53	#3622	#107	1	4	yes	—
54	#1845	—	17	2	no	—
55	#1834	—	17	4	no	—
56	C₂×C₈²	—	17	1	no	—
57	C₂²×C₁₇	#384	1	1	yes	characteristic
58	C₂×D₁₇	#493	1	1	yes	characteristic
59	C₂×D₁₇	#494	1	1	yes	characteristic
60	C₂×D₁₇	#495	1	4	yes	—
61	#595	—	16	4	no	—
62	C₄×C₁₇	C₂×#107	1	2	yes	—
63	C₄×C₁₇	#427	1	4	yes	—
64	Dic₁₇	#499	1	4	yes	—
65	Dic₁₇	#500	1	2	yes	—
66	C₂×#107	—	34	4	no	—
67	C₂×C₄×C₈	—	17	2	no	—
68	C₂×C₄×C₈	—	17	1	no	—
69	#384	—	17	4	no	—
70	#340	—	34	8	no	—
71	C₈²	—	17	2	no	—
72	C₈²	—	17	1	no	—
73	C₈²	#595	1	1	yes	characteristic, complemented
74	C₂×C₁₇	#1845	1	1	yes	characteristic
75	C₂×C₁₇	#1846	1	2	yes	—
76	D₁₇	#1709	1	2	yes	—
77	D₁₇	#1710	1	2	yes	—
78	C₂²×C₈	—	17	2	no	—
79	C₂²×C₈	—	17	4	no	—
80	C₂×C₄⋊C₄	—	34	8	no	—
81	C₂×C₄⋊C₄	—	34	4	no	—
82	C₂×C₄²	—	17	1	no	—
83	#107	—	68	8	no	—
84	C₄×C₈	—	17	2	no	—
85	C₄×C₈	—	17	4	no	—
86	C₄×C₈	—	17	1	no	—
87	C₄×C₈	—	17	2	no	—
88	C₄×C₈	C₂×#595	1	2	yes	—
89	C₄×C₈	C₂×#595	1	1	yes	characteristic
90	C₁₇	#8816	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄	—	17	1	no	—
92	C₂²×C₄	—	17	2	no	—
93	C₂²×C₄	—	68	4	no	—
94	C₂×C₈	—	17	2	no	—
95	C₂×C₈	—	17	4	no	—
96	C₂×C₈	—	17	4	no	—
97	C₂×C₈	—	17	4	no	—
98	C₂×C₈	—	34	8	no	—
99	C₂×C₈	C₄⋊#595	1	4	yes	—
100	C₂×C₈	C₄⋊#595	1	2	yes	—
101	C₄⋊C₄	—	68	8	no	—
102	C₄⋊C₄	—	68	16	no	—
103	C₄²	—	17	1	no	—
104	C₄²	—	17	2	no	—
105	C₄²	C₂²×#595	1	1	yes	characteristic
106	C₂³	—	17	1	no	—
107	C₂×C₄	—	17	1	no	—
108	C₂×C₄	—	17	2	no	—
109	C₂×C₄	—	17	2	no	—
110	C₂×C₄	—	17	2	no	—
111	C₂×C₄	—	17	8	no	—
112	C₂×C₄	—	136	4	no	—
113	C₂×C₄	—	136	8	no	—
114	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	17	4	no	—
118	C₈	—	34	4	no	—
119	C₈	#10587534	1	4	yes	—
120	C₂²	—	17	1	no	—
121	C₂²	—	17	1	no	—
122	C₂²	—	17	4	no	—
123	C₂²	#11763755	1	1	yes	characteristic
124	C₄	—	272	4	no	—
125	C₄	—	17	2	no	—
126	C₄	—	17	4	no	—
127	C₄	C₂×#10587534	1	2	yes	—
128	C₄	#11763819	1	4	yes	—
129	C₂	—	17	2	no	—
130	C₂	—	17	2	no	—
131	C₂	#14544720	1	1	yes	characteristic
132	C₂	#14544721	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544720

C₂

yes

maximal

#14544705

C₂

yes

maximal

C₈²×D₁₇

C₂

yes

maximal, characteristic

C₂×#10587534

—

#11763755

C₂²

yes

—

#11763683

—

C₄×C₈×D₁₇

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₈×#3622

C₄

yes

characteristic, complemented

C₈×#3622

C₄

yes

complemented

C₈²×C₁₇

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄⋊#595

—

C₂×C₈×D₁₇

D₄

yes

—

C₂×C₈×D₁₇

D₄

yes

—

C₄²×D₁₇

C₂³

yes

characteristic

#10587534

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

—

C₈×Dic₁₇

Q₈

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

C₂×D₄

yes

—

C₄⋊#595

—

C₄⋊#595

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

D₈

yes

—

C₈×D₁₇

Q₁₆

yes

—

—

C₂²×D₁₇

yes

—

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×Dic₁₇

C₄⋊Q₈

yes

—

C₂×Dic₁₇

C₄⋊Q₈

yes

characteristic

C₄×D₁₇

C₂×D₈

yes

—

C₄×D₁₇

C₂×Q₁₆

yes

—

C₄×D₁₇

#135

yes

—

C₈×C₁₇

—

C₈×C₁₇

#107

yes

—

#3622

—

#3622

#107

yes

—

#1845

—

#1834

—

C₂×C₈²

—

C₂²×C₁₇

yes

C₂×D₁₇

yes

C₂×D₁₇

yes

C₂×D₁₇

yes

—

#595

—

C₄×C₁₇

C₂×#107

yes

—

C₄×C₁₇

#427

yes

—

Dic₁₇

#499

yes

—

Dic₁₇

#500

yes

—

C₂×#107

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#384

—

#340

—

C₈²

—

C₈²

—

C₈²

#595

yes

characteristic, complemented

C₂×C₁₇

#1845

yes

characteristic

C₂×C₁₇

#1846

yes

—

D₁₇

#1709

yes

—

D₁₇

#1710

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#8816

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊#595

yes

—

100

C₂×C₈

C₄⋊#595

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×#595

yes

characteristic

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

136

—

113

C₂×C₄

—

136

—

114

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

115

C₂×C₄

C₂×C₄⋊#595

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

#10587534

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#11763755

yes

characteristic

124

C₄

—

272

—

125

C₄

—

126

C₄

—

127

C₄

C₂×#10587534

yes

—

128

C₄

#11763819

yes

—

129

C₂

—

130

C₂

—

131

C₂

#14544720

yes

characteristic

132

C₂

#14544721

yes

—

	subgroup	G/H
center	C₂²	#11763755
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	#384
prime radical	C₂²×D₁₇	#127
cosocle	C₄²×D₁₇	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	#8816
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11763755

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₈²×C₁₇

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄²×D₁₇

C₂³

C₈²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
9. C₈×#3622	C₄	semidirect
10. C₈×#3622	C₄	semidirect
11. C₈²×C₁₇	C₄	semidirect
73. C₈²	#595	semidirect
90. C₁₇	#8816	semidirect

H ≅ G/N

type

9. C₈×#3622

C₄

semidirect

10. C₈×#3622

C₄

semidirect

11. C₈²×C₁₇

C₄

semidirect

73. C₈²

#595

semidirect

90. C₁₇

#8816

semidirect

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

subgroup	index
#14544720	2
#14544705	2
C₈²×D₁₇	2
#8816	17

subgroup

index

#14544720

#14544705

C₈²×D₁₇

#8816

Subgroups — #17953831

Conjugacy classes of subgroups

Complements

Maximal subgroups