Group #17948485 — subgroups

Subgroups — #17948485

Order 4352.

202 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 62 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	#14543828	C₂	1	2	yes	maximal
2	#14529750	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587538	—	2	2	no	—
4	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
5	#11762675	C₄	1	1	yes	characteristic
6	#11762571	—	2	2	no	—
7	C₂×C₁₇×#125	C₄	1	1	yes	characteristic
8	C₁₇×#125	—	2	1	no	—
9	C₁₇×#125	—	4	1	no	—
10	C₁₇×#125	C₈	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₇	—	2	1	no	—
15	C₂²×#3622	D₄	1	2	yes	—
16	#10587538	—	2	4	no	—
17	#10587538	—	4	2	no	—
18	C₂×#63216	—	4	2	no	—
19	#10587502	—	4	1	no	—
20	C₄×#3622	—	4	1	no	—
21	C₂²×C₄×C₁₇	—	2	1	no	—
22	C₂²×C₄×C₁₇	—	4	1	no	—
23	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×Dic₁₇	—	4	1	no	—
25	C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₄⋊C₄×C₁₇	—	4	2	no	—
27	C₄⋊C₄×C₁₇	—	4	2	no	—
28	C₄⋊C₄×C₁₇	—	4	1	no	—
29	C₄²×C₁₇	—	2	1	no	—
30	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
32	C₄⋊Dic₁₇	—	4	2	no	—
33	C₂×#3622	—	2	2	no	—
34	C₂×#3622	—	2	2	no	—
35	C₂×#3622	SD₁₆	1	2	yes	—
36	C₂×#3622	Q₁₆	1	2	yes	—
37	#63216	—	4	4	no	—
38	#7135	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	#100	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	—	4	2	no	—
43	C₂×C₄×C₁₇	—	4	2	no	—
44	C₂×C₄×C₁₇	—	8	2	no	—
45	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#103	1	2	yes	—
48	C₂×C₄×C₁₇	#104	1	2	yes	—
49	C₂×Dic₁₇	—	8	2	no	—
50	C₂×Dic₁₇	—	8	1	no	—
51	#3622	—	2	2	no	—
52	#3622	—	2	2	no	—
53	#1991	—	17	1	no	—
54	#1376	—	17	2	no	—
55	C₂²×C₁₇	—	2	2	no	—
56	C₂²×C₁₇	#323	1	1	yes	characteristic
57	C₂²×C₁₇	#329	1	1	yes	characteristic
58	C₂²×C₁₇	#333	1	1	yes	characteristic
59	C₄×C₁₇	—	4	1	no	—
60	C₄×C₁₇	—	4	1	no	—
61	C₄×C₁₇	—	8	2	no	—
62	C₄×C₁₇	Q₈⋊C₈	1	2	yes	—
63	C₄×C₁₇	#328	1	2	yes	—
64	Dic₁₇	—	16	1	no	—
65	C₂×#125	#595	1	1	yes	characteristic
66	C₂×M₃₂	—	34	2	no	—
67	C₂×C₄×C₈	—	17	1	no	—
68	#382	—	17	1	no	—
69	#340	—	34	2	no	—
70	C₂×C₁₇	—	2	1	no	—
71	C₂×C₁₇	—	2	1	no	—
72	C₂×C₁₇	#1272	1	1	yes	characteristic
73	C₂×C₁₇	#1366	1	2	yes	—
74	C₂²×C₈	—	17	2	no	—
75	#125	—	2	1	no	—
76	#125	—	4	1	no	—
77	#125	#3624	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄⋊C₄	—	34	1	no	—
81	C₂×C₄²	C₂×#595	1	1	yes	characteristic
82	M₃₂	—	34	4	no	—
83	M₃₂	—	68	2	no	—
84	C₂×C₁₆	—	68	2	no	—
85	C₄×C₈	—	68	1	no	—
86	#95	—	68	1	no	—
87	C₁₇	#7135	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	68	1	no	—
91	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
92	C₂×C₈	—	17	2	no	—
93	C₂×C₈	—	17	2	no	—
94	C₂×C₈	—	34	2	no	—
95	C₂×C₈	—	34	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	68	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×#3624	1	1	yes	characteristic
103	C₄²	#63247	1	1	yes	characteristic
104	C₁₆	—	68	4	no	—
105	C₂³	#10587553	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	136	2	no	—
112	C₂×C₄	—	136	1	no	—
113	C₂×C₄	#10587549	1	2	yes	—
114	C₂×C₄	#10587550	1	2	yes	—
115	C₂×C₄	#10587551	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
117	C₈	—	34	2	no	—
118	C₈	—	34	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#11763652	1	1	yes	characteristic
121	C₂²	#11763665	1	1	yes	characteristic
122	C₂²	#11763669	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	272	1	no	—
127	C₄	Q₈⋊#3624	1	2	yes	—
128	C₄	#11763663	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#14543604	1	1	yes	characteristic
132	C₂	#14543805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14543828

C₂

yes

maximal

#14529750

C₂

yes

maximal, characteristic

C₂×#10587538

—

C₂×C₄×#3622

C₂²

yes

characteristic

#11762675

C₄

yes

characteristic

#11762571

—

C₂×C₁₇×#125

C₄

yes

characteristic

C₁₇×#125

—

C₁₇×#125

—

C₁₇×#125

C₈

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

characteristic

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

SD₁₆

yes

—

C₂×#3622

Q₁₆

yes

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#104

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

Q₈⋊C₈

yes

—

C₄×C₁₇

#328

yes

—

Dic₁₇

—

C₂×#125

#595

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

—

#382

—

#340

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1272

yes

characteristic

C₂×C₁₇

#1366

yes

—

C₂²×C₈

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

#95

—

C₁₇

#7135

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×#3624

yes

characteristic

103

C₄²

#63247

yes

characteristic

104

C₁₆

—

105

C₂³

#10587553

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

136

—

112

C₂×C₄

—

136

—

113

C₂×C₄

#10587549

yes

—

114

C₂×C₄

#10587550

yes

—

115

C₂×C₄

#10587551

yes

characteristic

116

C₂×C₄

C₂²⋊#3624

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

#11763652

yes

characteristic

121

C₂²

#11763665

yes

characteristic

122

C₂²

#11763669

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

272

—

127

C₄

Q₈⋊#3624

yes

—

128

C₄

#11763663

yes

—

129

C₂

—

130

C₂

—

131

C₂

#14543604

yes

characteristic

132

C₂

#14543805

yes

—

	subgroup	G/H
center	C₂²	#11763669
commutator	C₄²×C₁₇	C₂×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₁₇×#125	C₄
socle	C₂²×C₁₇	#333
prime radical	C₂³×C₁₇	#100
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₂×#125	#595
hypocenter	C₁₇	#7135
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

#11763669

commutator

C₄²×C₁₇

C₂×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₁₇×#125

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄×#3622

C₂²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
87. C₁₇	#7135	semidirect

H ≅ G/N

type

87. C₁₇

#7135

semidirect

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

subgroup	index
#14543828	2
#14529750	2
#7135	17

subgroup

index

#14543828

#14529750

#7135

Subgroups — #17948485

Conjugacy classes of subgroups

Complements

Maximal subgroups