Group #17948474 — subgroups

Subgroups — #17948474

Order 4352.

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

normal subgroups only

This group has 206 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	#14543825	C₂	1	2	yes	maximal
2	#14529754	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587538	—	2	2	no	—
4	C₂×#10587474	C₂²	1	1	yes	characteristic
5	#11762679	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	C₂×C₄⋊Dic₁₇	—	4	1	no	—
16	C₂²×#3622	D₄	1	2	yes	—
17	#10587538	—	4	2	no	—
18	#10587538	—	4	2	no	—
19	C₂×#63216	—	2	4	no	—
20	#10587474	—	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	C₄⋊Dic₁₇	—	8	2	no	—
34	C₂×#3622	—	2	2	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	SD₁₆	1	2	yes	—
37	C₂×#3622	Q₁₆	1	2	yes	—
38	#63216	—	4	4	no	—
39	#7130	—	17	1	no	maximal, Sylow
40	C₂³×C₁₇	#100	1	1	yes	characteristic
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	—	4	1	no	—
44	C₂×C₄×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	—	8	1	no	—
47	C₂×C₄×C₁₇	—	8	1	no	—
48	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#103	1	2	yes	—
51	C₂×C₄×C₁₇	#104	1	2	yes	—
52	C₂×Dic₁₇	—	8	2	no	—
53	C₂×Dic₁₇	—	8	1	no	—
54	#3622	—	2	2	no	—
55	#3622	—	2	2	no	—
56	#1992	—	17	1	no	—
57	#1375	—	17	2	no	—
58	C₂²×C₁₇	—	2	1	no	—
59	C₂²×C₁₇	—	2	1	no	—
60	C₂²×C₁₇	#323	1	1	yes	characteristic
61	C₂²×C₁₇	#329	1	1	yes	characteristic
62	C₂²×C₁₇	#333	1	1	yes	characteristic
63	C₄×C₁₇	—	4	1	no	—
64	C₄×C₁₇	—	4	1	no	—
65	C₄×C₁₇	—	8	2	no	—
66	C₄×C₁₇	Q₈⋊C₈	1	2	yes	—
67	C₄×C₁₇	#328	1	2	yes	—
68	Dic₁₇	—	16	1	no	—
69	C₂×#125	#595	1	1	yes	characteristic
70	C₂×M₃₂	—	34	2	no	—
71	C₂×#97	—	17	1	no	—
72	#384	—	17	1	no	—
73	#340	—	34	2	no	—
74	C₂×C₁₇	—	2	1	no	—
75	C₂×C₁₇	—	2	1	no	—
76	C₂×C₁₇	#1272	1	1	yes	characteristic
77	C₂×C₁₇	#1355	1	1	yes	characteristic
78	C₂×C₁₇	#1356	1	1	yes	characteristic
79	C₂²×C₈	—	17	2	no	—
80	#125	—	2	1	no	—
81	#125	—	4	1	no	—
82	#125	#3624	1	2	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	4	1	no	—
85	C₂×C₄⋊C₄	—	34	1	no	—
86	C₂×C₄⋊C₄	—	68	1	no	—
87	C₂×C₄²	C₂×#595	1	1	yes	characteristic
88	M₃₂	—	68	2	no	—
89	M₃₂	—	68	2	no	—
90	C₂×C₁₆	—	34	4	no	—
91	#97	—	68	1	no	—
92	C₁₇	#7130	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	—	68	1	no	—
96	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
97	C₂×C₈	—	17	2	no	—
98	C₂×C₈	—	17	2	no	—
99	C₂×C₈	—	34	2	no	—
100	C₂×C₈	—	34	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	68	2	no	—
106	C₄⋊C₄	—	136	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	C₂×#3624	1	1	yes	characteristic
109	C₄²	#63247	1	1	yes	characteristic
110	C₁₆	—	68	4	no	—
111	C₂³	#10587553	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	136	2	no	—
120	C₂×C₄	—	136	1	no	—
121	C₂×C₄	#10587549	1	2	yes	—
122	C₂×C₄	#10587550	1	2	yes	—
123	C₂×C₄	#10587551	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
125	C₈	—	34	2	no	—
126	C₈	—	34	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#11763652	1	1	yes	characteristic
130	C₂²	#11763665	1	1	yes	characteristic
131	C₂²	#11763669	1	1	yes	characteristic
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	272	1	no	—
136	C₄	Q₈⋊#3624	1	2	yes	—
137	C₄	#11763663	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#14543604	1	1	yes	characteristic
141	C₂	#14543787	1	1	yes	characteristic
142	C₂	#14543791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14543825

C₂

yes

maximal

#14529754

C₂

yes

maximal, characteristic

C₂×#10587538

—

C₂×#10587474

C₂²

yes

characteristic

#11762679

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

—

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

—

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

—

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

—

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×#97

—

#384

—

#340

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#97

—

C₁₇

#7130

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

136

—

107

C₄²

—

108

C₄²

C₂×#3624

yes

characteristic

109

C₄²

#63247

yes

characteristic

110

C₁₆

—

111

C₂³

#10587553

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

136

—

120

C₂×C₄

—

136

—

121

C₂×C₄

#10587549

yes

—

122

C₂×C₄

#10587550

yes

—

123

C₂×C₄

#10587551

yes

characteristic

124

C₂×C₄

C₂²⋊#3624

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#11763652

yes

characteristic

130

C₂²

#11763665

yes

characteristic

131

C₂²

#11763669

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

272

—

136

C₄

Q₈⋊#3624

yes

—

137

C₄

#11763663

yes

—

138

C₂

—

139

C₂

—

140

C₂

#14543604

yes

characteristic

141

C₂

#14543787

yes

characteristic

142

C₂

#14543791

yes

characteristic

	subgroup	G/H
center	C₂²	#11763665
commutator	C₄²×C₁₇	C₂×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₁₇×#125	C₄
socle	C₂²×C₁₇	#329
prime radical	C₂³×C₁₇	#100
cosocle	C₂×#10587474	C₂²
hypercenter	C₂×#125	#595
hypocenter	C₁₇	#7130
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

#11763665

commutator

C₄²×C₁₇

C₂×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₁₇×#125

C₄

C₂²×C₁₇

C₂³×C₁₇

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
92. C₁₇	#7130	semidirect

H ≅ G/N

type

92. C₁₇

#7130

semidirect

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

subgroup	index
#14543825	2
#14529754	2
#7130	17

subgroup

index

#14543825

#14529754

#7130

Subgroups — #17948474

Conjugacy classes of subgroups

Complements

Maximal subgroups