Group #16939346 — subgroups

Subgroups — #16939346

Order 4352.

316 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14529765	C₂	1	1	yes	maximal, characteristic
2	#14527558	C₂	1	2	yes	maximal
3	#14527186	C₂	1	2	yes	maximal
4	#14526613	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1843	C₂	1	1	yes	maximal, characteristic
6	#11763366	C₂²	1	1	yes	characteristic
7	#11762862	C₂²	1	2	yes	—
8	#11762833	C₂²	1	1	yes	characteristic
9	#11762569	—	2	2	no	—
10	#11762561	—	2	4	no	—
11	#11762539	—	2	4	no	—
12	C₂×C₁₇×C₄⋊C₈	—	2	2	no	—
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₁₇×#376	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	—	2	2	no	—
16	C₂×C₄²×C₁₇	D₄	1	2	yes	—
17	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
18	#10587612	—	2	1	no	—
19	#10587610	—	2	2	no	—
20	C₂×#63223	—	2	1	no	—
21	#10587556	—	2	1	no	—
22	#10587555	Q₈	1	2	yes	—
23	C₄×Dic₃₄	—	2	2	no	—
24	C₁₇×C₄⋊C₈	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	4	4	no	—
27	C₁₇×#95	—	2	2	no	—
28	C₂²⋊#3622	—	2	2	no	—
29	C₄⋊#3622	—	2	2	no	—
30	C₄⋊#3622	—	2	2	no	—
31	C₄×#3622	D₄	1	4	yes	—
32	C₂²×C₄×C₁₇	—	2	2	no	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	#41	1	2	yes	—
35	C₂×Dic₃₄	—	2	2	no	—
36	C₂×C₈×C₁₇	—	4	2	no	—
37	C₂×C₈×C₁₇	—	4	4	no	—
38	C₄²×C₁₇	—	2	2	no	—
39	C₄²×C₁₇	Q₁₆	1	4	yes	—
40	C₄²×C₁₇	C₂×D₄	1	2	yes	—
41	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₇	#41	1	1	yes	characteristic
43	C₂²⋊Dic₁₇	—	4	1	no	—
44	C₄⋊Dic₁₇	—	2	2	no	—
45	C₄⋊Dic₁₇	—	2	2	no	—
46	#63225	—	4	2	no	—
47	C₄×Dic₁₇	—	4	1	no	—
48	#63223	—	4	2	no	—
49	C₂×#3622	—	2	4	no	—
50	C₂×#3622	—	4	1	no	—
51	#10668	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	#123	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	4	no	—
54	C₂×C₄×C₁₇	—	4	2	no	—
55	C₂×C₄×C₁₇	—	4	4	no	—
56	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#123	1	2	yes	—
59	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
61	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₂×Dic₁₇	—	4	2	no	—
63	Dic₃₄	—	4	4	no	—
64	C₈×C₁₇	—	8	2	no	—
65	#3622	—	4	4	no	—
66	#3622	—	8	1	no	—
67	#1995	—	17	1	no	—
68	#1843	D₁₇	1	1	yes	characteristic
69	#1571	—	17	2	no	—
70	#1481	—	17	2	no	—
71	#1274	—	17	1	no	—
72	C₂²×C₁₇	—	2	2	no	—
73	C₂²×C₁₇	#399	1	1	yes	characteristic
74	C₂²×C₁₇	#484	1	2	yes	—
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	—	2	2	no	—
77	C₄×C₁₇	—	4	4	no	—
78	C₄×C₁₇	#477	1	2	yes	—
79	C₄×C₁₇	#494	1	1	yes	characteristic
80	C₄×C₁₇	#496	1	1	yes	characteristic
81	Dic₁₇	—	8	2	no	—
82	#533	—	17	1	no	—
83	#431	—	17	2	no	—
84	#423	—	17	1	no	—
85	C₂×C₄⋊C₈	—	2	2	no	—
86	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
87	#376	C₂×D₁₇	1	1	yes	characteristic
88	#339	—	34	2	no	—
89	#335	—	34	4	no	—
90	Q₈⋊C₈	—	34	4	no	—
91	C₂×C₁₇	—	2	2	no	—
92	C₂×C₁₇	#2095	1	1	yes	characteristic
93	C₂×C₁₇	#2098	1	2	yes	—
94	C₂×M₁₆	—	34	1	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₄⋊Q₈	—	17	2	no	—
97	#125	—	34	1	no	—
98	C₂²⋊Q₈	—	34	2	no	—
99	C₄×Q₈	—	34	2	no	—
100	#117	—	34	1	no	—
101	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
102	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	4	no	—
106	C₄⋊C₈	—	34	2	no	—
107	C₄⋊C₈	—	34	2	no	—
108	C₂²⋊C₈	—	34	2	no	—
109	C₄×C₈	—	17	4	no	—
110	#95	—	2	2	no	—
111	C₁₇	#10668	1	1	yes	characteristic, complemented, Sylow
112	C₂×Q₈	—	34	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	#63252	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
116	M₁₆	—	68	2	no	—
117	C₂×C₈	—	4	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	34	4	no	—
120	C₂×C₈	—	68	1	no	—
121	C₄⋊C₄	—	34	2	no	—
122	C₄⋊C₄	—	34	2	no	—
123	C₄⋊C₄	—	68	2	no	—
124	C₂²⋊C₄	—	68	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	68	1	no	—
127	C₄²	#63231	1	4	yes	—
128	C₄²	D₄×D₁₇	1	2	yes	—
129	C₄²	Q₈×D₁₇	1	1	yes	characteristic
130	C₄²	#63256	1	1	yes	characteristic
131	C₂³	#10587615	1	1	yes	characteristic
132	Q₈	—	68	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	68	2	no	—
137	C₂×C₄	#10587580	1	2	yes	—
138	C₂×C₄	#10587582	1	2	yes	—
139	C₂×C₄	#10587618	1	2	yes	—
140	C₂×C₄	#10587626	1	1	yes	characteristic
141	C₂×C₄	C₂×#63231	1	2	yes	—
142	C₂×C₄	#10587632	1	1	yes	characteristic
143	C₈	—	8	2	no	—
144	C₈	—	68	4	no	—
145	C₈	—	136	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#11762744	1	1	yes	characteristic
148	C₂²	#11763144	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	136	2	no	—
153	C₄	#11763101	1	2	yes	—
154	C₄	#11763188	1	1	yes	characteristic
155	C₄	#11763196	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#14530323	1	1	yes	characteristic
158	C₂	#14530340	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529765

C₂

yes

maximal, characteristic

#14527558

C₂

yes

maximal

#14527186

C₂

yes

maximal

#14526613

C₂

yes

maximal, characteristic

C₁₇×#1843

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

characteristic

#11762862

C₂²

yes

—

#11762833

C₂²

yes

—

—

—

C₂×C₁₇×C₄⋊C₈

—

C₂×C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×#376

C₂²

yes

characteristic

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

—

—

—

—

Q₈

yes

—

C₄×Dic₃₄

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#95

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄×#3622

D₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#399

yes

characteristic

C₂²×C₁₇

#484

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#477

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#533

—

#431

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₇

yes

—

#376

C₂×D₁₇

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₂×C₁₇

—

C₂×C₁₇

#2095

yes

characteristic

C₂×C₁₇

#2098

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄²

C₂²⋊D₁₇

yes

—

102

C₂×C₄²

C₂²×D₁₇

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

#95

—

111

C₁₇

#10668

yes

characteristic, complemented, Sylow

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

#63252

yes

—

115

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#63231

yes

—

128

C₄²

D₄×D₁₇

yes

—

129

C₄²

Q₈×D₁₇

yes

characteristic

130

C₄²

#63256

yes

characteristic

131

C₂³

#10587615

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#10587580

yes

—

138

C₂×C₄

#10587582

yes

—

139

C₂×C₄

#10587618

yes

—

140

C₂×C₄

#10587626

yes

characteristic

141

C₂×C₄

C₂×#63231

yes

—

142

C₂×C₄

#10587632

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

136

—

146

C₂²

—

147

C₂²

#11762744

yes

characteristic

148

C₂²

#11763144

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

136

—

153

C₄

#11763101

yes

—

154

C₄

#11763188

yes

characteristic

155

C₄

#11763196

yes

characteristic

156

C₂

—

157

C₂

#14530323

yes

characteristic

158

C₂

#14530340

yes

—

	subgroup	G/H
center	C₂×C₄	#10587626
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1843	C₂
socle	C₂²×C₁₇	#399
prime radical	C₂³×C₁₇	#123
hypercenter	#1843	D₁₇
hypocenter	C₁₇	#10668
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#10587626

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1843

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
111. C₁₇	#10668	semidirect

H ≅ G/N

type

111. C₁₇

#10668

semidirect

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

subgroup	index
#14529765	2
#14527558	2
#14527186	2
#14526613	2
C₁₇×#1843	2
#10668	17

subgroup

index

C₁₇×#1843

Subgroups — #16939346

Conjugacy classes of subgroups

Complements

Maximal subgroups