Group #16948945 — subgroups

Subgroups — #16948945

Order 4352.

212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14530923	C₂	1	2	yes	maximal
2	#14527682	C₂	1	1	yes	maximal, characteristic
3	#14526848	C₂	1	1	yes	maximal, characteristic
4	#14526839	C₂	1	2	yes	maximal
5	C₁₇×#1713	C₂	1	1	yes	maximal, characteristic
6	#11762939	—	2	1	no	—
7	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
8	#11762636	—	2	2	no	—
9	C₄⋊#63214	C₂²	1	2	yes	—
10	#11762612	—	2	2	no	—
11	#11762562	C₂²	1	1	yes	characteristic
12	#11762541	—	2	1	no	—
13	C₁₇×#500	—	2	1	no	—
14	C₁₇×#500	C₂²	1	2	yes	—
15	C₁₇×#494	—	2	1	no	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	—	4	1	no	—
18	C₂×Q₁₆×C₁₇	—	4	1	no	—
19	C₁₇×C₄⋊Q₈	—	2	2	no	—
20	C₂×Dic₆₈	D₄	1	1	yes	characteristic
21	C₄×Dic₃₄	—	2	1	no	—
22	C₁₇×#107	—	2	2	no	—
23	C₁₇×#107	—	2	2	no	—
24	C₁₇×#107	—	4	2	no	—
25	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	#10587489	D₄	1	1	yes	characteristic
29	#10587487	—	2	1	no	—
30	#10587486	—	4	1	no	—
31	#10587485	—	4	1	no	—
32	C₂×#63214	—	2	2	no	—
33	C₄⋊#3622	—	2	1	no	—
34	C₂×Q₈×C₁₇	—	4	2	no	—
35	C₂×Dic₃₄	—	2	1	no	—
36	Q₁₆×C₁₇	—	8	2	no	—
37	Q₁₆×C₁₇	—	8	2	no	—
38	C₂×C₈×C₁₇	—	2	1	no	—
39	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₄⋊C₄×C₁₇	—	4	2	no	—
45	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₁₇	—	2	1	no	—
47	#63225	—	4	1	no	—
48	C₄×Dic₁₇	—	4	1	no	—
49	C₂×#3622	—	4	1	no	—
50	Dic₆₈	—	2	2	no	—
51	#63214	—	4	2	no	—
52	#12498	—	17	1	no	maximal, Sylow
53	Q₈×C₁₇	—	8	2	no	—
54	Q₈×C₁₇	—	8	2	no	—
55	C₂×C₄×C₁₇	—	8	2	no	—
56	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂×Dic₁₇	—	4	1	no	—
60	C₂×Dic₁₇	—	4	1	no	—
61	Dic₃₄	—	4	1	no	—
62	Dic₃₄	—	4	1	no	—
63	C₈×C₁₇	—	2	2	no	—
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	2	1	no	—
66	C₈×C₁₇	SD₃₂	1	2	yes	—
67	C₈×C₁₇	#137	1	2	yes	—
68	#3622	—	8	1	no	—
69	#2225	—	17	2	no	—
70	#1713	D₁₇	1	1	yes	characteristic
71	#1591	—	17	1	no	—
72	#1367	—	17	1	no	—
73	#1364	—	17	2	no	—
74	C₂²×C₁₇	#459	1	1	yes	characteristic
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	—	16	2	no	—
77	C₄×C₁₇	#470	1	1	yes	characteristic
78	C₄×C₁₇	#473	1	1	yes	characteristic
79	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
80	C₄×C₁₇	#509	1	1	yes	characteristic
81	Dic₁₇	—	8	1	no	—
82	Dic₁₇	—	8	1	no	—
83	#500	—	2	1	no	—
84	#500	C₂×D₁₇	1	2	yes	—
85	#494	—	2	1	no	—
86	#446	—	34	1	no	—
87	C₄×Q₁₆	—	17	1	no	—
88	#367	—	34	2	no	—
89	C₄⋊C₁₆	—	17	2	no	—
90	#358	—	34	2	no	—
91	#335	—	17	1	no	—
92	Q₈⋊C₈	—	34	1	no	—
93	C₈²	C₂×D₁₇	1	1	yes	characteristic
94	C₂×C₁₇	#1689	1	1	yes	characteristic
95	C₂×C₁₇	#2207	1	1	yes	characteristic
96	C₂×C₁₇	#2214	1	1	yes	characteristic
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×Q₁₆	—	4	1	no	—
99	C₂×Q₁₆	—	17	1	no	—
100	C₄⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	34	1	no	—
102	C₂×C₁₆	—	34	2	no	—
103	#107	—	2	2	no	—
104	#107	—	2	2	no	—
105	#107	—	4	2	no	—
106	#107	—	17	1	no	—
107	C₄⋊C₈	—	34	1	no	—
108	C₄⋊C₈	—	68	1	no	—
109	#103	—	34	1	no	—
110	#97	—	68	1	no	—
111	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
114	C₁₇	#12498	1	1	yes	characteristic, complemented, Sylow
115	C₂×Q₈	—	4	2	no	—
116	C₂×Q₈	—	34	1	no	—
117	Q₁₆	—	8	2	no	—
118	Q₁₆	—	8	2	no	—
119	Q₁₆	—	34	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	68	1	no	—
122	C₂×C₈	#63228	1	1	yes	characteristic
123	C₂×C₈	#63228	1	1	yes	characteristic
124	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
125	C₂×C₈	#63254	1	1	yes	characteristic
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	34	1	no	—
129	C₄⋊C₄	—	68	1	no	—
130	C₄²	—	68	1	no	—
131	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
132	C₁₆	—	68	2	no	—
133	Q₈	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	68	1	no	—
136	Q₈	—	68	1	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	68	1	no	—
139	C₂×C₄	—	68	1	no	—
140	C₂×C₄	C₂×#63228	1	1	yes	characteristic
141	C₂×C₄	#10587618	1	1	yes	characteristic
142	C₂×C₄	#10587624	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	136	1	no	—
147	C₈	#10587498	1	2	yes	—
148	C₈	#10587604	1	2	yes	—
149	C₂²	#11763001	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	16	2	no	—
152	C₄	—	136	1	no	—
153	C₄	—	136	1	no	—
154	C₄	#11763064	1	1	yes	characteristic
155	C₄	#11763081	1	1	yes	characteristic
156	C₄	C₂×#10587498	1	1	yes	characteristic
157	C₄	#11763245	1	1	yes	characteristic
158	C₂	#14528270	1	1	yes	characteristic
159	C₂	#14530821	1	1	yes	characteristic
160	C₂	#14530862	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530923

C₂

yes

maximal

#14527682

C₂

yes

maximal, characteristic

#14526848

C₂

yes

maximal, characteristic

#14526839

C₂

yes

maximal

C₁₇×#1713

C₂

yes

maximal, characteristic

#11762939

—

C₄×Dic₆₈

C₂²

yes

characteristic

#11762636

—

C₄⋊#63214

C₂²

yes

—

#11762612

—

#11762562

C₂²

yes

characteristic

#11762541

—

C₁₇×#500

—

C₁₇×#500

C₂²

yes

—

C₁₇×#494

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₂×Dic₆₈

D₄

yes

characteristic

C₄×Dic₃₄

—

C₁₇×#107

—

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587489

D₄

yes

—

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

#63214

—

#12498

—

maximal, Sylow

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

SD₃₂

yes

—

C₈×C₁₇

#137

yes

—

#3622

—

#2225

—

#1713

D₁₇

yes

—

—

—

C₂²×C₁₇

#459

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×SD₃₂

yes

characteristic

C₄×C₁₇

#509

yes

characteristic

Dic₁₇

—

Dic₁₇

—

#500

—

#500

C₂×D₁₇

yes

—

#494

—

#446

—

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₄⋊Q₈

—

101

C₄×Q₈

—

102

C₂×C₁₆

—

103

#107

—

104

#107

—

105

#107

—

106

#107

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

110

#97

—

111

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

112

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

113

C₄×C₈

C₂²×D₁₇

yes

characteristic

114

C₁₇

#12498

yes

characteristic, complemented, Sylow

115

C₂×Q₈

—

116

C₂×Q₈

—

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

#63228

yes

characteristic

123

C₂×C₈

#63228

yes

characteristic

124

C₂×C₈

D₄×D₁₇

yes

characteristic

125

C₂×C₈

#63254

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

132

C₁₆

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×#63228

yes

characteristic

141

C₂×C₄

#10587618

yes

characteristic

142

C₂×C₄

#10587624

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

136

—

147

C₈

#10587498

yes

—

148

C₈

#10587604

yes

—

149

C₂²

#11763001

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

136

—

153

C₄

—

136

—

154

C₄

#11763064

yes

characteristic

155

C₄

#11763081

yes

characteristic

156

C₄

C₂×#10587498

yes

characteristic

157

C₄

#11763245

yes

characteristic

158

C₂

#14528270

yes

characteristic

159

C₂

#14530821

yes

characteristic

160

C₂

#14530862

yes

characteristic

	subgroup	G/H
center	C₂²	#11763001
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1713	C₂
socle, prime radical	C₂²×C₁₇	#459
hypercenter	#1713	D₁₇
hypocenter	C₁₇	#12498
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#11763001

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1713

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
114. C₁₇	#12498	semidirect

H ≅ G/N

type

114. C₁₇

#12498

semidirect

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

subgroup	index
#14530923	2
#14527682	2
#14526848	2
#14526839	2
C₁₇×#1713	2
#12498	17

subgroup

index

C₁₇×#1713

Subgroups — #16948945

Conjugacy classes of subgroups

Complements

Maximal subgroups