Finite Group Databasebeta

← #16947070

Subgroups — #16947070

Order 4352.

388 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14529951C211yesmaximal, characteristic
2#14527774C212yesmaximal
3#14526784C212yesmaximal
4#14526714C211yesmaximal, characteristic
5C17×#2817C211yesmaximal, characteristic
6#11763435C2211yescharacteristic
7#11763139—22no—
8#11762672—21no—
9#11762576C2211yescharacteristic
10#11762575—21no—
11#11762545—22no—
12#11762533C2212yes—
13C2×C17×C22⋊Q8—21no—
14C17×#400—22no—
15C17×#397—21no—
16C17×#397C2212yes—
17C17×#395—22no—
18C17×#381C2211yescharacteristic
19C22×Q8×C17—21no—
20C23×C4×C17D411yescharacteristic
21C17×C22⋊Q8—42no—
22C17×C22⋊Q8—42no—
23C2×C4⋊C4×C17—22no—
24C2×C4⋊C4×C17—21no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17D412yes—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—21no—
32#10587610—22no—
33#10587564—42no—
34C17×#95—22no—
35C17×#95—21no—
36C17×#95—22no—
37C17×#95—22no—
38C17×#95D412yes—
39#10587502—42no—
40C22⋊#3622—22no—
41#10587481—42no—
42#10587479—42no—
43C24×C17C2×D411yescharacteristic
44C2×Q8×C17—42no—
45C2×Q8×C17—81no—
46C2×Q8×C17—81no—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17—22no—
50C22×C4×C17—21no—
51C22×C4×C17—22no—
52C22×C4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17SD1612yes—
55C22×C4×C17C2×D412yes—
56C22×Dic17—22no—
57C22×Dic17—41no—
58C2×Dic34—22no—
59C4⋊C4×C17—42no—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—44no—
62C4⋊C4×C17—81no—
63C22⋊C4×C17—41no—
64C22⋊C4×C17—42no—
65C22⋊C4×C17—42no—
66C22⋊C4×C17—41no—
67C22⋊Dic17—41no—
68C22⋊Dic17—41no—
69C22⋊Dic17—41no—
70C22⋊Dic17—41no—
71C4⋊Dic17—22no—
72#63225—42no—
73C2×#3622—42no—
74#12207—171nomaximal, Sylow
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—42no—
78C23×C17#12011yescharacteristic
79C23×C17C2×SD1611yescharacteristic
80C23×C17AGL(1,ℤ/8)11yescharacteristic
81Q8×C17—84no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—41no—
85C2×C4×C17—41no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—44no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—41no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—81no—
96C2×Dic17—42no—
97C2×Dic17—82no—
98C2×Dic17—81no—
99C2×Dic17—82no—
100Dic34—82no—
101#3622—82no—
102#2817D1711yescharacteristic
103#2030—171no—
104#1608—172no—
105#1344—172no—
106#1317—171no—
107C22×C17—22no—
108C22×C17—21no—
109C22×C17—21no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—42no—
113C22×C17—81no—
114C22×C17#45011yescharacteristic
115C22×C17#45511yescharacteristic
116C22×C17B(4,𝔽2)11yescharacteristic
117C4×C17—42no—
118C4×C17—42no—
119C4×C17—82no—
120C4×C17—81no—
121C4×C17—82no—
122Dic17—82no—
123Dic17—161no—
124#543—171no—
125C2×C22⋊Q8—21no—
126#483—342no—
127#400—22no—
128#397—21no—
129#397C2×D1712yes—
130#395—22no—
131#381—341no—
132#381C2×D1711yescharacteristic
133#342—171no—
134#342—341no—
135#328—342no—
136#323—172no—
137C2×C17—21no—
138C2×C17—21no—
139C2×C17—41no—
140C2×C17—41no—
141C2×C17#160111yescharacteristic
142C2×C17#220112yes—
143C22×Q8—21no—
144C23×C4C22⋊D1711yescharacteristic
145C22⋊Q8—42no—
146C22⋊Q8—42no—
147C22⋊Q8—342no—
148C22⋊Q8—682no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4C22⋊D1712yes—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—341no—
155C2×C22⋊C4—341no—
156C2×C22⋊C4C22⋊D1711yescharacteristic
157C2×C22⋊C4C22×D1711yescharacteristic
158#106—682no—
159#103—682no—
160C22⋊C8—342no—
161#95—22no—
162#95—21no—
163#95—22no—
164#95—22no—
165#95—682no—
166#95C22⋊D1712yes—
167C17#1220711yescharacteristic, complemented, Sylow
168C24C2×C22⋊D1711yescharacteristic
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—342no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—342no—
181C22×C4—681no—
182C22×C4#6322912yes—
183C22×C4C2×C22⋊D1712yes—
184C2×C8—682no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—342no—
190C4⋊C4—682no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—681no—
196C22⋊C4—681no—
197C22⋊C4—681no—
198C22⋊C4—681no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1058761811yescharacteristic
203C23C2×#6322911yescharacteristic
204C23#1058763911yescharacteristic
205Q8—84no—
206Q8—1362no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—682no—
222C2×C4—1361no—
223C2×C4—1362no—
224C2×C4—1362no—
225C8—1362no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1176296011yescharacteristic
234C22#1176298711yescharacteristic
235C22#1176299611yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—2721no—
242C4—1362no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#1452773611yescharacteristic
248C2#1453079712yes—
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C22⋊C4×C17C23
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2817C2
socleC22×C17B(4,𝔽2)
prime radicalC24×C17C2×D4
hypercenter#2817D17
hypocenterC17#12207
LCSG ▷ C2×C22⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
167. C17#12207semidirect

Maximal subgroups

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

subgroupindex
#145299512
#145277742
#145267842
#145267142
C17×#28172
#1220717