Finite Group Databasebeta

← #16943994

Subgroups — #16943994

Order 4352.

374 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533989C211yesmaximal, characteristic
2#14527108C211yesmaximal, characteristic
3#14526729C214yesmaximal
4C17×#1467C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6#11762862—21no—
7#11762729—21no—
8#11762729—21no—
9#11762716C412yes—
10#11762707—21no—
11#11762707—21no—
12#11762678C2212yes—
13C4×C4⋊Dic17C411yescharacteristic
14#11762535—21no—
15#11762535C2212yes—
16C2×C4×Q8×C17C2211yescharacteristic
17C17×#432—21no—
18C17×#324—21no—
19C17×#324C2212yes—
20C22×Q8×C17C2×C411yescharacteristic
21C22×Dic34—21no—
22C4×Q8×C17—41no—
23C4×Q8×C17—41no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17C2×C411yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C412yes—
33C2×#63225—22no—
34C2×#63225—22no—
35C2×C4×Dic17C2×C412yes—
36#10587555—41no—
37#10587555—41no—
38C17×C4⋊C8—41no—
39C17×C22⋊C8—22no—
40C17×#97—41no—
41#10587502—22no—
42#10587502—22no—
43#10587502—21no—
44C22⋊#3622—22no—
45C4⋊#3622—41no—
46C4×#3622—41no—
47C2×Q8×C17—22no—
48C2×Q8×C17—22no—
49C2×Q8×C17—41no—
50C22×C4×C17C22⋊C411yescharacteristic
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22×C411yescharacteristic
55C22×C4×C17C2×D411yescharacteristic
56C22×C4×C17C2×D411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17—22no—
59C2×Dic34—42no—
60C2×Dic34—82no—
61C2×C8×C17—42no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—41no—
64C4⋊C4×C17—41no—
65C42×C17—21no—
66C42×C17—21no—
67C42×C17—21no—
68C42×C17—21no—
69C42×C17—21no—
70C42×C17—21no—
71C4⋊Dic17—24no—
72C4⋊Dic17—41no—
73C4⋊Dic17—41no—
74C4⋊Dic17—42no—
75#63225—82no—
76#63225—82no—
77C4×Dic17—24no—
78C2×#3622—42no—
79#11574—171nomaximal, Sylow
80C23×C17C2×C22⋊C411yescharacteristic
81Q8×C17—44no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17#9911yescharacteristic
97C2×C4×C17#9911yescharacteristic
98C2×C4×C17#10112yes—
99C2×C4×C17#13112yes—
100C2×Dic17—24no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106Dic34—84no—
107C8×C17—82no—
108#3622—82no—
109#2567—171no—
110#1467D1711yescharacteristic
111#1461—171no—
112#1325—174no—
113C22×C17—21no—
114C22×C17—21no—
115C22×C17#40811yescharacteristic
116C22×C17C2×#9911yescharacteristic
117C22×C17C2×#10111yescharacteristic
118C4×C17—21no—
119C4×C17—22no—
120C4×C17—22no—
121C4×C17—41no—
122C4×C17—41no—
123C4×C17—41no—
124C4×C17—41no—
125C4×C17#35612yes—
126Dic17—44no—
127Dic17—82no—
128C2×C4⋊Q8—171no—
129C2×C4×Q8C2×D1711yescharacteristic
130#432—21no—
131#431—341no—
132#395—341no—
133#395—341no—
134#391—172no—
135#389—341no—
136#389—341no—
137#384—172no—
138C4×C4⋊C4—171no—
139#324—21no—
140#324—172no—
141#324—341no—
142#324C2×D1712yes—
143C2×C17—21no—
144C2×C17—21no—
145C2×C17#147211yescharacteristic
146C2×C17#212811yescharacteristic
147C2×C17C2×#35611yescharacteristic
148C22×Q8—341no—
149C22×Q8C4×D1711yescharacteristic
150C4⋊Q8—681no—
151C4⋊Q8—681no—
152C4×Q8—41no—
153C4×Q8—41no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4—341no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4—342no—
158C2×C4⋊C4—342no—
159C2×C4⋊C4D6811yescharacteristic
160C2×C4⋊C4C22⋊D1711yescharacteristic
161C2×C4⋊C4C22×D1711yescharacteristic
162C2×C42—172no—
163C2×C42C4×D1711yescharacteristic
164C2×C42D6811yescharacteristic
165C2×C42C22⋊D1711yescharacteristic
166C4⋊C8—41no—
167C4⋊C8—681no—
168C22⋊C8—22no—
169C22⋊C8—342no—
170#97—41no—
171C4×C8—681no—
172#95—341no—
173#95—342no—
174#95—342no—
175C17#1157411yescharacteristic, complemented, Sylow
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—682no—
180C2×Q8—1362no—
181C22×C4—342no—
182C22×C4—342no—
183C22×C4#6322711yescharacteristic
184C22×C4#6322711yescharacteristic
185C22×C4#6322711yescharacteristic
186C22×C4#6322711yescharacteristic
187C22×C4C2×C4×D1711yescharacteristic
188C22×C4C2×D6811yescharacteristic
189C22×C4C2×C22⋊D1711yescharacteristic
190C2×C8—42no—
191C2×C8—682no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—344no—
196C4⋊C4—682no—
197C4⋊C4—681no—
198C4⋊C4—681no—
199C4⋊C4—1362no—
200C4⋊C4—1362no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—344no—
208C23C2×#6322711yescharacteristic
209Q8—44no—
210Q8—1364no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—344no—
226C2×C4—682no—
227C2×C4—682no—
228C2×C4—682no—
229C2×C4—682no—
230C2×C4—682no—
231C2×C4#1058747711yescharacteristic
232C2×C4#1058747711yescharacteristic
233C2×C4#1058749512yes—
234C2×C4#1058759312yes—
235C8—82no—
236C8—1362no—
237C22—21no—
238C22—21no—
239C22#1176277711yescharacteristic
240C22C2×#1058747711yescharacteristic
241C22C2×#1058749511yescharacteristic
242C4—21no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—684no—
250C4—1362no—
251C4#1176260712yes—
252C2—21no—
253C2—21no—
254C2#1452714911yescharacteristic
255C2#1453045511yescharacteristic
256C2C2×#1176260711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1467C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1467D17
hypocenterC17#11574
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
175. C17#11574semidirect

Maximal subgroups

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

subgroupindex
#145339892
#145271082
#145267292
C17×#14672
#1157417