Finite Group Databasebeta

← #16943976

Subgroups — #16943976

Order 4352.

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

normal subgroups only

This group has 480 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#14533737C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526729C214yesmaximal, complemented
4C17×#1465C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6C2×#10587610—21no—
7C2×#10587610—21no—
8C2×C22⋊#3622—21no—
9#11762745—21no—
10#11762745—21no—
11#11762712C412yes—
12#11762678C2212yes—
13C4×C4⋊Dic17C411yescharacteristic
14#11762535C2212yes—
15#11762533—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#407—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C22×D4×C17C2×C411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C2×M16×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C412yes—
36#10587610—44no—
37#10587610—44no—
38C2×#63225—22no—
39C2×#63225—22no—
40C2×C4×Dic17C2×C412yes—
41C22×#3622—41no—
42#10587555—41no—
43#10587555—41no—
44C17×C22⋊C8—22no—
45C17×C22⋊C8—41no—
46#10587502—21no—
47C22⋊#3622—22no—
48C22⋊#3622—41no—
49C24×C17C22⋊C411yescharacteristic
50C24×C17C22⋊C411yescharacteristic
51C2×D4×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—22no—
66C22×Dic17—22no—
67C2×Dic34—42no—
68C2×Dic34—42no—
69C2×Dic34—41no—
70C2×Dic34—41no—
71M16×C17—82no—
72C2×C8×C17—42no—
73C4⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C42×C17—21no—
77C42×C17—21no—
78C22⋊Dic17—44no—
79C22⋊Dic17—44no—
80C4⋊Dic17—24no—
81C4⋊Dic17—42no—
82C4⋊Dic17—42no—
83#63225—44no—
84#63225—44no—
85C4×Dic17—24no—
86C2×#3622—42no—
87C2×#3622—82no—
88#11570—171nomaximal, Sylow
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17#10112yes—
99C23×C17C2×C22⋊C411yescharacteristic
100D4×C17—44no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—81no—
113C2×C4×C17—81no—
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#13112yes—
117C2×Dic17—24no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123Dic34—82no—
124Dic34—82no—
125C8×C17—82no—
126#3622—82no—
127#2542—171no—
128#1465D1711yescharacteristic
129#1459—171no—
130#1325—174no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—22no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—81no—
142C22×C17—81no—
143C22×C17#40811yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17C2×#10111yescharacteristic
146C4×C17—21no—
147C4×C17—22no—
148C4×C17—41no—
149C4×C17—41no—
150C4×C17—41no—
151C4×C17#35312yes—
152Dic17—44no—
153Dic17—82no—
154C2×C4⋊Q8—171no—
155C2×C22⋊Q8—341no—
156C2×C22⋊Q8—341no—
157C2×C4×D4C2×D1711yescharacteristic
158#407—21no—
159C2×C22⋊C8—341no—
160#399—341no—
161#399—341no—
162#390—172no—
163#384—172no—
164C4×C4⋊C4—171no—
165#324—172no—
166#324C2×D1712yes—
167#323—21no—
168#323—341no—
169C2×C17—21no—
170C2×C17—21no—
171C2×C17—42no—
172C2×C17—81no—
173C2×C17#147211yescharacteristic
174C2×C17C2×#35311yescharacteristic
175C2×C17#213511yescharacteristic
176C22×Q8—341no—
177C22×D4C4×D1711yescharacteristic
178C23×C4D6811yescharacteristic
179C23×C4C22⋊D1711yescharacteristic
180C2×M16—41no—
181C22×C8—681no—
182C4⋊Q8—681no—
183C4⋊Q8—681no—
184C22⋊Q8—684no—
185C22⋊Q8—684no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—172no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—342no—
191C2×C4⋊C4—342no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4C22×D1711yescharacteristic
194C2×C22⋊C4—342no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4D6811yescharacteristic
197C2×C22⋊C4C22⋊D1711yescharacteristic
198C2×C42—172no—
199C2×C42C4×D1711yescharacteristic
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—342no—
203C22⋊C8—681no—
204#95—341no—
205C17#1157011yescharacteristic, complemented, Sylow
206C24#6322711yescharacteristic
207C24#6322711yescharacteristic
208C2×Q8—682no—
209C2×Q8—682no—
210C2×Q8—681no—
211C2×Q8—681no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—342no—
222C22×C4—342no—
223C22×C4#6322711yescharacteristic
224C22×C4#6322711yescharacteristic
225C22×C4C2×C4×D1711yescharacteristic
226C22×C4C2×D6811yescharacteristic
227C22×C4C2×C22⋊D1711yescharacteristic
228M16—82no—
229C2×C8—42no—
230C2×C8—682no—
231C2×C8—1362no—
232C4⋊C4—41no—
233C4⋊C4—344no—
234C4⋊C4—682no—
235C4⋊C4—682no—
236C4⋊C4—684no—
237C4⋊C4—684no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—684no—
241C22⋊C4—684no—
242C42—21no—
243C42—21no—
244C42—344no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1058749512yes—
255C23C2×#6322711yescharacteristic
256Q8—1362no—
257Q8—1362no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—344no—
273C2×C4—682no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—682no—
277C2×C4—682no—
278C2×C4#1058747711yescharacteristic
279C2×C4#1058747711yescharacteristic
280C2×C4#1058759312yes—
281C8—82no—
282C8—1362no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22#1176277711yescharacteristic
296C22C2×#1058747711yescharacteristic
297C22C2×#1058749511yescharacteristic
298C4—21no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—684no—
304C4—1362no—
305C4#1176260112yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#1452714911yescharacteristic
311C2C2×#1176260111yescharacteristic
312C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11570
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526729C2semidirect
205. C17#11570semidirect

Maximal subgroups

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

subgroupindex
#145337372
#145271022
#145267292
C17×#14652
#1157017