Finite Group Databasebeta

← #16945253

Subgroups — #16945253

Order 4352.

368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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#14534482C211yesmaximal, characteristic
2#14528867C211yesmaximal, characteristic
3#14526812C212yesmaximal, complemented
4#14526791C212yesmaximal, complemented
5C17×#1458C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic
7#11762773—21no—
8#11762748C2211yescharacteristic
9#11762745C411yescharacteristic
10#11762739C2211yescharacteristic
11#11762734C411yescharacteristic
12#11762704—21no—
13#11762703—21no—
14#11762693—21no—
15#11762682—21no—
16#11762655C411yescharacteristic
17#11762578—21no—
18#11762578C2212yes—
19C2×C17×#117C2211yescharacteristic
20C17×#431—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C23×C4×C17D411yescharacteristic
24C17×#117—41no—
25C17×#117—41no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17—21no—
41C2×#63223—22no—
42#10587556—41no—
43#10587556—41no—
44C17×C4⋊C8—41no—
45C17×C22⋊C8—22no—
46C4×C8×C17—41no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—41no—
55C22⋊#3622—41no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—41no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C2×C8×C17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C22⋊Dic17—42no—
81C22⋊Dic17—42no—
82C4⋊Dic17—41no—
83C4⋊Dic17—41no—
84#63225—42no—
85#63225—42no—
86#63225—41no—
87#63225—41no—
88#63225—42no—
89C4×Dic17—42no—
90#63223—42no—
91#63223—42no—
92C2×#3622—42no—
93#11853—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17C2×C22⋊C411yescharacteristic
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#13512yes—
117C2×Dic17—22no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C8×C17—82no—
130#3622—82no—
131#2611—171no—
132#1822—171no—
133#1458D1711yescharacteristic
134#1353—172no—
135#1346—172no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—42no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—81no—
143C22×C17C2×#9911yescharacteristic
144C22×C17#41611yescharacteristic
145C22×C17#42111yescharacteristic
146C4×C17—22no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—41no—
150C4×C17—41no—
151C4×C17—41no—
152C4×C17—41no—
153Dic17—42no—
154Dic17—81no—
155Dic17—81no—
156Dic17—81no—
157C2×#125—171no—
158C2×#117C2×D1711yescharacteristic
159#431—21no—
160#407—341no—
161#400—171no—
162#399—171no—
163#398—171no—
164#397—171no—
165#389—341no—
166#388—341no—
167#387—341no—
168#385—341no—
169#376—171no—
170#343—172no—
171#343—341no—
172#324C2×D1712yes—
173#323—21no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—81no—
177C2×C17#151611yescharacteristic
178C2×C17#212311yescharacteristic
179C2×C17#213211yescharacteristic
180C23×C4D6811yescharacteristic
181C2×M16—342no—
182#125—681no—
183#125—681no—
184#117—41no—
185#117—41no—
186C2×C4⋊C4—171no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4—341no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4D6811yescharacteristic
193C2×C4⋊C4C22×D1711yescharacteristic
194C2×C22⋊C4—341no—
195C2×C22⋊C4—341no—
196C2×C22⋊C4C4×D1711yescharacteristic
197C2×C22⋊C4C22⋊D1711yescharacteristic
198C2×C42—341no—
199C2×C42C4×D1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—41no—
202C22⋊C8—22no—
203C22⋊C8—681no—
204C22⋊C8—681no—
205C4×C8—41no—
206#95—171no—
207#95—171no—
208#95—341no—
209#95—341no—
210#95—341no—
211#95—341no—
212#95—341no—
213C17#1185311yescharacteristic, complemented, Sylow
214C24#6322711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—341no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4#6322711yescharacteristic
224C22×C4#6322711yescharacteristic
225C22×C4#6322711yescharacteristic
226C22×C4C2×C4×D1711yescharacteristic
227C22×C4C2×D6811yescharacteristic
228C22×C4C2×C22⋊D1711yescharacteristic
229M16—682no—
230M16—682no—
231C2×C8—42no—
232C2×C8—682no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—681no—
236C4⋊C4—681no—
237C4⋊C4—682no—
238C4⋊C4—682no—
239C4⋊C4—681no—
240C4⋊C4—681no—
241C4⋊C4—682no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—682no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×#6322711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—41no—
271C2×C4—342no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—681no—
279C2×C4—681no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4#1058747711yescharacteristic
284C2×C4#1058747711yescharacteristic
285C2×C4#1058759012yes—
286C8—82no—
287C8—1362no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1058747711yescharacteristic
296C22#1176280611yescharacteristic
297C22#1176282511yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—682no—
306C4—1361no—
307C4—1361no—
308C4—1361no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#1452734111yescharacteristic
313C2#1453043711yescharacteristic
314C2#1453046711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1458C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1458D17
hypocenterC17#11853
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #14526812C2semidirect
4. #14526791C2semidirect
213. C17#11853semidirect

Maximal subgroups

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

subgroupindex
#145344822
#145288672
#145268122
#145267912
C17×#14582
#1185317