Finite Group Databasebeta

← #16927935

Subgroups — #16927935

Order 4352.

446 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#10587504C211yesmaximal, characteristic
2#14527125C211yesmaximal, characteristic
3#14526677C212yesmaximal, complemented
4#14526648C212yesmaximal, complemented
5C17×#2353C211yesmaximal, characteristic
6#11762866C2212yes—
7C2×#10587504—21no—
8C4×#63223C2212yes—
9#11762578—21no—
10#11762533—21no—
11#11762533—21no—
12C2×C4×D4×C17C2211yescharacteristic
13C17×#388—21no—
14C17×#387—21no—
15C17×#386—21no—
16C17×#383C2211yescharacteristic
17C17×#381C411yescharacteristic
18C17×#380C411yescharacteristic
19C4×C22⋊C4×C17—21no—
20C4×C22⋊C4×C17C411yescharacteristic
21C17×#376C2211yescharacteristic
22C2×C17×#95C411yescharacteristic
23C22×D4×C17D411yescharacteristic
24C23×C4×C17—21no—
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17—21no—
37C2×C42×C17—21no—
38C2×C42×C17C2311yescharacteristic
39C2×#63223—22no—
40C17×#95—21no—
41C17×#95—21no—
42C17×#95—21no—
43C17×#95—21no—
44C17×#95—21no—
45C17×#95C2×C411yescharacteristic
46C17×#95C2×C411yescharacteristic
47C17×#95C2×C411yescharacteristic
48C17×#95C2×C411yescharacteristic
49#10587504—24no—
50C22⋊#3622—22no—
51C4⋊#3622—22no—
52#10587474—22no—
53#10587474—22no—
54C4×#3622—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C22⋊C411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17—81no—
71C22×C4×C17—81no—
72C22×C4×C17—81no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22×C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C4⋊C4×C17—22no—
79C4⋊C4×C17—81no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—81no—
85C42×C17—42no—
86C42×C17—81no—
87C42×C17#4112yes—
88C42×C17#4112yes—
89#63223—24no—
90#63223—42no—
91C2×#3622—42no—
92C2×#3622—42no—
93#8773—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17#10412yes—
104C23×C17C2×C22⋊C411yescharacteristic
105D4×C17—44no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—81no—
127C2×C4×C17—81no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17—84no—
131C2×C4×C17#11711yescharacteristic
132C2×C4×C17#11711yescharacteristic
133C2×C4×C17#12312yes—
134C2×C4×C17#12312yes—
135#3622—44no—
136#3622—82no—
137#2353D1711yescharacteristic
138C4×#100—171no—
139#1466—171no—
140#1300—172no—
141#1288—172no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—22no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—81no—
153C22×C17—81no—
154C22×C17#38111yescharacteristic
155C22×C17C2×#10411yescharacteristic
156C22×C17#42111yescharacteristic
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—42no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—81no—
165C4×C17—81no—
166C4×C17—81no—
167C2×C4×D4C2×D1711yescharacteristic
168#432—172no—
169C2×#100—341no—
170C4×M16—172no—
171#388—21no—
172#387—21no—
173#386—21no—
174#383C2×D1711yescharacteristic
175#381Dic1711yescharacteristic
176#380Dic1711yescharacteristic
177C4×C22⋊C4—21no—
178C4×C22⋊C4Dic1711yescharacteristic
179#376C2×D1711yescharacteristic
180C2×#95Dic1711yescharacteristic
181#343—341no—
182#323—341no—
183#323—341no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17—42no—
187C2×C17—81no—
188C2×C17#182611yescharacteristic
189C2×C17#182711yescharacteristic
190C2×C17#182911yescharacteristic
191C22×D4C22⋊D1711yescharacteristic
192C23×C4—21no—
193C23×C4C2×Dic1711yescharacteristic
194C23×C4C2×Dic1711yescharacteristic
195C2×M16—342no—
196C4×D4—22no—
197C4×D4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C22⋊D1711yescharacteristic
204C2×C22⋊C4C22⋊D1711yescharacteristic
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22×D1711yescharacteristic
208C4⋊C8—342no—
209#100—344no—
210C22⋊C8—342no—
211#97—342no—
212#97—342no—
213C4×C8—342no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95C2×Dic1711yescharacteristic
220#95C2×Dic1711yescharacteristic
221#95C2×Dic1711yescharacteristic
222#95C2×Dic1711yescharacteristic
223C17#877311yescharacteristic, complemented, Sylow
224C24C22⋊Dic1711yescharacteristic
225C24C22⋊Dic1711yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C22×Dic1711yescharacteristic
245C22×C4C2×C22⋊D1711yescharacteristic
246C22×C4C2×C22⋊D1711yescharacteristic
247M16—344no—
248M16—682no—
249C2×C8—682no—
250C2×C8—682no—
251C4⋊C4—22no—
252C4⋊C4—81no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—81no—
258C42—42no—
259C42—81no—
260C42#6325212yes—
261C42#6325212yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1058750812yes—
272C23C2×C22⋊Dic1711yescharacteristic
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—84no—
299C2×C4#1058761211yescharacteristic
300C2×C4#1058761211yescharacteristic
301C2×C4#1058761512yes—
302C2×C4#1058761512yes—
303C8—684no—
304C8—1362no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22#1176267211yescharacteristic
318C22C2×#1058750811yescharacteristic
319C22#1176282611yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—81no—
334C2#1452888311yescharacteristic
335C2#1452888811yescharacteristic
336C2#1452889511yescharacteristic
subgroupG/H
centerC22#11762672
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2353C2
socleC22×C17#381
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#2353D17
hypocenterC17#8773
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #14526677C2semidirect
4. #14526648C2semidirect
223. C17#8773semidirect

Maximal subgroups

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

subgroupindex
C4×#105875042
#145271252
#145266772
#145266482
C17×#23532
#877317