Finite Group Databasebeta

← #16944713

Subgroups — #16944713

Order 4352.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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×#10587507C211yesmaximal, characteristic, complemented
2#14527108C211yesmaximal, characteristic, complemented
3#14526814C212yesmaximal, complemented
4#14526787C212yesmaximal
5C17×#2632C211yesmaximal, characteristic
6#11762862—21no—
7C2×#10587507—21no—
8C4×#63223—21no—
9#11762578C2212yes—
10#11762535—21no—
11#11762535C2212yes—
12C2×C17×#124C411yescharacteristic
13C2×C4×Q8×C17C2211yescharacteristic
14C17×#399—21no—
15C17×#395C411yescharacteristic
16C17×#395C2211yescharacteristic
17C17×#394C411yescharacteristic
18C17×#394C2211yescharacteristic
19C17×#393—21no—
20C17×#388—21no—
21C17×#388—21no—
22C17×#383C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×Q8×C17D411yescharacteristic
25C22×D4×C17C2×C411yescharacteristic
26C17×#124—41no—
27C17×#124—41no—
28C4×Q8×C17—41no—
29C4×Q8×C17—41no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17C2311yescharacteristic
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×#63223—22no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51#10587507—24no—
52C22⋊#3622—22no—
53C4⋊#3622—41no—
54#10587474—41no—
55C4×#3622—41no—
56C4×#3622—41no—
57C24×C17—21no—
58C2×Q8×C17—22no—
59C2×Q8×C17—22no—
60C2×Q8×C17—41no—
61C2×Q8×C17—41no—
62C2×Q8×C17—81no—
63C2×D4×C17—41no—
64C2×D4×C17—81no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—81no—
79C4⋊C4×C17—81no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—81no—
86C22⋊C4×C17—81no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—21no—
92C42×C17—21no—
93#63223—24no—
94#63223—42no—
95C2×#3622—42no—
96C2×#3622—42no—
97#11738—171nomaximal, Sylow
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106Q8×C17—44no—
107Q8×C17—82no—
108D4×C17—82no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
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—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17#9911yescharacteristic
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17#10412yes—
135C2×C4×C17#13512yes—
136#3622—44no—
137#3622—82no—
138#2632D1711yescharacteristic
139C4×#101—171no—
140#1461—171no—
141#1354—172no—
142#1345—172no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—81no—
146C22×C17—81no—
147C22×C17—81no—
148C22×C17—81no—
149C22×C17—81no—
150C22×C17—81no—
151C22×C17—81no—
152C22×C17C2×#9911yescharacteristic
153C22×C17#41611yescharacteristic
154C22×C17C2×#10411yescharacteristic
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17—81no—
164C4×C17—81no—
165C4×C17—81no—
166C2×#124Dic1711yescharacteristic
167C2×C4×Q8C2×D1711yescharacteristic
168#431—341no—
169C2×#101—341no—
170C4×M16—341no—
171#399—21no—
172#395Dic1711yescharacteristic
173#395C2×D1711yescharacteristic
174#394Dic1711yescharacteristic
175#394C2×D1711yescharacteristic
176#393—21no—
177#388—21no—
178#388—21no—
179#383Dic1711yescharacteristic
180#343—172no—
181#324—172no—
182#324—341no—
183C2×C17—21no—
184C2×C17—21no—
185C2×C17—161no—
186C2×C17#150111yescharacteristic
187C2×C17#212211yescharacteristic
188C2×C17#213311yescharacteristic
189C22×Q8C2×Dic1711yescharacteristic
190C22×Q8C22⋊D1711yescharacteristic
191C22×D4C2×Dic1711yescharacteristic
192C2×M16—342no—
193#124—41no—
194#124—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×Dic1711yescharacteristic
200C2×C4⋊C4C22⋊D1711yescharacteristic
201C2×C4⋊C4C22×D1711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42C2×Dic1711yescharacteristic
209C2×C42C22⋊D1711yescharacteristic
210C2×C42C22⋊D1711yescharacteristic
211C4⋊C8—681no—
212#101—344no—
213C22⋊C8—342no—
214#97—681no—
215C4×C8—681no—
216C4×C8—681no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C2×Dic1711yescharacteristic
222#95C2×Dic1711yescharacteristic
223C17#1173811yescharacteristic, complemented, Sylow
224C24—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—81no—
230C2×D4—41no—
231C2×D4—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C22⋊Dic1711yescharacteristic
236C22×C4C22⋊Dic1711yescharacteristic
237C22×C4C22⋊Dic1711yescharacteristic
238C22×C4C22⋊Dic1711yescharacteristic
239C22×C4C22×Dic1711yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—344no—
243M16—682no—
244C2×C8—682no—
245C2×C8—682no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C22⋊C4—42no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊Dic1711yescharacteristic
272Q8—44no—
273Q8—82no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
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—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4#1058750511yescharacteristic
299C2×C4#1058750511yescharacteristic
300C2×C4#1058750812yes—
301C2×C4#1058763612yes—
302C8—684no—
303C8—1362no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×#1058750511yescharacteristic
314C22#1176280511yescharacteristic
315C22C2×#1058750811yescharacteristic
316C4—22no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C2—21no—
328C2—21no—
329C2—161no—
330C2#1452727511yescharacteristic
331C2#1453043411yescharacteristic
332C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2632C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2632D17
hypocenterC17#11738
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. C4×#10587507C2semidirect
2. #14527108C2semidirect
3. #14526814C2semidirect
223. C17#11738semidirect

Maximal subgroups

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

subgroupindex
C4×#105875072
#145271082
#145268142
#145267872
C17×#26322
#1173817