Finite Group Databasebeta

← #16945460

Subgroups — #16945460

Order 4352.

508 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528772C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526808C214yesmaximal, complemented
4C17×#2542C211yesmaximal, characteristic
5C2×#10587505—21no—
6C2×C22⋊#3622—21no—
7#11762669—21no—
8#11762576C2212yes—
9#11762535—21no—
10#11762533C2212yes—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C17×C22⋊Q8—21no—
13C2×C17×C22⋊Q8—21no—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#399C412yescomplemented
16C17×#399C2212yes—
17C17×#390—21no—
18C17×#384—21no—
19C4×C4⋊C4×C17C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×C4⋊Q8—41no—
25C17×C4⋊Q8—41no—
26C17×C22⋊Q8—44no—
27C17×C22⋊Q8—44no—
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17C2×C412yes—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17C2311yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C22⋊Dic17—22no—
43C2×C4⋊Dic17—41no—
44C22×#3622—41no—
45C17×#95C2×C412yes—
46#10587505—42no—
47#10587502—41no—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×Q8×C17—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17SD1612yes—
59C2×D4×C17Q1612yes—
60C22×C4×C17—22no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—42no—
74C4⋊C4×C17—24no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—44no—
78C4⋊C4×C17—44no—
79C4⋊C4×C17—42no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—44no—
83C22⋊C4×C17—44no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C4⋊Dic17—82no—
90C2×#3622—42no—
91C2×#3622—82no—
92#11901—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17#10312yes—
102C23×C17C2×C22⋊C411yescharacteristic
103C23×C17C2×SD1611yescharacteristic
104C23×C17C2×Q1611yescharacteristic
105Q8×C17—82no—
106Q8×C17—82no—
107D4×C17—42no—
108D4×C17—42no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—24no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—81no—
127C2×C4×C17—81no—
128C2×C4×C17#9911yescharacteristic
129C2×C4×C17#9911yescharacteristic
130C2×C4×C17#10312yes—
131C2×Dic17—82no—
132C2×Dic17—82no—
133C2×Dic17—82no—
134#3622—82no—
135#2542D1711yescharacteristic
136#1799—171no—
137#1459—171no—
138#1352—174no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—22no—
143C22×C17—21no—
144C22×C17—42no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17C2×#9911yescharacteristic
154C22×C17C2×#10311yescharacteristic
155C22×C17#42111yescharacteristic
156C4×C17—21no—
157C4×C17—22no—
158C4×C17—44no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—82no—
163C4×C17#35312yes—
164Dic17—162no—
165C2×C4⋊Q8Dic1711yescharacteristic, complemented
166C2×C22⋊Q8—21no—
167C2×C22⋊Q8—21no—
168C2×C4×D4C2×D1711yescharacteristic
169C2×#99—341no—
170C2×C22⋊C8—341no—
171#399Dic1712yescomplemented
172#399C2×D1712yes—
173#390—21no—
174#384—21no—
175#380—341no—
176C4×C4⋊C4Dic1711yescharacteristic, complemented
177#342—172no—
178#324—341no—
179#323—172no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—41no—
183C2×C17—42no—
184C2×C17—41no—
185C2×C17#152011yescharacteristic
186C2×C17#212311yescharacteristic
187C2×C17C2×#35311yescharacteristic
188C22×Q8—21no—
189C22×D4C22⋊D1711yescharacteristic
190C23×C4C22⋊D1711yescharacteristic
191C23×C4C22⋊D1711yescharacteristic
192C22×C8—681no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—681no—
204C2×C4⋊C4C2×Dic1712yes—
205C2×C4⋊C4C22⋊D1711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4C2×Dic1711yescharacteristic
210C2×C22⋊C4C22×D1711yescharacteristic
211C2×C42—21no—
212C2×C42C2×Dic1711yescharacteristic
213#99—682no—
214C22⋊C8—342no—
215C22⋊C8—681no—
216#95—681no—
217#95C2×Dic1712yes—
218C17#1190111yescharacteristic, complemented, Sylow
219C24C22⋊Dic1711yescharacteristic
220C24C2×C22⋊D1711yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#6323012yes—
228C2×D4#6323112yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—682no—
238C22×C4C22⋊Dic1711yescharacteristic
239C22×C4C22⋊Dic1711yescharacteristic
240C22×C4C22⋊Dic1711yescharacteristic
241C22×C4C22×Dic1711yescharacteristic
242C22×C4C2×C22⋊D1711yescharacteristic
243C2×C8—682no—
244C2×C8—1362no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—1362no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—682no—
257C22⋊C4—682no—
258C42—21no—
259C42—21no—
260C42—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1058750612yes—
270C23C2×#6323011yescharacteristic
271C23C2×#6323111yescharacteristic
272C23C2×C22⋊Dic1711yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—1362no—
297C2×C4—1362no—
298C2×C4—1362no—
299C2×C4#1058750511yescharacteristic
300C2×C4#1058750511yescharacteristic
301C2×C4#1058750612yes—
302C8—1362no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1058750511yescharacteristic
318C22C2×#1058750611yescharacteristic
319C22#1176282611yescharacteristic
320C4—22no—
321C4—21no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—2722no—
328C4#1176260212yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#1452735611yescharacteristic
335C2#1453043811yescharacteristic
336C2C2×#1176260211yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2542C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2542D17
hypocenterC17#11901
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #14526808C2semidirect
11. C2×C17×C4⋊Q8C4semidirect
15. C17×#399C4semidirect
19. C4×C4⋊C4×C17C4semidirect
165. C2×C4⋊Q8Dic17semidirect
171. #399Dic17semidirect
176. C4×C4⋊C4Dic17semidirect
218. C17#11901semidirect

Maximal subgroups

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

subgroupindex
#145287722
#145271022
#145268082
C17×#25422
#1190117