Finite Group Databasebeta

← #16945461

Subgroups — #16945461

Order 4352.

508 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 99 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#14528772C211yesmaximal, characteristic
3#14526809C214yesmaximal, complemented
4C17×#1459C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587610—21no—
7C2×#10587610—21no—
8C2×#10587505—21no—
9#11762745C412yescomplemented
10#11762745C2212yes—
11#11762712—21no—
12#11762678—21no—
13#11762669—21no—
14C4×C4⋊Dic17C411yescharacteristic, complemented
15#11762576C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×C22⋊C8—21no—
18C17×#324—21no—
19C17×#323C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C22×C8×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—41no—
37C2×C4⋊Dic17C2×C412yes—
38#10587610—44no—
39#10587610—44no—
40C2×#63225—22no—
41C2×#63225—22no—
42C2×C4×Dic17—21no—
43#10587555—41no—
44#10587555—41no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—41no—
47#10587505—42no—
48#10587502—41no—
49#10587502C2×C412yes—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17SD1612yes—
55C2×D4×C17Q1612yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—42no—
70C2×Dic34—42no—
71C2×Dic34—42no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×C8×C17—42no—
75C2×C8×C17—82no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—41no—
79C42×C17—21no—
80C42×C17—21no—
81C22⋊Dic17—44no—
82C22⋊Dic17—44no—
83C22⋊Dic17—42no—
84C22⋊Dic17—42no—
85C4⋊Dic17—24no—
86C4⋊Dic17—42no—
87C4⋊Dic17—42no—
88C4⋊Dic17—82no—
89#63225—44no—
90#63225—44no—
91C4×Dic17—42no—
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
105D4×C17—42no—
106D4×C17—42no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—81no—
119C2×C4×C17—81no—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#10312yes—
123C2×Dic17—24no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—82no—
130C2×Dic17—82no—
131C2×Dic17—82no—
132Dic34—82no—
133Dic34—82no—
134C8×C17—82no—
135#2542—171no—
136#1799—171no—
137#1459D1711yescharacteristic
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—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17#35312yes—
162Dic17—44no—
163Dic17—82no—
164Dic17—162no—
165C2×C4⋊Q8—171no—
166C2×C22⋊Q8—341no—
167C2×C22⋊Q8—341no—
168C2×C4×D4C2×D1711yescharacteristic
169C2×#99—341no—
170C2×C22⋊C8—21no—
171#399—172no—
172#399—172no—
173#390—341no—
174#384—341no—
175#380—341no—
176C4×C4⋊C4—171no—
177#342—172no—
178#324—21no—
179#323C2×D1712yes—
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—341no—
189C22×D4D6811yescharacteristic
190C23×C4D6811yescharacteristic
191C23×C4C22⋊D1711yescharacteristic
192C22×C8—41no—
193C4⋊Q8—681no—
194C4⋊Q8—681no—
195C22⋊Q8—684no—
196C22⋊Q8—684no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—172no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—681no—
205C2×C4⋊C4C22⋊D1711yescharacteristic
206C2×C22⋊C4—342no—
207C2×C22⋊C4—342no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4C4×D1711yescharacteristic
210C2×C22⋊C4C22×D1711yescharacteristic
211C2×C42—341no—
212C2×C42C4×D1711yescharacteristic
213#99—682no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216#95—172no—
217#95—681no—
218C17#1190111yescharacteristic, complemented, Sylow
219C24#6322711yescharacteristic
220C24C2×D6811yescharacteristic
221C2×Q8—682no—
222C2×Q8—682no—
223C2×Q8—681no—
224C2×Q8—681no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#6321912yes—
228C2×D4Dic6812yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—682no—
238C22×C4#6322711yescharacteristic
239C22×C4#6322711yescharacteristic
240C22×C4#6322711yescharacteristic
241C22×C4C2×C4×D1711yescharacteristic
242C22×C4C2×C22⋊D1711yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—41no—
246C4⋊C4—344no—
247C4⋊C4—682no—
248C4⋊C4—682no—
249C4⋊C4—684no—
250C4⋊C4—684no—
251C4⋊C4—1362no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—684no—
255C22⋊C4—684no—
256C22⋊C4—682no—
257C22⋊C4—682no—
258C42—21no—
259C42—21no—
260C42—682no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1058748712yes—
270C23C2×#6321911yescharacteristic
271C23C2×Dic6811yescharacteristic
272C23C2×#6322711yescharacteristic
273Q8—1362no—
274Q8—1362no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—344no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—1362no—
297C2×C4—1362no—
298C2×C4—1362no—
299C2×C4#1058747711yescharacteristic
300C2×C4#1058747711yescharacteristic
301C2×C4#1058748712yes—
302C8—82no—
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×#1058747711yescharacteristic
318C22C2×#1058748711yescharacteristic
319C22#1176282511yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—2722no—
326C4—684no—
327C4—1362no—
328C4#1176260112yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#1452735711yescharacteristic
335C2#1453043911yescharacteristic
336C2C2×#1176260111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1459C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1459D17
hypocenterC17#11901
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #14526809C2semidirect
5. C2×#10587555C4semidirect
9. #11762745C4semidirect
14. C4×C4⋊Dic17C4semidirect
218. C17#11901semidirect

Maximal subgroups

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

subgroupindex
#145337372
#145287722
#145268092
C17×#14592
#1190117