Finite Group Databasebeta

← #16952995

Subgroups — #16952995

Order 4352.

556 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C4×#10587505C211yesmaximal, characteristic
3#14526909C214yesmaximal
4C17×#2542C211yesmaximal, characteristic
5C2×#10587505C2212yes—
6C2×#10587505C2212yes—
7#11762669—21no—
8C4×C22⋊Dic17—21no—
9#11762576—21no—
10#11762576—21no—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C17×C22⋊Q8—21no—
13C2×C17×C22⋊Q8—21no—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#399—21no—
16C17×#399—21no—
17C17×#390C2212yes—
18C17×#384C412yescomplemented
19C4×C4⋊C4×C17C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D4×C17C2311yescharacteristic
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—22no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C412yes—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C42×C17C2×C411yescharacteristic
41C2×C42×C17C2×C412yes—
42C2×C22⋊Dic17—22no—
43C2×C22⋊Dic17—22no—
44C2×C4⋊Dic17—41no—
45C2×C4×Dic17—41no—
46C17×#95—21no—
47#10587505—22no—
48#10587505—24no—
49#10587505Q814yescomplemented
50#10587502—41no—
51#10587502—41no—
52C24×C17C2×D411yescharacteristic
53C24×C17C2×D411yescharacteristic
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C2×Q8×C17—41no—
57C2×Q8×C17—41no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17C2×Q812yes—
61C2×D4×C17#4112yes—
62C22×C4×C17—22no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—22no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22×C411yescharacteristic
75C22×Dic17—42no—
76C22×Dic17—42no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—44no—
81C4⋊C4×C17—44no—
82C4⋊C4×C17—42no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—44no—
86C22⋊C4×C17—44no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—42no—
90C22⋊Dic17—24no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C4⋊Dic17—82no—
95C4×Dic17—82no—
96#13298—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17C22⋊Q812yes—
107C23×C17#12411yescharacteristic
108C23×C17C4⋊Q811yescharacteristic
109Q8×C17—82no—
110Q8×C17—82no—
111D4×C17—42no—
112D4×C17—42no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—22no—
116C2×C4×C17—24no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17#9911yescharacteristic
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17C4×Q812yes—
135C2×Dic17—44no—
136C2×Dic17—82no—
137C2×Dic17—82no—
138C2×Dic17—82no—
139C2×Dic17—82no—
140C2×Dic17—82no—
141#2542D1711yescharacteristic
142#1799—171no—
143C4×#99—171no—
144#1393—174no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17#39111yescharacteristic
160C22×C17C2×#9911yescharacteristic
161C22×C17#41011yescharacteristic
162C4×C17—21no—
163C4×C17—22no—
164C4×C17—44no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—82no—
169C4×C17#35312yes—
170Dic17—84no—
171Dic17—162no—
172C2×C4⋊Q8Dic1711yescharacteristic, complemented
173C2×C22⋊Q8—21no—
174C2×C22⋊Q8—21no—
175C2×C4×D4C2×D1711yescharacteristic
176C2×#99—172no—
177C2×#99—172no—
178#399—21no—
179#399—21no—
180#390C2×D1712yes—
181#384Dic1712yescomplemented
182#380—341no—
183C4×C4⋊C4Dic1711yescharacteristic, complemented
184C4×C22⋊C4—341no—
185#342—341no—
186#342—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—42no—
190C2×C17—41no—
191C2×C17—41no—
192C2×C17#199611yescharacteristic
193C2×C17C2×#35311yescharacteristic
194C2×C17#212911yescharacteristic
195C22×Q8—21no—
196C22×D4C22×D1711yescharacteristic
197C23×C4C22⋊D1711yescharacteristic
198C23×C4C22⋊D1711yescharacteristic
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—681no—
210C2×C4⋊C4C2×Dic1712yes—
211C2×C4⋊C4C2×Dic1711yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—342no—
215C2×C22⋊C4—342no—
216C2×C22⋊C4C22⋊D1711yescharacteristic
217C2×C22⋊C4C22⋊D1711yescharacteristic
218C2×C42—681no—
219C2×C42C2×Dic1711yescharacteristic
220C2×C42C2×Dic1712yes—
221#99—174no—
222#99—342no—
223#99—344no—
224#95—21no—
225#95—681no—
226#95—681no—
227C17#1329811yescharacteristic, complemented, Sylow
228C24C2×C22⋊D1711yescharacteristic
229C24C2×C22⋊D1711yescharacteristic
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4Q8×D1712yes—
237C2×D4#6325612yes—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—682no—
247C22×C4—682no—
248C22×C4C22⋊Dic1711yescharacteristic
249C22×C4C22⋊Dic1711yescharacteristic
250C22×C4C22⋊Dic1711yescharacteristic
251C22×C4C22⋊Dic1711yescharacteristic
252C22×C4C22×Dic1711yescharacteristic
253C4⋊C4—24no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—1362no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—344no—
265C22⋊C4—682no—
266C22⋊C4—682no—
267C22⋊C4—682no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—1362no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1058763011yescharacteristic
281C23#1058763212yes—
282C23#1058763311yescharacteristic
283C23C2×C22⋊Dic1711yescharacteristic
284Q8—82no—
285Q8—82no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—684no—
308C2×C4—1362no—
309C2×C4—1362no—
310C2×C4—1362no—
311C2×C4—1362no—
312C2×C4—1362no—
313C2×C4#1058750511yescharacteristic
314C2×C4#1058750511yescharacteristic
315C2×C4Q8×Dic1712yes—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22#1176271611yescharacteristic
331C22C2×#1058750511yescharacteristic
332C22#1176278311yescharacteristic
333C4—21no—
334C4—22no—
335C4—44no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—82no—
340C4—2722no—
341C4—1364no—
342C4#1176260212yes—
343C2—21no—
344C2—21no—
345C2—41no—
346C2—42no—
347C2—41no—
348C2#1452977011yescharacteristic
349C2C2×#1176260211yescharacteristic
350C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2542C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2542D17
hypocenterC17#13298
LCSG ▷ C22×C4×C17 ▷ C22×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
11. C2×C17×C4⋊Q8C4semidirect
18. C17×#384C4semidirect
19. C4×C4⋊C4×C17C4semidirect
49. #10587505Q8semidirect
172. C2×C4⋊Q8Dic17semidirect
181. #384Dic17semidirect
183. C4×C4⋊C4Dic17semidirect
227. C17#13298semidirect

Maximal subgroups

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

subgroupindex
#145287722
C4×#105875052
#145269092
C17×#25422
#1329817