Finite Group Databasebeta

← #16952638

Subgroups — #16952638

Order 4352.

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

normal subgroups only

This group has 552 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
1C2×#11762604C214yesmaximal
2#14528772C211yesmaximal, characteristic
3C4×#10587505C211yesmaximal, characteristic
4C17×#2547C211yesmaximal, characteristic
5C2×#10587505C2212yes—
6C2×#10587505C2212yes—
7#11762669—21no—
8C4×C22⋊Dic17—21no—
9#11762604—28no—
10#11762576—21no—
11#11762576—21no—
12C2×C17×C4⋊Q8C411yescharacteristic, complemented
13C2×C17×#124C2212yes—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#397—21no—
16C17×#397—21no—
17C17×#391C412yescomplemented
18C17×#380—21no—
19C17×#380—21no—
20C4×C4⋊C4×C17C411yescharacteristic, complemented
21C22×Q8×C17C2×C412yes—
22C22×D4×C17C2311yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C17×C4⋊Q8—41no—
26C17×C4⋊Q8—41no—
27C17×#124—24no—
28C17×#124—24no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C42×C17C2×C412yes—
40C2×C42×C17C2×C411yescharacteristic
41C2×C22⋊Dic17—22no—
42C2×C22⋊Dic17—22no—
43C2×C4⋊Dic17—41no—
44C2×C4×Dic17—41no—
45C17×#95—21no—
46C17×#95—22no—
47C17×#95—22no—
48#10587505—22no—
49#10587505—24no—
50#10587505D414yes—
51#10587502—41no—
52#10587502—41no—
53C24×C17C2×D411yescharacteristic
54C24×C17C2×D411yescharacteristic
55C2×Q8×C17—24no—
56C2×Q8×C17—42no—
57C2×Q8×C17—42no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17C2×D412yes—
61C2×D4×C17#4112yes—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—22no—
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—41no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—44no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—24no—
90C22⋊Dic17—24no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C4⋊Dic17—82no—
95C4×Dic17—82no—
96#13224—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×C17#12112yes—
107C23×C17#12411yescharacteristic
108C23×C17#12711yescharacteristic
109Q8×C17—84no—
110D4×C17—42no—
111D4×C17—42no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
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—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—81no—
130C2×C4×C17—81no—
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17#9911yescharacteristic
133C2×C4×C17C4×D412yes—
134C2×Dic17—44no—
135C2×Dic17—82no—
136C2×Dic17—82no—
137C2×Dic17—82no—
138C2×Dic17—82no—
139C2×Dic17—82no—
140#2547D1711yescharacteristic
141C2×#354—174no—
142#1799—171no—
143C4×#99—171no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—22no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—42no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—81no—
157C22×C17—81no—
158C22×C17#39011yescharacteristic
159C22×C17C2×#9911yescharacteristic
160C22×C17#41011yescharacteristic
161C4×C17—21no—
162C4×C17—22no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—82no—
167C4×C17—82no—
168C4×C17#35412yescomplemented
169Dic17—84no—
170Dic17—162no—
171C2×C4⋊Q8Dic1711yescharacteristic, complemented
172C2×#124C2×D1712yes—
173C2×C4×D4C2×D1711yescharacteristic
174C2×#99—172no—
175C2×#99—172no—
176#397—21no—
177#397—21no—
178#391Dic1712yescomplemented
179#380—21no—
180#380—21no—
181#380—341no—
182C4×C4⋊C4Dic1711yescharacteristic, complemented
183C4×C22⋊C4—341no—
184#354—348no—
185#342—341no—
186#342—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—42no—
190C2×C17—41no—
191C2×C17—41no—
192C2×C17#196411yescharacteristic
193C2×C17C2×#35411yescharacteristic
194C2×C17#212911yescharacteristic
195C22×Q8C2×Dic1712yes—
196C22×D4C22×D1711yescharacteristic
197C23×C4C22⋊D1711yescharacteristic
198C23×C4C22⋊D1711yescharacteristic
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201#124—24no—
202#124—24no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—681no—
209C2×C4⋊C4C2×Dic1711yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4C22⋊D1711yescharacteristic
215C2×C22⋊C4C22⋊D1711yescharacteristic
216C2×C42—681no—
217C2×C42C2×Dic1712yes—
218C2×C42C2×Dic1711yescharacteristic
219#99—174no—
220#99—342no—
221#99—344no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—681no—
226#95—681no—
227C17#1322411yescharacteristic, complemented, Sylow
228C24C2×C22⋊D1711yescharacteristic
229C24C2×C22⋊D1711yescharacteristic
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4D4×D1712yes—
236C2×D4#6325412yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—682no—
246C22×C4—682no—
247C22×C4C22⋊Dic1711yescharacteristic
248C22×C4C22⋊Dic1711yescharacteristic
249C22×C4C22⋊Dic1711yescharacteristic
250C22×C4C22⋊Dic1711yescharacteristic
251C22×C4C22×Dic1711yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—1362no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—344no—
264C22⋊C4—682no—
265C22⋊C4—682no—
266C22⋊C4—682no—
267C42—21no—
268C42—21no—
269C42—24no—
270C42—1362no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23#1058762211yescharacteristic
280C23#1058762412yes—
281C23#1058762611yescharacteristic
282C23C2×C22⋊Dic1711yescharacteristic
283Q8—84no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—684no—
306C2×C4—1362no—
307C2×C4—1362no—
308C2×C4—1362no—
309C2×C4—1362no—
310C2×C4—1362no—
311C2×C4#1058750511yescharacteristic
312C2×C4#1058750511yescharacteristic
313C2×C4D4×Dic1712yes—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22#1176271211yescharacteristic
329C22C2×#1058750511yescharacteristic
330C22#1176278311yescharacteristic
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—82no—
338C4—2722no—
339C4—1364no—
340C4#1176260412yescomplemented
341C2—21no—
342C2—21no—
343C2—41no—
344C2—42no—
345C2—41no—
346C2#1452962111yescharacteristic
347C2C2×#1176260411yescharacteristic
348C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2547C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2547D17
hypocenterC17#13224
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
12. C2×C17×C4⋊Q8C4semidirect
17. C17×#391C4semidirect
20. C4×C4⋊C4×C17C4semidirect
168. C4×C17#354semidirect
171. C2×C4⋊Q8Dic17semidirect
178. #391Dic17semidirect
182. C4×C4⋊C4Dic17semidirect
227. C17#13224semidirect
340. C4#11762604semidirect

Maximal subgroups

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

subgroupindex
C2×#117626042
#145287722
C4×#105875052
C17×#25472
#1322417