Finite Group Databasebeta

← #16950313

Subgroups — #16950313

Order 4352.

588 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 101 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#14531510C211yesmaximal, characteristic
2C2×#11762602C211yesmaximal, characteristic
3#14528770C212yesmaximal, complemented
4#14526908C212yesmaximal, complemented
5C2×C17×#353C211yesmaximal, characteristic
6C2×D4×Dic17C411yescharacteristic, complemented
7C2×D4×Dic17C2211yescharacteristic
8C2×#10587505C2212yes—
9#11762712C411yescharacteristic, complemented
10#11762712C2211yescharacteristic
11#11762689—22no—
12#11762689—22no—
13#11762675C411yescharacteristic, complemented
14#11762667—22no—
15#11762655C411yescharacteristic, complemented
16#11762602—22no—
17#11762575—22no—
18C2×C17×#127C2211yescharacteristic
19C2×C17×#99C2212yes—
20C17×#353—22no—
21C22×D4×C17—22no—
22C22×D4×C17C2×C411yescharacteristic
23C22×D4×C17C2311yescharacteristic
24C23×Dic17—21no—
25C23×Dic17D412yes—
26C17×#127—22no—
27C17×#127D412yes—
28C17×#127Q812yes—
29C2×C22⋊C4×C17—22no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35C2×C22⋊Dic17D412yes—
36D4×Dic17—42no—
37D4×Dic17—42no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—42no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43C17×#99—22no—
44C17×#99—22no—
45#10587505—22no—
46#10587505—22no—
47#10587502—21no—
48#10587502—21no—
49#10587502—22no—
50#10587502—42no—
51C24×C17—21no—
52C24×C17C2×D412yes—
53C2×D4×C17—24no—
54C2×D4×C17—24no—
55C2×D4×C17—21no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17—82no—
61C2×D4×C17—82no—
62C2×D4×C17C4⋊C412yescomplemented
63C2×D4×C17C2×D411yescharacteristic
64C2×D4×C17C2×Q811yescharacteristic
65C2×D4×C17#4112yes—
66C22×C4×C17—21no—
67C22×C4×C17—42no—
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—42no—
74C22×Dic17—42no—
75C22×Dic17—82no—
76C22×Dic17—82no—
77C22×Dic17C22⋊C411yescharacteristic
78C22×Dic17C22⋊C412yes—
79C22×Dic17C22⋊C411yescharacteristic
80C22⋊C4×C17—42no—
81C22⋊C4×C17—42no—
82C42×C17—21no—
83C42×C17C4⋊C412yescomplemented
84C22⋊Dic17—42no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—82no—
89C22⋊Dic17—82no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92#63225—84no—
93C4×Dic17—22no—
94C4×Dic17—41no—
95#12799—171nomaximal, Sylow
96C23×C17—22no—
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17#12112yes—
107C23×C17C22⋊Q812yes—
108D4×C17—41no—
109D4×C17—41no—
110D4×C17—42no—
111D4×C17—42no—
112D4×C17—42no—
113D4×C17—84no—
114D4×C17—84no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—22no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—82no—
121C2×C4×C17—82no—
122C2×C4×C17—82no—
123C2×C4×C17C2×C4⋊C411yescharacteristic
124C2×C4×C17#11711yescharacteristic
125C2×Dic17—21no—
126C2×Dic17—22no—
127C2×Dic17—41no—
128C2×Dic17—44no—
129C2×Dic17—42no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—42no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—82no—
137C2×Dic17—84no—
138C2×Dic17—82no—
139C2×Dic17—82no—
140C2×Dic17—82no—
141C2×Dic17#9912yes—
142#2328—171no—
143C2×#353—171no—
144C2×#353D1711yescharacteristic
145#1799—172no—
146#1393—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17—44no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—82no—
158C22×C17—82no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17#38011yescharacteristic
162C22×C17C2×#9911yescharacteristic
163C22×C17#41011yescharacteristic
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—42no—
167C4×C17—162no—
168Dic17—22no—
169Dic17—42no—
170Dic17—41no—
171Dic17—81no—
172Dic17—81no—
173Dic17—162no—
174C2×#127C2×D1711yescharacteristic
175C2×C4×D4—171no—
176C2×C4×D4—171no—
177C2×#99—172no—
178C2×#99C2×D1712yes—
179#390—171no—
180#390—171no—
181#386—342no—
182#386—342no—
183#382—171no—
184#380—342no—
185#376—171no—
186#353—22no—
187#353—342no—
188#342—342no—
189C2×C17—21no—
190C2×C17—21no—
191C2×C17—42no—
192C2×C17—42no—
193C2×C17—82no—
194C2×C17#179911yescharacteristic
195C2×C17#212711yescharacteristic
196C2×C17#212811yescharacteristic
197C22×D4—22no—
198C22×D4C4×D1711yescharacteristic
199C22×D4C22×D1711yescharacteristic
200C23×C4—172no—
201C23×C4—341no—
202#127—22no—
203#127Dic3412yes—
204#127C22⋊D1712yes—
205C4×D4—682no—
206C4×D4—682no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—682no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—172no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4—342no—
215C2×C22⋊C4—342no—
216C2×C42—171no—
217C2×C42—171no—
218C2×C42C4×D1711yescharacteristic
219#99—22no—
220#99—22no—
221#99—342no—
222#99—342no—
223#95—341no—
224#95—341no—
225#95—342no—
226#95—682no—
227C17#1279911yescharacteristic, complemented, Sylow
228C24—21no—
229C24D4×D1712yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4#6322512yescomplemented
240C2×D4C2×Dic3411yescharacteristic
241C2×D4#6325412yes—
242C2×D4C2×C22⋊D1711yescharacteristic
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—172no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—682no—
253C22×C4—682no—
254C22×C4—1362no—
255C22×C4—1362no—
256C22×C4C2×C4×D1711yescharacteristic
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—1364no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C22⋊C4—682no—
265C22⋊C4—682no—
266C22⋊C4—1362no—
267C22⋊C4—1362no—
268C42—21no—
269C42—342no—
270C42—681no—
271C42#6322512yescomplemented
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23#1058756412yes—
282C23C22⋊C4×D1711yescharacteristic
283C23#1058762512yes—
284D4—41no—
285D4—42no—
286D4—41no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—172no—
300C2×C4—341no—
301C2×C4—342no—
302C2×C4—681no—
303C2×C4—684no—
304C2×C4—682no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—682no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—1362no—
312C2×C4—1362no—
313C2×C4—1362no—
314C2×C4—1362no—
315C2×C4—1364no—
316C2×C4#1058756311yescharacteristic
317C2×C4C2×#6322511yescharacteristic
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22#1176266711yescharacteristic
333C22#99×D1711yescharacteristic
334C22#1176278111yescharacteristic
335C4—21no—
336C4—21no—
337C4—42no—
338C4—162no—
339C4—2722no—
340C4—342no—
341C4—682no—
342C4—681no—
343C4—1361no—
344C4—1361no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#1452877011yescharacteristic
351C2#1453045011yescharacteristic
352C2#1453045411yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC2×C17×#353C2
socleC22×C17C2×#99
prime radicalC2×C17×#127C22
cosocleC22×D4×C17C23
hypercenterC2×#353D17
hypocenterC17#12799
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #14528770C2semidirect
4. #14526908C2semidirect
6. C2×D4×Dic17C4semidirect
9. #11762712C4semidirect
13. #11762675C4semidirect
15. #11762655C4semidirect
62. C2×D4×C17C4⋊C4semidirect
83. C42×C17C4⋊C4semidirect
227. C17#12799semidirect
239. C2×D4#63225semidirect
271. C42#63225semidirect

Maximal subgroups

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

subgroupindex
#145315102
C2×#117626022
#145287702
#145269082
C2×C17×#3532
#1279917