Finite Group Databasebeta

← #16950309

Subgroups — #16950309

Order 4352.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#124×Dic17C211yesmaximal, characteristic
2C2×#11762604C211yesmaximal, characteristic
3#14528770C212yesmaximal
4#14526908C212yesmaximal
5C2×C17×#354C211yesmaximal, characteristic
6C2×Q8×Dic17C411yescharacteristic, complemented
7C2×D4×Dic17C2211yescharacteristic
8C2×#10587505C2212yes—
9#11762716C411yescharacteristic, complemented
10#11762712C2211yescharacteristic
11#11762702—22no—
12#11762675C411yescharacteristic, complemented
13#11762667—22no—
14C22⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic, complemented
16#11762604—22no—
17#11762575—22no—
18C2×C17×#124C2211yescharacteristic
19C2×C17×#99C2212yes—
20C17×#354—22no—
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17C2311yescharacteristic
23C23×Dic17D412yes—
24C17×#124—22no—
25C17×#124D412yes—
26C17×#124Q812yes—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17D412yes—
33Q8×Dic17—41no—
34Q8×Dic17—41no—
35D4×Dic17—41no—
36D4×Dic17—41no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—42no—
41C2×C4×Dic17—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17C2×C411yescharacteristic
46C17×#99—22no—
47C17×#99—22no—
48#10587505—22no—
49#10587505—22no—
50#10587502—22no—
51#10587502—22no—
52#10587502—42no—
53C24×C17C2×D412yes—
54C2×Q8×C17—21no—
55C2×Q8×C17—41no—
56C2×Q8×C17—41no—
57C2×Q8×C17C4⋊C412yescomplemented
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17C2×D411yescharacteristic
61C2×D4×C17C2×Q811yescharacteristic
62C2×D4×C17#4112yes—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17C22×C411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—21no—
71C22×Dic17—44no—
72C22×Dic17—42no—
73C22×Dic17C22⋊C411yescharacteristic
74C22×Dic17C22⋊C412yes—
75C22×Dic17C22⋊C411yescharacteristic
76C22⋊C4×C17—24no—
77C22⋊C4×C17—24no—
78C22⋊C4×C17—42no—
79C22⋊C4×C17—42no—
80C42×C17—21no—
81C42×C17C4⋊C412yescomplemented
82C22⋊Dic17—42no—
83C22⋊Dic17—44no—
84C22⋊Dic17—42no—
85C22⋊Dic17—42no—
86C4⋊Dic17—41no—
87C4⋊Dic17—42no—
88C4⋊Dic17—41no—
89#63225—84no—
90C4×Dic17—21no—
91C4×Dic17—21no—
92C4×Dic17—42no—
93C4×Dic17—42no—
94C4×Dic17—41no—
95C4×Dic17—41no—
96C4×Dic17—41no—
97#12798—171nomaximal, Sylow
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—42no—
101C23×C17—42no—
102C23×C17C2×C22⋊C411yescharacteristic
103C23×C17#12112yes—
104C23×C17C22⋊Q812yes—
105Q8×C17—42no—
106Q8×C17—42no—
107D4×C17—41no—
108D4×C17—41no—
109D4×C17—42no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—82no—
119C2×C4×C17—82no—
120C2×C4×C17—82no—
121C2×C4×C17C2×C4⋊C411yescharacteristic
122C2×C4×C17#11711yescharacteristic
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—21no—
126C2×Dic17—22no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—44no—
130C2×Dic17—42no—
131C2×Dic17—41no—
132C2×Dic17—42no—
133C2×Dic17—41no—
134C2×Dic17—82no—
135C2×Dic17—82no—
136C2×Dic17—82no—
137C2×Dic17—82no—
138C2×Dic17#9911yescharacteristic
139C2×Dic17#9911yescharacteristic
140C4×#124—171no—
141C2×#354—171no—
142C2×#354D1711yescharacteristic
143#1799—172no—
144#1393—172no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—22no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17—82no—
154C22×C17#38011yescharacteristic
155C22×C17C2×#9911yescharacteristic
156C22×C17#41011yescharacteristic
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—42no—
160C4×C17—42no—
161C4×C17—162no—
162Dic17—21no—
163Dic17—42no—
164Dic17—42no—
165Dic17—41no—
166Dic17—81no—
167Dic17—162no—
168Dic17#35412yescomplemented
169C2×#124C2×D1711yescharacteristic
170C2×C4×Q8—171no—
171C2×C4×D4—171no—
172C2×#99—172no—
173C2×#99C2×D1712yes—
174#391—171no—
175#390—171no—
176#388—342no—
177#382—171no—
178#380—342no—
179C4×C22⋊C4—342no—
180C43—171no—
181#354—22no—
182#354—342no—
183#342—342no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17—42no—
187C2×C17—42no—
188C2×C17#179911yescharacteristic
189C2×C17C2×#35411yescharacteristic
190C2×C17#212611yescharacteristic
191C22×Q8C4×D1711yescharacteristic
192C22×D4C22×D1711yescharacteristic
193C23×C4—172no—
194#124—22no—
195#124Dic3412yes—
196#124C22⋊D1712yes—
197C4×Q8—681no—
198C4×Q8—681no—
199C4×D4—681no—
200C4×D4—681no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—682no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—172no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4—342no—
210C2×C42—171no—
211C2×C42—171no—
212C2×C42—341no—
213C2×C42—341no—
214C2×C42—341no—
215C2×C42C4×D1711yescharacteristic
216#99—22no—
217#99—22no—
218#99—342no—
219#99—342no—
220#95—342no—
221#95—342no—
222#95—682no—
223C17#1279811yescharacteristic, complemented, Sylow
224C24D4×D1712yes—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8#6322512yescomplemented
229C2×D4—41no—
230C2×D4—41no—
231C2×D4C2×Dic3411yescharacteristic
232C2×D4#6325412yes—
233C2×D4C2×C22⋊D1711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—171no—
238C22×C4—172no—
239C22×C4—171no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—684no—
245C22×C4—682no—
246C22×C4C2×C4×D1711yescharacteristic
247C4⋊C4—681no—
248C4⋊C4—681no—
249C4⋊C4—682no—
250C4⋊C4—1364no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—682no—
256C22⋊C4—684no—
257C22⋊C4—682no—
258C22⋊C4—682no—
259C42—21no—
260C42—341no—
261C42—341no—
262C42—681no—
263C42—681no—
264C42—681no—
265C42—682no—
266C42—682no—
267C42#6322512yescomplemented
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23#1058756412yes—
273C23C22⋊C4×D1711yescharacteristic
274C23#1058762512yes—
275Q8—42no—
276Q8—42no—
277D4—41no—
278D4—42no—
279D4—41no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—171no—
292C2×C4—171no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—341no—
296C2×C4—342no—
297C2×C4—684no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4—681no—
304C2×C4—1362no—
305C2×C4—1362no—
306C2×C4—1362no—
307C2×C4—1362no—
308C2×C4#1058756311yescharacteristic
309C2×C4C2×#6322511yescharacteristic
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22#1176266711yescharacteristic
320C22#99×D1711yescharacteristic
321C22#1176278111yescharacteristic
322C4—21no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—162no—
327C4—2722no—
328C4—172no—
329C4—341no—
330C4—682no—
331C4—682no—
332C4—681no—
333C4—1361no—
334C2—21no—
335C2—21no—
336C2—42no—
337C2—42no—
338C2#1452877011yescharacteristic
339C2D17×#35411yescharacteristic
340C2#1453044811yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC2×C17×#354C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenterC2×#354D17
hypocenterC17#12798
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic17C4semidirect
9. #11762716C4semidirect
12. #11762675C4semidirect
15. C42×Dic17C4semidirect
57. C2×Q8×C17C4⋊C4semidirect
81. C42×C17C4⋊C4semidirect
168. Dic17#354semidirect
223. C17#12798semidirect
228. C2×Q8#63225semidirect
267. C42#63225semidirect

Maximal subgroups

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

subgroupindex
#124×Dic172
C2×#117626042
#145287702
#145269082
C2×C17×#3542
#1279817