Finite Group Databasebeta

← #16950305

Subgroups — #16950305

Order 4352.

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

normal subgroups only

This group has 680 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#127×Dic17C211yesmaximal, characteristic
2C2×#11762602C211yesmaximal, characteristic
3#14528770C212yesmaximal, complemented
4#14526908C212yesmaximal, complemented
5C2×C17×#353C211yesmaximal, characteristic
6C2×D4×Dic17—22no—
7C2×D4×Dic17C411yescharacteristic, complemented
8C2×D4×Dic17C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762712—22no—
11#11762712C411yescharacteristic, complemented
12#11762712C2211yescharacteristic
13#11762679C411yescharacteristic, complemented
14#11762667—22no—
15C42×Dic17C411yescharacteristic, 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—22no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35C2×C22⋊Dic17D412yes—
36D4×Dic17—41no—
37D4×Dic17—41no—
38D4×Dic17—48no—
39D4×Dic17—41no—
40D4×Dic17—41no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×#63225—42no—
45C2×C4×Dic17—21no—
46C2×C4×Dic17—21no—
47C2×C4×Dic17C2×C411yescharacteristic
48C2×C4×Dic17C2×C411yescharacteristic
49C17×#99—22no—
50C17×#99—22no—
51#10587505—22no—
52#10587505—22no—
53#10587502—42no—
54C24×C17—21no—
55C24×C17C2×D412yes—
56C2×D4×C17—24no—
57C2×D4×C17—24no—
58C2×D4×C17—21no—
59C2×D4×C17—41no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17—44no—
64C2×D4×C17—44no—
65C2×D4×C17C4⋊C412yescomplemented
66C2×D4×C17C2×D411yescharacteristic
67C2×D4×C17C2×Q811yescharacteristic
68C2×D4×C17#4112yes—
69C22×C4×C17—21no—
70C22×C4×C17—42no—
71C22×C4×C17C22×C411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—22no—
74C22×Dic17—22no—
75C22×Dic17—21no—
76C22×Dic17—44no—
77C22×Dic17—44no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—42no—
81C22×Dic17C22⋊C411yescharacteristic
82C22×Dic17C22⋊C412yes—
83C22×Dic17C22⋊C411yescharacteristic
84C22⋊C4×C17—42no—
85C22⋊C4×C17—42no—
86C42×C17—21no—
87C42×C17C4⋊C412yescomplemented
88C22⋊Dic17—42no—
89C22⋊Dic17—44no—
90C22⋊Dic17—44no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C4⋊Dic17—41no—
95C4⋊Dic17—44no—
96C4⋊Dic17—41no—
97#63225—84no—
98C4×Dic17—24no—
99C4×Dic17—21no—
100C4×Dic17—21no—
101C4×Dic17—42no—
102C4×Dic17—41no—
103#12797—171nomaximal, Sylow
104C23×C17—22no—
105C23×C17—22no—
106C23×C17—22no—
107C23×C17—42no—
108C23×C17—42no—
109C23×C17—42no—
110C23×C17—42no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17C2×C22⋊C411yescharacteristic
114C23×C17#12112yes—
115C23×C17C22⋊Q812yes—
116D4×C17—41no—
117D4×C17—41no—
118D4×C17—42no—
119D4×C17—42no—
120D4×C17—48no—
121D4×C17—48no—
122D4×C17—42no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—22no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—82no—
129C2×C4×C17—82no—
130C2×C4×C17—82no—
131C2×C4×C17C2×C4⋊C411yescharacteristic
132C2×C4×C17#11711yescharacteristic
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—21no—
136C2×Dic17—22no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—44no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—42no—
143C2×Dic17—41no—
144C2×Dic17—82no—
145C2×Dic17—82no—
146C2×Dic17—82no—
147C2×Dic17—82no—
148C2×Dic17—82no—
149C2×Dic17—82no—
150C2×Dic17#9911yescharacteristic
151C2×Dic17#9911yescharacteristic
152C4×#127—171no—
153C2×#353—171no—
154C2×#353D1711yescharacteristic
155#1799—172no—
156#1393—172no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—42no—
162C22×C17—42no—
163C22×C17—44no—
164C22×C17—42no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17—82no—
168C22×C17—82no—
169C22×C17—81no—
170C22×C17—81no—
171C22×C17#38011yescharacteristic
172C22×C17C2×#9911yescharacteristic
173C22×C17#41011yescharacteristic
174C4×C17—24no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—162no—
178Dic17—21no—
179Dic17—42no—
180Dic17—42no—
181Dic17—41no—
182Dic17—81no—
183Dic17—162no—
184Dic17#35312yescomplemented
185C2×#127C2×D1711yescharacteristic
186C2×C4×D4—171no—
187C2×C4×D4—171no—
188C2×C4×D4—342no—
189C2×#99—172no—
190C2×#99C2×D1712yes—
191#390—171no—
192#390—171no—
193#390—342no—
194#384—171no—
195#380—342no—
196C43—171no—
197#353—22no—
198#353—342no—
199#342—342no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—42no—
203C2×C17—42no—
204C2×C17—82no—
205C2×C17#179911yescharacteristic
206C2×C17C2×#35311yescharacteristic
207C2×C17#212611yescharacteristic
208C22×D4—22no—
209C22×D4C4×D1711yescharacteristic
210C22×D4C22×D1711yescharacteristic
211C23×C4—172no—
212C23×C4—341no—
213#127—22no—
214#127Dic3412yes—
215#127C22⋊D1712yes—
216C4×D4—681no—
217C4×D4—681no—
218C4×D4—688no—
219C4×D4—681no—
220C4×D4—681no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—342no—
224C2×C4⋊C4—682no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—172no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4—342no—
229C2×C22⋊C4—342no—
230C2×C22⋊C4—342no—
231C2×C42—171no—
232C2×C42—171no—
233C2×C42—341no—
234C2×C42—341no—
235C2×C42C4×D1711yescharacteristic
236#99—22no—
237#99—22no—
238#99—342no—
239#99—342no—
240#95—682no—
241C17#1279711yescharacteristic, complemented, Sylow
242C24—21no—
243C24D4×D1712yes—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4#6322512yescomplemented
254C2×D4C2×Dic3411yescharacteristic
255C2×D4#6325412yes—
256C2×D4C2×C22⋊D1711yescharacteristic
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—172no—
262C22×C4—342no—
263C22×C4—342no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4—684no—
269C22×C4—684no—
270C22×C4—682no—
271C22×C4C2×C4×D1711yescharacteristic
272C4⋊C4—681no—
273C4⋊C4—681no—
274C4⋊C4—684no—
275C4⋊C4—1364no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—682no—
279C22⋊C4—684no—
280C22⋊C4—684no—
281C22⋊C4—682no—
282C22⋊C4—682no—
283C22⋊C4—682no—
284C42—21no—
285C42—341no—
286C42—341no—
287C42—344no—
288C42—681no—
289C42—682no—
290C42#6322512yescomplemented
291C23—22no—
292C23—22no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—41no—
299C23—41no—
300C23#1058756412yes—
301C23C22⋊C4×D1711yescharacteristic
302C23#1058762512yes—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—42no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—171no—
319C2×C4—171no—
320C2×C4—342no—
321C2×C4—342no—
322C2×C4—341no—
323C2×C4—342no—
324C2×C4—684no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—682no—
329C2×C4—682no—
330C2×C4—681no—
331C2×C4—1362no—
332C2×C4—1362no—
333C2×C4—1362no—
334C2×C4—1362no—
335C2×C4—1362no—
336C2×C4—1362no—
337C2×C4#1058756311yescharacteristic
338C2×C4C2×#6322511yescharacteristic
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22#1176266711yescharacteristic
354C22#99×D1711yescharacteristic
355C22#1176278111yescharacteristic
356C4—24no—
357C4—21no—
358C4—21no—
359C4—162no—
360C4—2722no—
361C4—172no—
362C4—341no—
363C4—682no—
364C4—682no—
365C4—681no—
366C4—1361no—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—42no—
371C2—82no—
372C2#1452877011yescharacteristic
373C2D17×#35311yescharacteristic
374C2#1453044611yescharacteristic
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#12797
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
7. C2×D4×Dic17C4semidirect
11. #11762712C4semidirect
13. #11762679C4semidirect
15. C42×Dic17C4semidirect
65. C2×D4×C17C4⋊C4semidirect
87. C42×C17C4⋊C4semidirect
184. Dic17#353semidirect
241. C17#12797semidirect
253. C2×D4#63225semidirect
290. C42#63225semidirect

Maximal subgroups

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

subgroupindex
#127×Dic172
C2×#117626022
#145287702
#145269082
C2×C17×#3532
#1279717