Finite Group Databasebeta

← #17019681

Subgroups — #17019681

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538585C211yesmaximal, characteristic
2C2×#11763101C211yesmaximal, characteristic
3#14537864C211yesmaximal, characteristic
4#14528239C214yesmaximal, complemented
5#14528185C214yesmaximal, complemented
6#14527559C211yesmaximal, characteristic
7#14527465C211yesmaximal, characteristic
8#14527441C211yesmaximal, characteristic
9C17×#3230C211yesmaximal, characteristic
10C2×C4×Dic34C2211yescharacteristic
11#11763113—22no—
12#11763101—22no—
13#11763101C2214yes—
14#11763101C2212yes—
15#11763091C2212yes—
16C8×Dic34C2214yes—
17#11762933—22no—
18#11762862C2211yescharacteristic
19#11762859—22no—
20#11762852—21no—
21C2×#10587478—21no—
22#11762833C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24#11762817—21no—
25C2×#10587481—21no—
26#11762562C2214yes—
27#11762558C2212yes—
28#11762558C2214yes—
29#11762555C2212yes—
30C17×#533C2211yescharacteristic
31C2×C17×C4⋊Q8C2211yescharacteristic
32C17×#500C2212yes—
33C17×#487C2212yes—
34C17×#484—22no—
35C17×#477—22no—
36C17×#431C2211yescharacteristic
37C22×Q8×C17—21no—
38C22×Dic34D411yescharacteristic
39C17×C4⋊Q8—21no—
40C17×C4⋊Q8—21no—
41C17×C4⋊Q8—21no—
42C17×C4⋊Q8C2311yescharacteristic
43C17×C4⋊Q8C2312yes—
44C17×C4⋊Q8C2311yescharacteristic
45C17×#125—21no—
46C17×C22⋊Q8—21no—
47C17×C22⋊Q8—21no—
48C4×Q8×C17—21no—
49C4×Q8×C17—21no—
50C17×#117—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C42×C17C2311yescharacteristic
54C2×C4⋊Dic17D411yescharacteristic
55C2×#63225—21no—
56C2×C4×Dic17—21no—
57C2×#63223—21no—
58C22×#3622—21no—
59C4×Dic34—21no—
60C4×Dic34—21no—
61C4×Dic34D412yes—
62C4×Dic34C2312yes—
63C17×#107—24no—
64C17×C4⋊C8D412yes—
65C17×#103—24no—
66C17×C22⋊C8D412yes—
67C4×C8×C17C2312yes—
68C22⋊#3622—21no—
69#10587485—24no—
70C8×Dic17—24no—
71#10587481—24no—
72#10587481—22no—
73#10587481—22no—
74#10587479—22no—
75#10587478—24no—
76#10587478—22no—
77C4⋊#3622—21no—
78C4⋊#3622—21no—
79C4⋊#3622C2312yes—
80C4⋊#3622C2312yes—
81C4⋊#3622C2312yes—
82C4×#3622—21no—
83C2×Q8×C17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17—22no—
86C2×Q8×C17—41no—
87C2×Q8×C17—82no—
88C22×C4×C17—41no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17#4111yescharacteristic
92C22×Dic17—21no—
93C22×Dic17—21no—
94C2×Dic34—21no—
95C2×Dic34—22no—
96C2×Dic34Q1614yes—
97C2×Dic34C2×D412yes—
98C2×C8×C17C2×D414yes—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—22no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C4⋊C4×C17—41no—
109C22⋊C4×C17—41no—
110C42×C17—41no—
111C42×C17C2×D411yescharacteristic
112C42×C17C2×D411yescharacteristic
113C42×C17#4111yescharacteristic
114C42×C17C2411yescharacteristic
115C4⋊Dic17—21no—
116C4⋊Dic17C2×D412yes—
117#63225—22no—
118#63225—22no—
119C4×Dic17—22no—
120C4×Dic17—22no—
121#63223—42no—
122C2×#3622—22no—
123C2×#3622—22no—
124C2×#3622—22no—
125C2×#3622—42no—
126#19405—171nomaximal, Sylow
127C23×C17#12111yescharacteristic
128Q8×C17—42no—
129Q8×C17—81no—
130Q8×C17—82no—
131Q8×C17—82no—
132C2×C4×C17—22no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—42no—
137C2×C4×C17—82no—
138C2×C4×C17#12111yescharacteristic
139C2×C4×C17#12111yescharacteristic
140C2×C4×C17#12111yescharacteristic
141C2×C4×C17C2×Q1612yes—
142C2×C4×C17#13512yes—
143C2×C4×C17C22×D411yescharacteristic
144C2×C4×C17C22×D411yescharacteristic
145C2×C4×C17C2×#4111yescharacteristic
146C2×Dic17—22no—
147C2×Dic17—24no—
148C2×Dic17—22no—
149C2×Dic17—81no—
150Dic34—22no—
151Dic34—24no—
152Dic34C2×Q1614yes—
153C8×C17—24no—
154#3622—42no—
155#3622—42no—
156#3622—42no—
157#3230D1711yescharacteristic
158#3078—171no—
159C2×#477—171no—
160#2982—171no—
161#1684—174no—
162#1676—174no—
163#1571—171no—
164#1548—171no—
165#1542—171no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17#46712yes—
169C22×C17C2×#12111yescharacteristic
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—81no—
174C4×C17—82no—
175C4×C17—81no—
176C4×C17#46712yes—
177C4×C17C22×Q1611yescharacteristic
178C4×C17C2×#13511yescharacteristic
179C4×C17#57812yes—
180Dic17—24no—
181Dic17—42no—
182#533C2×D1711yescharacteristic
183C2×C4⋊Q8C2×D1711yescharacteristic
184C2×C4×Q8—171no—
185#500C2×D1712yes—
186#487C2×D1712yes—
187#484—22no—
188#479—342no—
189#477—22no—
190#477—174no—
191#477—172no—
192#477—342no—
193#475—172no—
194C8×Q8—174no—
195C8×Q8—342no—
196#431—171no—
197#431—342no—
198#431C2×D1711yescharacteristic
199#428—341no—
200C2×#107—341no—
201#423—171no—
202C2×C4⋊C8—171no—
203#419—341no—
204C2×#103—341no—
205#335—174no—
206#333—172no—
207#333—174no—
208#332—172no—
209C2×C17—22no—
210C2×C17#304411yescharacteristic
211C2×C17C2×#46711yescharacteristic
212C2×C17#306811yescharacteristic
213C22×Q8—21no—
214C22×Q8—171no—
215C2×M16—341no—
216C22×C8—341no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8C22×D1712yes—
221C4⋊Q8C22×D1711yescharacteristic
222C4⋊Q8C22×D1711yescharacteristic
223#125—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—172no—
229C4×Q8—172no—
230C4×Q8—341no—
231C4×Q8—341no—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—341no—
237C2×C42—341no—
238C2×C42C22×D1711yescharacteristic
239#107—24no—
240#107—344no—
241#107—342no—
242#106—342no—
243C4⋊C8—172no—
244C4⋊C8—172no—
245C4⋊C8—172no—
246C4⋊C8—344no—
247C4⋊C8—341no—
248C4⋊C8—341no—
249C4⋊C8C22⋊D1712yes—
250#103—24no—
251#103—344no—
252#103—342no—
253#103—342no—
254C22⋊C8—341no—
255C22⋊C8C22⋊D1712yes—
256C4×C8—344no—
257C4×C8—341no—
258C4×C8C22×D1712yes—
259C17#1940511yescharacteristic, complemented, Sylow
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—82no—
265C2×Q8—172no—
266C2×Q8—174no—
267C2×Q8—342no—
268C2×Q8—341no—
269C22×C4—41no—
270C22×C4—341no—
271C22×C4—341no—
272C22×C4D4×D1711yescharacteristic
273C22×C4#6325411yescharacteristic
274C22×C4C2×C22⋊D1711yescharacteristic
275M16—682no—
276C2×C8—342no—
277C2×C8—342no—
278C2×C8—342no—
279C2×C8—682no—
280C2×C8C2×C22⋊D1714yes—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—172no—
292C4⋊C4—342no—
293C4⋊C4—341no—
294C4⋊C4—342no—
295C22⋊C4—41no—
296C42—41no—
297C42—342no—
298C42—342no—
299C42D4×D1711yescharacteristic
300C42#6325411yescharacteristic
301C42C2×C22⋊D1711yescharacteristic
302C42C23×D1711yescharacteristic
303C23#1058762411yescharacteristic
304Q8—42no—
305Q8—82no—
306Q8—81no—
307Q8—82no—
308Q8—174no—
309Q8—344no—
310Q8—342no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4—342no—
318C2×C4—344no—
319C2×C4—342no—
320C2×C4—1361no—
321C2×C4Q16×D1712yes—
322C2×C4#1058760512yes—
323C2×C4#1058762411yescharacteristic
324C2×C4#1058762411yescharacteristic
325C2×C4#1058762411yescharacteristic
326C2×C4C2×D4×D1711yescharacteristic
327C2×C4C2×#6325411yescharacteristic
328C2×C4C22×C22⋊D1711yescharacteristic
329C8—24no—
330C8—682no—
331C8—682no—
332C8—682no—
333C22—21no—
334C22—21no—
335C22#1176304712yes—
336C22C2×#1058762411yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—82no—
341C4—81no—
342C4—81no—
343C4—344no—
344C4—682no—
345C4#1176304712yes—
346C4C2×Q16×D1711yescharacteristic
347C4C2×#1058760511yescharacteristic
348C4#1176362412yes—
349C2—22no—
350C2#1453831011yescharacteristic
351C2C2×#1176304711yescharacteristic
352C2#1453850811yescharacteristic
subgroupG/H
centerC22C2×#10587624
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3230C2
socleC22×C17C2×#121
prime radicalC23×C17#121
hypercenter#3230D17
hypocenterC17#19405
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
4. #14528239C2semidirect
5. #14528185C2semidirect
259. C17#19405semidirect

Maximal subgroups

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

subgroupindex
#145385852
C2×#117631012
#145378642
#145282392
#145281852
#145275592
#145274652
#145274412
C17×#32302
#1940517