Finite Group Databasebeta

← #17019687

Subgroups — #17019687

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#14539932C211yesmaximal, characteristic
2C2×#11763104C211yesmaximal, characteristic
3#14537865C211yesmaximal, characteristic
4#14528240C214yesmaximal, complemented
5#14528186C214yesmaximal, complemented
6#14527560C211yesmaximal, characteristic
7#14527466C211yesmaximal, characteristic
8#14527441C211yesmaximal, characteristic
9C17×#3078C211yesmaximal, characteristic
10C2×#10587555C2211yescharacteristic
11#11763212C2212yes—
12#11763159C2212yes—
13#11763142—22no—
14#11763104—22no—
15#11763104—22no—
16#11763104C2214yes—
17Q8×#3622—22no—
18Q8×#3622C2214yes—
19#11762862—22no—
20#11762862C2211yescharacteristic
21#11762862C2211yescharacteristic
22C2×#10587478—21no—
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587506—21no—
25#11762561C2214yes—
26#11762558C2212yes—
27#11762557C2214yes—
28#11762555C2212yes—
29C17×#533C2211yescharacteristic
30C2×C4×Q8×C17C2211yescharacteristic
31C17×#479—22no—
32C17×#477C2212yes—
33C17×#475C2212yes—
34C17×#428—21no—
35C17×#423C2211yescharacteristic
36C17×#419—21no—
37C22×Q8×C17D411yescharacteristic
38C22×Dic34—21no—
39C2×M16×C17—21no—
40C17×C4⋊Q8C2311yescharacteristic
41C17×C4⋊Q8C2311yescharacteristic
42C17×#125—21no—
43C17×C22⋊Q8—21no—
44C17×C22⋊Q8—21no—
45C4×Q8×C17—21no—
46C4×Q8×C17—21no—
47C4×Q8×C17—21no—
48C4×Q8×C17—21no—
49C4×Q8×C17D412yes—
50C4×Q8×C17C2312yes—
51C17×#117—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17D411yescharacteristic
54C2×C42×C17—21no—
55C2×C42×C17C2311yescharacteristic
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17—21no—
58C22×#3622—21no—
59#10587555—21no—
60#10587555—21no—
61#10587555—21no—
62#10587555C2312yes—
63C17×#107—22no—
64C17×#106—22no—
65C17×C4⋊C8—21no—
66C17×C4⋊C8C2312yes—
67C17×#103—22no—
68C17×#103—22no—
69#10587506—24no—
70#10587506—24no—
71C22⋊#3622—21no—
72C22⋊#3622D412yes—
73#10587478—24no—
74#10587478—24no—
75C4⋊#3622—24no—
76C4⋊#3622—21no—
77C4⋊#3622D412yes—
78C4⋊#3622C2312yes—
79C4⋊#3622C2312yes—
80C4×#3622—24no—
81C4×#3622—21no—
82C4×#3622C2312yes—
83C2×Q8×C17—21no—
84C2×Q8×C17—22no—
85C2×Q8×C17—21no—
86C2×Q8×C17—21no—
87C2×Q8×C17Q1614yes—
88C2×Q8×C17C2×D412yes—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17#4111yescharacteristic
94C22×Dic17—41no—
95C2×Dic34—22no—
96C2×Dic34—41no—
97C2×Dic34—82no—
98M16×C17—42no—
99C2×C8×C17—22no—
100C4⋊C4×C17—22no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—41no—
108C4⋊C4×C17—41no—
109C4⋊C4×C17C2×D412yes—
110C22⋊C4×C17—41no—
111C42×C17—22no—
112C42×C17—22no—
113C42×C17—41no—
114C42×C17C2×D411yescharacteristic
115C42×C17C2×D411yescharacteristic
116C42×C17#4111yescharacteristic
117C42×C17C2411yescharacteristic
118C4⋊Dic17—22no—
119C4⋊Dic17—22no—
120C4⋊Dic17—41no—
121C4⋊Dic17—41no—
122C2×#3622—22no—
123C2×#3622—22no—
124C2×#3622—42no—
125C2×#3622C2×D414yes—
126#19405—171nomaximal, Sylow
127C23×C17#12111yescharacteristic
128Q8×C17—22no—
129Q8×C17—24no—
130Q8×C17—42no—
131Q8×C17—81no—
132Q8×C17C2×Q1614yes—
133C2×C4×C17—22no—
134C2×C4×C17—24no—
135C2×C4×C17—22no—
136C2×C4×C17—22no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—81no—
141C2×C4×C17#12111yescharacteristic
142C2×C4×C17#12111yescharacteristic
143C2×C4×C17#12111yescharacteristic
144C2×C4×C17C2×Q1612yes—
145C2×C4×C17#13512yes—
146C2×C4×C17C22×D411yescharacteristic
147C2×C4×C17C22×D411yescharacteristic
148C2×C4×C17C2×#4111yescharacteristic
149C2×Dic17—42no—
150C2×Dic17—82no—
151Dic34—82no—
152Dic34—82no—
153C8×C17—42no—
154#3622—24no—
155#3622—42no—
156#3622—42no—
157#3230—171no—
158#3078D1711yescharacteristic
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—24no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—42no—
175C4×C17—81no—
176C4×C17—81no—
177C4×C17#46712yes—
178C4×C17C22×Q1611yescharacteristic
179C4×C17C2×#13511yescharacteristic
180C4×C17#57812yes—
181Dic17—82no—
182#533C2×D1711yescharacteristic
183C2×C4⋊Q8—171no—
184C2×C4×Q8C2×D1711yescharacteristic
185#500—172no—
186#487—172no—
187#484—342no—
188#479—22no—
189#477—174no—
190#477—342no—
191#477—342no—
192#477C2×D1712yes—
193#475C2×D1712yes—
194C8×Q8—174no—
195C8×Q8—342no—
196#431—171no—
197#431—171no—
198#431—342no—
199#428—21no—
200C2×#107—341no—
201#423C2×D1711yescharacteristic
202C2×C4⋊C8—171no—
203#419—21no—
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—341no—
214C22×Q8C22⋊D1711yescharacteristic
215C2×M16—21no—
216C22×C8—341no—
217C4⋊Q8—172no—
218C4⋊Q8—341no—
219C4⋊Q8—341no—
220C4⋊Q8—341no—
221C4⋊Q8C22×D1711yescharacteristic
222C4⋊Q8C22×D1711yescharacteristic
223#125—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C22⋊D1712yes—
231C4×Q8C22×D1712yes—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4C22⋊D1711yescharacteristic
237C2×C42—21no—
238C2×C42C22×D1711yescharacteristic
239#107—22no—
240#107—344no—
241#107—344no—
242#106—22no—
243C4⋊C8—21no—
244C4⋊C8—172no—
245C4⋊C8—172no—
246C4⋊C8—172no—
247C4⋊C8—344no—
248C4⋊C8—341no—
249C4⋊C8C22×D1712yes—
250#103—22no—
251#103—22no—
252#103—344no—
253#103—344no—
254C22⋊C8—172no—
255C22⋊C8—341no—
256C4×C8—172no—
257C4×C8—344no—
258C4×C8—341no—
259C17#1940511yescharacteristic, complemented, Sylow
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—342no—
265C2×Q8—681no—
266C2×Q8—1362no—
267C2×Q8#6323114yes—
268C2×Q8C2×C22⋊D1712yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—681no—
272C22×C4D4×D1711yescharacteristic
273C22×C4#6325411yescharacteristic
274C22×C4C2×C22⋊D1711yescharacteristic
275M16—42no—
276C2×C8—22no—
277C2×C8—174no—
278C2×C8—342no—
279C2×C8—342no—
280C2×C8—682no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—342no—
291C4⋊C4—342no—
292C4⋊C4—681no—
293C4⋊C4—681no—
294C4⋊C4C2×C22⋊D1712yes—
295C22⋊C4—41no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42D4×D1711yescharacteristic
300C42#6325411yescharacteristic
301C42C2×C22⋊D1711yescharacteristic
302C42C23×D1711yescharacteristic
303C23#1058762511yescharacteristic
304Q8—24no—
305Q8—22no—
306Q8—42no—
307Q8—81no—
308Q8—1362no—
309Q8—1362no—
310Q8C2×#6323114yes—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—682no—
320C2×C4—1362no—
321C2×C4#1058762511yescharacteristic
322C2×C4#1058762511yescharacteristic
323C2×C4#1058762511yescharacteristic
324C2×C4C2×#6323112yes—
325C2×C4#1058763612yes—
326C2×C4C2×D4×D1711yescharacteristic
327C2×C4C2×#6325411yescharacteristic
328C2×C4C22×C22⋊D1711yescharacteristic
329C8—42no—
330C8—344no—
331C8—682no—
332C8—682no—
333C22—21no—
334C22—21no—
335C22#1176304812yes—
336C22C2×#1058762511yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—24no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—1362no—
345C4#1176304812yes—
346C4C22×#6323111yescharacteristic
347C4C2×#1058763611yescharacteristic
348C4#1176362512yes—
349C2—22no—
350C2#1453831611yescharacteristic
351C2C2×#1176304811yescharacteristic
352C2#1453851411yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3078C2
socleC22×C17C2×#121
prime radicalC23×C17#121
hypercenter#3078D17
hypocenterC17#19405
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
4. #14528240C2semidirect
5. #14528186C2semidirect
259. C17#19405semidirect

Maximal subgroups

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

subgroupindex
#145399322
C2×#117631042
#145378652
#145282402
#145281862
#145275602
#145274662
#145274412
C17×#30782
#1940517