Finite Group Databasebeta

← #16927402

Subgroups — #16927402

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762933C211yesmaximal, characteristic
2#14528911C211yesmaximal, characteristic
3#14528790C211yesmaximal, characteristic
4#14528786C211yesmaximal, characteristic
5C2×#11762561C211yesmaximal, characteristic
6#14526668C211yesmaximal, characteristic
7C17×#1296C211yesmaximal, characteristic
8C2×C4×Dic34C2211yescharacteristic
9#11762933—22no—
10#11762933C414yescomplemented
11C2×C4⋊#3622C2211yescharacteristic
12C2×#10587485—21no—
13C2×#10587487—21no—
14C2×#10587481—21no—
15C2×C4×#3622C2211yescharacteristic
16C2×C8×Dic17—21no—
17#11762679C2211yescharacteristic
18#11762572—21no—
19#11762571—21no—
20#11762571—21no—
21#11762571—21no—
22#11762561—22no—
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C17×#384C2211yescharacteristic
26C22×Dic34D411yescharacteristic
27C22×C8×C17—21no—
28C22×C8×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C42×C17C2311yescharacteristic
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17D411yescharacteristic
37C2×#63225—21no—
38C2×C4×Dic17—21no—
39C22×#3622—21no—
40C22×#3622D411yescharacteristic
41C22×#3622D411yescharacteristic
42C4×Dic34—21no—
43C4×Dic34—21no—
44C4×Dic34C2×C412yes—
45C4×Dic34Q812yes—
46C17×C4⋊C8—21no—
47C17×C4⋊C8—21no—
48C17×C4⋊C8—21no—
49C17×C4⋊C8C2×C412yes—
50#10587487—42no—
51#10587485—24no—
52C8×Dic17—24no—
53#10587481—42no—
54C4⋊#3622—21no—
55C4⋊#3622—21no—
56C4×#3622C2×C412yes—
57C4×#3622D412yes—
58C22×C4×C17—41no—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17#4111yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—41no—
65C2×Dic34—21no—
66C2×Dic34—21no—
67C2×Dic34—21no—
68C2×Dic34C4⋊C412yes—
69C2×Dic34D812yes—
70C2×Dic34Q1612yes—
71C2×C8×C17—22no—
72C2×C8×C17—41no—
73C2×C8×C17—41no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
76C2×C8×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—81no—
82C42×C17C22×C411yescharacteristic
83C42×C17C2×D411yescharacteristic
84C42×C17C2×Q811yescharacteristic
85C42×C17#4111yescharacteristic
86C4⋊Dic17—21no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—81no—
92C4⋊Dic17C4⋊C412yes—
93#63225—22no—
94#63225—22no—
95C4×Dic17—22no—
96C4×Dic17—22no—
97C2×#3622—21no—
98C2×#3622—21no—
99C2×#3622—21no—
100C2×#3622—21no—
101C2×#3622—41no—
102C2×#3622—41no—
103C2×#3622—41no—
104C2×#3622C22⋊C412yes—
105C2×#3622C22⋊C412yes—
106#8649—171nomaximal, Sylow
107C23×C17#12111yescharacteristic
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—81no—
111C2×C4×C17—81no—
112C2×C4×C17—81no—
113C2×C4×C17C2×C22⋊C411yescharacteristic
114C2×C4×C17C2×C4⋊C411yescharacteristic
115C2×C4×C17#11711yescharacteristic
116C2×C4×C17#12111yescharacteristic
117C2×C4×C17C22⋊Q811yescharacteristic
118C2×C4×C17C22⋊Q811yescharacteristic
119C2×C4×C17C2×D811yescharacteristic
120C2×C4×C17C2×Q1611yescharacteristic
121C2×C4×C17#13511yescharacteristic
122C2×C4×C17#13511yescharacteristic
123C2×C4×C17#13511yescharacteristic
124C2×C4×C17#13511yescharacteristic
125C2×C4×C17#13711yescharacteristic
126C2×C4×C17#13711yescharacteristic
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—22no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—81no—
134C2×Dic17—81no—
135C2×Dic17—81no—
136Dic34—22no—
137Dic34—22no—
138Dic34—22no—
139Dic34#10714yes—
140C8×C17—42no—
141C8×C17—81no—
142#3622—22no—
143#3622—22no—
144#3622—81no—
145C2×C8×Q8—171no—
146#1834—171no—
147#1803—171no—
148#1803—171no—
149C2×#335—171no—
150#1296—171no—
151#1296D1711yescharacteristic
152C22×C17#38011yescharacteristic
153C22×C17#46411yescharacteristic
154C22×C17#46411yescharacteristic
155C22×C17#46611yescharacteristic
156C22×C17#46711yescharacteristic
157C22×C17#47111yescharacteristic
158C22×C17#47111yescharacteristic
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—161no—
162C4×C17#41611yescharacteristic
163C4×C17#41911yescharacteristic
164C4×C17C2×#10711yescharacteristic
165C4×C17#42711yescharacteristic
166C4×C17#46611yescharacteristic
167C4×C17#46711yescharacteristic
168C4×C17#47911yescharacteristic
169C4×C17#47911yescharacteristic
170Dic17—24no—
171Dic17—42no—
172Dic17—161no—
173C2×C4×Q8—171no—
174C8×Q8—174no—
175C8×Q8—342no—
176C2×C4⋊C8—171no—
177C2×C4⋊C8—341no—
178C2×C4⋊C8C2×D1711yescharacteristic
179C2×C4⋊C8C2×D1711yescharacteristic
180C2×#103—341no—
181C2×#103—341no—
182C2×C4×C8—171no—
183C2×C4×C8—341no—
184#384—171no—
185#384C2×D1711yescharacteristic
186#340—341no—
187#340—341no—
188#340—341no—
189#340—341no—
190#335—342no—
191C2×C17#168211yescharacteristic
192C2×C17#168211yescharacteristic
193C2×C17#168411yescharacteristic
194C2×C17#168411yescharacteristic
195C2×C17#180511yescharacteristic
196C2×C17#180811yescharacteristic
197C2×C17#181511yescharacteristic
198C22×Q8—171no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—171no—
202C22×C8—171no—
203C22×C8—341no—
204C4×Q8—172no—
205C4×Q8—172no—
206C4×Q8—341no—
207C4×Q8—341no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C42—341no—
217C2×C42C22×D1711yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—344no—
222C4⋊C8—341no—
223C4⋊C8—341no—
224C4⋊C8C4×D1712yes—
225#103—682no—
226#103—682no—
227C4×C8—172no—
228C4×C8—172no—
229C4×C8—344no—
230C17#864911yescharacteristic, complemented, Sylow
231C2×Q8—172no—
232C2×Q8—172no—
233C2×Q8—172no—
234C2×Q8—341no—
235C2×Q8—341no—
236C2×Q8—341no—
237C22×C4—41no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—681no—
241C22×C4#6325211yescharacteristic
242C22×C4D4×D1711yescharacteristic
243C22×C4D4×D1711yescharacteristic
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—172no—
251C2×C8—172no—
252C2×C8—341no—
253C2×C8—341no—
254C2×C8—341no—
255C2×C8—341no—
256C2×C8—681no—
257C2×C8—681no—
258C2×C8—681no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—172no—
265C4⋊C4—342no—
266C4⋊C4—341no—
267C4⋊C4—342no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—1361no—
273C42—342no—
274C42—342no—
275C42C2×C4×D1711yescharacteristic
276C42#6325211yescharacteristic
277C42D4×D1711yescharacteristic
278C42Q8×D1711yescharacteristic
279C23#1058757011yescharacteristic
280Q8—174no—
281Q8—342no—
282Q8—342no—
283Q8—342no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—342no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—1361no—
296C2×C4—1361no—
297C2×C4—1361no—
298C2×C4#1058755811yescharacteristic
299C2×C4C22⋊C4×D1711yescharacteristic
300C2×C4#1058757011yescharacteristic
301C2×C4C4⋊C4×D1711yescharacteristic
302C2×C4#1058758211yescharacteristic
303C2×C4#1058758211yescharacteristic
304C2×C4D8×D1711yescharacteristic
305C2×C4#1058759811yescharacteristic
306C2×C4#1058760111yescharacteristic
307C2×C4#1058760211yescharacteristic
308C2×C4#1058760211yescharacteristic
309C2×C4#1058760211yescharacteristic
310C2×C4Q16×D1711yescharacteristic
311C2×C4#1058760411yescharacteristic
312C8—42no—
313C8—81no—
314C8—342no—
315C8—342no—
316C8—1361no—
317C22#1176266811yescharacteristic
318C22#1176303011yescharacteristic
319C22#1176303411yescharacteristic
320C22#1176304411yescharacteristic
321C22#1176304911yescharacteristic
322C22#1176307111yescharacteristic
323C22#1176307511yescharacteristic
324C4—21no—
325C4—21no—
326C4—161no—
327C4—2721no—
328C4—344no—
329C4—682no—
330C4#1176280211yescharacteristic
331C4#1176281611yescharacteristic
332C4#107×D1711yescharacteristic
333C4#1176284411yescharacteristic
334C4#1176304411yescharacteristic
335C4#1176304911yescharacteristic
336C4#1176311511yescharacteristic
337C4#1176311911yescharacteristic
338C2#1452822711yescharacteristic
339C2#1452823111yescharacteristic
340C2#1452824111yescharacteristic
341C2#1452824511yescharacteristic
342C2#1452879811yescharacteristic
343C2#1452881211yescharacteristic
344C2#1452884211yescharacteristic
subgroupG/H
centerC23#10587570
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1296C2
socle, prime radicalC23×C17#121
cosocleC2×C42×C17C23
hypercenter#1296D17
hypocenterC17#8649
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
10. #11762933C4semidirect
230. C17#8649semidirect

Maximal subgroups

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

subgroupindex
C2×#117629332
#145289112
#145287902
#145287862
C2×#117625612
#145266682
C17×#12962
#864917