Finite Group Databasebeta

← #17959742

Subgroups — #17959742

Order 4352.

418 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 88 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#14545009C211yesmaximal, characteristic
2#14544661C211yesmaximal, characteristic
3#14543773C214yesmaximal, complemented
4C2×#11763051C211yesmaximal, characteristic
5C2×#10587529—21no—
6C2×#10587534—21no—
7#11763751—21no—
8#11763704C2212yes—
9C22⋊#63216—21no—
10#11763683C2212yes—
11C22×Dic68C411yescharacteristic, complemented
12C22×C8×D17C2211yescharacteristic
13C2×#10587583—21no—
14#11763051—21no—
15#11763051—21no—
16#11763051C412yescomplemented
17#11763051C2212yes—
18C2×#10587481—21no—
19C2×C17×#107C411yescharacteristic, complemented
20C22×C4×D17D411yescharacteristic
21C22×Dic34—21no—
22C2×C22⋊#595—41no—
23C2×C4⋊#595—22no—
24C22×C8×C17C2×C411yescharacteristic
25C2×C4⋊C4×C17—21no—
26C2×#63227—41no—
27C2×C4⋊Dic17—41no—
28C22×#3622D411yescharacteristic
29C2×Dic68—21no—
30C2×Dic68—42no—
31C2×Dic68C2×C412yes—
32C2×C8×D17—42no—
33C2×C8×D17C2×C411yescharacteristic
34C2×C8×D17D411yescharacteristic
35C2×C8×D17D411yescharacteristic
36C2×C8×D17C2311yescharacteristic
37#10587583—22no—
38#10587583—22no—
39#10587583—41no—
40#10587583—41no—
41(C2×#595)⋊C4—41no—
42C2×#63216—22no—
43#10587534—42no—
44#10587529—42no—
45C17×#107—21no—
46C17×#107C2×C412yes—
47#10587481—22no—
48#10587481—22no—
49C23×D17D811yescharacteristic
50C22×#595—42no—
51C22×C4×C17—41no—
52C22×C4×C17C22⋊C411yescharacteristic
53C22×Dic17—41no—
54C22×Dic17SD1611yescharacteristic
55C2×C4×D17—41no—
56C2×C4×D17—41no—
57C2×C4×D17C22⋊C411yescharacteristic
58C2×C4×D17D811yescharacteristic
59C2×C4×D17SD1611yescharacteristic
60C2×C4×D17C2×D411yescharacteristic
61C2×Dic34—22no—
62C2×Dic34—42no—
63C2×Dic34—42no—
64C2×Dic34—41no—
65C22⋊#595—82no—
66C4⋊#595—42no—
67C4⋊#595—42no—
68C2×C8×C17—22no—
69C2×C8×C17C22⋊C411yescharacteristic
70C2×C8×C17C22×C411yescharacteristic
71C4⋊C4×C17—22no—
72C4⋊C4×C17—41no—
73#63227—42no—
74#63227—42no—
75C4⋊Dic17—42no—
76C4⋊Dic17—42no—
77C2×#3622—41no—
78C2×#3622C22⋊C411yescharacteristic
79C2×#3622C2×D411yescharacteristic
80Dic68—42no—
81Dic68—42no—
82C8×D17—22no—
83C8×D17—22no—
84C8×D17—41no—
85C8×D17—41no—
86#63216—42no—
87#11887—171nomaximal, Sylow
88C23×C17#10211yescharacteristic
89C22×D17—41no—
90C22×D17—41no—
91C22×D17#10211yescharacteristic
92C22×D17SD3212yes—
93C22×D17Q3212yes—
94C22×D17C2×D811yescharacteristic
95C2×#595—82no—
96C2×#595—82no—
97C2×#595—82no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—42no—
101C2×C4×C17—82no—
102C2×C4×C17#10211yescharacteristic
103C2×C4×C17C2×C22⋊C411yescharacteristic
104C2×Dic17—41no—
105C2×Dic17—42no—
106C2×Dic17—82no—
107C2×Dic17#10211yescharacteristic
108C2×Dic17C2×SD1611yescharacteristic
109C4×D17—21no—
110C4×D17—21no—
111C4×D17—21no—
112C4×D17—21no—
113C4×D17—41no—
114C4×D17—41no—
115Dic34—42no—
116Dic34—42no—
117Dic34—82no—
118C8×C17—21no—
119C8×C17#9912yes—
120#3622—21no—
121#3622—21no—
122C2×#467—171no—
123C2×C22⋊C16—171no—
124#1808—171no—
125#1348—174no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17#35812yes—
129C22×C17C2×#10211yescharacteristic
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17#35812yes—
137C2×D17C2×SD3211yescharacteristic
138C2×D17C2×Q3211yescharacteristic
139#595—162no—
140C4×C17—21no—
141C4×C17—82no—
142C4×C17C2×#9911yescharacteristic
143C4×C17#42111yescharacteristic
144Dic17—21no—
145Dic17—21no—
146Dic17—82no—
147C22×Q16—171no—
148C23×C8—171no—
149C2×C22⋊Q8—341no—
150C22×C16—341no—
151#467—172no—
152#467—172no—
153#467—341no—
154#467—341no—
155C2×#107—341no—
156C2×#107#59511yescharacteristic, complemented
157C2×#103—341no—
158#380—341no—
159C22⋊C16—172no—
160C22⋊C16—341no—
161#340—172no—
162C2×C17—22no—
163C2×C17#151911yescharacteristic
164C2×C17C2×#35811yescharacteristic
165C2×C17#214511yescharacteristic
166D17—21no—
167D17—22no—
168D17—21no—
169C22×Q8—341no—
170C23×C4—171no—
171C2×Q16—172no—
172C2×Q16—341no—
173C2×Q16—682no—
174C22×C8—171no—
175C22×C8—171no—
176C22×C8—171no—
177C22×C8—171no—
178C22×C8—171no—
179C22×C8—682no—
180C22×C8C2×#59511yescharacteristic
181C22⋊Q8—342no—
182C22⋊Q8—342no—
183C22⋊Q8—681no—
184C22⋊Q8—681no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—342no—
187C2×C4⋊C4—681no—
188C2×C22⋊C4—681no—
189C2×C22⋊C4—681no—
190C2×C16—342no—
191C2×C16—682no—
192#107—21no—
193#107—682no—
194#107C2×#59512yes—
195#103—342no—
196#103—342no—
197#95—681no—
198C17#1188711yescharacteristic, complemented, Sylow
199C24—171no—
200C2×Q8—342no—
201C2×Q8—682no—
202C2×Q8—682no—
203C2×Q8—681no—
204C22×C4—41no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—681no—
211C22×C4—681no—
212C22×C4—681no—
213C22×C4—682no—
214C22×C4C22⋊#59511yescharacteristic
215Q16—682no—
216Q16—682no—
217C2×C8—22no—
218C2×C8—171no—
219C2×C8—171no—
220C2×C8—342no—
221C2×C8—342no—
222C2×C8—681no—
223C2×C8—681no—
224C2×C8—681no—
225C2×C8C22⋊#59511yescharacteristic
226C2×C8C22×#59511yescharacteristic
227C4⋊C4—22no—
228C4⋊C4—41no—
229C4⋊C4—682no—
230C4⋊C4—682no—
231C4⋊C4—682no—
232C4⋊C4—682no—
233C22⋊C4—682no—
234C22⋊C4—682no—
235C22⋊C4—1362no—
236C16—682no—
237C23—171no—
238C23—171no—
239C23—172no—
240C23—172no—
241C23—681no—
242C23—681no—
243C23#1058754711yescharacteristic
244Q8—682no—
245Q8—682no—
246Q8—1362no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—171no—
252C2×C4—171no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—341no—
256C2×C4—341no—
257C2×C4—681no—
258C2×C4—681no—
259C2×C4—681no—
260C2×C4—682no—
261C2×C4—1362no—
262C2×C4—1362no—
263C2×C4—1362no—
264C2×C4—1362no—
265C2×C4#1058754711yescharacteristic
266C2×C4C2×C22⋊#59511yescharacteristic
267C8—21no—
268C8—341no—
269C8—341no—
270C8#1058753912yes—
271C22—21no—
272C22—21no—
273C22—171no—
274C22—172no—
275C22—171no—
276C22—341no—
277C22—341no—
278C22—341no—
279C22—341no—
280C22—682no—
281C22—682no—
282C22#1176372212yes—
283C22C2×#1058754711yescharacteristic
284C4—21no—
285C4—82no—
286C4—2722no—
287C4—341no—
288C4—341no—
289C4—1362no—
290C4C2×#1058753911yescharacteristic
291C4#1176380811yescharacteristic
292C2—22no—
293C2—341no—
294C2—342no—
295C2—341no—
296C2#1454422111yescharacteristic
297C2C2×#1176372211yescharacteristic
298C2#1454510511yescharacteristic
subgroupG/H
centerC22C2×#10587547
commutatorC2×C8×C17C22×C4
FrattiniC2×C8C22×#595
FittingC2×C17×#107C4
socleC22×C17C2×#102
prime radicalC23×D17D8
cosocleC2×C8×D17C23
hypercenterC2×#107#595
hypocenterC17#11887
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
3. #14543773C2semidirect
11. C22×Dic68C4semidirect
16. #11763051C4semidirect
19. C2×C17×#107C4semidirect
156. C2×#107#595semidirect
198. C17#11887semidirect

Maximal subgroups

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

subgroupindex
#145450092
#145446612
#145437732
C2×#117630512
#1188717