Finite Group Databasebeta

← #17959136

Subgroups — #17959136

Order 4352.

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

normal subgroups only

This group has 448 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#14544483C211yesmaximal, characteristic
2#14544017C211yesmaximal, characteristic
3#14543781C214yesmaximal
4#14534606C211yesmaximal, characteristic
5C2×#10587677—21no—
6#11763825—21no—
7#11763687—21no—
8#11763687C2212yes—
9#11763655—21no—
10#11763653C2212yes—
11C2×#10587558C2211yescharacteristic
12#11762747C412yescomplemented
13#11762744C2212yes—
14#11762728—21no—
15#11762720—21no—
16#11762705—21no—
17#11762675C411yescharacteristic, complemented
18#11762666—21no—
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×C4×D17D411yescharacteristic
22C2×C22⋊#595—22no—
23#10587677—24no—
24#10587677—42no—
25C2×C4⋊#595—41no—
26C2×C4×#595—41no—
27C17×C4⋊Q8—41no—
28C17×C4⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—22no—
33C2×#63227D411yescharacteristic
34C2×#63227C2311yescharacteristic
35C2×C4⋊Dic17—21no—
36C2×#63225—22no—
37C2×#63225C2×C411yescharacteristic
38C2×#63225D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40#10587558—41no—
41#10587558—41no—
42C22⋊#3624—22no—
43C4⋊#3624—41no—
44C4×#3624—41no—
45#10587502—21no—
46#10587502—22no—
47#10587502—22no—
48#10587502C2×C412yes—
49C23×D17C2×D411yescharacteristic
50C22×#595—42no—
51C2×Q8×C17—42no—
52C2×Q8×C17—82no—
53C22×C4×C17—22no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17C22⋊C411yescharacteristic
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C2×D411yescharacteristic
61C2×C4×D17—21no—
62C2×C4×D17—21no—
63C2×C4×D17—22no—
64C2×C4×D17SD1612yes—
65C2×C4×D17Q1612yes—
66C22⋊#595—24no—
67C22⋊#595—42no—
68C2×#3624—42no—
69C4⋊#595—44no—
70C4×#595—44no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C42×C17—21no—
75C42×C17—21no—
76#63227—41no—
77#63227—41no—
78#63227—82no—
79C4⋊Dic17—42no—
80#63225—41no—
81#63225—41no—
82#63225—82no—
83C4×Dic17—21no—
84C4×Dic17—21no—
85#11718—171nomaximal, Sylow
86C23×C17C2×C22⋊C411yescharacteristic
87C22×D17—21no—
88C22×D17—21no—
89C22×D17—41no—
90C22×D17—41no—
91C22×D17C2×SD1611yescharacteristic
92C22×D17C2×Q1611yescharacteristic
93C2×#595—44no—
94C2×#595—82no—
95C2×#595—82no—
96Q8×C17—84no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17#9911yescharacteristic
105C2×C4×C17#9911yescharacteristic
106C2×C4×C17#10312yes—
107C2×C4×C17#10412yes—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—41no—
113C2×Dic17—41no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17#10312yes—
118C4×D17—22no—
119C4×D17—22no—
120C4×D17—22no—
121C4×D17—22no—
122#3624—82no—
123#2628—171no—
124#1740—171no—
125#1458—171no—
126#1352—174no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17C2×#9911yescharacteristic
130C22×C17C2×#10311yescharacteristic
131C22×C17C2×#10411yescharacteristic
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—41no—
137C2×D17—41no—
138C2×D17—81no—
139#595—84no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—82no—
143C4×C17#32812yes—
144C4×C17#32812yes—
145Dic17—22no—
146Dic17—41no—
147Dic17—41no—
148Dic17—82no—
149C2×C4⋊Q8#59511yescharacteristic, complemented
150C2×#117—171no—
151C2×#117—341no—
152#431—341no—
153#400—172no—
154#399—172no—
155#395—341no—
156#393—341no—
157#389—341no—
158#382—171no—
159#380—341no—
160#342—172no—
161#342—341no—
162#324—341no—
163#323—172no—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17C2×#32811yescharacteristic
167C2×C17#212311yescharacteristic
168C2×C17#212811yescharacteristic
169D17—41no—
170D17—41no—
171C22×Q8—21no—
172C23×C4—171no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175#117—344no—
176#117—681no—
177#117—681no—
178#117—682no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—171no—
181C2×C4⋊C4—171no—
182C2×C4⋊C4—341no—
183C2×C4⋊C4—342no—
184C2×C4⋊C4—681no—
185C2×C4⋊C4C2×#59512yes—
186C2×C22⋊C4—171no—
187C2×C22⋊C4—171no—
188C2×C22⋊C4—342no—
189C2×C22⋊C4—342no—
190C2×C42—171no—
191C2×C42—681no—
192C2×C42C2×#59511yescharacteristic
193C4⋊C8—681no—
194C22⋊C8—342no—
195C4×C8—681no—
196#95—172no—
197#95—342no—
198#95—342no—
199#95—341no—
200C17#1171811yescharacteristic, complemented, Sylow
201C24—171no—
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—22no—
205C22×C4—171no—
206C22×C4—172no—
207C22×C4—172no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—342no—
211C22×C4—341no—
212C22×C4—341no—
213C22×C4—342no—
214C22×C4—682no—
215C22×C4C22⋊#59511yescharacteristic
216C22×C4C22⋊#59511yescharacteristic
217C22×C4C22×#59511yescharacteristic
218C2×C8—682no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—681no—
223C4⋊C4—681no—
224C4⋊C4—684no—
225C4⋊C4—682no—
226C4⋊C4—1362no—
227C22⋊C4—344no—
228C22⋊C4—681no—
229C22⋊C4—681no—
230C22⋊C4—682no—
231C22⋊C4—1362no—
232C42—21no—
233C42—21no—
234C42—341no—
235C42—341no—
236C42—684no—
237C23—171no—
238C23—171no—
239C23—341no—
240C23—341no—
241C23—681no—
242C23—681no—
243C23C2×C22⋊#59511yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—172no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—341no—
260C2×C4—341no—
261C2×C4—681no—
262C2×C4—681no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—684no—
266C2×C4—682no—
267C2×C4—1362no—
268C2×C4—1362no—
269C2×C4#1058754912yes—
270C2×C4#1058755012yes—
271C2×C4#1058755111yescharacteristic
272C2×C4#1058755111yescharacteristic
273C8—1362no—
274C22—21no—
275C22—21no—
276C22—341no—
277C22—341no—
278C22—681no—
279C22—681no—
280C22—681no—
281C22—681no—
282C22—1361no—
283C22C2×#1058755111yescharacteristic
284C22C2×#1058754911yescharacteristic
285C22C2×#1058755011yescharacteristic
286C4—41no—
287C4—41no—
288C4—82no—
289C4—342no—
290C4—681no—
291C4—681no—
292C4—1362no—
293C4—1364no—
294C4#1176366312yes—
295C4#1176366312yes—
296C2—21no—
297C2—21no—
298C2—681no—
299C2—681no—
300C2C2×#1176366311yescharacteristic
301C2#1454504611yescharacteristic
302C2#1454506311yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC23×D17C2×D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11718
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. #11762747C4semidirect
17. #11762675C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
149. C2×C4⋊Q8#595semidirect
200. C17#11718semidirect

Maximal subgroups

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

subgroupindex
#145444832
#145440172
#145437812
#145346062
#1171817