Finite Group Databasebeta

← #17953573

Subgroups — #17953573

Order 4352.

588 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 110 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#14544676C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C214yesmaximal
4#14531189C211yesmaximal, characteristic
5C2×#10587543C2212yes—
6C2×(C2×#595)⋊C4—21no—
7C2×(C2×#595)⋊C4—21no—
8C2×(C2×#595)⋊C4C2212yes—
9#11763675—21no—
10#11763675—21no—
11C2×#10587578—21no—
12C2×C42×D17C2211yescharacteristic
13#11762679C411yescharacteristic, complemented
14#11762674—21no—
15#11762666—21no—
16#11762666C2212yes—
17C4⋊C4×Dic17—21no—
18C4⋊C4×Dic17C412yescomplemented
19C17×#384C411yescharacteristic, complemented
20C23×#595—22no—
21C22×C4×D17D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17C2311yescharacteristic
24C22×#3624—22no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17C2×C412yes—
28C2×C42×C17C2×C411yescharacteristic
29C2×#63227—22no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C412yes—
33C2×C4×Dic17—22no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36C2×C4×Dic17D411yescharacteristic
37#10587578—42no—
38#10587578—42no—
39C42×D17—24no—
40(C2×#595)⋊C4—24no—
41(C2×#595)⋊C4—24no—
42(C2×#595)⋊C4—42no—
43#10587543—24no—
44#10587502—22no—
45#10587502—22no—
46C23×D17C2×D411yescharacteristic
47C22×#595—24no—
48C22×#595—42no—
49C22×#595—42no—
50C22×#595—42no—
51C22×#595—42no—
52C22×#595—42no—
53C22×#595—42no—
54C22×C4×C17—22no—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—22no—
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C2×D411yescharacteristic
62C2×C4×D17—22no—
63C2×C4×D17—22no—
64C2×C4×D17—22no—
65C2×C4×D17—22no—
66C2×C4×D17—41no—
67C2×C4×D17—41no—
68C2×C4×D17—41no—
69C2×C4×D17D812yes—
70C2×C4×D17SD1612yes—
71C2×C4×D17#4114yes—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—42no—
78C42×C17—22no—
79#63227—42no—
80#63227—42no—
81C4⋊Dic17—42no—
82C4⋊Dic17—42no—
83C4⋊Dic17—42no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—22no—
87C4×Dic17—44no—
88#8716—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—21no—
94C22×D17#12312yes—
95C22×D17C2×D811yescharacteristic
96C22×D17C2×SD1611yescharacteristic
97C22×D17#13712yes—
98C2×#595—44no—
99C2×#595—44no—
100C2×#595—44no—
101C2×#595—82no—
102C2×#595—82no—
103C2×#595—82no—
104C2×#595—82no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17#10212yes—
115C2×C4×C17#10412yes—
116C2×C4×C17#11712yes—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17#10212yes—
128C2×Dic17#12312yes—
129C4×D17—24no—
130C4×D17—24no—
131C4×D17—44no—
132C4×D17—44no—
133#3624—82no—
134#2285—171no—
135#1817—171no—
136#1434—171no—
137#1294—174no—
138C22×C17#38111yescharacteristic
139C22×C17#41011yescharacteristic
140C22×C17#41011yescharacteristic
141C22×C17C2×#10211yescharacteristic
142C22×C17#41911yescharacteristic
143C22×C17C2×#10411yescharacteristic
144C22×C17#42111yescharacteristic
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—21no—
148C2×D17—21no—
149C2×D17—22no—
150C2×D17—22no—
151C2×D17—21no—
152C2×D17—21no—
153C2×D17#48012yes—
154C2×D17#48312yes—
155#595—84no—
156C4×C17—22no—
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—82no—
160Dic17—22no—
161Dic17—22no—
162Dic17—41no—
163Dic17—82no—
164C2×#117—341no—
165C22×C42—171no—
166C2×C22⋊C8—172no—
167#384—171no—
168#384#59511yescharacteristic, complemented
169#382—341no—
170#380—172no—
171#380—341no—
172C4×C4⋊C4—172no—
173C4×C4⋊C4—341no—
174C2×#95—172no—
175C2×#95—341no—
176C2×#95—341no—
177#336—341no—
178#336—341no—
179C2×C17#150211yescharacteristic
180C2×C17#150611yescharacteristic
181C2×C17#151511yescharacteristic
182C2×C17#151711yescharacteristic
183C2×C17#182011yescharacteristic
184C2×C17#182311yescharacteristic
185C2×C17#182611yescharacteristic
186D17—22no—
187D17—22no—
188C23×C4—171no—
189C23×C4—171no—
190C23×C4—171no—
191C23×C4—342no—
192C22×C8—342no—
193#117—682no—
194#117—682no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—172no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4C2×#59512yes—
201C2×C22⋊C4—342no—
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—344no—
206C2×C42—342no—
207C2×C42C2×#59511yescharacteristic
208C22⋊C8—344no—
209#95—342no—
210#95—342no—
211#95—344no—
212#95—344no—
213#95—682no—
214C17#871611yescharacteristic, complemented, Sylow
215C24—171no—
216C22×C4—22no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—172no—
220C22×C4—172no—
221C22×C4—174no—
222C22×C4—171no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—344no—
228C22×C4—342no—
229C22×C4—681no—
230C22×C4—681no—
231C22×C4—681no—
232C22×C4—682no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4—682no—
238C22×C4C22⋊#59511yescharacteristic
239C22×C4C22⋊#59511yescharacteristic
240C22×C4C22×#59511yescharacteristic
241C2×C8—682no—
242C2×C8—682no—
243C2×C8—682no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—682no—
248C4⋊C4—682no—
249C4⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4—682no—
252C42—22no—
253C42—342no—
254C42—342no—
255C42—342no—
256C42—684no—
257C23—172no—
258C23—171no—
259C23—171no—
260C23—172no—
261C23—341no—
262C23—341no—
263C23—341no—
264C23—341no—
265C23C2×C22⋊#59511yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—172no—
276C2×C4—172no—
277C2×C4—344no—
278C2×C4—344no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—684no—
287C2×C4—684no—
288C2×C4—684no—
289C2×C4—684no—
290C2×C4—684no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—1362no—
295C2×C4—1362no—
296C2×C4—1362no—
297C2×C4—1362no—
298C2×C4#1058754712yes—
299C2×C4#1058754812yes—
300C2×C4#1058767712yes—
301C8—1362no—
302C22—172no—
303C22—172no—
304C22—341no—
305C22—341no—
306C22—342no—
307C22—342no—
308C22—342no—
309C22—342no—
310C22—341no—
311C22—341no—
312C22#1176375211yescharacteristic
313C22#1176378911yescharacteristic
314C22#1176379011yescharacteristic
315C22C2×#1058754711yescharacteristic
316C22#1176380211yescharacteristic
317C22C2×#1058754811yescharacteristic
318C22#1176380811yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—82no—
323C4—342no—
324C4—342no—
325C4—681no—
326C4—1364no—
327C4—1362no—
328C2—342no—
329C2—342no—
330C2#1454416411yescharacteristic
331C2#1454417611yescharacteristic
332C2#1454420811yescharacteristic
333C2#1454421511yescharacteristic
334C2#1454468211yescharacteristic
335C2#1454468711yescharacteristic
336C2#1454469011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenter#384#595
hypocenterC17#8716
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
13. #11762679C4semidirect
18. C4⋊C4×Dic17C4semidirect
19. C17×#384C4semidirect
168. #384#595semidirect
214. C17#8716semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145439602
#145436572
#145311892
#871617