Finite Group Databasebeta

← #17953577

Subgroups — #17953577

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—44no—
38C42×D17—24no—
39(C2×#595)⋊C4—24no—
40(C2×#595)⋊C4—24no—
41(C2×#595)⋊C4—42no—
42#10587543—24no—
43#10587502—22no—
44#10587502—22no—
45C23×D17C2×D411yescharacteristic
46C22×#595—24no—
47C22×#595—42no—
48C22×#595—42no—
49C22×#595—42no—
50C22×#595—42no—
51C22×#595—42no—
52C22×#595—42no—
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—22no—
62C2×C4×D17—22no—
63C2×C4×D17—22no—
64C2×C4×D17—22no—
65C2×C4×D17—41no—
66C2×C4×D17—41no—
67C2×C4×D17—41no—
68C2×C4×D17SD1612yes—
69C2×C4×D17Q1612yes—
70C2×C4×D17#4114yes—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—42no—
77C42×C17—22no—
78#63227—44no—
79C4⋊Dic17—42no—
80C4⋊Dic17—42no—
81C4⋊Dic17—42no—
82C4×Dic17—22no—
83C4×Dic17—22no—
84C4×Dic17—22no—
85C4×Dic17—44no—
86#8717—171nomaximal, Sylow
87C23×C17C2×C22⋊C411yescharacteristic
88C22×D17—21no—
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17#12312yes—
93C22×D17C2×SD1611yescharacteristic
94C22×D17C2×Q1611yescharacteristic
95C22×D17AGL(1,ℤ/8)12yes—
96C2×#595—44no—
97C2×#595—44no—
98C2×#595—44no—
99C2×#595—82no—
100C2×#595—82no—
101C2×#595—82no—
102C2×#595—82no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17#10312yes—
113C2×C4×C17#10412yes—
114C2×C4×C17#11712yes—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17#10312yes—
126C2×Dic17#12312yes—
127C4×D17—24no—
128C4×D17—24no—
129C4×D17—44no—
130C4×D17—44no—
131#3624—82no—
132#2285—171no—
133#1817—171no—
134#1434—171no—
135#1294—174no—
136C22×C17#38111yescharacteristic
137C22×C17#41011yescharacteristic
138C22×C17#41011yescharacteristic
139C22×C17C2×#10311yescharacteristic
140C22×C17#41811yescharacteristic
141C22×C17C2×#10411yescharacteristic
142C22×C17#42111yescharacteristic
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—21no—
146C2×D17—21no—
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—21no—
150C2×D17—21no—
151C2×D17#48112yes—
152C2×D17#48412yes—
153#595—84no—
154C4×C17—22no—
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—82no—
158Dic17—22no—
159Dic17—22no—
160Dic17—41no—
161Dic17—82no—
162C2×#117—341no—
163C22×C42—171no—
164C2×C22⋊C8—172no—
165#384—171no—
166#384#59511yescharacteristic, complemented
167#382—341no—
168#380—172no—
169#380—341no—
170C4×C4⋊C4—172no—
171C4×C4⋊C4—341no—
172C2×#95—172no—
173C2×#95—341no—
174C2×#95—341no—
175#336—341no—
176#336—341no—
177C2×C17#150311yescharacteristic
178C2×C17#150511yescharacteristic
179C2×C17#151411yescharacteristic
180C2×C17#151811yescharacteristic
181C2×C17#182011yescharacteristic
182C2×C17#182411yescharacteristic
183C2×C17#182611yescharacteristic
184D17—22no—
185D17—22no—
186C23×C4—171no—
187C23×C4—171no—
188C23×C4—171no—
189C23×C4—342no—
190C22×C8—342no—
191#117—684no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—172no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4C2×#59512yes—
198C2×C22⋊C4—342no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—344no—
203C2×C42—342no—
204C2×C42C2×#59511yescharacteristic
205C22⋊C8—344no—
206#95—342no—
207#95—342no—
208#95—344no—
209#95—344no—
210#95—682no—
211C17#871711yescharacteristic, complemented, Sylow
212C24—171no—
213C22×C4—22no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—172no—
217C22×C4—172no—
218C22×C4—174no—
219C22×C4—171no—
220C22×C4—342no—
221C22×C4—342no—
222C22×C4—342no—
223C22×C4—342no—
224C22×C4—344no—
225C22×C4—342no—
226C22×C4—681no—
227C22×C4—681no—
228C22×C4—681no—
229C22×C4—682no—
230C22×C4—682no—
231C22×C4—682no—
232C22×C4—682no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4C22⋊#59511yescharacteristic
236C22×C4C22⋊#59511yescharacteristic
237C22×C4C22×#59511yescharacteristic
238C2×C8—682no—
239C2×C8—682no—
240C2×C8—682no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—682no—
245C4⋊C4—682no—
246C4⋊C4—682no—
247C22⋊C4—684no—
248C42—22no—
249C42—342no—
250C42—342no—
251C42—342no—
252C42—684no—
253C23—172no—
254C23—171no—
255C23—171no—
256C23—172no—
257C23—341no—
258C23—341no—
259C23—341no—
260C23—341no—
261C23C2×C22⋊#59511yescharacteristic
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—172no—
272C2×C4—172no—
273C2×C4—344no—
274C2×C4—344no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—684no—
283C2×C4—684no—
284C2×C4—684no—
285C2×C4—684no—
286C2×C4—684no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—1362no—
291C2×C4—1362no—
292C2×C4—1362no—
293C2×C4—1362no—
294C2×C4#1058754912yes—
295C2×C4#1058755012yes—
296C2×C4#1058767712yes—
297C8—1362no—
298C22—172no—
299C22—172no—
300C22—341no—
301C22—341no—
302C22—342no—
303C22—342no—
304C22—342no—
305C22—342no—
306C22—341no—
307C22—341no—
308C22#1176375211yescharacteristic
309C22#1176378911yescharacteristic
310C22#1176379011yescharacteristic
311C22C2×#1058754911yescharacteristic
312C22#1176380111yescharacteristic
313C22C2×#1058755011yescharacteristic
314C22#1176380711yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—82no—
319C4—342no—
320C4—342no—
321C4—681no—
322C4—1364no—
323C4—1362no—
324C2—342no—
325C2—342no—
326C2#1454416811yescharacteristic
327C2#1454417511yescharacteristic
328C2#1454420411yescharacteristic
329C2#1454421611yescharacteristic
330C2#1454468211yescharacteristic
331C2#1454468811yescharacteristic
332C2#1454469111yescharacteristic
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#8717
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
166. #384#595semidirect
211. C17#8717semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145439602
#145436572
#145311892
#871717