Finite Group Databasebeta

← #17952481

Subgroups — #17952481

Order 4352.

836 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 146 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#14544639C211yesmaximal, characteristic
2C4×#10587543C211yesmaximal, characteristic
3C2×#11763659C214yesmaximal, complemented
4C2×#11763364C211yesmaximal, characteristic
5C2×C4⋊#3624C2212yes—
6C2×#10587543—21no—
7C2×#10587543—21no—
8C2×C4×#3624C2212yes—
9#11763675—21no—
10#11763675—21no—
11#11763659—28no—
12#11763364—28no—
13C2×#10587555C411yescharacteristic
14C2×#10587576—21no—
15C2×#10587583C412yes—
16C2×#10587632—21no—
17C2×Q8×Dic17—21no—
18C2×#10587573C2212yes—
19C2×#10587578—21no—
20C2×C42×D17C2211yescharacteristic
21C2×C17×C4⋊Q8C411yescharacteristic
22C22×Q8×C17—21no—
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×Dic34C2×C412yes—
27C22×#3624—22no—
28C22×#3624—22no—
29C17×C4⋊Q8—24no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C42×C17C2×C411yescharacteristic
33#10587632—28no—
34Q8×Dic17—28no—
35C2×#63227—22no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—22no—
39C2×#63225—22no—
40C2×C4×Dic17—22no—
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44#10587583—24no—
45#10587583—24no—
46#10587578—28no—
47#10587576—28no—
48#10587573—24no—
49#10587573D418yes—
50C42×D17—24no—
51#10587555—24no—
52#10587543—24no—
53#10587543—42no—
54C4⋊#3624—24no—
55C4×#3624—24no—
56C23×D17C22⋊C411yescharacteristic
57C2×Q8×C17—24no—
58C2×Q8×C17—42no—
59C2×Q8×C17—42no—
60C22×C4×C17—22no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—22no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—41no—
76C2×C4×D17SD1612yes—
77C2×C4×D17Q1612yes—
78C2×Dic34—24no—
79C2×Dic34—24no—
80C2×Dic34—22no—
81C2×Dic34C22⋊C414yes—
82C2×#3624—24no—
83C2×#3624—42no—
84C2×#3624—42no—
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—42no—
88C4⋊C4×C17—24no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17C22⋊C414yes—
91C42×C17—22no—
92#63227—28no—
93C4⋊Dic17—24no—
94C4⋊Dic17—24no—
95C4⋊Dic17—28no—
96#63225—28no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99C4×Dic17—28no—
100C4×Dic17C2×D414yes—
101#8314—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17—21no—
109C22×D17#10312yes—
110Q8×C17—48no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—24no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17#10312yes—
119C2×C4×C17#10412yes—
120C2×C4×C17C2×C22⋊C412yes—
121C2×C4×C17#13112yes—
122C2×C4×C17#13112yes—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—24no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17#12012yes—
132C2×Dic17#12012yes—
133C2×Dic17C2×SD1611yescharacteristic
134C2×Dic17C2×Q1611yescharacteristic
135C2×Dic17#13712yes—
136C4×D17—24no—
137C4×D17—24no—
138C4×D17—44no—
139C4×D17—44no—
140Dic34—28no—
141Dic34—44no—
142#3624—44no—
143#3624—82no—
144C2×#533—171no—
145#1797—171no—
146C4×C22⋊C8—171no—
147C2×Q8⋊C8—174no—
148C22×C17#37911yescharacteristic
149C22×C17#40811yescharacteristic
150C22×C17#40811yescharacteristic
151C22×C17C2×#10311yescharacteristic
152C22×C17#41911yescharacteristic
153C22×C17C2×#10411yescharacteristic
154C22×C17#42111yescharacteristic
155C2×D17—22no—
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—41no—
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—41no—
162C4×C17—22no—
163C4×C17—22no—
164C4×C17—44no—
165C4×C17#40814yes—
166Dic17—22no—
167Dic17—44no—
168Dic17—41no—
169Dic17#44812yes—
170Dic17#45112yes—
171#533—348no—
172C2×C4⋊Q8—171no—
173C2×C4⋊Q8#59511yescharacteristic
174C2×#125—341no—
175C2×C22⋊Q8—172no—
176C2×C22⋊Q8—341no—
177C2×C4×Q8—172no—
178C2×C4×Q8—341no—
179C2×#117—341no—
180C22×C42—171no—
181C2×C4⋊C8—172no—
182C2×C22⋊C8—341no—
183C2×C22⋊C8—341no—
184C2×C4×C8—172no—
185#336—341no—
186#336—341no—
187Q8⋊C8—348no—
188C2×C17#148111yescharacteristic
189C2×C17#148511yescharacteristic
190C2×C17#149311yescharacteristic
191C2×C17#149711yescharacteristic
192C2×C17#178011yescharacteristic
193C2×C17#178711yescharacteristic
194C2×C17#178911yescharacteristic
195D17—42no—
196C22×Q8—21no—
197C22×Q8—172no—
198C23×C4—171no—
199C23×C4—171no—
200C23×C4—171no—
201C22×C8—342no—
202C22×C8—342no—
203C4⋊Q8—24no—
204C4⋊Q8—344no—
205#125—348no—
206C22⋊Q8—344no—
207C22⋊Q8—348no—
208C22⋊Q8—344no—
209C4×Q8—178no—
210C4×Q8—344no—
211C4×Q8—348no—
212#117—348no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—342no—
215C2×C4⋊C4—342no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4C2×#59512yes—
219C2×C22⋊C4—342no—
220C2×C42—171no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—344no—
224C2×C42—342no—
225C2×C42C2×#59511yescharacteristic
226C4⋊C8—344no—
227C22⋊C8—344no—
228C22⋊C8—682no—
229C4×C8—344no—
230C17#831411yescharacteristic, complemented, Sylow
231C24—171no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—174no—
236C2×Q8—342no—
237C2×Q8—344no—
238C2×Q8—344no—
239C22×C4—22no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—171no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—342no—
252C22×C4—342no—
253C22×C4—681no—
254C22×C4C22⋊#59511yescharacteristic
255C22×C4C22⋊#59511yescharacteristic
256C22×C4C22×#59511yescharacteristic
257C2×C8—344no—
258C2×C8—682no—
259C2×C8—682no—
260C2×C8—682no—
261C2×C8—682no—
262C2×C8—682no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—348no—
266C4⋊C4—344no—
267C4⋊C4—348no—
268C4⋊C4—344no—
269C4⋊C4C22⋊#59514yes—
270C22⋊C4—348no—
271C42—22no—
272C42—174no—
273C42—342no—
274C42—342no—
275C42—348no—
276C23—172no—
277C23—341no—
278C23—341no—
279C23—341no—
280C23—341no—
281C23—341no—
282C23—341no—
283C23C2×C22⋊#59511yescharacteristic
284Q8—48no—
285Q8—348no—
286Q8—684no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—172no—
295C2×C4—171no—
296C2×C4—171no—
297C2×C4—172no—
298C2×C4—172no—
299C2×C4—344no—
300C2×C4—344no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—344no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—684no—
308C2×C4—682no—
309C2×C4—682no—
310C2×C4—684no—
311C2×C4#1058754812yes—
312C2×C4#1058754912yes—
313C2×C4#1058768012yes—
314C2×C4#1058768012yes—
315C2×C4C2×C22⋊#59512yes—
316C8—684no—
317C8—1362no—
318C22—342no—
319C22—342no—
320C22—342no—
321C22—681no—
322C22—681no—
323C22—681no—
324C22—681no—
325C22#1176375011yescharacteristic
326C22#1176378411yescharacteristic
327C22#1176378411yescharacteristic
328C22C2×#1058754911yescharacteristic
329C22#1176380211yescharacteristic
330C22C2×#1058754811yescharacteristic
331C22#1176380711yescharacteristic
332C4—22no—
333C4—22no—
334C4—44no—
335C4—172no—
336C4—172no—
337C4—342no—
338C4—681no—
339C4—684no—
340C4#1176378414yes—
341C2—682no—
342C2#1454409411yescharacteristic
343C2#1454410811yescharacteristic
344C2#1454413411yescharacteristic
345C2#1454414811yescharacteristic
346C2#1454460911yescharacteristic
347C2#1454462311yescharacteristic
348C2#1454462511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#8314
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#11763659C2semidirect
230. C17#8314semidirect

Maximal subgroups

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

subgroupindex
#145446392
C4×#105875432
C2×#117636592
C2×#117633642
#831417