Finite Group Databasebeta

← #17952555

Subgroups — #17952555

Order 4352.

814 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 120 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#14544511C211yesmaximal, characteristic
2#14544511C211yesmaximal, characteristic
3C2×#11763666C214yesmaximal, complemented
4C2×#11763364C211yesmaximal, characteristic
5C2×#10587543—21no—
6C2×#10587543—21no—
7C2×#10587542C2212yes—
8C2×#10587542C2212yes—
9#11763675—21no—
10#11763675—21no—
11#11763666—28no—
12#11763364—28no—
13C2×#10587555C411yescharacteristic
14C2×#10587576C2212yes—
15C2×#10587583—21no—
16C2×#10587632—21no—
17C2×Q8×Dic17—21no—
18C2×#10587573—21no—
19C2×#10587578C412yes—
20C2×C42×D17C2211yescharacteristic
21C2×C17×C4⋊Q8C411yescharacteristic
22C22×Q8×C17—21no—
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×Dic34—21no—
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—22no—
38C2×C4⋊Dic17C2×C412yes—
39C2×#63225—22no—
40C2×C4×Dic17—22no—
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44#10587583—28no—
45#10587578—24no—
46#10587578—24no—
47#10587576—24no—
48#10587576D418yes—
49#10587573—28no—
50C42×D17—24no—
51#10587555—24no—
52#10587543—24no—
53#10587543—24no—
54#10587542—24no—
55#10587542—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×D17—41no—
77C2×Dic34—24no—
78C2×Dic34—42no—
79C2×Dic34—42no—
80C2×#3624—24no—
81C2×#3624—24no—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C2×#3624—42no—
86C4⋊C4×C17—24no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17C22⋊C414yes—
89C42×C17—22no—
90#63227—28no—
91C4⋊Dic17—24no—
92C4⋊Dic17—22no—
93C4⋊Dic17—28no—
94C4⋊Dic17C22⋊C414yes—
95#63225—28no—
96C4×Dic17—28no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99C4×Dic17C2×D414yes—
100#8338—171nomaximal, Sylow
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17#10112yes—
109Q8×C17—48no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—24no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17#10412yes—
120C2×C4×C17#10412yes—
121C2×C4×C17C2×C22⋊C412yes—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—24no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17#10112yes—
132C2×Dic17#12012yes—
133C2×Dic17#12012yes—
134C4×D17—28no—
135C4×D17—44no—
136C4×D17—44no—
137Dic34—48no—
138#3624—44no—
139#3624—44no—
140C2×#533—171no—
141#1749—171no—
142#1749—171no—
143C2×#330—174no—
144C22×C17#37911yescharacteristic
145C22×C17C2×#10111yescharacteristic
146C22×C17C2×#10111yescharacteristic
147C22×C17C2×#10411yescharacteristic
148C22×C17C2×#10411yescharacteristic
149C22×C17#42111yescharacteristic
150C22×C17#42111yescharacteristic
151C2×D17—22no—
152C2×D17—22no—
153C2×D17—22no—
154C2×D17—41no—
155C2×D17—41no—
156C2×D17—41no—
157C2×D17—41no—
158C4×C17—22no—
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—44no—
162Dic17—22no—
163Dic17—22no—
164Dic17—22no—
165Dic17—44no—
166#533—348no—
167C2×C4⋊Q8—171no—
168C2×C4⋊Q8#59511yescharacteristic
169C2×#125—172no—
170C2×C22⋊Q8—341no—
171C2×C22⋊Q8—341no—
172C2×C4×Q8—341no—
173C2×C4×Q8—341no—
174C2×#117—172no—
175C22×C42—171no—
176C2×C22⋊C8—341no—
177C2×C22⋊C8—341no—
178C2×#97—172no—
179C2×#97—172no—
180#336—341no—
181#336—341no—
182#330—348no—
183C2×C17#152611yescharacteristic
184C2×C17#153111yescharacteristic
185C2×C17#153111yescharacteristic
186C2×C17#153711yescharacteristic
187C2×C17#178411yescharacteristic
188C2×C17#178911yescharacteristic
189C2×C17#178911yescharacteristic
190D17—42no—
191C22×Q8—21no—
192C22×Q8—341no—
193C23×C4—171no—
194C23×C4—171no—
195C23×C4—171no—
196C22×C8—342no—
197C22×C8—342no—
198C4⋊Q8—24no—
199C4⋊Q8—344no—
200#125—178no—
201#125—344no—
202C22⋊Q8—348no—
203C22⋊Q8—348no—
204C4×Q8—348no—
205C4×Q8—348no—
206#117—344no—
207#117—344no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—172no—
210C2×C4⋊C4—342no—
211C2×C4⋊C4—342no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4C2×#59512yes—
214C2×C22⋊C4—342no—
215C2×C42—171no—
216C2×C42—171no—
217C2×C42—171no—
218C2×C42—344no—
219C2×C42—342no—
220C2×C42C2×#59511yescharacteristic
221C22⋊C8—344no—
222C22⋊C8—344no—
223#97—344no—
224#97—344no—
225C17#833811yescharacteristic, complemented, Sylow
226C24—171no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—344no—
231C2×Q8—682no—
232C2×Q8—682no—
233C22×C4—22no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—342no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—681no—
246C22×C4—681no—
247C22×C4C22⋊#59511yescharacteristic
248C22×C4C22⋊#59511yescharacteristic
249C22×C4C22×#59511yescharacteristic
250C2×C8—344no—
251C2×C8—344no—
252C2×C8—682no—
253C2×C8—682no—
254C2×C8—682no—
255C2×C8—682no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—174no—
259C4⋊C4—348no—
260C4⋊C4—348no—
261C4⋊C4—344no—
262C4⋊C4—342no—
263C4⋊C4C22⋊#59514yes—
264C22⋊C4—348no—
265C42—22no—
266C42—174no—
267C42—342no—
268C42—342no—
269C42—348no—
270C23—172no—
271C23—341no—
272C23—341no—
273C23—341no—
274C23—341no—
275C23—341no—
276C23—341no—
277C23C2×C22⋊#59511yescharacteristic
278Q8—48no—
279Q8—688no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—172no—
290C2×C4—172no—
291C2×C4—172no—
292C2×C4—348no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—344no—
300C2×C4—684no—
301C2×C4—684no—
302C2×C4—682no—
303C2×C4—682no—
304C2×C4#1058754812yes—
305C2×C4#1058755012yes—
306C2×C4C2×C22⋊#59512yes—
307C8—684no—
308C8—684no—
309C22—342no—
310C22—342no—
311C22—342no—
312C22—681no—
313C22—681no—
314C22—681no—
315C22—681no—
316C22#1176375011yescharacteristic
317C22D17⋊#10111yescharacteristic
318C22D17⋊#10111yescharacteristic
319C22C2×#1058754811yescharacteristic
320C22C2×#1058755011yescharacteristic
321C22#1176380711yescharacteristic
322C22#1176380811yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—342no—
328C4—342no—
329C4—342no—
330C4—684no—
331C2—682no—
332C2#1454424211yescharacteristic
333C2#1454425711yescharacteristic
334C2#1454425811yescharacteristic
335C2#1454427611yescharacteristic
336C2#1454461911yescharacteristic
337C2#1454462511yescharacteristic
338C2#1454462711yescharacteristic
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#8338
LCSG ▷ C22×C4×C17 ▷ C22×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×#11763666C2semidirect
225. C17#8338semidirect

Maximal subgroups

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

subgroupindex
#145445112
C2×#117636662
C2×#117633642
#833817