Finite Group Databasebeta

← #17966504

Subgroups — #17966504

Order 4352.

610 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545484C212yesmaximal
2#14544356C214yesmaximal, complemented
3#14544352C212yesmaximal, complemented
4#14544345C212yesmaximal, complemented
5#14544342C211yesmaximal, characteristic, complemented
6#14544334C212yesmaximal, complemented
7#14544329C211yesmaximal, characteristic, complemented
8C8×#63256C211yesmaximal, characteristic
9#11763834—24no—
10#11763831C2214yes—
11#11763828C2214yes—
12#11763813C2212yes—
13#11763810C2212yes—
14#11763784—22no—
15#11763774—22no—
16#11763673C2212yes—
17#11763672C2211yescharacteristic
18#11763672C2211yescharacteristic
19#11763672C2212yes—
20#11763672C2212yes—
21#11763671C2214yes—
22#11763670C2214yes—
23C4×#63256C2211yescharacteristic
24Q8×#3622C411yescharacteristic
25C8×D68C411yescharacteristic
26C8×D68C412yes—
27#11762874C411yescharacteristic
28#11762874C412yes—
29#11762861C2211yescharacteristic
30#11762861C2212yes—
31C4×C8×D17C2211yescharacteristic
32C4×C8×D17C2212yes—
33C8×Q8×C17C411yescharacteristic
34C2×#63256D411yescharacteristic
35C2×#63243—22no—
36C2×#63242—22no—
37C4×Q8×C17C2×C411yescharacteristic
38Q8×Dic17Q811yescharacteristic
39C2×C8×D17C2×C411yescharacteristic
40C2×C8×D17C2×C412yes—
41C2×C8×D17C2×C411yescharacteristic
42C2×C8×D17C2×C412yes—
43#10587579Q812yes—
44#10587579Q811yescharacteristic
45#10587578D411yescharacteristic
46#10587578D412yes—
47C4×D68C2×C412yes—
48C4×D68C2×C411yescharacteristic
49C42×D17C2311yescharacteristic
50C42×D17C2312yes—
51#10587544—22no—
52#10587543—24no—
53#10587543—24no—
54#10587542—24no—
55C4⋊#3624—22no—
56C4⋊#3624C2312yes—
57C4⋊#3624C2312yes—
58C4⋊#3624C2314yes—
59C4×#3624—24no—
60C17×C4⋊C8C2×C411yescharacteristic
61C17×C4⋊C8C2×C412yes—
62C4×C8×C17C2×C411yescharacteristic
63C4×C8×C17C2×C412yes—
64#10587491C2×C412yes—
65#10587491C2×C412yes—
66#10587491C2×C412yes—
67C8×Dic17C2311yescharacteristic
68C8×Dic17C2312yes—
69C8×Dic17C2311yescharacteristic
70C4⋊#3622C2×C411yescharacteristic
71C4⋊#3622C2×C412yes—
72C4×#3622C2×C411yescharacteristic
73C4×#3622C2×C412yes—
74C2×Q8×C17C4⋊C411yescharacteristic
75#63256—22no—
76#63256—22no—
77C2×D68C4⋊C412yes—
78C2×D68C4⋊C411yescharacteristic
79C2×C4×D17C22×C412yes—
80C2×C4×D17C22×C411yescharacteristic
81C2×C4×D17C2×D412yes—
82C2×C4×D17C2×D411yescharacteristic
83C2×C4×D17C2×Q812yes—
84C2×C4×D17C2×Q811yescharacteristic
85C2×#3624—22no—
86C2×#3624—22no—
87C2×#3624—24no—
88#63243—44no—
89#63242—44no—
90C2×C8×C17C22×C412yes—
91C2×C8×C17C22×C411yescharacteristic
92C2×C8×C17C22×C411yescharacteristic
93C4⋊C4×C17C4⋊C412yes—
94C4⋊C4×C17C4⋊C411yescharacteristic
95C42×C17C22×C412yes—
96C42×C17C22×C411yescharacteristic
97#63227C4⋊C412yes—
98#63227C4⋊C412yes—
99#63227C4⋊C412yes—
100C4⋊Dic17C4⋊C412yes—
101C4⋊Dic17C4⋊C411yescharacteristic
102C4×Dic17C2×D412yes—
103C4×Dic17C2×D411yescharacteristic
104C4×Dic17C2×Q812yes—
105C4×Dic17C2×Q811yescharacteristic
106C4×Dic17C2411yescharacteristic
107C2×#3622C22×C411yescharacteristic
108C2×#3622C22×C412yes—
109C2×#3622C22×C411yescharacteristic
110C8×D17—22no—
111C8×D17—24no—
112C8×D17—41no—
113C8×D17—42no—
114#17829—171nomaximal, Sylow
115C22×D17C2×C4⋊C412yes—
116C22×D17C2×C4⋊C411yescharacteristic
117Q8×C17—22no—
118C2×C4×C17C2×C4⋊C412yes—
119C2×C4×C17C2×C4⋊C411yescharacteristic
120C2×C4×C17C2×C4⋊C412yes—
121C2×C4×C17C2×C4⋊C411yescharacteristic
122C2×C4×C17C23×C411yescharacteristic
123C2×Dic17C2×C4⋊C412yes—
124C2×Dic17C2×C4⋊C411yescharacteristic
125C2×Dic17C22×D411yescharacteristic
126C2×Dic17C22×Q811yescharacteristic
127D68—24no—
128D68—22no—
129C4×D17—21no—
130C4×D17—22no—
131C4×D17—21no—
132C4×D17—22no—
133C4×D17—22no—
134C4×D17—22no—
135C4×D17—41no—
136C4×D17—42no—
137C4×D17#13512yes—
138C4×D17#13512yes—
139#3624—42no—
140#3624—42no—
141#3624—44no—
142C8×C17—22no—
143C8×C17—21no—
144C8×C17#13112yes—
145#3622—22no—
146#3622—21no—
147#3622—21no—
148C8×#41—171no—
149#2925—172no—
150#1572—174no—
151#1570—172no—
152#1568—172no—
153#1567—171no—
154#1564—172no—
155C2×#335—171no—
156C22×C17C22×C4⋊C411yescharacteristic
157C2×D17—21no—
158C2×D17—21no—
159C2×D17—22no—
160C2×D17—22no—
161C2×D17—22no—
162C2×D17#42712yes—
163C4×C17—22no—
164C4×C17—21no—
165C4×C17—22no—
166C4×C17#42712yes—
167C4×C17C2×#13111yescharacteristic
168C4×C17#56811yescharacteristic
169Dic17—21no—
170Dic17—22no—
171Dic17—21no—
172Dic17C2×#13511yescharacteristic
173Dic17C2×#13511yescharacteristic
174C4×#41—171no—
175C8×Q8—171no—
176C8×Q8#59511yescharacteristic
177C8×D4—171no—
178C8×D4—172no—
179C8×D4—171no—
180C8×D4—172no—
181#433—344no—
182#432—174no—
183#431—171no—
184#431—172no—
185#431—174no—
186#423—172no—
187C2×C4⋊C8—172no—
188#408—342no—
189#405—342no—
190C2×C4×C8—171no—
191C2×C4×C8—172no—
192#335—171no—
193#335—171no—
194#335—172no—
195#335—172no—
196#335—172no—
197#334—174no—
198#334—174no—
199C2×C17#290611yescharacteristic
200C2×C17C2×#42711yescharacteristic
201C2×C17#291611yescharacteristic
202D17—22no—
203D17—42no—
204C2×#41—171no—
205C2×M16—342no—
206C22×C8—171no—
207C22×C8—172no—
208C22×C8—171no—
209C22×C8—172no—
210C22×C8—342no—
211C4×Q8—171no—
212C4×Q8C2×#59511yescharacteristic
213C4×D4—172no—
214C4×D4—171no—
215C4×D4—172no—
216C4×D4—171no—
217#117—171no—
218#117—172no—
219C2×C42—171no—
220C2×C42—172no—
221C4⋊C8—172no—
222C4⋊C8—172no—
223C4⋊C8—174no—
224C4⋊C8—171no—
225C4⋊C8—172no—
226C4⋊C8—342no—
227C4⋊C8—342no—
228C4⋊C8C2×#59511yescharacteristic
229C4⋊C8C2×#59512yes—
230C22⋊C8—172no—
231C22⋊C8—172no—
232C22⋊C8—172no—
233C22⋊C8—344no—
234C22⋊C8—344no—
235#97—344no—
236C4×C8—172no—
237C4×C8—171no—
238C4×C8—171no—
239C4×C8—171no—
240C4×C8—172no—
241C4×C8—344no—
242C4×C8C2×#59511yescharacteristic
243C4×C8C2×#59512yes—
244C17#1782911yescharacteristic, complemented, Sylow
245#41—342no—
246#41—342no—
247C2×Q8C4⋊#59511yescharacteristic
248C2×D4—172no—
249C2×D4—171no—
250C22×C4—171no—
251C22×C4—172no—
252C22×C4—171no—
253C22×C4—172no—
254C22×C4—171no—
255C22×C4—172no—
256M16—684no—
257C2×C8—171no—
258C2×C8—172no—
259C2×C8—171no—
260C2×C8—342no—
261C2×C8—344no—
262C2×C8—342no—
263C2×C8—344no—
264C2×C8—342no—
265C2×C8—681no—
266C2×C8—682no—
267C2×C8—684no—
268C2×C8C22×#59512yes—
269C2×C8C22×#59511yescharacteristic
270C2×C8C22×#59511yescharacteristic
271C4⋊C4—172no—
272C4⋊C4—171no—
273C4⋊C4C4⋊#59512yes—
274C4⋊C4C4⋊#59511yescharacteristic
275C22⋊C4—172no—
276C22⋊C4—172no—
277C22⋊C4—172no—
278C42—172no—
279C42—171no—
280C42—172no—
281C42—171no—
282C42—171no—
283C42C22×#59512yes—
284C42C22×#59511yescharacteristic
285C23—171no—
286C23—172no—
287Q8—22no—
288D4—344no—
289D4—342no—
290C2×C4—171no—
291C2×C4—172no—
292C2×C4—172no—
293C2×C4—171no—
294C2×C4—172no—
295C2×C4—171no—
296C2×C4—342no—
297C2×C4—342no—
298C2×C4—341no—
299C2×C4—342no—
300C2×C4—341no—
301C2×C4—342no—
302C2×C4—681no—
303C2×C4—682no—
304C2×C4C2×C4⋊#59512yes—
305C2×C4C2×C4⋊#59511yescharacteristic
306C2×C4C2×C4⋊#59512yes—
307C2×C4C2×C4⋊#59511yescharacteristic
308C2×C4C23×#59511yescharacteristic
309C8—22no—
310C8—21no—
311C8—342no—
312C8—341no—
313C8—341no—
314C8—682no—
315C8—682no—
316C8—684no—
317C8#1058768012yes—
318C22—172no—
319C22—341no—
320C22—341no—
321C22—342no—
322C22—342no—
323C22—342no—
324C22C22×C4⋊#59511yescharacteristic
325C4—22no—
326C4—21no—
327C4—22no—
328C4—171no—
329C4—171no—
330C4—342no—
331C4—341no—
332C4—341no—
333C4#1176381912yes—
334C4C2×#1058768011yescharacteristic
335C4#1176388811yescharacteristic
336C2—342no—
337C2—682no—
338C2#1454543011yescharacteristic
339C2C2×#1176381911yescharacteristic
340C2#1454544611yescharacteristic
subgroupG/H
centerC22C22×C4⋊#595
commutatorC2×C4×C17C23×C4
FrattiniC2×C4C23×#595
FittingC8×Q8×C17C4
socleC22×C17C22×C4⋊C4
prime radicalC2×#63256D4
cosocleC4×Dic17C24
hypercenterC8×Q8#595
hypocenterC17#17829
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
2. #14544356C2semidirect
3. #14544352C2semidirect
4. #14544345C2semidirect
5. #14544342C2semidirect
6. #14544334C2semidirect
7. #14544329C2semidirect
244. C17#17829semidirect

Maximal subgroups

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

subgroupindex
#145454842
#145443562
#145443522
#145443452
#145443422
#145443342
#145443292
C8×#632562
#1782917