Finite Group Databasebeta

← #17899306

Subgroups — #17899306

Order 4352.

1702 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537890C214yesmaximal, complemented
2#14537886C214yesmaximal, complemented
3#14537879C218yesmaximal, complemented
4Q8×#63223C218yesmaximal, complemented
5#14537422C214yesmaximal
6#14537417C212yesmaximal
7C2×C17×#533C211yesmaximal, characteristic
8#11762940C22116yescomplemented
9#11762940C22132yescomplemented
10Q8×#3622C22116yescomplemented
11#11762870C22116yes—
12#11762870C22116yes—
13#11762866C2214yes—
14#11762866C2214yes—
15#11762862C2214yes—
16#11762862C2214yes—
17#11762833C2214yes—
18#11762833C2218yes—
19#11762773C2214yes—
20C4×#63223C2214yes—
21C4×#63223C2218yes—
22C17×#533C412yes—
23C17×#533C414yes—
24C17×#533C412yes—
25C17×#533C2214yes—
26C17×#533C2212yes—
27C17×#533C2212yes—
28C2×C17×C4⋊Q8C412yes—
29C2×C17×#125C412yes—
30C2×C17×C22⋊Q8C414yes—
31C2×C4×Q8×C17C2214yes—
32C2×C17×#117C2212yes—
33C22×C42×C17C2211yescharacteristic
34C22×Q8×C17C2×C414yes—
35C23×C4×C17C2×C412yes—
36C23×C4×C17C2311yescharacteristic
37C17×C4⋊Q8C2×C414yes—
38C17×C4⋊Q8C2×C414yes—
39C17×C4⋊Q8C2×C414yes—
40C17×C4⋊Q8C2×C414yes—
41C17×#125C2×C418yes—
42C17×#125C2×C414yes—
43C17×#125C2×C414yes—
44C17×C22⋊Q8C2×C418yes—
45C17×C22⋊Q8C2×C418yes—
46C17×C22⋊Q8C2×C418yes—
47C17×C22⋊Q8C2×C418yes—
48C4×Q8×C17C2×C418yes—
49C4×Q8×C17C2×C418yes—
50C4×Q8×C17C2318yes—
51C4×Q8×C17C2318yes—
52C17×#117C2×C414yes—
53C17×#117C2×C414yes—
54C17×#117C2314yes—
55C17×#117C2314yes—
56C2×C4⋊C4×C17C2×C414yes—
57C2×C4⋊C4×C17C2×C418yes—
58C2×C4⋊C4×C17C2×C414yes—
59C2×C22⋊C4×C17C2×C414yes—
60C2×C42×C17C2×C411yescharacteristic
61C2×C42×C17C2×C411yescharacteristic
62C2×C42×C17C2×C411yescharacteristic
63C2×C42×C17C2×C412yes—
64C2×C42×C17C2×C411yescharacteristic
65C2×C42×C17C2312yes—
66C2×C42×C17C2311yescharacteristic
67C2×C42×C17C2312yes—
68C2×C42×C17C2311yescharacteristic
69C2×C42×C17C2314yes—
70C2×#63223C2318yes—
71C22⋊#3622C2318yes—
72C22⋊#3622C2318yes—
73C4⋊#3622C2318yes—
74C4⋊#3622C2318yes—
75C4⋊#3622C23132yes—
76#10587474C2318yes—
77#10587474C23116yes—
78C4×#3622C2318yes—
79C4×#3622C23116yes—
80C24×C17C22×C411yescharacteristic
81C2×Q8×C17—48no—
82C2×Q8×C17C22×C418yes—
83C2×Q8×C17C22×C418yes—
84C22×C4×C17—24no—
85C22×C4×C17—44no—
86C22×C4×C17C22×C412yes—
87C22×C4×C17C22×C412yes—
88C22×C4×C17C22×C412yes—
89C22×C4×C17C22×C411yescharacteristic
90C22×C4×C17C22×C411yescharacteristic
91C22×C4×C17C22×C411yescharacteristic
92C22×C4×C17C22×C412yes—
93C22×C4×C17C22×C412yes—
94C22×C4×C17C22×C412yes—
95C22×C4×C17C22×C418yes—
96C22×C4×C17C2411yescharacteristic
97C22×C4×C17C2411yescharacteristic
98C22×C4×C17C2411yescharacteristic
99C4⋊C4×C17C22×C4116yes—
100C4⋊C4×C17C22×C4116yes—
101C4⋊C4×C17C22×C418yes—
102C4⋊C4×C17C22×C418yes—
103C4⋊C4×C17C22×C418yes—
104C4⋊C4×C17C22×C418yes—
105C22⋊C4×C17C22×C418yes—
106C22⋊C4×C17C22×C418yes—
107C42×C17C22×C411yescharacteristic
108C42×C17C22×C412yes—
109C42×C17C22×C411yescharacteristic
110C42×C17C22×C411yescharacteristic
111C42×C17C22×C412yes—
112C42×C17C22×C411yescharacteristic
113C42×C17C22×C412yes—
114C42×C17C22×C412yes—
115C42×C17C22×C418yes—
116C42×C17C2412yes—
117C42×C17C2418yes—
118C42×C17C2412yes—
119#63223—216no—
120C2×#3622C24116yes—
121#60249—171nomaximal, Sylow
122C23×C17—22no—
123C23×C17—42no—
124C23×C17C23×C411yescharacteristic
125C23×C17C23×C411yescharacteristic
126C23×C17C23×C411yescharacteristic
127Q8×C17—216no—
128Q8×C17—48no—
129C2×C4×C17—22no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—24no—
133C2×C4×C17—24no—
134C2×C4×C17—24no—
135C2×C4×C17—22no—
136C2×C4×C17—22no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17—48no—
141C2×C4×C17C23×C411yescharacteristic
142C2×C4×C17C23×C411yescharacteristic
143C2×C4×C17C23×C412yes—
144C2×C4×C17C23×C412yes—
145C2×C4×C17C23×C412yes—
146C2×C4×C17C23×C4116yes—
147C2×C4×C17C23×C411yescharacteristic
148C2×C4×C17C23×C412yes—
149C2×C4×C172−1+414yes—
150C2×C4×C17C2511yescharacteristic
151#3622—216no—
152C2×#533D1711yescharacteristic
153#2987—174no—
154#2986—174no—
155#2985—178no—
156Q8×M16—178no—
157#2921—174no—
158#2920—172no—
159C22×C17—22no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—44no—
165C22×C17#51912yes—
166C22×C17C24×C411yescharacteristic
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—22no—
170C4×C17—22no—
171C4×C17—22no—
172C4×C17—216no—
173C4×C17#51912yes—
174C4×C17#56814yes—
175C4×C17C2×2−1+412yes—
176#533Dic1712yes—
177#533Dic1714yes—
178#533Dic1712yes—
179#533C2×D1714yes—
180#533C2×D1712yes—
181#533C2×D1712yes—
182C2×C4⋊Q8Dic1712yes—
183C2×#125Dic1712yes—
184C2×C22⋊Q8Dic1714yes—
185C2×C4×Q8C2×D1714yes—
186C2×#117C2×D1712yes—
187C22×C42C2×D1711yescharacteristic
188#446—1716no—
189#446—1732no—
190C8×Q8—1716no—
191#433—1716no—
192#433—1716no—
193#432—174no—
194#432—174no—
195#431—174no—
196#431—174no—
197#423—174no—
198#423—178no—
199#407—174no—
200C4×M16—174no—
201C4×M16—178no—
202C2×C17—22no—
203C2×C17—41no—
204C2×C17—41no—
205C2×C17C2×#51911yescharacteristic
206C2×C17C2×#56812yes—
207C22×Q8C2×Dic1714yes—
208C23×C4C2×Dic1712yes—
209C23×C4C22×D1711yescharacteristic
210C2×M16—178no—
211C4⋊Q8C2×Dic1714yes—
212C4⋊Q8C2×Dic1714yes—
213C4⋊Q8C2×Dic1714yes—
214C4⋊Q8C2×Dic1714yes—
215#125C2×Dic1718yes—
216#125C2×Dic1714yes—
217#125C2×Dic1714yes—
218C22⋊Q8C2×Dic1718yes—
219C22⋊Q8C2×Dic1718yes—
220C22⋊Q8C2×Dic1718yes—
221C22⋊Q8C2×Dic1718yes—
222C4×Q8C2×Dic1718yes—
223C4×Q8C2×Dic1718yes—
224C4×Q8C22×D1718yes—
225C4×Q8C22×D1718yes—
226#117C2×Dic1714yes—
227#117C2×Dic1714yes—
228#117C22×D1714yes—
229#117C22×D1714yes—
230C2×C4⋊C4C2×Dic1714yes—
231C2×C4⋊C4C2×Dic1718yes—
232C2×C4⋊C4C2×Dic1714yes—
233C2×C22⋊C4C2×Dic1714yes—
234C2×C42C2×Dic1711yescharacteristic
235C2×C42C2×Dic1711yescharacteristic
236C2×C42C2×Dic1711yescharacteristic
237C2×C42C2×Dic1712yes—
238C2×C42C2×Dic1711yescharacteristic
239C2×C42C22×D1714yes—
240C2×C42C22×D1712yes—
241C2×C42C22×D1711yescharacteristic
242C2×C42C22×D1712yes—
243C2×C42C22×D1711yescharacteristic
244C4⋊C8—178no—
245C4⋊C8—178no—
246C4⋊C8—1732no—
247C22⋊C8—178no—
248C22⋊C8—178no—
249#97—178no—
250#97—1716no—
251C4×C8—178no—
252C4×C8—1716no—
253C17#6024911yescharacteristic, complemented, Sylow
254C24C22×Dic1711yescharacteristic
255C2×Q8—48no—
256C2×Q8C22×Dic1718yes—
257C2×Q8C22×Dic1718yes—
258C22×C4—24no—
259C22×C4—44no—
260C22×C4C22×Dic1711yescharacteristic
261C22×C4C22×Dic1711yescharacteristic
262C22×C4C22×Dic1711yescharacteristic
263C22×C4C22×Dic1712yes—
264C22×C4C22×Dic1712yes—
265C22×C4C22×Dic1712yes—
266C22×C4C22×Dic1712yes—
267C22×C4C22×Dic1712yes—
268C22×C4C22×Dic1712yes—
269C22×C4C22×Dic1718yes—
270C22×C4C23×D1711yescharacteristic
271C22×C4C23×D1711yescharacteristic
272C22×C4C23×D1711yescharacteristic
273M16—3416no—
274C2×C8—1716no—
275C4⋊C4C22×Dic1718yes—
276C4⋊C4C22×Dic1718yes—
277C4⋊C4C22×Dic17116yes—
278C4⋊C4C22×Dic17116yes—
279C4⋊C4C22×Dic1718yes—
280C4⋊C4C22×Dic1718yes—
281C22⋊C4C22×Dic1718yes—
282C22⋊C4C22×Dic1718yes—
283C42C22×Dic1712yes—
284C42C22×Dic1712yes—
285C42C22×Dic1711yescharacteristic
286C42C22×Dic1712yes—
287C42C22×Dic1711yescharacteristic
288C42C22×Dic1711yescharacteristic
289C42C22×Dic1712yes—
290C42C22×Dic1711yescharacteristic
291C42C22×Dic1718yes—
292C42C23×D1712yes—
293C42C23×D1718yes—
294C42C23×D1712yes—
295C23—22no—
296C23—42no—
297C23C23×Dic1711yescharacteristic
298C23C23×Dic1711yescharacteristic
299C23C23×Dic1711yescharacteristic
300Q8—216no—
301Q8—48no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—48no—
314C2×C4#1058769414yes—
315C2×C4C23×Dic1711yescharacteristic
316C2×C4C23×Dic1711yescharacteristic
317C2×C4C23×Dic1712yes—
318C2×C4C23×Dic1712yes—
319C2×C4C23×Dic1712yes—
320C2×C4C23×Dic17116yes—
321C2×C4C23×Dic1711yescharacteristic
322C2×C4C23×Dic1712yes—
323C2×C4C24×D1711yescharacteristic
324C8—3416no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—44no—
331C22#1176328712yes—
332C22C24×Dic1711yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—216no—
337C4—21no—
338C4—21no—
339C4#1176328712yes—
340C4#1176357614yes—
341C4C2×#1058769412yes—
342C2—22no—
343C2—41no—
344C2—41no—
345C2C2×#1176328711yescharacteristic
346C2C2×#1176357612yes—
subgroupG/H
center, FrattiniC2×C4C24×D17
commutator, socleC22×C17C24×C4
FittingC2×C17×#533C2
prime radicalC24×C17C22×C4
cosocleC2×C4×C17C25
hypercenterC2×#533D17
hypocenterC17#60249
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. #14537890C2semidirect
2. #14537886C2semidirect
3. #14537879C2semidirect
4. Q8×#63223C2semidirect
8. #11762940C22semidirect
9. #11762940C22semidirect
10. Q8×#3622C22semidirect
253. C17#60249semidirect

Maximal subgroups

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

subgroupindex
#145378902
#145378862
#145378792
Q8×#632232
#145374222
#145374172
C2×C17×#5332
#6024917