Finite Group Databasebeta

← #16929409

Subgroups — #16929409

Order 4352.

516 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 111 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#14537869C211yesmaximal, characteristic, complemented
2C4×#10587507C211yesmaximal, characteristic, complemented
3#14527227C212yesmaximal
4#14526677C212yesmaximal, complemented
5C17×#2398C211yesmaximal, characteristic
6#11762940—24no—
7#11762866—21no—
8#11762866C2212yes—
9#11762862—21no—
10C2×#10587507—21no—
11C4×#63223C2212yes—
12#11762578—21no—
13D4⋊#3622—24no—
14C2×C4×Q8×C17C2211yescharacteristic
15C2×C4×D4×C17C2211yescharacteristic
16C17×#400C411yescharacteristic
17C17×#400C411yescharacteristic
18C17×#397—21no—
19C17×#394C411yescharacteristic
20C17×#394C411yescharacteristic
21C17×#393—21no—
22C4×C22⋊C4×C17—21no—
23C4×C22⋊C4×C17—21no—
24C17×#376C2211yescharacteristic
25C22×Q8×C17D411yescharacteristic
26C22×D4×C17C2×C411yescharacteristic
27C23×C4×C17—21no—
28C4×Q8×C17—22no—
29C4×Q8×C17—22no—
30C4×D4×C17—22no—
31C4×D4×C17—22no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C42×C17—21no—
42C2×C42×C17D411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×C42×C17C2311yescharacteristic
45C2×#63223—22no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52C17×#95C2×C411yescharacteristic
53#10587507—24no—
54C22⋊#3622—22no—
55C4⋊#3622—22no—
56C4⋊#3622—22no—
57C4⋊#3622D414yes—
58#10587474—22no—
59#10587474—22no—
60#10587474—22no—
61C4×#3622—22no—
62C4×#3622—22no—
63C24×C17—21no—
64C2×Q8×C17—22no—
65C2×Q8×C17—22no—
66C2×Q8×C17—41no—
67C2×D4×C17—22no—
68C2×D4×C17—42no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—42no—
73C22×C4×C17—42no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C4⋊C4×C17—22no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—22no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—42no—
91C22⋊C4×C17—42no—
92C42×C17—21no—
93C42×C17—21no—
94C42×C17—21no—
95C42×C17—21no—
96C42×C17—42no—
97C42×C17C2×D412yes—
98C42×C17#4112yes—
99#63223—24no—
100#63223—42no—
101C2×#3622—24no—
102C2×#3622—42no—
103#9032—171nomaximal, Sylow
104C23×C17—22no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112Q8×C17—44no—
113D4×C17—24no—
114D4×C17—42no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—82no—
138C2×C4×C17—82no—
139C2×C4×C17#10412yes—
140C2×C4×C17C4×D412yes—
141C2×C4×C17#12011yescharacteristic
142C2×C4×C17#12112yes—
143C2×C4×C17#12311yescharacteristic
144#3622—44no—
145#3622—44no—
146#2983—171no—
147#2398D1711yescharacteristic
148C4×#101—171no—
149#1490—172no—
150#1300—172no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—81no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17—81no—
160C22×C17#38611yescharacteristic
161C22×C17C2×#10411yescharacteristic
162C22×C17#42111yescharacteristic
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—21no—
166C4×C17—22no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—81no—
171C4×C17—81no—
172C4×C17—81no—
173C4×C17#45412yes—
174C2×C4×Q8C2×D1711yescharacteristic
175C2×C4×D4C2×D1711yescharacteristic
176#446—344no—
177#432—172no—
178#432—341no—
179#431—341no—
180C2×#101—341no—
181C4×M16—172no—
182#400Dic1711yescharacteristic
183#400Dic1711yescharacteristic
184#397—21no—
185#394Dic1711yescharacteristic
186#394Dic1711yescharacteristic
187#393—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190#376C2×D1711yescharacteristic
191#343—341no—
192D4⋊C8—344no—
193C2×C17—21no—
194C2×C17—21no—
195C2×C17—82no—
196C2×C17#187111yescharacteristic
197C2×C17#188811yescharacteristic
198C2×C17#189111yescharacteristic
199C22×Q8C22⋊D1711yescharacteristic
200C22×D4C2×Dic1711yescharacteristic
201C23×C4—21no—
202C2×M16—342no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4C2×Dic1711yescharacteristic
208C2×C4⋊C4C2×Dic1711yescharacteristic
209C2×C4⋊C4C2×Dic1711yescharacteristic
210C2×C4⋊C4C22⋊D1711yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42C22⋊D1711yescharacteristic
218C2×C42C22⋊D1711yescharacteristic
219C2×C42C22×D1711yescharacteristic
220C4⋊C8—174no—
221C4⋊C8—342no—
222C4⋊C8—342no—
223#101—344no—
224C22⋊C8—342no—
225#97—342no—
226#97—342no—
227#97—342no—
228C4×C8—342no—
229C4×C8—342no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C2×Dic1711yescharacteristic
236#95C2×Dic1711yescharacteristic
237C17#903211yescharacteristic, complemented, Sylow
238C24—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C22⋊Dic1711yescharacteristic
252C22×C4C22⋊Dic1711yescharacteristic
253C22×C4C22⋊Dic1711yescharacteristic
254C22×C4C22⋊Dic1711yescharacteristic
255C22×C4C22×Dic1711yescharacteristic
256C22×C4C2×C22⋊D1711yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—344no—
259M16—682no—
260C2×C8—344no—
261C2×C8—682no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42D4×D1712yes—
277C42#6325412yes—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊Dic1711yescharacteristic
286Q8—44no—
287D4—24no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#1058750812yes—
314C2×C4D4×Dic1712yes—
315C2×C4#1058762111yescharacteristic
316C2×C4#1058762311yescharacteristic
317C2×C4#1058762512yes—
318C8—684no—
319C8—684no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#1176268911yescharacteristic
330C22C2×#1058750811yescharacteristic
331C22#1176282611yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#1176298112yes—
343C2—21no—
344C2—21no—
345C2—82no—
346C2#1452908711yescharacteristic
347C2#1452918811yescharacteristic
348C2#1452920311yescharacteristic
subgroupG/H
centerC2×C4#10587623
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2398C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#2398D17
hypocenterC17#9032
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. #14537869C2semidirect
2. C4×#10587507C2semidirect
4. #14526677C2semidirect
237. C17#9032semidirect

Maximal subgroups

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

subgroupindex
#145378692
C4×#105875072
#145272272
#145266772
C17×#23982
#903217