Finite Group Databasebeta

← #16945222

Subgroups — #16945222

Order 4352.

432 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 87 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#14528867C211yesmaximal, characteristic, complemented
2#14527098C211yesmaximal, characteristic, complemented
3#14526811C212yesmaximal, complemented
4#14526787C212yesmaximal, complemented
5C17×#2598C211yesmaximal, characteristic
6#11762862—21no—
7#11762773—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535C2212yes—
11#11762533—21no—
12C2×C4×D4×C17C411yescharacteristic
13C2×C17×#117C2211yescharacteristic
14C17×#400C2211yescharacteristic
15C17×#399C411yescharacteristic
16C17×#394C2211yescharacteristic
17C17×#392C411yescharacteristic
18C17×#388—21no—
19C17×#386—21no—
20C17×#386—21no—
21C17×#385—21no—
22C17×#383C411yescharacteristic
23C22×D4×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C23×C4×C17—21no—
26C23×C4×C17D411yescharacteristic
27C4×D4×C17—41no—
28C4×D4×C17—41no—
29C17×#117—41no—
30C17×#117—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17C2311yescharacteristic
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C42×C17C2×C411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×#63223—22no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51C22⋊#3622—22no—
52C22⋊#3622—41no—
53C22⋊#3622—41no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17—21no—
57C24×C17C22⋊C411yescharacteristic
58C2×D4×C17—41no—
59C2×D4×C17—42no—
60C2×D4×C17—42no—
61C2×D4×C17—42no—
62C2×D4×C17—81no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—41no—
70C22×C4×C17—42no—
71C22×C4×C17—81no—
72C22×C4×C17—81no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—81no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—81no—
90C22⋊C4×C17—81no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—21no—
94C42×C17—21no—
95#63223—42no—
96#63223—42no—
97C2×#3622—42no—
98C2×#3622—42no—
99#11847—171nomaximal, Sylow
100C23×C17—22no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17C2×C22⋊C411yescharacteristic
113D4×C17—42no—
114D4×C17—42no—
115D4×C17—84no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
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—41no—
138C2×C4×C17—41no—
139C2×C4×C17—84no—
140C2×C4×C17#9911yescharacteristic
141C2×C4×C17#9911yescharacteristic
142C2×C4×C17#13512yes—
143#3622—82no—
144#3622—82no—
145#2598D1711yescharacteristic
146#1822—171no—
147#1458—171no—
148#1353—172no—
149#1345—172no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17C2×#9911yescharacteristic
165C22×C17#41611yescharacteristic
166C22×C17#42111yescharacteristic
167C4×C17—22no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—81no—
175C4×C17—81no—
176C4×C17—81no—
177C2×C4×D4Dic1711yescharacteristic
178C2×#117C2×D1711yescharacteristic
179#431—341no—
180#407—341no—
181#400C2×D1711yescharacteristic
182#399Dic1711yescharacteristic
183#394C2×D1711yescharacteristic
184#392Dic1711yescharacteristic
185#388—21no—
186#386—21no—
187#386—21no—
188#385—21no—
189#383Dic1711yescharacteristic
190#343—172no—
191#343—341no—
192#324—172no—
193#323—341no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17—81no—
197C2×C17—82no—
198C2×C17#151611yescharacteristic
199C2×C17#212011yescharacteristic
200C2×C17#213411yescharacteristic
201C22×D4—21no—
202C22×D4C2×Dic1711yescharacteristic
203C23×C4—21no—
204C23×C4C22⋊D1711yescharacteristic
205C2×M16—342no—
206C4×D4—41no—
207C4×D4—41no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic1711yescharacteristic
212C2×C4⋊C4C22⋊D1711yescharacteristic
213C2×C4⋊C4C22×D1711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×Dic1711yescharacteristic
220C2×C22⋊C4C22⋊D1711yescharacteristic
221C2×C42C2×Dic1711yescharacteristic
222C2×C42C22⋊D1711yescharacteristic
223C4⋊C8—681no—
224C22⋊C8—342no—
225C22⋊C8—681no—
226C22⋊C8—681no—
227C4×C8—681no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic1711yescharacteristic
233#95C2×Dic1711yescharacteristic
234C17#1184711yescharacteristic, complemented, Sylow
235C24—21no—
236C24C22⋊Dic1711yescharacteristic
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4C22⋊Dic1711yescharacteristic
253C22×C4C22⋊Dic1711yescharacteristic
254C22×C4C22⋊Dic1711yescharacteristic
255C22×C4C22×Dic1711yescharacteristic
256C22×C4C2×C22⋊D1711yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—682no—
259M16—682no—
260C2×C8—682no—
261C2×C8—682no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C2×C22⋊Dic1711yescharacteristic
291D4—42no—
292D4—42no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—84no—
318C2×C4#1058750511yescharacteristic
319C2×C4#1058750511yescharacteristic
320C2×C4#1058763612yes—
321C8—1362no—
322C8—1362no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22C2×#1058750511yescharacteristic
338C22#1176280511yescharacteristic
339C22#1176282611yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—82no—
354C2#1452734011yescharacteristic
355C2#1453042511yescharacteristic
356C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2598C2
socleC22×C17C2×#99
prime radicalC17×#392C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2598D17
hypocenterC17#11847
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
1. #14528867C2semidirect
2. #14527098C2semidirect
3. #14526811C2semidirect
4. #14526787C2semidirect
234. C17#11847semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145270982
#145268112
#145267872
C17×#25982
#1184717