Finite Group Databasebeta

← #16945155

Subgroups — #16945155

Order 4352.

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

normal subgroups only

This group has 400 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#14528776C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526814C212yesmaximal, complemented
4#14526790C212yesmaximal, complemented
5C17×#2638C211yesmaximal, characteristic
6#11762833—21no—
7C2×#10587504—21no—
8#11762773—21no—
9#11762578C2212yes—
10#11762535C2212yes—
11#11762533—21no—
12C2×C17×C22⋊Q8C411yescharacteristic
13C2×C4×D4×C17C2211yescharacteristic
14C17×#398C2211yescharacteristic
15C17×#397—21no—
16C17×#397C411yescharacteristic
17C17×#395C2211yescharacteristic
18C17×#394—21no—
19C17×#393C411yescharacteristic
20C17×#387—21no—
21C17×#385—21no—
22C17×#381C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17D411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C17×C22⋊Q8—41no—
28C17×C22⋊Q8—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2311yescharacteristic
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×C17D411yescharacteristic
43C2×#63223—22no—
44C2×#63223—41no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95C2×C411yescharacteristic
51C17×#95C2×C411yescharacteristic
52#10587504—42no—
53C22⋊#3622—22no—
54C22⋊#3622—41no—
55C4⋊#3622—41no—
56C4⋊#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22⋊C411yescharacteristic
59C2×Q8×C17—41no—
60C2×Q8×C17—81no—
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—81no—
74C22×C4×C17—81no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—81no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—42no—
91C22⋊C4×C17—81no—
92C22⋊C4×C17—81no—
93C42×C17—21no—
94C42×C17—21no—
95#63223—42no—
96#63223—42no—
97#63223—82no—
98C2×#3622—42no—
99C2×#3622—42no—
100#11834—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112Q8×C17—82no—
113D4×C17—42no—
114D4×C17—42no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
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—81no—
138C2×C4×C17—81no—
139C2×C4×C17—81no—
140C2×C4×C17#9911yescharacteristic
141C2×C4×C17#9911yescharacteristic
142C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C17#13711yescharacteristic
144#3622—82no—
145#3622—82no—
146#2638D1711yescharacteristic
147#1800—171no—
148#1465—171no—
149#1354—172no—
150#1346—172no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—42no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17C2×#9911yescharacteristic
164C22×C17#41711yescharacteristic
165C22×C17#42111yescharacteristic
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—81no—
174C4×C17—81no—
175C4×C17—81no—
176C4×C17—81no—
177C2×C22⋊Q8Dic1711yescharacteristic
178C2×C4×D4C2×D1711yescharacteristic
179#423—341no—
180C2×#100—341no—
181#407—341no—
182#398C2×D1711yescharacteristic
183#397—21no—
184#397Dic1711yescharacteristic
185#395C2×D1711yescharacteristic
186#394—21no—
187#393Dic1711yescharacteristic
188#387—21no—
189#385—21no—
190#381Dic1711yescharacteristic
191#343—172no—
192#324—172no—
193#323—341no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17—81no—
197C2×C17—81no—
198C2×C17#151311yescharacteristic
199C2×C17#212111yescharacteristic
200C2×C17#213511yescharacteristic
201C22×Q8C2×Dic1711yescharacteristic
202C22×D4C22⋊D1711yescharacteristic
203C23×C4C2×Dic1711yescharacteristic
204C23×C4C22⋊D1711yescharacteristic
205C2×M16—342no—
206C2×M16—681no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic1711yescharacteristic
215C2×C4⋊C4C22×D1711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4C2×Dic1711yescharacteristic
221C2×C22⋊C4C22⋊D1711yescharacteristic
222C2×C42C22⋊D1711yescharacteristic
223C4⋊C8—681no—
224C4⋊C8—681no—
225#100—682no—
226C22⋊C8—342no—
227C22⋊C8—681no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×Dic1711yescharacteristic
234#95C2×Dic1711yescharacteristic
235C17#1183411yescharacteristic, complemented, Sylow
236C24C22⋊Dic1711yescharacteristic
237C24C22⋊Dic1711yescharacteristic
238C2×Q8—41no—
239C2×Q8—81no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C22⋊Dic1711yescharacteristic
255C22×C4C22⋊Dic1711yescharacteristic
256C22×C4C22×Dic1711yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258C22×C4C2×C22⋊D1711yescharacteristic
259M16—682no—
260M16—682no—
261M16—1362no—
262C2×C8—682no—
263C2×C8—682no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C42—21no—
278C42—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic1711yescharacteristic
290Q8—82no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
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—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#1058750511yescharacteristic
319C2×C4#1058750511yescharacteristic
320C2×C4#1058763511yescharacteristic
321C2×C4#1058763711yescharacteristic
322C8—1362no—
323C8—1362no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×#1058750511yescharacteristic
337C22#1176281011yescharacteristic
338C22#1176282611yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—81no—
354C2#1452732711yescharacteristic
355C2#1453043011yescharacteristic
356C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2638C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#2638D17
hypocenterC17#11834
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
4. #14526790C2semidirect
235. C17#11834semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145271222
#145268142
#145267902
C17×#26382
#1183417