Finite Group Databasebeta

← #16945015

Subgroups — #16945015

Order 4352.

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

normal subgroups only

This group has 472 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#14527118C211yesmaximal, characteristic, complemented
3#14526790C212yesmaximal, complemented
4#14526787C212yesmaximal, complemented
5C17×#2392C211yesmaximal, characteristic
6#11762866—21no—
7#11762773—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535C2212yes—
11#11762533—21no—
12C2×C4×D4×C17—21no—
13C2×C4×D4×C17C411yescharacteristic
14C2×C17×#117C2211yescharacteristic
15C17×#398C2211yescharacteristic
16C17×#397C411yescharacteristic
17C17×#396—21no—
18C17×#394—21no—
19C17×#394C2211yescharacteristic
20C17×#392C411yescharacteristic
21C4×C4⋊C4×C17C411yescharacteristic
22C4×C22⋊C4×C17—21no—
23C22×D4×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C23×C4×C17—21no—
26C23×C4×C17D411yescharacteristic
27C4×D4×C17—41no—
28C4×D4×C17—41no—
29C4×D4×C17—44no—
30C17×#117—41no—
31C17×#117—41no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17C2×C411yescharacteristic
43C2×C22⋊C4×C17D411yescharacteristic
44C2×C42×C17—21no—
45C2×C42×C17C2×C411yescharacteristic
46C2×C42×C17D411yescharacteristic
47C2×#63223—22no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95C2×C411yescharacteristic
51C17×#95C2×C411yescharacteristic
52C22⋊#3622—22no—
53C22⋊#3622—41no—
54C22⋊#3622—41no—
55C4⋊#3622—41no—
56#10587474—41no—
57C24×C17—21no—
58C24×C17C22⋊C411yescharacteristic
59C2×D4×C17—41no—
60C2×D4×C17—42no—
61C2×D4×C17—42no—
62C2×D4×C17—42no—
63C2×D4×C17—42no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—22no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—41no—
71C22×C4×C17—42no—
72C22×C4×C17—42no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—42no—
85C4⋊C4×C17—42no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—42no—
91C22⋊C4×C17—42no—
92C22⋊C4×C17—42no—
93C42×C17—21no—
94C42×C17—21no—
95C42×C17—21no—
96C42×C17—21no—
97C42×C17—42no—
98#63223—42no—
99#63223—42no—
100C2×#3622—42no—
101C2×#3622—42no—
102#11804—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17C2×C22⋊C411yescharacteristic
116D4×C17—42no—
117D4×C17—48no—
118D4×C17—42no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
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—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—82no—
143C2×C4×C17—82no—
144C2×C4×C17#9911yescharacteristic
145C2×C4×C17#9911yescharacteristic
146C2×C4×C17AGL(1,ℤ/8)12yes—
147#3622—82no—
148#3622—82no—
149#2392D1711yescharacteristic
150#1822—171no—
151#1464—171no—
152#1346—172no—
153#1345—172no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17—42no—
159C22×C17—42no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17C2×#9911yescharacteristic
169C22×C17#41811yescharacteristic
170C22×C17#42111yescharacteristic
171C4×C17—22no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—42no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17—81no—
180C4×C17—81no—
181C2×C4×D4—21no—
182C2×C4×D4Dic1711yescharacteristic
183C2×#117C2×D1711yescharacteristic
184#432—341no—
185#407—341no—
186#398C2×D1711yescharacteristic
187#397Dic1711yescharacteristic
188#396—21no—
189#394—21no—
190#394C2×D1711yescharacteristic
191#392Dic1711yescharacteristic
192C4×C4⋊C4Dic1711yescharacteristic
193C4×C22⋊C4—21no—
194#343—172no—
195#343—341no—
196#324—172no—
197#323—341no—
198C2×C17—21no—
199C2×C17—21no—
200C2×C17—81no—
201C2×C17—82no—
202C2×C17#151011yescharacteristic
203C2×C17#212011yescharacteristic
204C2×C17#213211yescharacteristic
205C22×D4—21no—
206C22×D4C2×Dic1711yescharacteristic
207C23×C4—21no—
208C23×C4C22⋊D1711yescharacteristic
209C2×M16—342no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—44no—
213#117—41no—
214#117—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic1711yescharacteristic
218C2×C4⋊C4C22⋊D1711yescharacteristic
219C2×C4⋊C4C22×D1711yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic1711yescharacteristic
226C2×C22⋊C4C22⋊D1711yescharacteristic
227C2×C42—21no—
228C2×C42C2×Dic1711yescharacteristic
229C2×C42C22⋊D1711yescharacteristic
230C4⋊C8—681no—
231C22⋊C8—342no—
232C22⋊C8—681no—
233C22⋊C8—681no—
234#97—681no—
235#95—21no—
236#95—21no—
237#95C2×Dic1711yescharacteristic
238#95C2×Dic1711yescharacteristic
239C17#1180411yescharacteristic, complemented, Sylow
240C24—21no—
241C24C22⋊Dic1711yescharacteristic
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊Dic1711yescharacteristic
259C22×C4C22⋊Dic1711yescharacteristic
260C22×C4C22⋊Dic1711yescharacteristic
261C22×C4C22×Dic1711yescharacteristic
262C22×C4C2×C22⋊D1711yescharacteristic
263C22×C4C2×C22⋊D1711yescharacteristic
264M16—682no—
265M16—682no—
266C2×C8—682no—
267C2×C8—682no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—42no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic1711yescharacteristic
298D4—42no—
299D4—48no—
300D4—42no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#1058750511yescharacteristic
327C2×C4#1058750511yescharacteristic
328C2×C4#1058761812yes—
329C8—1362no—
330C8—1362no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1058750511yescharacteristic
346C22#1176281411yescharacteristic
347C22#1176282611yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—82no—
362C2#1452731311yescharacteristic
363C2#1453042511yescharacteristic
364C2#1453046911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2392C2
socleC22×C17C2×#99
prime radicalC17×#392C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2392D17
hypocenterC17#11804
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #14528867C2semidirect
2. #14527118C2semidirect
3. #14526790C2semidirect
4. #14526787C2semidirect
239. C17#11804semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145271182
#145267902
#145267872
C17×#23922
#1180417