Finite Group Databasebeta

← #16944915

Subgroups — #16944915

Order 4352.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#10587507C211yesmaximal, characteristic, complemented
2#14527128C211yesmaximal, characteristic, complemented
3#14526787C214yesmaximal, complemented
4C17×#2548C211yesmaximal, characteristic
5#11762866—21no—
6C2×#10587507—21no—
7C4×#63223—21no—
8#11762578C2212yes—
9#11762535—21no—
10#11762535C2212yes—
11C2×C17×#127C411yescharacteristic
12C2×C17×#124—21no—
13C2×C4×Q8×C17C2211yescharacteristic
14C17×#394C412yes—
15C17×#394C2212yes—
16C17×#390—21no—
17C17×#390—21no—
18C17×#390—21no—
19C4×C4⋊C4×C17C411yescharacteristic
20C22×Q8×C17D411yescharacteristic
21C22×D4×C17—21no—
22C22×D4×C17—21no—
23C22×D4×C17C2×C412yes—
24C17×#127—41no—
25C17×#127—41no—
26C17×#124—44no—
27C4×Q8×C17—41no—
28C4×Q8×C17—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—22no—
37C2×C42×C17—21no—
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×#63223—22no—
42C17×#95C2×C412yes—
43#10587507—24no—
44C22⋊#3622—22no—
45C4⋊#3622—41no—
46#10587474—41no—
47#10587474—41no—
48C4×#3622—41no—
49C24×C17—22no—
50C2×Q8×C17—22no—
51C2×Q8×C17—22no—
52C2×Q8×C17—41no—
53C2×D4×C17—42no—
54C2×D4×C17—42no—
55C2×D4×C17—42no—
56C2×D4×C17—42no—
57C2×D4×C17—42no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17—42no—
61C2×D4×C17—82no—
62C22×C4×C17—22no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C4⋊C4×C17—41no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—42no—
74C22⋊C4×C17—44no—
75C22⋊C4×C17—44no—
76C22⋊C4×C17—82no—
77C22⋊C4×C17—82no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—42no—
85#63223—24no—
86#63223—42no—
87C2×#3622—42no—
88C2×#3622—42no—
89#11781—171nomaximal, Sylow
90C23×C17—42no—
91C23×C17—42no—
92C23×C17—42no—
93C23×C17—42no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17C2×C22⋊C411yescharacteristic
98Q8×C17—44no—
99D4×C17—82no—
100D4×C17—84no—
101D4×C17—84no—
102D4×C17—82no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#10412yes—
123C2×C4×C17AGL(1,ℤ/8)12yes—
124#3622—44no—
125#3622—82no—
126#2548D1711yescharacteristic
127C4×#101—171no—
128#1467—171no—
129#1345—174no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—82no—
133C22×C17—82no—
134C22×C17—82no—
135C22×C17—82no—
136C22×C17—82no—
137C22×C17—82no—
138C22×C17—82no—
139C22×C17C2×#9911yescharacteristic
140C22×C17#41811yescharacteristic
141C22×C17C2×#10411yescharacteristic
142C4×C17—21no—
143C4×C17—22no—
144C4×C17—22no—
145C4×C17—41no—
146C4×C17—41no—
147C4×C17—41no—
148C4×C17—41no—
149C4×C17—82no—
150C4×C17#35512yes—
151C2×#127Dic1711yescharacteristic
152C2×#124—21no—
153C2×C4×Q8C2×D1711yescharacteristic
154#432—341no—
155C2×#101—341no—
156C4×M16—341no—
157#394Dic1712yes—
158#394C2×D1712yes—
159#390—21no—
160#390—21no—
161#390—21no—
162C4×C4⋊C4Dic1711yescharacteristic
163#343—172no—
164#324—172no—
165#324—341no—
166C2×C17—21no—
167C2×C17—21no—
168C2×C17—162no—
169C2×C17#150711yescharacteristic
170C2×C17#212211yescharacteristic
171C2×C17C2×#35511yescharacteristic
172C22×Q8C22⋊D1711yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C2×Dic1712yes—
176C2×M16—342no—
177#127—41no—
178#127—41no—
179#124—44no—
180C4×Q8—41no—
181C4×Q8—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×Dic1711yescharacteristic
184C2×C4⋊C4C22⋊D1711yescharacteristic
185C2×C4⋊C4C22×D1711yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—21no—
191C2×C42C2×Dic1711yescharacteristic
192C2×C42C22⋊D1711yescharacteristic
193C2×C42C22⋊D1711yescharacteristic
194C4⋊C8—681no—
195#101—344no—
196C22⋊C8—342no—
197#97—681no—
198#97—681no—
199C4×C8—681no—
200#95C2×Dic1712yes—
201C17#1178111yescharacteristic, complemented, Sylow
202C24—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C22×C4—22no—
216C22×C4C22⋊Dic1711yescharacteristic
217C22×C4C22⋊Dic1711yescharacteristic
218C22×C4C22⋊Dic1711yescharacteristic
219C22×C4C22⋊Dic1711yescharacteristic
220C22×C4C22×Dic1711yescharacteristic
221C22×C4C2×C22⋊D1711yescharacteristic
222C22×C4C2×C22⋊D1711yescharacteristic
223M16—344no—
224M16—682no—
225C2×C8—682no—
226C2×C8—682no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊Dic1711yescharacteristic
250Q8—44no—
251D4—84no—
252D4—82no—
253D4—84no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#1058750511yescharacteristic
273C2×C4#1058750511yescharacteristic
274C2×C4#1058750812yes—
275C2×C4#1058761812yes—
276C8—684no—
277C8—1362no—
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#1058750511yescharacteristic
288C22#1176281411yescharacteristic
289C22C2×#1058750811yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4#1176260612yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2#1452730011yescharacteristic
303C2#1453043411yescharacteristic
304C2C2×#1176260611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2548C2
socleC22×C17C2×#99
prime radicalC2×C17×#127C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2548D17
hypocenterC17#11781
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C4×#10587507C2semidirect
2. #14527128C2semidirect
3. #14526787C2semidirect
201. C17#11781semidirect

Maximal subgroups

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

subgroupindex
C4×#105875072
#145271282
#145267872
C17×#25482
#1178117