Finite Group Databasebeta

← #16945024

Subgroups — #16945024

Order 4352.

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

normal subgroups only

This group has 488 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#14526787C214yesmaximal, complemented
4C17×#2619C211yesmaximal, characteristic
5#11762866—21no—
6#11762773—21no—
7#11762578—21no—
8#11762578C2212yes—
9#11762535C2212yes—
10#11762533—21no—
11C2×C17×#127C411yescharacteristic
12C2×C17×#121—21no—
13C2×C17×#117C2211yescharacteristic
14C17×#394—21no—
15C17×#394C2212yes—
16C17×#392—21no—
17C17×#392C412yes—
18C17×#390—21no—
19C17×#382C411yescharacteristic
20C22×D4×C17—22no—
21C22×D4×C17—21no—
22C22×D4×C17—21no—
23C22×D4×C17C2×C412yes—
24C23×C4×C17D411yescharacteristic
25C17×#127—41no—
26C17×#127—41no—
27C17×#121—44no—
28C17×#117—41no—
29C17×#117—41no—
30C2×C4⋊C4×C17—21no—
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×C17C2×C411yescharacteristic
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×#63223—22no—
41C17×#95—21no—
42C17×#95C2×C412yes—
43C22⋊#3622—22no—
44C22⋊#3622—41no—
45C22⋊#3622—41no—
46C4⋊#3622—41no—
47#10587474—41no—
48C24×C17—22no—
49C24×C17C22⋊C411yescharacteristic
50C2×D4×C17—42no—
51C2×D4×C17—42no—
52C2×D4×C17—42no—
53C2×D4×C17—42no—
54C2×D4×C17—44no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17—82no—
58C2×D4×C17—82no—
59C2×D4×C17—82no—
60C2×D4×C17—82no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—22no—
65C22×C4×C17—41no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—42no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—44no—
78C22⋊C4×C17—82no—
79C22⋊C4×C17—82no—
80C42×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84#63223—42no—
85#63223—42no—
86C2×#3622—42no—
87C2×#3622—42no—
88#11806—171nomaximal, Sylow
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17C2×C22⋊C411yescharacteristic
102D4×C17—82no—
103D4×C17—82no—
104D4×C17—84no—
105D4×C17—84no—
106D4×C17—88no—
107C2×C4×C17—21no—
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—21no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17AGL(1,ℤ/8)12yes—
127#3622—82no—
128#3622—82no—
129#2619D1711yescharacteristic
130#1822—171no—
131#1464—171no—
132#1345—174no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—42no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—81no—
140C22×C17—82no—
141C22×C17—82no—
142C22×C17—82no—
143C22×C17—82no—
144C22×C17—82no—
145C22×C17—82no—
146C22×C17—82no—
147C22×C17C2×#9911yescharacteristic
148C22×C17#41811yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—82no—
158C2×#127Dic1711yescharacteristic
159C2×#121—21no—
160C2×#117C2×D1711yescharacteristic
161#432—341no—
162#407—341no—
163#394—21no—
164#394C2×D1712yes—
165#392—21no—
166#392Dic1712yes—
167#390—21no—
168#382Dic1711yescharacteristic
169#343—172no—
170#343—341no—
171#324—172no—
172#323—341no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—81no—
176C2×C17—162no—
177C2×C17#151011yescharacteristic
178C2×C17#212211yescharacteristic
179C2×C17#213411yescharacteristic
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C2×Dic1712yes—
184C23×C4C22⋊D1711yescharacteristic
185C2×M16—342no—
186#127—41no—
187#127—41no—
188#121—44no—
189#117—41no—
190#117—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22⋊D1711yescharacteristic
193C2×C4⋊C4C22×D1711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C2×Dic1711yescharacteristic
198C2×C22⋊C4C22⋊D1711yescharacteristic
199C2×C42C2×Dic1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—681no—
202C22⋊C8—342no—
203C22⋊C8—681no—
204C22⋊C8—681no—
205#97—681no—
206#95—21no—
207#95C2×Dic1712yes—
208C17#1180611yescharacteristic, complemented, Sylow
209C24—22no—
210C24C22⋊Dic1711yescharacteristic
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4C22⋊Dic1711yescharacteristic
230C22×C4C22×Dic1711yescharacteristic
231C22×C4C2×C22⋊D1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233M16—682no—
234M16—682no—
235C2×C8—682no—
236C2×C8—682no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23C2×C22⋊Dic1711yescharacteristic
262D4—82no—
263D4—82no—
264D4—84no—
265D4—84no—
266D4—88no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4#1058750511yescharacteristic
285C2×C4#1058750511yescharacteristic
286C2×C4#1058761812yes—
287C8—1362no—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22C2×#1058750511yescharacteristic
304C22#1176281411yescharacteristic
305C22#1176282611yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—162no—
318C2#1452731311yescharacteristic
319C2#1453043411yescharacteristic
320C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
Fitting, prime radicalC17×#2619C2
socleC22×C17C2×#99
cosocleC2×C4⋊C4×C17C23
hypercenter#2619D17
hypocenterC17#11806
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #14528867C2semidirect
2. #14527118C2semidirect
3. #14526787C2semidirect
208. C17#11806semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145271182
#145267872
C17×#26192
#1180617