Finite Group Databasebeta

← #16927818

Subgroups — #16927818

Order 4352.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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#14531282C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526641C212yesmaximal, complemented
4#14526639C212yesmaximal, complemented
5C17×#1464C211yesmaximal, characteristic
6#11762866C2212yes—
7#11762702—21no—
8#11762702—21no—
9#11762695—21no—
10#11762695—21no—
11C4×C4⋊Dic17C2211yescharacteristic
12C4×C4⋊Dic17C2211yescharacteristic
13C4×C22⋊Dic17C411yescharacteristic
14C4×C22⋊Dic17C411yescharacteristic
15#11762655C411yescharacteristic
16C42×Dic17C411yescharacteristic
17#11762535—21no—
18#11762533—21no—
19C2×C17×#117C2211yescharacteristic
20C17×#432C2212yes—
21C17×#324—21no—
22C17×#323—21no—
23C23×C4×C17C2×C411yescharacteristic
24C17×#117—22no—
25C17×#117—22no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17C2311yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17C2×C411yescharacteristic
42C17×C4⋊C8—22no—
43C17×C22⋊C8—22no—
44C17×#97—22no—
45#10587502—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51C22⋊#3622—22no—
52C4⋊#3622—22no—
53#10587474—22no—
54C24×C17C22⋊C411yescharacteristic
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C2×C8×C17—42no—
70C4⋊C4×C17—22no—
71C4⋊C4×C17—22no—
72C22⋊C4×C17—22no—
73C22⋊C4×C17—22no—
74C42×C17—21no—
75C42×C17—21no—
76C42×C17#4112yes—
77C42×C17#4112yes—
78C22⋊Dic17—42no—
79C22⋊Dic17—42no—
80C4⋊Dic17—42no—
81C4⋊Dic17—42no—
82C4×Dic17—42no—
83C4×Dic17—42no—
84C4×Dic17—41no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C4×Dic17—41no—
88C2×#3622—42no—
89#8745—171nomaximal, Sylow
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17C2×C22⋊C411yescharacteristic
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17#11711yescharacteristic
110C2×C4×C17#11711yescharacteristic
111C2×C4×C17#12312yes—
112C2×C4×C17#12312yes—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C8×C17—82no—
126#3622—82no—
127#2298—171no—
128#1464—171no—
129#1464D1711yescharacteristic
130#1285—172no—
131#1285—172no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—81no—
138C22×C17—81no—
139C22×C17#38111yescharacteristic
140C22×C17#41311yescharacteristic
141C22×C17#41311yescharacteristic
142C4×C17—21no—
143C4×C17—21no—
144C4×C17—22no—
145C4×C17—22no—
146C4×C17—41no—
147C4×C17—41no—
148C4×C17—41no—
149Dic17—42no—
150Dic17—42no—
151Dic17—81no—
152Dic17—81no—
153C2×#117C2×D1711yescharacteristic
154#432—172no—
155#432C2×D1712yes—
156#388—341no—
157#388—341no—
158#387—341no—
159#387—341no—
160C4×C4⋊C4—171no—
161C4×C4⋊C4—171no—
162C4×C22⋊C4—171no—
163C4×C22⋊C4—171no—
164#376—171no—
165C43—171no—
166#324—21no—
167#324—341no—
168#323—21no—
169#323—341no—
170C2×C17—21no—
171C2×C17—21no—
172C2×C17—81no—
173C2×C17#182211yescharacteristic
174C2×C17#182711yescharacteristic
175C2×C17#182711yescharacteristic
176C23×C4C4×D1711yescharacteristic
177#117—22no—
178#117—22no—
179C2×C4⋊C4—341no—
180C2×C4⋊C4—341no—
181C2×C4⋊C4D6811yescharacteristic
182C2×C4⋊C4C22⋊D1711yescharacteristic
183C2×C22⋊C4—341no—
184C2×C22⋊C4—341no—
185C2×C22⋊C4D6811yescharacteristic
186C2×C22⋊C4C22⋊D1711yescharacteristic
187C2×C42—171no—
188C2×C42—171no—
189C2×C42—171no—
190C2×C42—171no—
191C2×C42—341no—
192C2×C42—341no—
193C2×C42C4×D1711yescharacteristic
194C2×C42C22×D1711yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—342no—
197C22⋊C8—22no—
198C22⋊C8—342no—
199#97—22no—
200#97—342no—
201#95—341no—
202#95—341no—
203#95—341no—
204#95—341no—
205#95—341no—
206#95—341no—
207C17#874511yescharacteristic, complemented, Sylow
208C24#6322711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—341no—
214C22×C4—341no—
215C22×C4—341no—
216C22×C4—341no—
217C22×C4#6322711yescharacteristic
218C22×C4#6322711yescharacteristic
219C22×C4#6322711yescharacteristic
220C22×C4C2×C4×D1711yescharacteristic
221C22×C4C2×D6811yescharacteristic
222C22×C4C2×C22⋊D1711yescharacteristic
223C2×C8—42no—
224C2×C8—682no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—682no—
228C4⋊C4—682no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—682no—
232C22⋊C4—682no—
233C42—21no—
234C42—21no—
235C42—681no—
236C42—681no—
237C42—681no—
238C42—681no—
239C42—682no—
240C42—682no—
241C42#6325412yes—
242C42#6325412yes—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23C2×#6322711yescharacteristic
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—681no—
265C2×C4—681no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—681no—
271C2×C4—681no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4#1058756311yescharacteristic
275C2×C4#1058756311yescharacteristic
276C2×C4#1058757212yes—
277C2×C4#1058762112yes—
278C8—82no—
279C8—1362no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1176267011yescharacteristic
288C22#1176279111yescharacteristic
289C22#1176279111yescharacteristic
290C4—21no—
291C4—21no—
292C4—22no—
293C4—22no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—682no—
298C4—682no—
299C4—1361no—
300C4—1361no—
301C2—21no—
302C2—21no—
303C2—81no—
304C2#1452886511yescharacteristic
305C2#1452888511yescharacteristic
306C2#1452888911yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1464C2
socleC22×C17#381
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1464D17
hypocenterC17#8745
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #14526641C2semidirect
4. #14526639C2semidirect
207. C17#8745semidirect

Maximal subgroups

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

subgroupindex
#145312822
#145271182
#145266412
#145266392
C17×#14642
#874517