Finite Group Databasebeta

← #16950315

Subgroups — #16950315

Order 4352.

588 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#11762602C212yesmaximal
2#14528772C212yesmaximal, complemented
3#14526909C212yesmaximal, complemented
4C17×#2328C211yesmaximal, characteristic
5C2×#10587505C2214yes—
6#11762669—22no—
7#11762602—24no—
8#11762576—22no—
9C2×C17×#127C2211yescharacteristic
10C2×C4×D4×C17C411yescharacteristic, complemented
11C2×C4×D4×C17C2211yescharacteristic
12C17×#390C411yescharacteristic, complemented
13C17×#390C2211yescharacteristic
14C17×#386—22no—
15C17×#386—22no—
16C17×#382C411yescharacteristic, complemented
17C17×#376C411yescharacteristic, complemented
18C22×D4×C17—22no—
19C22×D4×C17C2×C411yescharacteristic
20C22×D4×C17C2311yescharacteristic
21C23×C4×C17—21no—
22C23×C4×C17D412yes—
23C17×#127—22no—
24C17×#127D412yes—
25C17×#127Q812yes—
26C4×D4×C17—42no—
27C4×D4×C17—42no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17—21no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17D412yes—
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×C42×C17C2×C411yescharacteristic
37C2×C22⋊Dic17—24no—
38C2×C4⋊Dic17—42no—
39C17×#95—21no—
40C17×#95—21no—
41C17×#95—22no—
42#10587505—24no—
43#10587505—24no—
44#10587502—42no—
45C24×C17—21no—
46C24×C17C2×D412yes—
47C2×D4×C17—24no—
48C2×D4×C17—24no—
49C2×D4×C17—21no—
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17—82no—
55C2×D4×C17—82no—
56C2×D4×C17C4⋊C412yescomplemented
57C2×D4×C17C2×D411yescharacteristic
58C2×D4×C17C2×Q811yescharacteristic
59C2×D4×C17#4112yes—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—82no—
67C22×C4×C17—82no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C412yes—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×Dic17—44no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C22⋊C4×C17—42no—
76C22⋊C4×C17—42no—
77C22⋊C4×C17—82no—
78C22⋊C4×C17—82no—
79C42×C17—22no—
80C42×C17—21no—
81C42×C17—41no—
82C42×C17C4⋊C412yescomplemented
83C22⋊Dic17—44no—
84C22⋊Dic17—44no—
85C4⋊Dic17—84no—
86#12799—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—22no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17—42no—
93C23×C17—42no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17#12112yes—
98C23×C17C22⋊Q812yes—
99D4×C17—41no—
100D4×C17—41no—
101D4×C17—42no—
102D4×C17—42no—
103D4×C17—42no—
104D4×C17—84no—
105D4×C17—84no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—41no—
112C2×C4×C17—44no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—82no—
123C2×C4×C17—84no—
124C2×C4×C17#9912yes—
125C2×C4×C17C2×C4⋊C411yescharacteristic
126C2×C4×C17#11711yescharacteristic
127C2×Dic17—84no—
128C2×Dic17—84no—
129C2×Dic17—84no—
130#2328D1711yescharacteristic
131C2×#353—172no—
132#1799—172no—
133#1393—172no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—42no—
139C22×C17—42no—
140C22×C17—44no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—82no—
145C22×C17—82no—
146C22×C17—81no—
147C22×C17—81no—
148C22×C17#38011yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17#41011yescharacteristic
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—42no—
155C4×C17—42no—
156C4×C17—41no—
157C4×C17—81no—
158C4×C17—81no—
159Dic17—164no—
160C2×#127C2×D1711yescharacteristic
161C2×C4×D4Dic1711yescharacteristic, complemented
162C2×C4×D4C2×D1711yescharacteristic
163C2×#99—174no—
164#390Dic1711yescharacteristic, complemented
165#390C2×D1711yescharacteristic
166#386—22no—
167#386—22no—
168#382Dic1711yescharacteristic, complemented
169#380—342no—
170#376Dic1711yescharacteristic, complemented
171#353—344no—
172#342—342no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—42no—
176C2×C17—42no—
177C2×C17—82no—
178C2×C17#179911yescharacteristic
179C2×C17#212711yescharacteristic
180C2×C17#212811yescharacteristic
181C22×D4—22no—
182C22×D4C2×Dic1711yescharacteristic
183C22×D4C22×D1711yescharacteristic
184C23×C4—21no—
185C23×C4C22⋊D1712yes—
186#127—22no—
187#127Dic3412yes—
188#127D6812yes—
189C4×D4—42no—
190C4×D4—42no—
191C2×C4⋊C4—682no—
192C2×C4⋊C4C2×Dic1711yescharacteristic
193C2×C4⋊C4C2×Dic1711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—344no—
198C2×C22⋊C4C22⋊D1712yes—
199C2×C42C2×Dic1711yescharacteristic
200C2×C42C2×Dic1711yescharacteristic
201C2×C42C2×Dic1711yescharacteristic
202#99—344no—
203#99—344no—
204#95—21no—
205#95—22no—
206#95—21no—
207#95—682no—
208C17#1279911yescharacteristic, complemented, Sylow
209C24—21no—
210C24C2×C22⋊D1712yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C4⋊Dic1712yescomplemented
221C2×D4C2×Dic3411yescharacteristic
222C2×D4C2×D6811yescharacteristic
223C2×D4#6325212yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—684no—
233C22×C4C22⋊Dic1711yescharacteristic
234C22×C4C22⋊Dic1712yes—
235C22×C4C22⋊Dic1711yescharacteristic
236C22×C4C22×Dic1711yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—1364no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C22⋊C4—684no—
245C22⋊C4—684no—
246C42—22no—
247C42—21no—
248C42—41no—
249C42C4⋊Dic1712yescomplemented
250C23—22no—
251C23—22no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23#1058761012yes—
260C23#1058761612yes—
261C23C2×C22⋊Dic1711yescharacteristic
262D4—41no—
263D4—41no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—1364no—
288C2×C4—1364no—
289C2×C4—1364no—
290C2×C4#1058750512yes—
291C2×C4C2×C4⋊Dic1711yescharacteristic
292C2×C4#1058761211yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22#1176266911yescharacteristic
308C22C2×#1058750511yescharacteristic
309C22#1176278311yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—2724no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2—42no—
323C2—82no—
324C2#1452877211yescharacteristic
325C2#1453045311yescharacteristic
326C2#1453045611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2328C2
socleC22×C17C2×#99
prime radicalC2×C17×#127C22
cosocleC22×D4×C17C23
hypercenter#2328D17
hypocenterC17#12799
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #14528772C2semidirect
3. #14526909C2semidirect
10. C2×C4×D4×C17C4semidirect
12. C17×#390C4semidirect
16. C17×#382C4semidirect
17. C17×#376C4semidirect
56. C2×D4×C17C4⋊C4semidirect
82. C42×C17C4⋊C4semidirect
161. C2×C4×D4Dic17semidirect
164. #390Dic17semidirect
168. #382Dic17semidirect
170. #376Dic17semidirect
208. C17#12799semidirect
220. C2×D4C4⋊Dic17semidirect
249. C42C4⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626022
#145287722
#145269092
C17×#23282
#1279917