Finite Group Databasebeta

← #16945063

Subgroups — #16945063

Order 4352.

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

normal subgroups only

This group has 586 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
1C2×#11762729C211yesmaximal, characteristic
2#14528864C211yesmaximal, characteristic
3#14526815C214yesmaximal, complemented
4C17×#1463C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic
6C22×#63225—21no—
7#11762773—21no—
8#11762729—21no—
9#11762729—21no—
10#11762729C412yes—
11#11762729C2212yes—
12C2×#10587502—21no—
13C2×#10587502C411yescharacteristic
14#11762578—21no—
15#11762578C2212yes—
16C22×C4⋊C4×C17C2211yescharacteristic
17C17×#407—21no—
18C17×#324—21no—
19C17×#324C2212yes—
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C22×Dic34—21no—
26C22×Dic34—41no—
27C22×Dic34—41no—
28C22×Dic34—44no—
29C22×Dic34C2×C412yes—
30C2×M16×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17—41no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×#63225—22no—
38C2×#63225—22no—
39C2×#63225—44no—
40C2×#63223—22no—
41C17×C22⋊C8—22no—
42C17×C22⋊C8—41no—
43#10587502—21no—
44#10587502—22no—
45#10587502—22no—
46#10587502C2×C412yes—
47C22⋊#3622—41no—
48C22⋊#3622—41no—
49C24×C17C22⋊C411yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—22no—
57C22×C4×C17—41no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—24no—
67C22×Dic17—24no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—42no—
71C22×Dic17—42no—
72C22×Dic17—42no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C2×Dic34—44no—
76C2×Dic34—42no—
77C2×Dic34—44no—
78C2×Dic34—48no—
79C2×Dic34—42no—
80C2×Dic34—44no—
81C2×Dic34—44no—
82M16×C17—82no—
83C2×C8×C17—42no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88#63225—44no—
89#63225—44no—
90#63223—42no—
91#63223—42no—
92C2×#3622—42no—
93#11815—171nomaximal, Sylow
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17#9911yescharacteristic
99C23×C17#9911yescharacteristic
100C23×C17C2×C22⋊C411yescharacteristic
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—81no—
120C2×C4×C17#13712yes—
121C2×Dic17—24no—
122C2×Dic17—24no—
123C2×Dic17—42no—
124C2×Dic17—44no—
125C2×Dic17—48no—
126C2×Dic17—42no—
127C2×Dic17—44no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130Dic34—48no—
131Dic34—44no—
132Dic34—44no—
133C8×C17—82no—
134#3622—82no—
135C2×#395—171no—
136#1821—171no—
137#1463D1711yescharacteristic
138#1354—174no—
139C22×C17—24no—
140C22×C17—24no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17AΣL(1,𝔽16)12yes—
148C22×C17#35612yes—
149C22×C17C2×#9911yescharacteristic
150C22×C17#41911yescharacteristic
151C22×C17#42111yescharacteristic
152C4×C17—22no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158Dic17—44no—
159Dic17—44no—
160C23×Q8—171no—
161C22×C4⋊C4—341no—
162C22×C4⋊C4C2×D1711yescharacteristic
163#407—21no—
164#407—341no—
165#395—172no—
166#395—172no—
167#395—341no—
168#395—341no—
169C2×#95—171no—
170C2×#95—341no—
171#343—172no—
172#343—341no—
173#324—21no—
174#324C2×D1712yes—
175C2×C17—21no—
176C2×C17—22no—
177C2×C17—22no—
178C2×C17—21no—
179C2×C17#151111yescharacteristic
180C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
181C2×C17C2×#35611yescharacteristic
182C22×Q8—172no—
183C22×Q8—341no—
184C22×Q8—681no—
185C22×Q8—681no—
186C22×Q8—684no—
187C23×C4—341no—
188C23×C4—341no—
189C23×C4C4×D1711yescharacteristic
190C23×C4D6811yescharacteristic
191C23×C4C22⋊D1711yescharacteristic
192C2×M16—41no—
193C2×M16—342no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—342no—
198C2×C4⋊C4—684no—
199C2×C4⋊C4C4×D1711yescharacteristic
200C2×C4⋊C4D6811yescharacteristic
201C2×C4⋊C4C22⋊D1711yescharacteristic
202C2×C4⋊C4C22×D1711yescharacteristic
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—681no—
206C22⋊C8—681no—
207#95—172no—
208#95—342no—
209#95—341no—
210#95—342no—
211C17#1181511yescharacteristic, complemented, Sylow
212C24#6322711yescharacteristic
213C2×Q8—682no—
214C2×Q8—682no—
215C2×Q8—684no—
216C2×Q8—684no—
217C2×Q8—688no—
218C2×Q8—684no—
219C2×Q8—684no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—344no—
231C22×C4—344no—
232C22×C4—342no—
233C22×C4—342no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4—681no—
238C22×C4—681no—
239C22×C4#6322711yescharacteristic
240C22×C4#6322711yescharacteristic
241C22×C4#6322711yescharacteristic
242C22×C4C2×C4×D1711yescharacteristic
243C22×C4C2×D6811yescharacteristic
244C22×C4C2×C22⋊D1711yescharacteristic
245M16—82no—
246M16—682no—
247M16—682no—
248C2×C8—42no—
249C2×C8—682no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—684no—
255C4⋊C4—684no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1058747711yescharacteristic
261C23#1058747711yescharacteristic
262C23C2×#6322711yescharacteristic
263Q8—684no—
264Q8—688no—
265Q8—684no—
266C2×C4—21no—
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—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—344no—
286C2×C4—344no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—684no—
290C2×C4—688no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—684no—
294C2×C4#1058759512yes—
295C8—82no—
296C8—1362no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1176259512yes—
306C22#1176260712yes—
307C22C2×#1058747711yescharacteristic
308C22#1176281911yescharacteristic
309C22#1176282511yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—684no—
317C4—684no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#1452731811yescharacteristic
323C2C2×#1176259511yescharacteristic
324C2C2×#1176260711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1463C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1463D17
hypocenterC17#11815
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #14526815C2semidirect
211. C17#11815semidirect

Maximal subgroups

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

subgroupindex
C2×#117627292
#145288642
#145268152
C17×#14632
#1181517