Finite Group Databasebeta

← #17949895

Subgroups — #17949895

Order 4352.

730 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544447C212yesmaximal, complemented
2#14543605C212yesmaximal, complemented
3#14543601C212yesmaximal, complemented
4#14531272C211yesmaximal, characteristic
5C22×#63247—22no—
6#11763675—22no—
7#11763675C2214yes—
8C22×#63252C411yescharacteristic
9C22×C4×Dic17C2211yescharacteristic
10#11762714C411yescharacteristic
11#11762714C2211yescharacteristic
12#11762710C411yescharacteristic
13#11762710C2211yescharacteristic
14#11762686—22no—
15C4×#63227—22no—
16C2×C17×#117C411yescharacteristic
17C23×C4×C17C2×C411yescharacteristic
18C22×C22⋊D17—21no—
19C23×Dic17D412yes—
20C2×#63252—22no—
21C2×#63252—22no—
22C2×#63252—42no—
23C2×#63252—42no—
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C22×Dic34C2×C411yescharacteristic
27C2×#63247—28no—
28C2×#63247—44no—
29C2×#63247—44no—
30C22×#3624—24no—
31C17×#117—42no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C42×C17—21no—
36C2×#63227—22no—
37C2×#63227—22no—
38C2×C4×Dic17—42no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D412yes—
41C2×C4×Dic17C2311yescharacteristic
42#10587502—22no—
43#10587502—22no—
44C24×C17C22⋊C411yescharacteristic
45C23×D17—21no—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22×C411yescharacteristic
54C2×C22⋊D17—24no—
55C2×C22⋊D17—42no—
56C2×C22⋊D17—42no—
57C22×Dic17—24no—
58C22×Dic17—24no—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—21no—
62C22×Dic17C22⋊C412yes—
63C22×Dic17C2×D412yes—
64#63252—48no—
65#63252—48no—
66C2×D68—21no—
67C2×D68—41no—
68C2×D68—41no—
69C2×D68M1612yescomplemented
70C2×C4×D17—24no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C2×Dic34—21no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34M1612yescomplemented
77#63247—416no—
78C2×#3624—28no—
79C2×#3624—44no—
80C2×#3624—44no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C42×C17—42no—
85#63227—44no—
86#63227—42no—
87#63227—42no—
88C4×Dic17—41no—
89C4×Dic17—42no—
90C4×Dic17—41no—
91#7631—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17#9912yes—
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—22no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17—42no—
102C22×D17—42no—
103C2×C4×C17—24no—
104C2×C4×C17—24no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17#10012yes—
114C2×C4×C17C2×M1611yescharacteristic
115C2×C4×C17C2×M1611yescharacteristic
116C22⋊D17—48no—
117C2×Dic17—28no—
118C2×Dic17—28no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17#10414yes—
127C2×Dic17AGL(1,ℤ/8)12yes—
128C2×Dic17#13712yes—
129D68—42no—
130D68—42no—
131C4×D17—48no—
132Dic34—42no—
133Dic34—42no—
134#3624—48no—
135#2297—171no—
136#1728—172no—
137#1272—172no—
138#1270—172no—
139C22×C17—22no—
140C22×C17—24no—
141C22×C17—41no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17#40711yescharacteristic
145C22×C17C2×#9911yescharacteristic
146C22×C17C2×#10011yescharacteristic
147C22×C17#41712yes—
148C22×C17C2×#10412yes—
149C2×D17—42no—
150C2×D17—44no—
151C2×D17—81no—
152C2×D17—82no—
153C2×D17—81no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—22no—
157C4×C17—81no—
158C4×C17—81no—
159Dic17—22no—
160Dic17—24no—
161Dic17—22no—
162Dic17—42no—
163C22×#41—171no—
164C22×M16—342no—
165C2×#117#59511yescharacteristic
166C22×C42—171no—
167#391—171no—
168#391—171no—
169#390—171no—
170#390—171no—
171#386—342no—
172C4×C22⋊C4—342no—
173#336—174no—
174#336—342no—
175C2×C17—42no—
176C2×C17#146211yescharacteristic
177C2×C17#148912yes—
178C2×C17#150412yes—
179C2×C17#152511yescharacteristic
180C2×C17#156011yescharacteristic
181D17—82no—
182C2×#41—342no—
183C2×#41—342no—
184C2×#41—682no—
185C2×#41—682no—
186C22×Q8—171no—
187C22×D4—171no—
188C22×D4—341no—
189C23×C4—172no—
190C23×C4—341no—
191C23×C4C2×#59511yescharacteristic
192C2×M16—348no—
193C2×M16—684no—
194C2×M16—684no—
195C22×C8—344no—
196#117—42no—
197C2×C4⋊C4C2×#59511yescharacteristic
198C2×C4⋊C4C2×#59511yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4—342no—
202C2×C42—21no—
203C2×C42—172no—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42—682no—
207#95—342no—
208#95—342no—
209C17#763111yescharacteristic, complemented, Sylow
210C24—341no—
211C24C22⋊#59511yescharacteristic
212#41—688no—
213#41—688no—
214C2×Q8—172no—
215C2×Q8—341no—
216C2×Q8—681no—
217C2×Q8—681no—
218C2×D4—172no—
219C2×D4—344no—
220C2×D4—341no—
221C2×D4—681no—
222C2×D4—681no—
223C2×D4—682no—
224C2×D4—682no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—172no—
232C22×C4—172no—
233C22×C4—344no—
234C22×C4—344no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—344no—
238C22×C4—341no—
239C22×C4—682no—
240C22×C4—682no—
241C22×C4C22⋊#59511yescharacteristic
242C22×C4C22×#59511yescharacteristic
243M16—6816no—
244C2×C8—348no—
245C2×C8—684no—
246C2×C8—684no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—682no—
251C22⋊C4—682no—
252C22⋊C4—684no—
253C42—42no—
254C42—682no—
255C42—681no—
256C42—681no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—342no—
262C23—681no—
263C23—681no—
264C23—682no—
265C23—682no—
266C23#1058753912yes—
267C23C2×C22⋊#59511yescharacteristic
268Q8—682no—
269Q8—682no—
270D4—682no—
271D4—682no—
272D4—688no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—172no—
284C2×C4—172no—
285C2×C4—174no—
286C2×C4—348no—
287C2×C4—348no—
288C2×C4—342no—
289C2×C4—342no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—688no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4#1058755312yes—
297C2×C4D17⋊M1611yescharacteristic
298C2×C4D17⋊M1611yescharacteristic
299C8—688no—
300C22—24no—
301C22—22no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—682no—
306C22—684no—
307C22—1362no—
308C22—1361no—
309C22—1361no—
310C22#1176378111yescharacteristic
311C22C2×#1058753911yescharacteristic
312C22C2×#1058755311yescharacteristic
313C22#1176380012yes—
314C22D17⋊#10412yes—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—81no—
319C4—81no—
320C4—342no—
321C4—342no—
322C4—344no—
323C4—682no—
324C2—42no—
325C2—1362no—
326C2#1454403111yescharacteristic
327C2#1454412112yes—
328C2#1454417212yes—
329C2#1454423811yescharacteristic
330C2#1454432111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC23×C17C2×C22⋊C4
prime radicalC22×#63252C4
cosocleC2×C4×Dic17C23
hypercenterC2×#117#595
hypocenterC17#7631
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #14544447C2semidirect
2. #14543605C2semidirect
3. #14543601C2semidirect
69. C2×D68M16semidirect
76. C2×Dic34M16semidirect
209. C17#7631semidirect

Maximal subgroups

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

subgroupindex
#145444472
#145436052
#145436012
#145312722
#763117