Finite Group Databasebeta

← #17949893

Subgroups — #17949893

Order 4352.

730 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 104 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#14543604C212yesmaximal, complemented
3#14543600C212yesmaximal, complemented
4#14531275C211yesmaximal, characteristic
5C22×#63247—22no—
6#11763675—22no—
7#11763675C2214yes—
8C2×#10587612C411yescharacteristic
9C22×C4×Dic17C2211yescharacteristic
10#11762716C411yescharacteristic
11#11762716C2211yescharacteristic
12#11762712C411yescharacteristic
13#11762712C2211yescharacteristic
14#11762689—22no—
15C4×C22⋊Dic17—22no—
16C22×#41×C17C411yescharacteristic
17C2×#41×C17—22no—
18C2×#41×C17—22no—
19C2×#41×C17—42no—
20C2×#41×C17—42no—
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17—21no—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17—21no—
25C23×C4×C17C2×C411yescharacteristic
26C23×Dic17D412yes—
27C2×#63247—28no—
28C2×#63247—44no—
29C2×#63247—44no—
30C22×#3624—24no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34#10587612—42no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4×Dic17—21no—
38C2×C4×Dic17—42no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D412yes—
41C2×C4×Dic17C2311yescharacteristic
42#10587502—22no—
43#10587502—22no—
44C24×C17—21no—
45C24×C17C22⋊C411yescharacteristic
46#41×C17—48no—
47#41×C17—48no—
48C2×Q8×C17—21no—
49C2×Q8×C17—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17M1612yescomplemented
52C2×D4×C17—24no—
53C2×D4×C17—21no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—42no—
57C2×D4×C17—42no—
58C2×D4×C17M1612yescomplemented
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—24no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—24no—
70C22×Dic17—24no—
71C22×Dic17—22no—
72C22×Dic17—22no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17C22⋊C412yes—
76C22×Dic17C2×D412yes—
77#63247—416no—
78C2×#3624—28no—
79C2×#3624—44no—
80C2×#3624—44no—
81C22⋊Dic17—42no—
82C22⋊Dic17—44no—
83C22⋊Dic17—42no—
84C22⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4×Dic17—41no—
88C4×Dic17—42no—
89C4×Dic17—41no—
90C4×Dic17—42no—
91#7631—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17#9912yes—
102C23×C17C2×C22⋊C411yescharacteristic
103Q8×C17—42no—
104Q8×C17—42no—
105D4×C17—42no—
106D4×C17—42no—
107D4×C17—48no—
108C2×C4×C17—24no—
109C2×C4×C17—24no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—48no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17#10012yes—
119C2×C4×C17C2×M1611yescharacteristic
120C2×C4×C17C2×M1611yescharacteristic
121C2×Dic17—28no—
122C2×Dic17—28no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17#10414yes—
132C2×Dic17AGL(1,ℤ/8)12yes—
133C2×Dic17#13712yes—
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—42no—
145C22×C17—44no—
146C22×C17—81no—
147C22×C17—82no—
148C22×C17—81no—
149C22×C17#40711yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#10011yescharacteristic
152C22×C17#41712yes—
153C22×C17C2×#10412yes—
154C4×C17—22no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—42no—
158Dic17—22no—
159Dic17—24no—
160Dic17—22no—
161Dic17—81no—
162Dic17—81no—
163C22×#41#59511yescharacteristic
164C22×M16—342no—
165C2×#117—171no—
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—82no—
177C2×C17#146211yescharacteristic
178C2×C17#148912yes—
179C2×C17#150412yes—
180C2×C17#152511yescharacteristic
181C2×C17#156011yescharacteristic
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—42no—
185C2×#41—42no—
186C22×Q8C2×#59511yescharacteristic
187C22×D4—21no—
188C22×D4C2×#59511yescharacteristic
189C23×C4—21no—
190C23×C4—172no—
191C23×C4C2×#59511yescharacteristic
192C2×M16—348no—
193C2×M16—684no—
194C2×M16—684no—
195C22×C8—344no—
196#117—682no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4—341no—
202C2×C42—172no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—341no—
206C2×C42—682no—
207#95—342no—
208#95—342no—
209C17#763111yescharacteristic, complemented, Sylow
210C24—21no—
211C24C22⋊#59511yescharacteristic
212#41—48no—
213#41—48no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8#6324712yescomplemented
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4#6324712yescomplemented
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—172no—
234C22×C4—172no—
235C22×C4—344no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—344no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4C22⋊#59511yescharacteristic
242C22×C4C22×#59511yescharacteristic
243M16—6816no—
244C2×C8—348no—
245C2×C8—684no—
246C2×C8—684no—
247C4⋊C4—681no—
248C4⋊C4—681no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4—684no—
252C22⋊C4—682no—
253C42—682no—
254C42—681no—
255C42—681no—
256C42—682no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—41no—
263C23—41no—
264C23—42no—
265C23—42no—
266C23#1058755112yes—
267C23C2×C22⋊#59511yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—42no—
271D4—42no—
272D4—48no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—48no—
281C2×C4—41no—
282C2×C4—41no—
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—681no—
293C2×C4—681no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4#1058755312yes—
297C2×C4C2×#6324711yescharacteristic
298C2×C4C2×#6324711yescharacteristic
299C8—688no—
300C22—24no—
301C22—22no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—44no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22#1176377911yescharacteristic
311C22C2×#1058755111yescharacteristic
312C22C2×#1058755311yescharacteristic
313C22#1176380012yes—
314C22D17⋊#10412yes—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—342no—
320C4—342no—
321C4—344no—
322C4—1361no—
323C4—1361no—
324C2—42no—
325C2—82no—
326C2#1454402911yescharacteristic
327C2#1454411912yes—
328C2#1454417012yes—
329C2#1454423811yescharacteristic
330C2#1454432111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
Fitting, prime radicalC22×#41×C17C4
socleC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×#41#595
hypocenterC17#7631
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #14544447C2semidirect
2. #14543604C2semidirect
3. #14543600C2semidirect
51. C2×Q8×C17M16semidirect
58. C2×D4×C17M16semidirect
209. C17#7631semidirect
217. C2×Q8#63247semidirect
224. C2×D4#63247semidirect

Maximal subgroups

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

subgroupindex
#145444472
#145436042
#145436002
#145312752
#763117