Finite Group Databasebeta

← #17949894

Subgroups — #17949894

Order 4352.

730 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 102 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#14544448C212yesmaximal, complemented
2#14543605C212yesmaximal, complemented
3#14543600C212yesmaximal, complemented
4#14531271C211yesmaximal, characteristic
5C22×#63243—22no—
6#11763675—22no—
7#11763675C2214yes—
8C22×#63254C411yescharacteristic
9C2×#10587578C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11#11762715C411yescharacteristic
12#11762714C2211yescharacteristic
13#11762712C2211yescharacteristic
14#11762685—22no—
15C22⋊C4×Dic17—22no—
16C17×#390C411yescharacteristic
17C22×D4×C17C2×C411yescharacteristic
18C22×C22⋊D17—21no—
19C23×Dic17—21no—
20C2×#63254—22no—
21C2×#63254—22no—
22C2×#63254—42no—
23C2×#63254—42no—
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D412yes—
26C22×Dic34C2×C411yescharacteristic
27C22×#3624—24no—
28C2×#63243—28no—
29C2×#63243—44no—
30C2×#63243—44no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C42×C17C2×C411yescharacteristic
34C2×C22⋊Dic17—22no—
35C2×#63227—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4×Dic17—21no—
38C2×C4×Dic17D412yes—
39C2×C4×Dic17C2311yescharacteristic
40#10587578—42no—
41C42×D17—42no—
42#10587502—22no—
43#10587502—22no—
44C24×C17—21no—
45C23×D17C22⋊C411yescharacteristic
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17M1612yescomplemented
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C2×C22⋊D17—24no—
54C2×C22⋊D17—42no—
55C2×C22⋊D17—42no—
56C22×Dic17—24no—
57C22×Dic17—21no—
58C22×Dic17—21no—
59C22×Dic17—42no—
60C22×Dic17—42no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C2×D412yes—
63#63254—48no—
64#63254—48no—
65C2×C4×D17—24no—
66C2×C4×D17—22no—
67C2×C4×D17—24no—
68C2×C4×D17—21no—
69C2×C4×D17—21no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×Dic34—21no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34M1612yescomplemented
77C2×#3624—28no—
78C2×#3624—44no—
79C2×#3624—44no—
80#63243—416no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—44no—
83C42×C17—41no—
84C22⋊Dic17—42no—
85C22⋊Dic17—42no—
86#63227—42no—
87C4⋊Dic17—41no—
88C4×Dic17—42no—
89C4×Dic17—41no—
90C4×Dic17—42no—
91#7631—171nomaximal, Sylow
92C23×C17—22no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—22no—
101C22×D17—22no—
102C22×D17#9912yes—
103D4×C17—42no—
104D4×C17—42no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17#10414yes—
112C2×C4×C17C2×M1611yescharacteristic
113C2×C4×C17C2×M1611yescharacteristic
114C22⋊D17—48no—
115C2×Dic17—21no—
116C2×Dic17—22no—
117C2×Dic17—21no—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—48no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—42no—
125C2×Dic17#10012yes—
126C2×Dic17AGL(1,ℤ/8)12yes—
127C2×Dic17#13712yes—
128C4×D17—28no—
129C4×D17—28no—
130C4×D17—24no—
131C4×D17—24no—
132Dic34—42no—
133Dic34—42no—
134#3624—48no—
135#2297—171no—
136#1728—172no—
137#1272—172no—
138#1270—172no—
139C22×C17—42no—
140C22×C17—44no—
141C22×C17—81no—
142C22×C17—82no—
143C22×C17—81no—
144C22×C17#40711yescharacteristic
145C22×C17C2×#9911yescharacteristic
146C22×C17C2×#10011yescharacteristic
147C22×C17#41712yes—
148C22×C17C2×#10412yes—
149C2×D17—22no—
150C2×D17—24no—
151C2×D17—41no—
152C2×D17—42no—
153C2×D17—41no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—24no—
157C4×C17—81no—
158Dic17—22no—
159Dic17—22no—
160Dic17—22no—
161Dic17—42no—
162Dic17—81no—
163C22×#41—171no—
164C22×M16—342no—
165C2×#117—171no—
166C22×C42—171no—
167#391—171no—
168#391—171no—
169#390—171no—
170#390#59511yescharacteristic
171#386—342no—
172C4×C22⋊C4—342no—
173#336—174no—
174#336—342no—
175C2×C17—82no—
176C2×C17#146211yescharacteristic
177C2×C17#148912yes—
178C2×C17#150412yes—
179C2×C17#152511yescharacteristic
180C2×C17#156011yescharacteristic
181D17—42no—
182C2×#41—342no—
183C2×#41—342no—
184C2×#41—682no—
185C2×#41—682no—
186C22×Q8—171no—
187C22×D4—341no—
188C22×D4C2×#59511yescharacteristic
189C23×C4—172no—
190C23×C4—171no—
191C23×C4—341no—
192C2×M16—348no—
193C2×M16—684no—
194C2×M16—684no—
195C22×C8—344no—
196#117—682no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4C2×#59511yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—341no—
201C2×C22⋊C4—342no—
202C2×C42—172no—
203C2×C42—171no—
204C2×C42—341no—
205C2×C42—682no—
206C2×C42C2×#59511yescharacteristic
207#95—342no—
208#95—342no—
209C17#763111yescharacteristic, complemented, Sylow
210C24—21no—
211C24—171no—
212#41—688no—
213#41—688no—
214C2×Q8—172no—
215C2×Q8—341no—
216C2×Q8—681no—
217C2×Q8—681no—
218C2×D4—21no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—344no—
222C2×D4—682no—
223C2×D4—682no—
224C2×D4#6324312yescomplemented
225C22×C4—21no—
226C22×C4—171no—
227C22×C4—172no—
228C22×C4—342no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4—344no—
232C22×C4—342no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—344no—
236C22×C4—344no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—682no—
240C22×C4—682no—
241C22×C4C22⋊#59512yes—
242C22×C4C22×#59511yescharacteristic
243M16—6816no—
244C2×C8—348no—
245C2×C8—684no—
246C2×C8—684no—
247C4⋊C4—41no—
248C4⋊C4—681no—
249C22⋊C4—44no—
250C22⋊C4—682no—
251C22⋊C4—682no—
252C22⋊C4—682no—
253C42—41no—
254C42—682no—
255C42—681no—
256C42—682no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—172no—
263C23—341no—
264C23—341no—
265C23—342no—
266C23—342no—
267C23C2×C22⋊#59511yescharacteristic
268Q8—682no—
269Q8—682no—
270D4—42no—
271D4—42no—
272D4—688no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—172no—
280C2×C4—172no—
281C2×C4—172no—
282C2×C4—348no—
283C2×C4—344no—
284C2×C4—344no—
285C2×C4—348no—
286C2×C4—341no—
287C2×C4—342no—
288C2×C4—341no—
289C2×C4—342no—
290C2×C4—342no—
291C2×C4—688no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—682no—
296C2×C4#1058754814yes—
297C2×C4C2×#6324311yescharacteristic
298C2×C4D17⋊M1611yescharacteristic
299C8—688no—
300C22—42no—
301C22—44no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22—344no—
306C22—342no—
307C22—681no—
308C22—681no—
309C22—682no—
310C22#1176378011yescharacteristic
311C22D17⋊#9911yescharacteristic
312C22D17⋊#10011yescharacteristic
313C22#1176380012yes—
314C22C2×#1058754812yes—
315C4—24no—
316C4—21no—
317C4—21no—
318C4—81no—
319C4—342no—
320C4—342no—
321C4—342no—
322C4—682no—
323C4—1361no—
324C2—82no—
325C2—682no—
326C2#1454403011yescharacteristic
327C2#1454412012yes—
328C2#1454416912yes—
329C2#1454423911yescharacteristic
330C2#1454432211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC22×#63254C4
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#7631
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #14544448C2semidirect
2. #14543605C2semidirect
3. #14543600C2semidirect
49. C2×D4×C17M16semidirect
76. C2×Dic34M16semidirect
209. C17#7631semidirect
224. C2×D4#63243semidirect

Maximal subgroups

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

subgroupindex
#145444482
#145436052
#145436002
#145312712
#763117