Finite Group Databasebeta

← #17949776

Subgroups — #17949776

Order 4352.

654 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 110 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#14544641C212yesmaximal, complemented
2#14543604C212yesmaximal, complemented
3#14543601C212yesmaximal, complemented
4#14531273C211yesmaximal, characteristic
5C2×#10587544—22no—
6C2×#10587543—22no—
7#11763675C2214yes—
8C22×#63256C411yescharacteristic
9C2×#10587578C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11#11762716C2211yescharacteristic
12#11762711C411yescharacteristic
13#11762710C2211yescharacteristic
14#11762687—22no—
15#11762656—22no—
16C17×#391C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C2×#63256—22no—
19C2×#63256—42no—
20C2×#63256—42no—
21C2×#63256C814yescomplemented
22C22×D68—21no—
23C22×D68C2×C411yescharacteristic
24C22×C4×D17—21no—
25C22×C4×D17C2×C411yescharacteristic
26C22×C4×D17D412yes—
27C22×#3624—24no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×#63227—21no—
31C2×#63227—22no—
32C2×#63227—22no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4×Dic17—21no—
35C2×C4×Dic17D412yes—
36C2×C4×Dic17C2311yescharacteristic
37#10587578—42no—
38C42×D17—42no—
39#10587544—44no—
40#10587543—44no—
41C17×#95—22no—
42#10587502—22no—
43C23×D17—21no—
44C23×D17C22⋊C411yescharacteristic
45C2×Q8×C17—21no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17C2×C812yes—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D412yes—
56#63256—48no—
57#63256—48no—
58C2×D68—24no—
59C2×D68—21no—
60C2×D68—41no—
61C2×D68—41no—
62C2×D68—42no—
63C2×D68—42no—
64C2×D68C2×C812yes—
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17—24no—
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C2×C4×D17—42no—
74C2×C4×D17C2×C812yes—
75C2×C4×D17M1612yes—
76C2×#3624—44no—
77C2×#3624—44no—
78C2×#3624—44no—
79C4⋊C4×C17—41no—
80C42×C17—41no—
81#63227—44no—
82#63227—42no—
83#63227—42no—
84#63227—42no—
85C4⋊Dic17—41no—
86C4×Dic17—24no—
87C4×Dic17—41no—
88C4×Dic17—42no—
89#7597—171nomaximal, Sylow
90C23×C17C2×C22⋊C411yescharacteristic
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—41no—
97C22×D17—41no—
98C22×D17—42no—
99C22×D17—42no—
100C22×D17C22⋊C812yes—
101Q8×C17—42no—
102Q8×C17—42no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17C22×C811yescharacteristic
114C2×C4×C17C2×M1611yescharacteristic
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17—41no—
118C2×Dic17—42no—
119C2×Dic17C22⋊C812yes—
120C2×Dic17#9912yes—
121C2×Dic17#10012yes—
122C2×Dic17AGL(1,ℤ/8)12yes—
123C2×Dic17#13712yes—
124D68—42no—
125D68—42no—
126D68—48no—
127C4×D17—24no—
128C4×D17—24no—
129C4×D17—44no—
130C4×D17—44no—
131C4×D17—48no—
132#3624—84no—
133#2297—171no—
134#1797—172no—
135#1272—172no—
136#1270—172no—
137C22×C17C2×C22⋊C811yescharacteristic
138C22×C17C2×#9911yescharacteristic
139C22×C17C2×#10011yescharacteristic
140C22×C17#41712yes—
141C22×C17#42112yes—
142C2×D17—22no—
143C2×D17—24no—
144C2×D17—41no—
145C2×D17—41no—
146C2×D17—42no—
147C2×D17—42no—
148C2×D17—44no—
149C2×D17—81no—
150C2×D17—81no—
151C2×D17—82no—
152C4×C17—21no—
153C4×C17—21no—
154C4×C17—42no—
155C4×C17—42no—
156C4×C17—81no—
157Dic17—22no—
158Dic17—22no—
159Dic17—81no—
160Dic17#32314yes—
161C22×#41—171no—
162C2×#117—171no—
163C22×C42—171no—
164C2×C4⋊C8—342no—
165C2×C22⋊C8—342no—
166#391—171no—
167#391#59511yescharacteristic
168#390—171no—
169#390—171no—
170#386—342no—
171C4×C22⋊C4—342no—
172#336—174no—
173C2×C17C2×#32311yescharacteristic
174C2×C17#147712yes—
175C2×C17#151312yes—
176C2×C17#153611yescharacteristic
177C2×C17#156011yescharacteristic
178D17—42no—
179D17—82no—
180C2×#41—174no—
181C2×#41—342no—
182C2×#41—682no—
183C2×#41—682no—
184C22×Q8C2×#59511yescharacteristic
185C22×D4—171no—
186C22×D4—341no—
187C23×C4—171no—
188C23×C4—172no—
189C23×C4—341no—
190C22×C8—344no—
191#117—682no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4C2×#59511yescharacteristic
194C2×C22⋊C4—342no—
195C2×C22⋊C4—341no—
196C2×C22⋊C4—342no—
197C2×C42—172no—
198C2×C42—171no—
199C2×C42—341no—
200C2×C42—682no—
201C2×C42C2×#59511yescharacteristic
202C4⋊C8—684no—
203C22⋊C8—684no—
204#95—22no—
205#95—342no—
206C17#759711yescharacteristic, complemented, Sylow
207C24—171no—
208C24—341no—
209#41—688no—
210#41—688no—
211C2×Q8—21no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8#6324212yes—
215C2×D4—172no—
216C2×D4—344no—
217C2×D4—341no—
218C2×D4—681no—
219C2×D4—681no—
220C2×D4—682no—
221C2×D4—682no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—171no—
225C22×C4—172no—
226C22×C4—172no—
227C22×C4—172no—
228C22×C4—342no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4—342no—
232C22×C4—344no—
233C22×C4—341no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4—682no—
238C22×C4C22⋊#59512yes—
239C22×C4C22×#59511yescharacteristic
240C2×C8—684no—
241C2×C8—684no—
242C2×C8—684no—
243C4⋊C4—41no—
244C4⋊C4—681no—
245C22⋊C4—684no—
246C22⋊C4—682no—
247C22⋊C4—682no—
248C22⋊C4—682no—
249C42—41no—
250C42—344no—
251C42—681no—
252C42—682no—
253C23—172no—
254C23—341no—
255C23—341no—
256C23—342no—
257C23—342no—
258C23—342no—
259C23—681no—
260C23—681no—
261C23—682no—
262C23—682no—
263C23C2×C22⋊#59511yescharacteristic
264Q8—42no—
265Q8—42no—
266D4—682no—
267D4—682no—
268D4—688no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—172no—
280C2×C4—172no—
281C2×C4—172no—
282C2×C4—172no—
283C2×C4—172no—
284C2×C4—344no—
285C2×C4—344no—
286C2×C4—342no—
287C2×C4—342no—
288C2×C4—688no—
289C2×C4—684no—
290C2×C4—684no—
291C2×C4—681no—
292C2×C4—682no—
293C2×C4C2×#6324211yescharacteristic
294C2×C4D17⋊M1611yescharacteristic
295C8—1364no—
296C22—342no—
297C22—344no—
298C22—682no—
299C22—681no—
300C22—681no—
301C22—682no—
302C22—684no—
303C22—1362no—
304C22—1361no—
305C22—1361no—
306C22#1176377711yescharacteristic
307C22D17⋊#9911yescharacteristic
308C22D17⋊#10011yescharacteristic
309C22#1176380012yes—
310C22#1176380812yes—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—174no—
317C4—342no—
318C4—342no—
319C4—1361no—
320C2—682no—
321C2—1362no—
322C2#1454401011yescharacteristic
323C2#1454408212yes—
324C2#1454420212yes—
325C2#1454427411yescharacteristic
326C2#1454432211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC23×C17C2×C22⋊C4
prime radicalC22×#63256C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7597
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #14544641C2semidirect
2. #14543604C2semidirect
3. #14543601C2semidirect
21. C2×#63256C8semidirect
206. C17#7597semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145436042
#145436012
#145312732
#759717