Finite Group Databasebeta

← #17955928

Subgroups — #17955928

Order 4352.

582 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 140 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#14544716C211yesmaximal, characteristic, complemented
2#14544498C211yesmaximal, characteristic, complemented
3C2×#11763659C212yesmaximal, complemented
4C2×#11763657C212yesmaximal
5#14531323C211yesmaximal, characteristic
6C2×C4⋊#3624—22no—
7C2×C4⋊#3624C2212yes—
8C2×C4×#3624C2212yes—
9#11763675—22no—
10#11763659—24no—
11#11763657—24no—
12C2×#10587561C411yescharacteristic
13C2×#10587573C2211yescharacteristic
14C2×#10587579C2211yescharacteristic
15#11762714C411yescharacteristic
16#11762710C411yescharacteristic
17#11762698—22no—
18#11762656—22no—
19C42×Dic17C2211yescharacteristic
20C17×#382C411yescharacteristic
21C22×D68C2×C411yescharacteristic
22C22×C4×D17—21no—
23C22×Dic34C2×C411yescharacteristic
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29C2×#63227—22no—
30C2×#63227—22no—
31C2×#63227—21no—
32C2×#63225—21no—
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D412yes—
36C2×C4×Dic17D412yes—
37C2×C4×Dic17C2311yescharacteristic
38#10587579—22no—
39#10587579—22no—
40#10587573—22no—
41#10587573—22no—
42#10587561—22no—
43#10587561C814yescomplemented
44C4⋊#3624—22no—
45C4⋊#3624—22no—
46C4⋊#3624—44no—
47C4×#3624—22no—
48C4×#3624D414yes—
49C17×#95—22no—
50#10587502—22no—
51C23×D17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C22⋊C412yes—
55C22×C4×C17C22×C411yescharacteristic
56C22×Dic17—21no—
57C22×Dic17C22⋊C412yes—
58C22×Dic17C2×D411yescharacteristic
59C22×Dic17C2×D411yescharacteristic
60C2×D68—22no—
61C2×D68—22no—
62C2×D68—21no—
63C2×D68C2×C812yes—
64C2×C4×D17—41no—
65C2×C4×D17—41no—
66C2×C4×D17—42no—
67C2×C4×D17—42no—
68C2×Dic34—22no—
69C2×Dic34—22no—
70C2×Dic34—21no—
71C2×Dic34C2×C812yes—
72C2×#3624—24no—
73C2×#3624—44no—
74C2×#3624—42no—
75C2×#3624—44no—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—22no—
78C42×C17C2×C812yes—
79C42×C17M1612yes—
80#63227—28no—
81#63227—44no—
82#63227—42no—
83#63225—42no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—22no—
87C4×Dic17—42no—
88C4×Dic17—41no—
89C4×Dic17—41no—
90C4×Dic17C2×D412yes—
91C4×Dic17#4112yes—
92#9640—171nomaximal, Sylow
93C23×C17C2×C22⋊C411yescharacteristic
94C22×D17—22no—
95C22×D17—41no—
96C22×D17—41no—
97C22×D17—42no—
98C22×D17—42no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17C22⋊C814yes—
106C2×C4×C17C4×D412yes—
107C2×C4×C17C22×C811yescharacteristic
108C2×C4×C17C2×M1611yescharacteristic
109C2×C4×C17#13112yes—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17#12112yes—
118C2×Dic17#12411yescharacteristic
119C2×Dic17#12711yescharacteristic
120C2×Dic17AGL(1,ℤ/8)12yes—
121C2×Dic17#13712yes—
122D68—24no—
123D68—42no—
124C4×D17—82no—
125C4×D17—81no—
126C4×D17—81no—
127Dic34—24no—
128Dic34—42no—
129#3624—44no—
130#3624—82no—
131C4×#124—171no—
132#1842—171no—
133#1744—171no—
134C2×Q8⋊C8—172no—
135C2×D4⋊C8—172no—
136C22×C17#39011yescharacteristic
137C22×C17C2×C22⋊C811yescharacteristic
138C22×C17#40811yescharacteristic
139C22×C17#41712yes—
140C22×C17#42112yes—
141C2×D17—42no—
142C2×D17—44no—
143C2×D17—81no—
144C2×D17—81no—
145C2×D17—82no—
146C4×C17—24no—
147C4×C17—81no—
148C4×C17—81no—
149C4×C17C8×D412yes—
150C4×C17#43612yes—
151Dic17—22no—
152Dic17—42no—
153Dic17—42no—
154Dic17#48912yes—
155Dic17#49611yescharacteristic
156Dic17#49711yescharacteristic
157C2×#124—171no—
158C2×C4×Q8—171no—
159C2×C4×D4—171no—
160C2×C4⋊C8—172no—
161C2×C4⋊C8—342no—
162C2×C4×C8—172no—
163#391—171no—
164#390—171no—
165#388—342no—
166#382#59511yescharacteristic
167C4×C22⋊C4—342no—
168C43—171no—
169#336—342no—
170Q8⋊C8—344no—
171D4⋊C8—344no—
172C2×C17#147712yes—
173C2×C17#149512yes—
174C2×C17#195411yescharacteristic
175C2×C17#195911yescharacteristic
176C2×C17#196311yescharacteristic
177D17—82no—
178C22×Q8—171no—
179C22×D4—171no—
180C23×C4—341no—
181C22×C8—342no—
182C22×C8—342no—
183#124—174no—
184#124—342no—
185C4×Q8—342no—
186C4×Q8—342no—
187C4×D4—342no—
188C4×D4—342no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4C2×#59511yescharacteristic
191C2×C4⋊C4C2×#59511yescharacteristic
192C2×C22⋊C4—342no—
193C2×C22⋊C4—341no—
194C2×C22⋊C4—342no—
195C2×C42—171no—
196C2×C42—172no—
197C2×C42—171no—
198C2×C42—172no—
199C2×C42—341no—
200C2×C42C2×#59511yescharacteristic
201C4⋊C8—342no—
202C4⋊C8—342no—
203C4⋊C8—684no—
204C4×C8—174no—
205C4×C8—342no—
206#95—22no—
207#95—342no—
208C17#964011yescharacteristic, complemented, Sylow
209C24—341no—
210C2×Q8—172no—
211C2×Q8—341no—
212C2×Q8—342no—
213C2×Q8—342no—
214C2×D4—172no—
215C2×D4—341no—
216C2×D4—342no—
217C2×D4—342no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—172no—
223C22×C4—341no—
224C22×C4—681no—
225C22×C4—681no—
226C22×C4—682no—
227C22×C4—682no—
228C22×C4C22⋊#59512yes—
229C22×C4C22×#59511yescharacteristic
230C2×C8—344no—
231C2×C8—684no—
232C2×C8—682no—
233C2×C8—684no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—682no—
237C22⋊C4—348no—
238C22⋊C4—682no—
239C22⋊C4—684no—
240C42—172no—
241C42—172no—
242C42—342no—
243C42—342no—
244C42—342no—
245C42—682no—
246C42—681no—
247C42—681no—
248C42#6324212yes—
249C42#6324312yes—
250C23—342no—
251C23—681no—
252C23—681no—
253C23—682no—
254C23—682no—
255C23C2×C22⋊#59511yescharacteristic
256Q8—344no—
257Q8—682no—
258D4—344no—
259D4—682no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—171no—
267C2×C4—171no—
268C2×C4—172no—
269C2×C4—172no—
270C2×C4—172no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—1361no—
279C2×C4—1362no—
280C2×C4—1361no—
281C2×C4#1058754314yes—
282C2×C4C2×#6324211yescharacteristic
283C2×C4C2×#6324311yescharacteristic
284C2×C4D4×#59512yes—
285C2×C4#1058768012yes—
286C8—684no—
287C8—1362no—
288C22—682no—
289C22—684no—
290C22—1362no—
291C22—1361no—
292C22—1361no—
293C22#1176376111yescharacteristic
294C22C2×#1058754311yescharacteristic
295C22#1176378411yescharacteristic
296C22#1176380012yes—
297C22#1176380612yes—
298C4—24no—
299C4—81no—
300C4—81no—
301C4—171no—
302C4—172no—
303C4—171no—
304C4—342no—
305C4—682no—
306C4—682no—
307C4#1176383712yes—
308C4#1176384312yes—
309C2—1362no—
310C2#1454408112yes—
311C2#1454414212yes—
312C2#1454490611yescharacteristic
313C2#1454491411yescharacteristic
314C2#1454492111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#382C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C4×Dic17C23
hypercenter#382#595
hypocenterC17#9640
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #14544716C2semidirect
2. #14544498C2semidirect
3. C2×#11763659C2semidirect
43. #10587561C8semidirect
208. C17#9640semidirect

Maximal subgroups

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

subgroupindex
#145447162
#145444982
C2×#117636592
C2×#117636572
#145313232
#964017