Finite Group Databasebeta

← #17958129

Subgroups — #17958129

Order 4352.

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

normal subgroups only

This group has 678 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#14544015C212yesmaximal, complemented
2#14543711C212yesmaximal, complemented
3#14543709C212yesmaximal
4#14531163C211yesmaximal, characteristic
5#11763825—22no—
6#11763654—22no—
7#11763652C2214yes—
8C23×C4×D17C411yescharacteristic
9C2×#10587612C2211yescharacteristic
10#11762670C411yescharacteristic
11#11762670C2211yescharacteristic
12#11762666—22no—
13#11762664C411yescharacteristic
14#11762664C2211yescharacteristic
15C4×#63227—22no—
16C2×C17×#117C411yescharacteristic
17C24×D17C2×C411yescharacteristic
18C23×C4×C17C2×C411yescharacteristic
19C23×Dic17C2×C411yescharacteristic
20C22×C4×D17—21no—
21C22×C4×D17—21no—
22C22×C4×D17—22no—
23C22×C4×D17—41no—
24C22×C4×D17—41no—
25C22×C4×D17—42no—
26C22×C4×D17C814yescomplemented
27C17×#117—42no—
28C2×C4⋊C4×C17—21no—
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17C2×C411yescharacteristic
31C2×C42×C17—21no—
32C2×C22⋊Dic17C2×C411yescharacteristic
33C2×C22⋊Dic17C2311yescharacteristic
34C2×#63227—22no—
35C2×#63227—22no—
36#10587612—41no—
37#10587612—41no—
38C2×C4⋊Dic17D412yes—
39C2×C4×Dic17D412yes—
40C22⋊#3624—24no—
41C4⋊#3624—42no—
42C4×#3624—42no—
43#10587502—22no—
44#10587502—22no—
45C24×C17C22×C411yescharacteristic
46C23×D17—21no—
47C23×D17—21no—
48C23×D17—44no—
49C23×D17—41no—
50C23×D17—41no—
51C23×D17C2×C812yes—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—22no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C2×C812yes—
60C22×C4×C17M1612yes—
61C22×Dic17—21no—
62C22×Dic17—21no—
63C22×Dic17—41no—
64C22×Dic17—41no—
65C22×Dic17C22⋊C412yes—
66C22×Dic17C2×C812yes—
67C22×Dic17C2×D412yes—
68C2×C4×D17—24no—
69C2×C4×D17—24no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C2×C4×D17—44no—
73C2×C4×D17—44no—
74C2×C4×D17—41no—
75C2×C4×D17—41no—
76C2×C4×D17—42no—
77C2×C4×D17—44no—
78C2×C4×D17—44no—
79C2×#3624—44no—
80C4⋊C4×C17—42no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C42×C17—42no—
84C22⋊Dic17—41no—
85C22⋊Dic17—41no—
86#63227—44no—
87#63227—44no—
88C4⋊Dic17—42no—
89C4×Dic17—22no—
90C4×Dic17—22no—
91#11425—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—42no—
95C23×C17C2×C22⋊C411yescharacteristic
96C23×C17C22×C811yescharacteristic
97C23×C17C2×M1611yescharacteristic
98C22×D17—24no—
99C22×D17—22no—
100C22×D17—44no—
101C22×D17—44no—
102C22×D17—42no—
103C22×D17—42no—
104C22×D17—48no—
105C22×D17—44no—
106C22×D17—41no—
107C22×D17—41no—
108C22×D17—42no—
109C22×D17—44no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17C22⋊C812yes—
121C2×C4×C17C22⋊C812yes—
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#10012yes—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C4×D17—48no—
135C4×D17—44no—
136C4×D17—44no—
137#3624—84no—
138#2280—171no—
139#1458—172no—
140#1317—172no—
141#1316—172no—
142C22×C17—22no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17C2×C22⋊C811yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#10011yescharacteristic
152C2×D17—24no—
153C2×D17—44no—
154C2×D17—42no—
155C2×D17—48no—
156C2×D17—44no—
157C2×D17—44no—
158C2×D17—42no—
159C2×D17—44no—
160C4×C17—22no—
161C4×C17—81no—
162C4×C17—81no—
163C4×C17#32312yes—
164C4×C17#32312yes—
165Dic17—42no—
166Dic17—42no—
167Dic17—42no—
168C24×C4—171no—
169C2×#117—171no—
170C2×#117#59511yescharacteristic
171#431—342no—
172#381—171no—
173#381—171no—
174#380—342no—
175#379—171no—
176#379—171no—
177C4×C22⋊C4—342no—
178#324—342no—
179#323—174no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—41no—
183C2×C17—41no—
184C2×C17C2×#32311yescharacteristic
185C2×C17#212012yes—
186D17—44no—
187C25—171no—
188C23×C4—171no—
189C23×C4—174no—
190C23×C4—342no—
191C23×C4—341no—
192C23×C4—341no—
193C23×C4—681no—
194C23×C4—681no—
195C23×C4—682no—
196C23×C4C2×#59511yescharacteristic
197#117—42no—
198#117—681no—
199#117—681no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—172no—
202C2×C22⋊C4—171no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4C2×#59511yescharacteristic
207C2×C22⋊C4C2×#59511yescharacteristic
208C2×C42—21no—
209C2×C42—172no—
210C4⋊C8—682no—
211C22⋊C8—344no—
212C4×C8—682no—
213#95—342no—
214#95—342no—
215C17#1142511yescharacteristic, complemented, Sylow
216C24—172no—
217C24—341no—
218C24—341no—
219C24—681no—
220C24—681no—
221C24—684no—
222C24C22×#59511yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—172no—
229C22×C4—172no—
230C22×C4—172no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4—344no—
234C22×C4—344no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4—684no—
238C22×C4—684no—
239C22×C4—684no—
240C22×C4—684no—
241C22×C4—681no—
242C22×C4—681no—
243C22×C4—682no—
244C22×C4—681no—
245C22×C4—681no—
246C22×C4#6324212yes—
247C22×C4#6324312yes—
248C22×C4C22⋊#59511yescharacteristic
249C22×C4C22⋊#59511yescharacteristic
250C2×C8—684no—
251C4⋊C4—42no—
252C4⋊C4—682no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—684no—
256C22⋊C4—684no—
257C22⋊C4—681no—
258C22⋊C4—681no—
259C42—42no—
260C42—342no—
261C42—342no—
262C23—21no—
263C23—21no—
264C23—42no—
265C23—344no—
266C23—342no—
267C23—682no—
268C23—682no—
269C23—682no—
270C23—684no—
271C23—684no—
272C23—684no—
273C23—688no—
274C23—684no—
275C23—681no—
276C23—681no—
277C23C2×#6324211yescharacteristic
278C23C2×#6324311yescharacteristic
279C23C2×C22⋊#59511yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—688no—
294C2×C4—682no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—684no—
298C2×C4—684no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4#1058753911yescharacteristic
303C2×C4#1058753911yescharacteristic
304C2×C4#1058754312yes—
305C2×C4#1058754312yes—
306C2×C4#1058755312yes—
307C8—1364no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—344no—
316C22—684no—
317C22—682no—
318C22—684no—
319C22—684no—
320C22—688no—
321C22—682no—
322C22—684no—
323C22C2×#1058754311yescharacteristic
324C22C2×#1058753911yescharacteristic
325C22C2×#1058755311yescharacteristic
326C4—22no—
327C4—81no—
328C4—81no—
329C4—682no—
330C4—682no—
331C4—682no—
332C4#1176365312yes—
333C4#1176365312yes—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—684no—
339C2C2×#1176365311yescharacteristic
340C2#1454503712yes—
subgroupG/H
centerC2×C4#10587539
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC2×C17×#117C4
socleC22×C17C2×#99
prime radicalC24×D17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenterC2×#117#595
hypocenterC17#11425
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #14544015C2semidirect
2. #14543711C2semidirect
26. C22×C4×D17C8semidirect
215. C17#11425semidirect

Maximal subgroups

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

subgroupindex
#145440152
#145437112
#145437092
#145311632
#1142517