Finite Group Databasebeta

← #17960622

Subgroups — #17960622

Order 4352.

818 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C2×D17⋊#101C212yesmaximal
2#14543882C212yesmaximal, complemented
3#14543881C212yesmaximal, complemented
4D17×#391C211yesmaximal, characteristic
5C2×D17⋊M16—22no—
6D17⋊#101—24no—
7D17⋊#101C418yescomplemented
8C2×#10587545—22no—
9C2×#10587545C2214yes—
10C22×Q8×D17C2211yescharacteristic
11C2×C4⋊C4×D17C411yescharacteristic
12C2×C42×D17C411yescharacteristic
13#11762716C411yescharacteristic
14#11762715C2211yescharacteristic
15#11762714C2211yescharacteristic
16#11762713—22no—
17#95×D17—22no—
18C17×#391C411yescharacteristic
19C22×Q8×C17C2×C411yescharacteristic
20C2×Q8×D17—41no—
21C2×Q8×D17—41no—
22C2×Q8×D17—42no—
23C2×Q8×D17C2×C414yes—
24C2×Q8×D17D412yes—
25C2×Q8×D17Q812yes—
26C22×C4×D17—21no—
27C22×C4×D17—21no—
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17D412yes—
30C22×Dic34D412yes—
31C22×Dic34C2311yescharacteristic
32C2×#63247—24no—
33D17⋊M16—28no—
34D17⋊M16—44no—
35C2×#63243—42no—
36C2×#63242—42no—
37C2×C4⋊C4×C17C2×C411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42C2×C4×Dic17C2×C411yescharacteristic
43C4⋊C4×D17—41no—
44C4⋊C4×D17—41no—
45C42×D17—42no—
46#10587545—28no—
47#10587545—24no—
48#10587545C2×C418yescomplemented
49C17×#95—22no—
50#10587502—22no—
51#10587502—22no—
52#10587502—22no—
53C23×D17C22⋊C411yescharacteristic
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17C4212yes—
57C2×Q8×C17C4⋊C412yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C2×D412yes—
66Q8×D17—44no—
67Q8×D17—44no—
68Q8×D17—42no—
69Q8×D17—42no—
70Q8×D17—42no—
71Q8×D17—42no—
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—24no—
75C2×C4×D17—22no—
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—24no—
79C2×C4×D17—42no—
80C2×C4×D17—42no—
81C2×C4×D17—44no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17C4212yes—
85C2×C4×D17C22⋊C414yes—
86C2×C4×D17C4⋊C412yes—
87C2×C4×D17C4⋊C414yes—
88C2×Dic34—42no—
89C2×Dic34—42no—
90C2×Dic34—41no—
91C2×Dic34—41no—
92C2×Dic34C22⋊C414yes—
93C2×Dic34C4⋊C414yes—
94C2×Dic34C22×C412yes—
95C2×Dic34C2×D411yescharacteristic
96C2×Dic34C2×Q811yescharacteristic
97#63247—28no—
98#63247—44no—
99C2×#3624—44no—
100#63243—48no—
101#63242—48no—
102C4⋊C4×C17—41no—
103C42×C17—41no—
104C4⋊Dic17—41no—
105#63225—42no—
106C4×Dic17—42no—
107C4×Dic17—41no—
108C2×#1394—171nomaximal, Sylow
109C23×C17C2×C22⋊C411yescharacteristic
110C22×D17—22no—
111C22×D17—22no—
112C22×D17#9512yes—
113C22×D17#9912yes—
114C22×D17#9911yescharacteristic
115C22×D17#9911yescharacteristic
116Q8×C17—42no—
117Q8×C17—42no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17#9514yes—
128C2×C4×C17C2×C4211yescharacteristic
129C2×C4×C17C2×C4⋊C411yescharacteristic
130C2×Dic17—22no—
131C2×Dic17—22no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—22no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—42no—
138C2×Dic17—41no—
139C2×Dic17#9512yes—
140C2×Dic17C2×C22⋊C412yes—
141C2×Dic17C2×C4⋊C412yes—
142C4×D17—24no—
143C4×D17—24no—
144C4×D17—22no—
145C4×D17—22no—
146C4×D17—22no—
147C4×D17—22no—
148C4×D17—24no—
149C4×D17—24no—
150C4×D17—48no—
151C4×D17—82no—
152C4×D17—82no—
153Dic34—44no—
154Dic34—44no—
155Dic34—42no—
156Dic34—41no—
157Dic34—41no—
158#3624—48no—
159C22×#101—172no—
160C2×#391—171no—
161#1394—172no—
162#1394—172no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17C2×#9511yescharacteristic
166C22×C17C2×#9911yescharacteristic
167C22×C17C2×#9911yescharacteristic
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—24no—
171C2×D17—24no—
172C2×D17#34212yes—
173C2×D17#34212yes—
174C2×D17#35512yes—
175C2×D17#35612yes—
176C4×C17—24no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—42no—
180C4×C17—81no—
181Dic17—22no—
182Dic17—22no—
183Dic17—22no—
184Dic17—42no—
185Dic17—81no—
186C23×Q8—171no—
187C22×M16—342no—
188C22×C4⋊C4—171no—
189C22×C42—171no—
190C2×#101—174no—
191C2×#101—178no—
192C2×#101—342no—
193C2×#101—344no—
194#391—171no—
195#391—171no—
196#391—171no—
197#391—342no—
198#391#59511yescharacteristic
199C2×#95—342no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17C2×#34211yescharacteristic
203C2×C17C2×#35511yescharacteristic
204C2×C17C2×#35611yescharacteristic
205D17—22no—
206D17#139414yescomplemented
207C22×Q8—174no—
208C22×Q8—172no—
209C22×Q8—172no—
210C22×Q8—172no—
211C22×Q8—171no—
212C22×Q8—682no—
213C22×Q8—681no—
214C22×Q8—681no—
215C22×Q8C2×#59511yescharacteristic
216C23×C4—172no—
217C23×C4—171no—
218C23×C4—341no—
219C23×C4—341no—
220C2×M16—348no—
221C2×M16—344no—
222C2×M16—682no—
223C2×M16—684no—
224C22×C8—682no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—681no—
228C2×C4⋊C4—681no—
229C2×C4⋊C4C2×#59511yescharacteristic
230C2×C42—171no—
231C2×C42—341no—
232C2×C42—682no—
233C2×C42C2×#59511yescharacteristic
234#101—178no—
235#101—344no—
236#101—348no—
237#95—22no—
238#95—342no—
239#95—342no—
240#95—342no—
241C17C2×#139411yescharacteristic, complemented, Sylow
242C24—171no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—172no—
246C2×Q8—171no—
247C2×Q8—171no—
248C2×Q8—174no—
249C2×Q8—174no—
250C2×Q8—681no—
251C2×Q8—682no—
252C2×Q8—681no—
253C2×Q8—682no—
254C2×Q8—682no—
255C2×Q8—684no—
256C2×Q8—682no—
257C2×Q8—684no—
258C2×Q8—682no—
259C2×Q8—682no—
260C2×Q8C4×#59512yes—
261C2×Q8C4⋊#59512yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—171no—
265C22×C4—172no—
266C22×C4—172no—
267C22×C4—172no—
268C22×C4—174no—
269C22×C4—174no—
270C22×C4—342no—
271C22×C4—341no—
272C22×C4—341no—
273C22×C4—342no—
274C22×C4—344no—
275C22×C4—342no—
276C22×C4—344no—
277C22×C4—341no—
278C22×C4—341no—
279C22×C4—682no—
280C22×C4—682no—
281C22×C4—681no—
282C22×C4—681no—
283C22×C4—684no—
284C22×C4C22⋊#59512yes—
285C22×C4C22×#59511yescharacteristic
286M16—348no—
287M16—688no—
288M16—684no—
289C2×C8—688no—
290C2×C8—684no—
291C4⋊C4—41no—
292C4⋊C4—681no—
293C4⋊C4—682no—
294C42—41no—
295C42—681no—
296C42—682no—
297C23—171no—
298C23—172no—
299C23—172no—
300C23—171no—
301C23—342no—
302C23—342no—
303C23C2×C22⋊#59511yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—682no—
307Q8—681no—
308Q8—681no—
309Q8—684no—
310Q8—684no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—172no—
321C2×C4—172no—
322C2×C4—172no—
323C2×C4—344no—
324C2×C4—344no—
325C2×C4—342no—
326C2×C4—342no—
327C2×C4—342no—
328C2×C4—342no—
329C2×C4—344no—
330C2×C4—344no—
331C2×C4—341no—
332C2×C4—342no—
333C2×C4—341no—
334C2×C4—342no—
335C2×C4—342no—
336C2×C4—688no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4—682no—
340C2×C4—681no—
341C2×C4—1362no—
342C2×C4—1362no—
343C2×C4(C2×#595)⋊C414yes—
344C2×C4C2×C4×#59511yescharacteristic
345C2×C4C2×C4⋊#59511yescharacteristic
346C8—688no—
347C22—21no—
348C22—21no—
349C22—172no—
350C22—172no—
351C22—172no—
352C22—172no—
353C22—344no—
354C22—344no—
355C22—341no—
356C22—341no—
357C22C2×(C2×#595)⋊C411yescharacteristic
358C22D17⋊#9911yescharacteristic
359C22D17⋊#9911yescharacteristic
360C4—21no—
361C4—21no—
362C4—24no—
363C4—42no—
364C4—81no—
365C4—342no—
366C4—342no—
367C4—342no—
368C4—682no—
369C4—1361no—
370C2—21no—
371C2—21no—
372C2—174no—
373C2—342no—
374C2#1454448111yescharacteristic
375C2D17⋊#35511yescharacteristic
376C2D17⋊#35611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC22×Dic34C23
hypercenter#391#595
hypocenterC17C2×#1394
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #14543882C2semidirect
3. #14543881C2semidirect
7. D17⋊#101C4semidirect
48. #10587545C2×C4semidirect
206. D17#1394semidirect
241. C17C2×#1394semidirect

Maximal subgroups

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

subgroupindex
C2×D17⋊#1012
#145438822
#145438812
D17×#3912
C2×#139417