Finite Group Databasebeta

← #17970432

Subgroups — #17970432

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763816C212yesmaximal
2C2×#11763810C211yesmaximal, characteristic
3C2×C4×#63243C211yesmaximal, characteristic
4#14544886C214yesmaximal, complemented
5#14544882C212yesmaximal, complemented
6#14544881C212yesmaximal, complemented
7#14544451C212yesmaximal
8#14531189C211yesmaximal, characteristic
9C22×#63243C2211yescharacteristic
10C22×#63243C2211yescharacteristic
11C22×#63242C2212yes—
12#11763816—28no—
13C2×#10587544C2212yes—
14C2×#10587544C2214yes—
15#11763810—24no—
16C2×C4⋊#3624C2211yescharacteristic
17C2×C4⋊#3624C2211yescharacteristic
18C2×C4⋊#3624C2214yes—
19C4×#63243—24no—
20C2×#10587542C2212yes—
21C2×C4×#3624C2212yes—
22#11763675C2214yes—
23#11763675C2214yes—
24C2×#10587578C2212yes—
25C2×C42×D17C2211yescharacteristic
26#11762679C411yescharacteristic
27#11762674C412yes—
28#11762666C412yes—
29#11762666C412yes—
30C4⋊C4×Dic17C2212yes—
31C4⋊C4×Dic17C2212yes—
32C17×#384C411yescharacteristic
33C22×C4×D17C2×C411yescharacteristic
34C22×C4×D17C2×C411yescharacteristic
35C22×C4×D17C2311yescharacteristic
36C22×#3624C2312yes—
37C22×#3624C2312yes—
38C22×#3624C2312yes—
39C22×#3624C2312yes—
40C2×#63243—22no—
41C2×#63243—22no—
42C2×#63243—22no—
43C2×#63243—22no—
44C2×#63243D414yes—
45C2×#63243Q814yes—
46C2×#63242—24no—
47C2×#63242—24no—
48C2×#63242—24no—
49C2×C4⋊C4×C17C2×C412yes—
50C2×C4⋊C4×C17C2×C412yes—
51C2×C4⋊C4×C17C2×C412yes—
52C2×C42×C17C2×C411yescharacteristic
53C2×#63227C2×C414yes—
54C2×C4⋊Dic17C2×C412yes—
55C2×C4⋊Dic17C2×C412yes—
56C2×C4⋊Dic17C2×C412yes—
57C2×C4×Dic17C2×C411yescharacteristic
58C2×C4×Dic17C2314yes—
59C2×C4×Dic17C2311yescharacteristic
60C2×C4×Dic17C2311yescharacteristic
61#10587578—24no—
62#10587578—24no—
63C42×D17D414yes—
64C42×D17Q814yes—
65#10587544—24no—
66#10587544—28no—
67#10587542—24no—
68C4⋊#3624—22no—
69C4⋊#3624—22no—
70C4⋊#3624—28no—
71C4×#3624—24no—
72#10587502C2×C414yes—
73#10587502C2×C414yes—
74C23×D17C22×C411yescharacteristic
75C22×C4×C17C22×C414yes—
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C22×C411yescharacteristic
79C22×Dic17C22×C414yes—
80C22×Dic17C22×C411yescharacteristic
81C22×Dic17C22×C411yescharacteristic
82C22×Dic17C2411yescharacteristic
83C2×C4×D17—22no—
84C2×C4×D17—22no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17C4⋊C414yes—
88C2×C4×D17C4⋊C414yes—
89C2×C4×D17C2×D412yes—
90C2×C4×D17C2×D412yes—
91C2×C4×D17C2×Q812yes—
92C2×C4×D17C2×Q812yes—
93C2×C4×D17#4112yes—
94C2×C4×D17#4112yes—
95C2×#3624—22no—
96C2×#3624—22no—
97C2×#3624—22no—
98C2×#3624—22no—
99C2×#3624—22no—
100C2×#3624—22no—
101C2×#3624—22no—
102C2×#3624—22no—
103C2×#3624—22no—
104C2×#3624—22no—
105C2×#3624C2×D414yes—
106C2×#3624C2×Q814yes—
107#63243—28no—
108#63243—28no—
109#63242—48no—
110C4⋊C4×C17—24no—
111C4⋊C4×C17—24no—
112C4⋊C4×C17—42no—
113C42×C17C4⋊C414yes—
114#63227—28no—
115C4⋊Dic17—24no—
116C4⋊Dic17—24no—
117C4⋊Dic17—42no—
118C4×Dic17—24no—
119C4×Dic17—24no—
120C4×Dic17C4⋊C414yes—
121C4×Dic17C2×D412yes—
122C4×Dic17C2×D412yes—
123C4×Dic17C2×Q812yes—
124C4×Dic17C2×Q812yes—
125#21145—171nomaximal, Sylow
126C23×C17C23×C411yescharacteristic
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17—21no—
131C22×D17C2×C4⋊C412yes—
132C22×D17C4×D412yes—
133C22×D17C4×Q812yes—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—24no—
137C2×C4×C17—24no—
138C2×C4×C17—24no—
139C2×C4×C17C2×C4⋊C412yes—
140C2×C4×C17C2×C4⋊C412yes—
141C2×C4×C17C2×C4⋊C412yes—
142C2×C4×C17C4×D412yes—
143C2×C4×C17C4×Q812yes—
144C2×C4×C17#13112yes—
145C2×C4×C17#13112yes—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—24no—
151C2×Dic17—24no—
152C2×Dic17—24no—
153C2×Dic17C2×C4⋊C412yes—
154C2×Dic17C2×C4⋊C412yes—
155C2×Dic17C22×D411yescharacteristic
156C2×Dic17C22×D411yescharacteristic
157C2×Dic17C22×Q811yescharacteristic
158C2×Dic17C22×Q811yescharacteristic
159C2×Dic17C2×#4111yescharacteristic
160C2×Dic17C2×#4111yescharacteristic
161C4×D17—28no—
162C4×D17—44no—
163C4×D17#12114yes—
164C4×D17C22⋊Q814yes—
165C4×D17#12514yes—
166C4×D17C4⋊Q814yes—
167#3624—24no—
168#3624—24no—
169#3624—42no—
170#3624—42no—
171C2×#424—172no—
172C22×C4⋊C8—171no—
173C2×C4×M16—171no—
174#2285—171no—
175#1940—174no—
176#1939—172no—
177#1939—172no—
178#1729—172no—
179C22×C17C22×C4⋊C411yescharacteristic
180C22×C17C2×C4×D411yescharacteristic
181C22×C17C2×C4×Q811yescharacteristic
182C22×C17C2×#13111yescharacteristic
183C22×C17C2×#13111yescharacteristic
184C22×C17#56811yescharacteristic
185C22×C17#56811yescharacteristic
186C2×D17—22no—
187C2×D17—22no—
188C2×D17—22no—
189C2×D17—22no—
190C2×D17—22no—
191C2×D17—22no—
192C2×D17#38914yes—
193C4×C17—22no—
194C4×C17—44no—
195C4×C17#38914yes—
196C4×C17#42414yescomplemented
197Dic17—22no—
198Dic17—22no—
199Dic17—44no—
200Dic17C2×#12111yescharacteristic
201Dic17C2×C22⋊Q811yescharacteristic
202Dic17C2×#12511yescharacteristic
203Dic17C2×C4⋊Q811yescharacteristic
204C22×M16—171no—
205C22×M16—171no—
206C23×C8—172no—
207C2×#117—172no—
208C22×C42—171no—
209#424—348no—
210C2×C4⋊C8—171no—
211C2×C4⋊C8—171no—
212C2×C4⋊C8—172no—
213C2×C4⋊C8—174no—
214C2×C4⋊C8—174no—
215C2×C4⋊C8—344no—
216C4×M16—344no—
217C2×#97—172no—
218C2×C4×C8—172no—
219#384—171no—
220#384#59511yescharacteristic
221#382—172no—
222#380—172no—
223#380—172no—
224C4×C4⋊C4—172no—
225C4×C4⋊C4—172no—
226#336—174no—
227#336—174no—
228C2×C17C2×#38911yescharacteristic
229C2×C17C2×#42411yescharacteristic
230C2×C17#290511yescharacteristic
231C2×C17#293511yescharacteristic
232C2×C17#293511yescharacteristic
233C2×C17#296211yescharacteristic
234C2×C17#296211yescharacteristic
235D17—24no—
236C23×C4—171no—
237C23×C4—171no—
238C23×C4—171no—
239C2×M16—174no—
240C2×M16—174no—
241C2×M16—342no—
242C2×M16—342no—
243C2×M16—342no—
244C2×M16—342no—
245C22×C8—172no—
246C22×C8—172no—
247C22×C8—172no—
248C22×C8—172no—
249C22×C8—344no—
250C22×C8—344no—
251C22×C8—344no—
252#117—344no—
253#117—344no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4—172no—
256C2×C4⋊C4—172no—
257C2×C4⋊C4C2×#59512yes—
258C2×C4⋊C4C2×#59512yes—
259C2×C4⋊C4C2×#59512yes—
260C2×C22⋊C4—174no—
261C2×C42—171no—
262C2×C42—171no—
263C2×C42—171no—
264C2×C42—174no—
265C2×C42—174no—
266C2×C42—174no—
267C2×C42C2×#59511yescharacteristic
268C4⋊C8—342no—
269C4⋊C8—342no—
270C4⋊C8—344no—
271C4⋊C8—348no—
272C4⋊C8—348no—
273#97—344no—
274C4×C8—344no—
275#95—174no—
276#95—174no—
277C17#2114511yescharacteristic, complemented, Sylow
278C24—171no—
279C22×C4—171no—
280C22×C4—172no—
281C22×C4—172no—
282C22×C4—172no—
283C22×C4—172no—
284C22×C4—172no—
285C22×C4—172no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—174no—
289C22×C4—174no—
290C22×C4—174no—
291C22×C4—342no—
292C22×C4—342no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4C22×#59514yes—
296C22×C4C22×#59511yescharacteristic
297C22×C4C22×#59511yescharacteristic
298C22×C4C22×#59511yescharacteristic
299M16—348no—
300M16—348no—
301C2×C8—174no—
302C2×C8—174no—
303C2×C8—342no—
304C2×C8—342no—
305C2×C8—342no—
306C2×C8—342no—
307C2×C8—342no—
308C2×C8—342no—
309C2×C8—342no—
310C2×C8—342no—
311C2×C8—342no—
312C2×C8—342no—
313C2×C8—688no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—42no—
317C4⋊C4—344no—
318C4⋊C4—344no—
319C4⋊C4—682no—
320C22⋊C4—348no—
321C42—172no—
322C42—172no—
323C42—174no—
324C42—172no—
325C42—172no—
326C42—344no—
327C42—344no—
328C42C4⋊#59514yes—
329C23—172no—
330C23—172no—
331C23—172no—
332C23—341no—
333C23—341no—
334C23—341no—
335C23—341no—
336C23C23×#59511yescharacteristic
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—171no—
343C2×C4—171no—
344C2×C4—171no—
345C2×C4—171no—
346C2×C4—171no—
347C2×C4—171no—
348C2×C4—174no—
349C2×C4—174no—
350C2×C4—174no—
351C2×C4—174no—
352C2×C4—172no—
353C2×C4—172no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—348no—
359C2×C4—344no—
360C2×C4—344no—
361C2×C4—344no—
362C2×C4—684no—
363C2×C4C2×C4⋊#59512yes—
364C2×C4C2×C4⋊#59512yes—
365C2×C4C2×C4⋊#59512yes—
366C2×C4#1058767812yes—
367C2×C4D4×#59512yes—
368C2×C4#1058768012yes—
369C2×C4Q8×#59512yes—
370C8—344no—
371C8—344no—
372C8—682no—
373C8—682no—
374C22—174no—
375C22—342no—
376C22—342no—
377C22—342no—
378C22—342no—
379C22—342no—
380C22—342no—
381C22C22×C4⋊#59511yescharacteristic
382C22C2×D4×#59511yescharacteristic
383C22C2×Q8×#59511yescharacteristic
384C22C2×#1058767811yescharacteristic
385C22C2×#1058768011yescharacteristic
386C22#1176388711yescharacteristic
387C22#1176388911yescharacteristic
388C4—22no—
389C4—44no—
390C4—171no—
391C4—171no—
392C4—171no—
393C4—171no—
394C4—342no—
395C4—342no—
396C4—684no—
397C4#1176376014yes—
398C4#1176381614yescomplemented
399C2—344no—
400C2C2×#1176376011yescharacteristic
401C2C2×#1176381611yescharacteristic
402C2#1454542811yescharacteristic
403C2#1454552711yescharacteristic
404C2#1454553011yescharacteristic
405C2#1454557811yescharacteristic
406C2#1454558011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC17×#384C4
prime radicalC23×D17C22×C4
cosocleC22×Dic17C24
hypercenter#384#595
hypocenterC17#21145
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #14544886C2semidirect
5. #14544882C2semidirect
6. #14544881C2semidirect
196. C4×C17#424semidirect
277. C17#21145semidirect
398. C4#11763816semidirect

Maximal subgroups

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

subgroupindex
C2×#117638162
C2×#117638102
C2×C4×#632432
#145448862
#145448822
#145448812
#145444512
#145311892
#2114517