Finite Group Databasebeta

← #14540340

Subgroups — #14540340

Order 2176.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587628C211yesmaximal, characteristic, complemented
2C2×#10587636C211yesmaximal, characteristic, complemented
3#11763464C211yesmaximal, characteristic, complemented
4#11763204C211yesmaximal, characteristic, complemented
5#11763053C211yesmaximal, characteristic, complemented
6#11763048C211yesmaximal, characteristic, complemented
7#11763021C211yesmaximal, characteristic, complemented
8#11763019C211yesmaximal, characteristic, complemented
9#11762971C211yesmaximal, characteristic, complemented
10#11762969C211yesmaximal, characteristic, complemented
11#11762965C211yesmaximal, characteristic, complemented
12#11762952C211yesmaximal, characteristic, complemented
13#11762907C211yesmaximal, characteristic, complemented
14#11762881C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×#63252C2211yescharacteristic
19C2×#63252C2211yescharacteristic
20C17×#124—21no—
21C17×#124C2211yescharacteristic
22C17×C22⋊Q8—21no—
23C17×C22⋊Q8C2211yescharacteristic
24C17×#121—21no—
25C17×#121C2211yescharacteristic
26C4×Q8×C17C2211yescharacteristic
27C4×D4×C17—21no—
28C4×D4×C17C2211yescharacteristic
29#10587636—24no—
30C2×#63231C2211yescharacteristic
31C2×#63231C2211yescharacteristic
32C2×#63231C2211yescharacteristic
33C2×#63231C2211yescharacteristic
34#10587628—24no—
35C2×#63230C2211yescharacteristic
36C2×#63230C2211yescharacteristic
37#10587624—21no—
38C2×#63229C2211yescharacteristic
39C2×#63229C2211yescharacteristic
40C2×#63228C2211yescharacteristic
41#10587610C2211yescharacteristic
42#10587610C2211yescharacteristic
43C2×#63223C2211yescharacteristic
44C22×#3622C2211yescharacteristic
45C4⋊C4×D17—21no—
46#10587569—21no—
47#10587569—21no—
48C4×D68C2211yescharacteristic
49#10587555C2211yescharacteristic
50#10587506C2211yescharacteristic
51#10587506C2211yescharacteristic
52#10587503C2211yescharacteristic
53C22⋊#3622C2211yescharacteristic
54C22⋊#3622C2211yescharacteristic
55#10587481C2211yescharacteristic
56#10587481C2211yescharacteristic
57#10587480C2211yescharacteristic
58#10587478C2211yescharacteristic
59C4⋊#3622C2211yescharacteristic
60#10587474C2211yescharacteristic
61#41×C17—22no—
62#41×C17—41no—
63C2×Q8×C17—21no—
64C2×Q8×C17—22no—
65C2×Q8×C17—41no—
66C2×Q8×C17C2311yescharacteristic
67C2×Q8×C17C2311yescharacteristic
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C2×C22⋊D17—21no—
77C2×C22⋊D17—21no—
78#63252—22no—
79#63252—22no—
80#63252—41no—
81#63252—41no—
82C2×D68C2311yescharacteristic
83C2×C4×D17—21no—
84C2×C4×D17—21no—
85C2×C4×D17—21no—
86C2×Dic34C2311yescharacteristic
87C2×Dic34C2311yescharacteristic
88C4⋊C4×C17—21no—
89C4⋊C4×C17—21no—
90C4⋊C4×C17D411yescharacteristic
91C4⋊C4×C17C2311yescharacteristic
92C22⋊C4×C17—21no—
93C22⋊C4×C17—21no—
94C22⋊C4×C17—21no—
95C22⋊C4×C17—21no—
96C22⋊C4×C17D411yescharacteristic
97C22⋊C4×C17D411yescharacteristic
98C42×C17—21no—
99C42×C17C2311yescharacteristic
100C22⋊Dic17—21no—
101C22⋊Dic17—21no—
102#63231—22no—
103#63231—22no—
104#63231—22no—
105#63231—41no—
106#63230—22no—
107#63230D414yescomplemented
108#63229—22no—
109#63229—41no—
110#63228—22no—
111#63227—21no—
112C4⋊Dic17—21no—
113C4⋊Dic17—21no—
114C4⋊Dic17C2311yescharacteristic
115#63225—21no—
116#63225—21no—
117#63223—22no—
118C2×#3622—22no—
119C2×#3622C2311yescharacteristic
120C2×#3622C2311yescharacteristic
121C2×#3622C2311yescharacteristic
122C2×#3622C2311yescharacteristic
123C23×C17—21no—
124C23×C17C2×D411yescharacteristic
125C23×C17C2×D411yescharacteristic
126C22×D17—21no—
127Q8×C17—21no—
128Q8×C17—21no—
129Q8×C17—21no—
130Q8×C17—42no—
131Q8×C17C2×D412yes—
132D4×C17—21no—
133D4×C17—21no—
134D4×C17—41no—
135D4×C17—42no—
136D4×C17—42no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17C2×D411yescharacteristic
147C2×C4×C17C2×D411yescharacteristic
148C2×C4×C17C2×D411yescharacteristic
149C2×C4×C17C2×D411yescharacteristic
150C2×C4×C17C2411yescharacteristic
151C22⋊D17—42no—
152C22⋊D17—42no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156D68—21no—
157D68C2×D412yes—
158C4×D17—42no—
159C4×D17—42no—
160C4×D17—42no—
161Dic34—21no—
162Dic34—21no—
163Dic34—21no—
164Dic34—21no—
165#3622—21no—
166#3622—21no—
167#3622—21no—
168#3622C2×D412yes—
169#3269—171nomaximal, Sylow
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—21no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17C22×D411yescharacteristic
180C2×D17—22no—
181C2×D17—41no—
182C2×D17—41no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—22no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17C22×D411yescharacteristic
191C4×C17H(5,𝔽2)11yescharacteristic
192Dic17—41no—
193Dic17—41no—
194Dic17—41no—
195C2×#137—171no—
196C2×#135—171no—
197#548D1711yescharacteristic, complemented
198#547—171no—
199#497—171no—
200#468—171no—
201#467—171no—
202#463—171no—
203#462—171no—
204#452—171no—
205#452—171no—
206#451—171no—
207#449—171no—
208#440—171no—
209#435—171no—
210C2×C17—41no—
211C2×C17—41no—
212C2×C17—81no—
213C2×C17D4211yescharacteristic
214C2×C17#57511yescharacteristic
215C2×C17#57811yescharacteristic
216D17—42no—
217C2×#41—171no—
218C2×#41—171no—
219C2×#41C2×D1711yescharacteristic
220C22×Q8C2×D1711yescharacteristic
221#137—344no—
222#135—344no—
223C2×Q16—171no—
224C2×Q16—171no—
225C2×Q16—171no—
226C2×Q16—171no—
227C2×SD16—171no—
228C2×SD16—171no—
229C2×SD16—171no—
230C2×SD16—171no—
231C2×D8—171no—
232C2×M16—171no—
233C22×C8—171no—
234C4⋊Q8—171no—
235#124—21no—
236#124C2×D1711yescharacteristic
237#123—341no—
238#123—341no—
239C22⋊Q8—21no—
240C22⋊Q8—171no—
241C22⋊Q8—171no—
242C22⋊Q8C2×D1711yescharacteristic
243#121—21no—
244#121—341no—
245#121C2×D1711yescharacteristic
246C4×Q8C2×D1711yescharacteristic
247C4×D4—21no—
248C4×D4—171no—
249C4×D4C2×D1711yescharacteristic
250C2×C4⋊C4—341no—
251#107—171no—
252C4⋊C8—171no—
253#103—171no—
254#103—171no—
255#103—171no—
256#103—171no—
257#102—171no—
258#102—171no—
259C22⋊C8—171no—
260C22⋊C8—171no—
261#97—171no—
262C17#326911yescharacteristic, complemented, Sylow
263#41—22no—
264#41—41no—
265#41—342no—
266#41—342no—
267#41—681no—
268#41—681no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—171no—
273C2×Q8—171no—
274C2×Q8C22×D1711yescharacteristic
275C2×Q8C22×D1711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—171no—
279C2×D4—341no—
280C2×D4—341no—
281C2×D4C22⋊D1711yescharacteristic
282C2×D4C22×D1711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—341no—
286C22×C4—341no—
287C22×C4—341no—
288C22×C4C22×D1711yescharacteristic
289C22×C4C22×D1711yescharacteristic
290Q16—342no—
291Q16—342no—
292Q16—342no—
293Q16—681no—
294SD16—174no—
295SD16—342no—
296SD16—342no—
297SD16—681no—
298D8—342no—
299M16—342no—
300C2×C8—171no—
301C2×C8—171no—
302C2×C8—171no—
303C2×C8—171no—
304C2×C8—342no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—171no—
308C4⋊C4—341no—
309C4⋊C4—341no—
310C4⋊C4—341no—
311C4⋊C4—341no—
312C4⋊C4C22⋊D1711yescharacteristic
313C4⋊C4C22×D1711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—341no—
319C22⋊C4—341no—
320C22⋊C4—341no—
321C22⋊C4C22⋊D1711yescharacteristic
322C22⋊C4C22⋊D1711yescharacteristic
323C42—21no—
324C42C22×D1711yescharacteristic
325C23—21no—
326C23—341no—
327C23C2×C22⋊D1711yescharacteristic
328C23C2×C22⋊D1711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—341no—
334Q8—341no—
335Q8—341no—
336Q8—341no—
337Q8D4×D1712yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—172no—
344D4—341no—
345D4—682no—
346D4—682no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—682no—
360C2×C4—682no—
361C2×C4—682no—
362C2×C4C2×C22⋊D1711yescharacteristic
363C2×C4C2×C22⋊D1711yescharacteristic
364C2×C4C2×C22⋊D1711yescharacteristic
365C2×C4C2×C22⋊D1711yescharacteristic
366C2×C4C23×D1711yescharacteristic
367C8—172no—
368C8—341no—
369C8—341no—
370C8—341no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—342no—
381C22—681no—
382C22—681no—
383C22C22×C22⋊D1711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—681no—
392C4—681no—
393C4—681no—
394C4C2×D4×D1711yescharacteristic
395C4#1058769111yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—682no—
400C2D4×C22⋊D1711yescharacteristic
401C2#1176360711yescharacteristic
402C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#548C2
socleC22×C17C22×D4
hypercenter#548D17
hypocenterC17#3269
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#10587628C2semidirect
2. C2×#10587636C2semidirect
3. #11763464C2semidirect
4. #11763204C2semidirect
5. #11763053C2semidirect
6. #11763048C2semidirect
7. #11763021C2semidirect
8. #11763019C2semidirect
9. #11762971C2semidirect
10. #11762969C2semidirect
11. #11762965C2semidirect
12. #11762952C2semidirect
13. #11762907C2semidirect
14. #11762881C2semidirect
15. C17×#548C2semidirect
107. #63230D4semidirect
197. #548D17semidirect
262. C17#3269semidirect

Maximal subgroups

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

subgroupindex
C2×#105876282
C2×#105876362
#117634642
#117632042
#117630532
#117630482
#117630212
#117630192
#117629712
#117629692
#117629652
#117629522
#117629072
#117628812
C17×#5482
#326917