Finite Group Databasebeta

← #14540131

Subgroups — #14540131

Order 2176.

360 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763343C211yesmaximal, characteristic, complemented
2#11763116C211yesmaximal, characteristic, complemented
3#11763113C211yesmaximal, characteristic, complemented
4#11763109C211yesmaximal, characteristic, complemented
5#11763106C211yesmaximal, characteristic, complemented
6#11763048C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic
8#11763040C211yesmaximal, characteristic, complemented
9#11763036C211yesmaximal, characteristic, complemented
10#11763031C211yesmaximal, characteristic, complemented
11#11763028C211yesmaximal, characteristic, complemented
12#11763024C211yesmaximal, characteristic, complemented
13#11763021C211yesmaximal, characteristic, complemented
14C2×C4⋊#3622C211yesmaximal, characteristic, complemented
15C17×#529C211yesmaximal, characteristic, complemented
16C17×#126—21no—
17C17×#125C2211yescharacteristic
18C17×#124C2211yescharacteristic
19C17×#123—21no—
20C17×C22⋊Q8C2211yescharacteristic
21C17×#121C2211yescharacteristic
22C4×Q8×C17C2211yescharacteristic
23C4×D4×C17—21no—
24C4×D4×C17C2211yescharacteristic
25C17×#117—21no—
26C2×C42×C17C2211yescharacteristic
27C2×#63231C2211yescharacteristic
28C2×#63230C2211yescharacteristic
29C2×#63229C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587615—21no—
33C4×C22⋊D17—21no—
34#10587610C2211yescharacteristic
35C22×#3622C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587562—21no—
38#10587561C2211yescharacteristic
39C4×D68C2211yescharacteristic
40#10587558—21no—
41#10587556C2211yescharacteristic
42C4×Dic34C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587481C2211yescharacteristic
48#10587481C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587479C2211yescharacteristic
52#10587479C2211yescharacteristic
53#10587478C2211yescharacteristic
54#10587478C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic
56C4⋊#3622C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58C4⋊#3622C2211yescharacteristic
59C2×Q8×C17C2311yescharacteristic
60C2×D4×C17—21no—
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17D411yescharacteristic
64C22×C4×C17D411yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C2×C22⋊D17—21no—
67C2×D68C2311yescharacteristic
68C2×C4×D17—21no—
69C2×Dic34C2311yescharacteristic
70C4⋊C4×C17—21no—
71C4⋊C4×C17—21no—
72C4⋊C4×C17—21no—
73C4⋊C4×C17—21no—
74C4⋊C4×C17C2311yescharacteristic
75C4⋊C4×C17C2311yescharacteristic
76C22⋊C4×C17—21no—
77C22⋊C4×C17—21no—
78C22⋊C4×C17—21no—
79C22⋊C4×C17—21no—
80C22⋊C4×C17—21no—
81C42×C17—21no—
82C42×C17D411yescharacteristic
83C42×C17D411yescharacteristic
84C42×C17C2311yescharacteristic
85C42×C17C2311yescharacteristic
86C22⋊Dic17—21no—
87#63231—41no—
88#63230—41no—
89#63229—41no—
90#63228—41no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95C4⋊Dic17C2311yescharacteristic
96C4⋊Dic17C2311yescharacteristic
97#63225—21no—
98#63225—21no—
99#63225—21no—
100#63225—21no—
101C4×Dic17—21no—
102C2×#3622—41no—
103C2×#3622—41no—
104C2×#3622C2311yescharacteristic
105C2×#3622C2311yescharacteristic
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C23×C17—21no—
109C23×C17C2×D411yescharacteristic
110C22×D17—21no—
111Q8×C17—21no—
112Q8×C17—21no—
113D4×C17—21no—
114D4×C17—21no—
115D4×C17—42no—
116C2×C4×C17—22no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C22⋊D17—42no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134D68—21no—
135D68—21no—
136C4×D17—42no—
137Dic34—21no—
138Dic34—21no—
139#3622—21no—
140#3622—21no—
141#3622—21no—
142#3622—21no—
143#3250—171nomaximal, Sylow
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17#13512yes—
150C22×C17C22×D411yescharacteristic
151C2×D17—41no—
152C2×D17—41no—
153C2×D17—41no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17H(5,𝔽2)11yescharacteristic
163C4×C172−1+411yescharacteristic
164Dic17—41no—
165Dic17—41no—
166Dic17—41no—
167#529—171no—
168#529D1711yescharacteristic, complemented
169#479—171no—
170#479—171no—
171#478—171no—
172#478—171no—
173#467—171no—
174#466—171no—
175#465—171no—
176#465—171no—
177#464—171no—
178#464—171no—
179#463—171no—
180#463—171no—
181C2×C4⋊C8—171no—
182C2×C17—22no—
183C2×C17—81no—
184C2×C17#53711yescharacteristic
185C2×C17C2×#13511yescharacteristic
186C2×C17#57511yescharacteristic
187D17—81no—
188C2×Q16—171no—
189C2×SD16—171no—
190C2×SD16—171no—
191C2×D8—171no—
192C22×C8—171no—
193C22×C8—171no—
194#126—21no—
195#126—341no—
196#125—171no—
197#125C2×D1711yescharacteristic
198#124—171no—
199#124C2×D1711yescharacteristic
200#123—21no—
201#123—341no—
202C22⋊Q8—171no—
203C22⋊Q8C2×D1711yescharacteristic
204#121—171no—
205#121C2×D1711yescharacteristic
206C4×Q8—171no—
207C4×Q8C2×D1711yescharacteristic
208C4×D4—21no—
209C4×D4—171no—
210C4×D4—341no—
211C4×D4C2×D1711yescharacteristic
212#117—21no—
213#117—341no—
214C2×C42C2×D1711yescharacteristic
215#107—171no—
216#107—171no—
217#106—171no—
218#106—171no—
219C4⋊C8—171no—
220C4⋊C8—171no—
221C4⋊C8—171no—
222C4⋊C8—171no—
223#103—171no—
224#103—171no—
225#103—171no—
226#103—171no—
227#102—171no—
228#102—171no—
229#102—171no—
230#102—171no—
231C17#325011yescharacteristic, complemented, Sylow
232C2×Q8—171no—
233C2×Q8C22×D1711yescharacteristic
234C2×D4—21no—
235C2×D4—171no—
236C2×D4—341no—
237C2×D4C22×D1711yescharacteristic
238C22×C4—21no—
239C22×C4—341no—
240C22×C4C22⋊D1711yescharacteristic
241C22×C4C22⋊D1711yescharacteristic
242C22×C4C22×D1711yescharacteristic
243Q16—681no—
244SD16—681no—
245SD16—681no—
246D8—681no—
247C2×C8—171no—
248C2×C8—171no—
249C2×C8—171no—
250C2×C8—171no—
251C2×C8—681no—
252C2×C8—681no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—171no—
258C4⋊C4—171no—
259C4⋊C4—341no—
260C4⋊C4—341no—
261C4⋊C4—341no—
262C4⋊C4—341no—
263C4⋊C4C22×D1711yescharacteristic
264C4⋊C4C22×D1711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—341no—
271C22⋊C4—341no—
272C22⋊C4—341no—
273C22⋊C4—341no—
274C22⋊C4—341no—
275C42—21no—
276C42—341no—
277C42C22⋊D1711yescharacteristic
278C42C22⋊D1711yescharacteristic
279C42C22×D1711yescharacteristic
280C42C22×D1711yescharacteristic
281C23—21no—
282C23—341no—
283C23C2×C22⋊D1711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—341no—
287Q8—341no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—341no—
292D4—341no—
293D4—682no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—682no—
306C2×C4C2×C22⋊D1711yescharacteristic
307C2×C4C2×C22⋊D1711yescharacteristic
308C2×C4C2×C22⋊D1711yescharacteristic
309C2×C4C2×C22⋊D1711yescharacteristic
310C2×C4C2×C22⋊D1711yescharacteristic
311C2×C4C23×D1711yescharacteristic
312C8—341no—
313C8—341no—
314C8—341no—
315C8—341no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—681no—
322C22—681no—
323C22—681no—
324C22#1058763612yes—
325C22C22×C22⋊D1711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—681no—
335C4—681no—
336C4—681no—
337C4#1058769611yescharacteristic
338C4#1058769711yescharacteristic
339C2—22no—
340C2—81no—
341C2—1361no—
342C2#1176338711yescharacteristic
343C2C2×#1058763611yescharacteristic
344C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#529C2
socleC22×C17C22×D4
prime radical#11763043C2
hypercenter#529D17
hypocenterC17#3250
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #11763343C2semidirect
2. #11763116C2semidirect
3. #11763113C2semidirect
4. #11763109C2semidirect
5. #11763106C2semidirect
6. #11763048C2semidirect
8. #11763040C2semidirect
9. #11763036C2semidirect
10. #11763031C2semidirect
11. #11763028C2semidirect
12. #11763024C2semidirect
13. #11763021C2semidirect
14. C2×C4⋊#3622C2semidirect
15. C17×#529C2semidirect
168. #529D17semidirect
231. C17#3250semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117631162
#117631132
#117631092
#117631062
#117630482
#117630432
#117630402
#117630362
#117630312
#117630282
#117630242
#117630212
C2×C4⋊#36222
C17×#5292
#325017