Finite Group Databasebeta

← #14540249

Subgroups — #14540249

Order 2176.

346 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#11763430C211yesmaximal, characteristic, complemented
2#11763099C211yesmaximal, characteristic, complemented
3#11763096C211yesmaximal, characteristic, complemented
4#11763094C211yesmaximal, characteristic, complemented
5#11763091C211yesmaximal, characteristic, complemented
6#11763065C211yesmaximal, characteristic, complemented
7#11763060C211yesmaximal, 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
14#11762836C211yesmaximal, characteristic, complemented
15C17×#542C211yesmaximal, characteristic, complemented
16C17×C4⋊Q8C2211yescharacteristic
17C17×#126—21no—
18C17×#124—21no—
19C17×#124C2211yescharacteristic
20C17×#123—21no—
21C17×C22⋊Q8—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×#121C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17C2211yescharacteristic
26C17×#117C2211yescharacteristic
27C2×#63231C2211yescharacteristic
28C2×#63230C2211yescharacteristic
29C2×#63229C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587610C2211yescharacteristic
33C2×#63223C2211yescharacteristic
34C22×#3622C2211yescharacteristic
35#10587584—21no—
36#10587582—21no—
37#10587571—21no—
38#10587569—21no—
39#10587561C2211yescharacteristic
40C4×D68C2211yescharacteristic
41#10587555C2211yescharacteristic
42C4×Dic34C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587481C2211yescharacteristic
48#10587481C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587479C2211yescharacteristic
52#10587479C2211yescharacteristic
53#10587479C2211yescharacteristic
54#10587479C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic
56C4⋊#3622C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58C4⋊#3622C2211yescharacteristic
59C2×Q8×C17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17—21no—
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×D68C2311yescharacteristic
67C2×C4×D17—21no—
68C2×Dic34—21no—
69C2×Dic34C2311yescharacteristic
70C4⋊C4×C17—21no—
71C4⋊C4×C17—21no—
72C4⋊C4×C17—21no—
73C4⋊C4×C17D411yescharacteristic
74C4⋊C4×C17D411yescharacteristic
75C4⋊C4×C17C2311yescharacteristic
76C4⋊C4×C17C2311yescharacteristic
77C22⋊C4×C17—21no—
78C22⋊C4×C17—21no—
79C22⋊C4×C17—21no—
80C22⋊C4×C17—21no—
81C22⋊C4×C17—21no—
82C22⋊C4×C17D411yescharacteristic
83C22⋊C4×C17D411yescharacteristic
84C42×C17—21no—
85C42×C17C2311yescharacteristic
86C42×C17C2311yescharacteristic
87C22⋊Dic17—21no—
88#63231—41no—
89#63230—41no—
90#63229—41no—
91#63228—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101C4×Dic17—21no—
102#63223—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—
113Q8×C17—81no—
114D4×C17—21no—
115D4×C17—21no—
116D4×C17—81no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—81no—
123C2×C4×C17C2×D411yescharacteristic
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C22⋊D17—81no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133D68—21no—
134D68—21no—
135C4×D17—81no—
136Dic34—21no—
137Dic34—21no—
138Dic34—81no—
139#3622—21no—
140#3622—21no—
141#3622—21no—
142#3622—21no—
143#3263—171nomaximal, Sylow
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
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#542—171no—
168#542D1711yescharacteristic, complemented
169#476—171no—
170#476—171no—
171#475—171no—
172#475—171no—
173#470—171no—
174#469—171no—
175#465—171no—
176#465—171no—
177#464—171no—
178#464—171no—
179#463—171no—
180#463—171no—
181#424—171no—
182C2×C17—41no—
183C2×C17—81no—
184C2×C17#53711yescharacteristic
185C2×C17#57711yescharacteristic
186C2×C17#57711yescharacteristic
187D17—81no—
188C2×Q16—171no—
189C2×SD16—171no—
190C2×SD16—171no—
191C2×D8—171no—
192C2×M16—171no—
193C22×C8—171no—
194C4⋊Q8—171no—
195C4⋊Q8C2×D1711yescharacteristic
196#126—21no—
197#126—341no—
198#124—21no—
199#124—171no—
200#124—341no—
201#124C2×D1711yescharacteristic
202#123—21no—
203#123—341no—
204C22⋊Q8—21no—
205C22⋊Q8—171no—
206C22⋊Q8—341no—
207C22⋊Q8C2×D1711yescharacteristic
208#121—171no—
209#121C2×D1711yescharacteristic
210C4×Q8—171no—
211C4×Q8C2×D1711yescharacteristic
212C4×D4—171no—
213C4×D4C2×D1711yescharacteristic
214#117C2×D1711yescharacteristic
215#106—171no—
216#106—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#326311yescharacteristic, complemented, Sylow
232C2×Q8—21no—
233C2×Q8—171no—
234C2×Q8—341no—
235C2×Q8C22×D1711yescharacteristic
236C2×D4—21no—
237C2×D4—171no—
238C2×D4—341no—
239C2×D4C22×D1711yescharacteristic
240C22×C4—21no—
241C22×C4—341no—
242C22×C4C22×D1711yescharacteristic
243Q16—681no—
244SD16—681no—
245SD16—681no—
246D8—681no—
247M16—681no—
248C2×C8—171no—
249C2×C8—171no—
250C2×C8—171no—
251C2×C8—171no—
252C2×C8—681no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—171no—
257C4⋊C4—171no—
258C4⋊C4—341no—
259C4⋊C4—341no—
260C4⋊C4—341no—
261C4⋊C4C22⋊D1711yescharacteristic
262C4⋊C4C22⋊D1711yescharacteristic
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—
275C22⋊C4C22⋊D1711yescharacteristic
276C22⋊C4C22⋊D1711yescharacteristic
277C42—21no—
278C42—341no—
279C42C22×D1711yescharacteristic
280C42C22×D1711yescharacteristic
281C23—21no—
282C23—341no—
283C23C2×C22⋊D1711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—341no—
288Q8—341no—
289Q8—1361no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—341no—
294D4—341no—
295D4—1361no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—1361no—
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—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—681no—
323C22—681no—
324C22—681no—
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—41no—
340C2—81no—
341C2—1361no—
342C2#1176338711yescharacteristic
343C2#1176361711yescharacteristic
344C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#542C2
socleC22×C17C22×D4
prime radical#11763060C2
hypercenter#542D17
hypocenterC17#3263
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #11763430C2semidirect
2. #11763099C2semidirect
3. #11763096C2semidirect
4. #11763094C2semidirect
5. #11763091C2semidirect
6. #11763065C2semidirect
8. #11763040C2semidirect
9. #11763036C2semidirect
10. #11763031C2semidirect
11. #11763028C2semidirect
12. #11763024C2semidirect
13. #11763021C2semidirect
14. #11762836C2semidirect
15. C17×#542C2semidirect
168. #542D17semidirect
231. C17#3263semidirect

Maximal subgroups

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

subgroupindex
#117634302
#117630992
#117630962
#117630942
#117630912
#117630652
#117630602
#117630402
#117630362
#117630312
#117630282
#117630242
#117630212
#117628362
C17×#5422
#326317