Finite Group Databasebeta

← #14533338

Subgroups — #14533338

Order 2176.

482 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17⋊#401C211yesmaximal, characteristic, complemented
2#11762750C211yesmaximal, characteristic, complemented
3#11762738C211yesmaximal, characteristic, complemented
4#11762738C211yesmaximal, characteristic, complemented
5#11762700C211yesmaximal, characteristic, complemented
6#11762700C211yesmaximal, characteristic, complemented
7#11762693C211yesmaximal, characteristic, complemented
8#11762688C211yesmaximal, characteristic
9#11762686C211yesmaximal, characteristic
10#11762686C211yesmaximal, characteristic
11#11762681C211yesmaximal, characteristic, complemented
12#11762677C211yesmaximal, characteristic, complemented
13#11762667C211yesmaximal, characteristic, complemented
14C2×#10587502C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C17C211yesmaximal, characteristic, complemented
16C23×C4×C17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C23×Dic17C2211yescharacteristic
19C23×Dic17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C42×C17C2211yescharacteristic
23C2×C42×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C17×#95C2211yescharacteristic
36C17×#95C2211yescharacteristic
37#10587502C2211yescharacteristic
38#10587502C2211yescharacteristic
39#10587502C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C23×D17C2311yescharacteristic
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—41no—
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17—22no—
63C2×C22⋊D17—22no—
64C2×C22⋊D17—22no—
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17—41no—
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22⋊C4×C17—22no—
79C22⋊C4×C17—22no—
80C42×C17—22no—
81C42×C17—41no—
82C22⋊Dic17—22no—
83#63227—22no—
84#63227—22no—
85#63227—41no—
86#63227—41no—
87#63227—41no—
88#63227—41no—
89#63225—22no—
90#63225—41no—
91#63225—41no—
92#63225—41no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17#4112yes—
98C23×C17#4112yes—
99C23×C17#4112yes—
100C23×C17C2411yescharacteristic
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—44no—
126C2×C4×C17#4112yes—
127C22⋊D17—44no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—44no—
149C2×Dic17—44no—
150C2×Dic17#4112yes—
151#2510—171nomaximal, Sylow
152C22×C17—22no—
153C22×C17—22no—
154C22×C17—22no—
155C22×C17—22no—
156C22×C17—22no—
157C22×C17—22no—
158C22×C17#12614yes—
159C22×C17C2×#4111yescharacteristic
160C22×C17C2×#4111yescharacteristic
161C22×C17C2×#4111yescharacteristic
162C22×C17C2×#4111yescharacteristic
163C22×C17C2×#4111yescharacteristic
164C22×C17H(5,𝔽2)11yescharacteristic
165C22×C17H(5,𝔽2)11yescharacteristic
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C4×C17—22no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179Dic17—22no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184Dic17—41no—
185Dic17—41no—
186#401—171no—
187#400—171no—
188#397—171no—
189#397—171no—
190#388—171no—
191#388—171no—
192#387—171no—
193#386—171no—
194#386—171no—
195#386—171no—
196#385—171no—
197#383—171no—
198#380—171no—
199C4×C22⋊C4D1711yescharacteristic, complemented
200C2×#95—171no—
201C2×C17—24no—
202C2×C17C2×#12611yescharacteristic
203C2×C17#52911yescharacteristic
204C2×C17#53811yescharacteristic
205C2×C17#55111yescharacteristic
206C2×C17#55111yescharacteristic
207C2×C17#55311yescharacteristic
208C2×C17#55311yescharacteristic
209D17—81no—
210C22×D4—171no—
211C23×C4—171no—
212C23×C4—171no—
213C23×C4C2×D1711yescharacteristic
214C2×C4⋊C4—171no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—171no—
218C2×C22⋊C4—171no—
219C2×C22⋊C4—171no—
220C2×C22⋊C4—171no—
221C2×C22⋊C4—171no—
222C2×C22⋊C4—171no—
223C2×C22⋊C4—171no—
224C2×C22⋊C4—171no—
225C2×C22⋊C4C2×D1711yescharacteristic
226C2×C22⋊C4C2×D1711yescharacteristic
227C2×C42C2×D1711yescharacteristic
228C2×C42C2×D1711yescharacteristic
229#95—171no—
230#95—171no—
231#95—171no—
232#95—171no—
233#95—171no—
234#95—171no—
235#95—171no—
236#95—171no—
237#95—171no—
238#95—171no—
239#95—171no—
240#95—171no—
241#95—171no—
242#95—171no—
243#95C2×D1711yescharacteristic
244#95C2×D1711yescharacteristic
245C17#251011yescharacteristic, complemented, Sylow
246C24—171no—
247C24C22×D1711yescharacteristic
248C2×D4—342no—
249C2×D4—342no—
250C2×D4—342no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—171no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—342no—
262C22×C4—342no—
263C22×C4—342no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4C22×D1711yescharacteristic
268C22×C4C22×D1711yescharacteristic
269C22×C4C22×D1711yescharacteristic
270C22×C4C22×D1711yescharacteristic
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C4⋊C4—342no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—342no—
280C22⋊C4—342no—
281C22⋊C4—342no—
282C22⋊C4—681no—
283C22⋊C4—681no—
284C22⋊C4—681no—
285C22⋊C4—681no—
286C42—22no—
287C42—41no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—341no—
293C23—341no—
294C23—341no—
295C23—341no—
296C23—341no—
297C23—341no—
298C23—341no—
299C23#6325212yes—
300C23#6325212yes—
301C23#6325212yes—
302C23C23×D1711yescharacteristic
303D4—684no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—44no—
322C2×C4—172no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—684no—
344C2×C4—684no—
345C2×C4#6325412yes—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—681no—
353C22—681no—
354C22—681no—
355C22—681no—
356C22—681no—
357C22—681no—
358C22—681no—
359C22#1058756214yes—
360C22C2×#6325211yescharacteristic
361C22C2×#6325211yescharacteristic
362C22C2×#6325211yescharacteristic
363C22C2×#6325411yescharacteristic
364C22#1058769111yescharacteristic
365C22#41×D1711yescharacteristic
366C22#1058769611yescharacteristic
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—342no—
374C4—681no—
375C4—681no—
376C4—681no—
377C4—681no—
378C4—681no—
379C4—681no—
380C2—24no—
381C2—1361no—
382C2C2×#1058756211yescharacteristic
383C2#1176333911yescharacteristic
384C2#1176339211yescharacteristic
385C2#1176348311yescharacteristic
386C2#1176348511yescharacteristic
387C2#1176350511yescharacteristic
388C2#1176350811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC4×C22⋊C4×C17C2
prime radicalC22×C22⋊D17C22
hypercenterC4×C22⋊C4D17
hypocenterC17#2510
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C17⋊#401C2semidirect
2. #11762750C2semidirect
3. #11762738C2semidirect
4. #11762738C2semidirect
5. #11762700C2semidirect
6. #11762700C2semidirect
7. #11762693C2semidirect
11. #11762681C2semidirect
12. #11762677C2semidirect
13. #11762667C2semidirect
14. C2×#10587502C2semidirect
15. C4×C22⋊C4×C17C2semidirect
199. C4×C22⋊C4D17semidirect
245. C17#2510semidirect

Maximal subgroups

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

subgroupindex
C17⋊#4012
#117627502
#117627382
#117627002
#117626932
#117626882
#117626862
#117626812
#117626772
#117626672
C2×#105875022
C4×C22⋊C4×C172
#251017