Finite Group Databasebeta

← #14536629

Subgroups — #14536629

Order 2176.

434 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762748C211yesmaximal, characteristic, complemented
2#11762744C211yesmaximal, characteristic, complemented
3#11762738C211yesmaximal, characteristic, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762736C211yesmaximal, characteristic, complemented
6#11762726C211yesmaximal, characteristic
7#11762711C211yesmaximal, characteristic
8#11762707C211yesmaximal, characteristic, complemented
9#11762701C211yesmaximal, characteristic, complemented
10#11762698C211yesmaximal, characteristic, complemented
11#11762692C211yesmaximal, characteristic, complemented
12#11762687C211yesmaximal, characteristic
13#11762683C211yesmaximal, characteristic, complemented
14#11762680C211yesmaximal, characteristic, complemented
15C17×#387C211yesmaximal, characteristic, complemented
16C22×D68C2211yescharacteristic
17C22×C4×D17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C42×C17C2211yescharacteristic
21C2×#63227C2211yescharacteristic
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×C4×Dic17C2211yescharacteristic
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
41C17×#95C2211yescharacteristic
42C17×#95C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C23×D17C2311yescharacteristic
52C23×D17C2311yescharacteristic
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C2×D68—22no—
67C2×D68—22no—
68C2×D68—41no—
69C2×C4×D17—41no—
70C2×C4×D17—41no—
71C2×C4×D17—41no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—41no—
74C42×C17—41no—
75#63227—22no—
76#63227—22no—
77#63227—22no—
78#63227—22no—
79#63227—41no—
80#63227—41no—
81#63227—41no—
82#63227—41no—
83#63227—41no—
84#63227—41no—
85#63227—41no—
86C4⋊Dic17—22no—
87C4⋊Dic17—22no—
88C4⋊Dic17—41no—
89#63225—41no—
90#63225—41no—
91#63225—41no—
92C4×Dic17—41no—
93C23×C17C2411yescharacteristic
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17#4112yes—
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—21no—
126C2×C4×C17—21no—
127C2×C4×C17#4112yes—
128C2×C4×C17#4112yes—
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—
147D68—44no—
148C4×D17—44no—
149#2796—171nomaximal, Sylow
150C22×C17C2×#4111yescharacteristic
151C22×C17C2×#4111yescharacteristic
152C22×C17C2×#4111yescharacteristic
153C22×C17H(5,𝔽2)11yescharacteristic
154C22×C17H(5,𝔽2)11yescharacteristic
155C22×C17H(5,𝔽2)11yescharacteristic
156C22×C172−1+411yescharacteristic
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C4×C17—22no—
172C4×C17—22no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#400—171no—
185#399—171no—
186#397—171no—
187#397—171no—
188#397—171no—
189#394—171no—
190#390—171no—
191#389—171no—
192#388—171no—
193#388—171no—
194#387—171no—
195#387D1711yescharacteristic, complemented
196#386—171no—
197#385—171no—
198#385—171no—
199C2×C17#53911yescharacteristic
200C2×C17#53911yescharacteristic
201C2×C17#54211yescharacteristic
202C2×C17#55111yescharacteristic
203C2×C17#55311yescharacteristic
204C2×C17#55911yescharacteristic
205C2×C17#56011yescharacteristic
206D17—42no—
207D17—81no—
208C22×D4—171no—
209C23×C4—171no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4C2×D1711yescharacteristic
217C2×C4⋊C4C2×D1711yescharacteristic
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⋊C4—171no—
226C2×C22⋊C4—171no—
227C2×C22⋊C4—171no—
228C2×C22⋊C4—171no—
229C2×C42—171no—
230C2×C42C2×D1711yescharacteristic
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#95C2×D1711yescharacteristic
240#95C2×D1711yescharacteristic
241#95C2×D1711yescharacteristic
242#95C2×D1711yescharacteristic
243C17#279611yescharacteristic, complemented, Sylow
244C24—171no—
245C24—171no—
246C2×D4—342no—
247C2×D4—342no—
248C2×D4—681no—
249C22×C4—171no—
250C22×C4—171no—
251C22×C4—171no—
252C22×C4—171no—
253C22×C4—171no—
254C22×C4—171no—
255C22×C4—681no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4C22×D1711yescharacteristic
259C22×C4C22×D1711yescharacteristic
260C22×C4C22×D1711yescharacteristic
261C22×C4C22×D1711yescharacteristic
262C22×C4C22×D1711yescharacteristic
263C22×C4C22×D1711yescharacteristic
264C22×C4C22×D1711yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—342no—
268C4⋊C4—342no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C22⋊C4—342no—
274C22⋊C4—342no—
275C22⋊C4—342no—
276C22⋊C4—342no—
277C22⋊C4—681no—
278C22⋊C4—681no—
279C22⋊C4—681no—
280C22⋊C4—681no—
281C22⋊C4—681no—
282C22⋊C4—681no—
283C22⋊C4—681no—
284C42—41no—
285C42—681no—
286C23—172no—
287C23—341no—
288C23—341no—
289C23—341no—
290C23—341no—
291C23—341no—
292C23—341no—
293C23—341no—
294C23—341no—
295C23—341no—
296C23—341no—
297C23—341no—
298C23—341no—
299C23—341no—
300C23C23×D1711yescharacteristic
301D4—684no—
302C2×C4—21no—
303C2×C4—21no—
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—341no—
322C2×C4—341no—
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—684no—
340C2×C4#6325212yes—
341C2×C4#6325612yes—
342C22—342no—
343C22—342no—
344C22—342no—
345C22—681no—
346C22—681no—
347C22—681no—
348C22—681no—
349C22—681no—
350C22—681no—
351C22—681no—
352C22—681no—
353C22—681no—
354C22—681no—
355C22—681no—
356C22C2×#6325211yescharacteristic
357C22#1058769111yescharacteristic
358C22C2×#6325611yescharacteristic
359C22#41×D1711yescharacteristic
360C22#1058769611yescharacteristic
361C22#1058769611yescharacteristic
362C22#1058769711yescharacteristic
363C4—22no—
364C4—22no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4—681no—
374C4—681no—
375C4—681no—
376C2—682no—
377C2—1361no—
378C2#1176340011yescharacteristic
379C2#1176340811yescharacteristic
380C2#1176343011yescharacteristic
381C2#1176348811yescharacteristic
382C2#1176351211yescharacteristic
383C2#1176354411yescharacteristic
384C2#1176354911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#387C2
prime radicalC22×D68C22
hypercenter#387D17
hypocenterC17#2796
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. #11762748C2semidirect
2. #11762744C2semidirect
3. #11762738C2semidirect
4. #11762736C2semidirect
5. #11762736C2semidirect
8. #11762707C2semidirect
9. #11762701C2semidirect
10. #11762698C2semidirect
11. #11762692C2semidirect
13. #11762683C2semidirect
14. #11762680C2semidirect
15. C17×#387C2semidirect
195. #387D17semidirect
243. C17#2796semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627442
#117627382
#117627362
#117627262
#117627112
#117627072
#117627012
#117626982
#117626922
#117626872
#117626832
#117626802
C17×#3872
#279617