Finite Group Databasebeta

← #14540790

Subgroups — #14540790

Order 2176.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763505C211yesmaximal, characteristic, complemented
2#11763173C211yesmaximal, characteristic, complemented
3#11763130C211yesmaximal, characteristic, complemented
4#11763120C211yesmaximal, characteristic, complemented
5#11763116C211yesmaximal, characteristic, complemented
6#11763060C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic, complemented
8#11763001C211yesmaximal, characteristic, complemented
9#11762954C211yesmaximal, characteristic, complemented
10#11762954C211yesmaximal, characteristic, complemented
11#11762907C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762810C211yesmaximal, characteristic, complemented
14#11762805C211yesmaximal, characteristic, complemented
15C17×#550C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic
18C17×#127—21no—
19C17×#127C2211yescharacteristic
20C17×#121—21no—
21C17×#121—21no—
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×#63230C2211yescharacteristic
28#10587625—21no—
29C2×#63228C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic
33#10587612C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587584—21no—
38#10587569—21no—
39#10587567—21no—
40C4×D68C2211yescharacteristic
41#10587556C2211yescharacteristic
42#10587506C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59#41×C17—22no—
60#41×C17—22no—
61#41×C17—41no—
62#41×C17—41no—
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C2×D4×C17C2311yescharacteristic
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C2×C22⋊D17—21no—
78C2×C22⋊D17—21no—
79C22×Dic17—21no—
80C2×D68C2311yescharacteristic
81C2×C4×D17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17D411yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17D411yescharacteristic
88C22⋊C4×C17D411yescharacteristic
89C42×C17—21no—
90C42×C17C2311yescharacteristic
91C22⋊Dic17—21no—
92#63230—22no—
93#63228—41no—
94#63228—41no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98C4⋊Dic17—21no—
99C4⋊Dic17C2311yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102#63225—21no—
103#63225—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C23×C17C2×D411yescharacteristic
115C22×D17—21no—
116Q8×C17—21no—
117Q8×C17#4112yes—
118D4×C17—21no—
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17—41no—
123D4×C17—42no—
124D4×C17—42no—
125D4×C17—42no—
126D4×C17—81no—
127D4×C17—81no—
128D4×C17—81no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—42no—
136C2×C4×C17—42no—
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2×D411yescharacteristic
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2411yescharacteristic
142C22⋊D17—42no—
143C22⋊D17—81no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—81no—
148D68—21no—
149D68—21no—
150C4×D17—81no—
151#3622—21no—
152#3622—21no—
153#3622—21no—
154#3622—21no—
155#3309—171nomaximal, Sylow
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17C22×D411yescharacteristic
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—22no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—41no—
178C4×C17C2×#4111yescharacteristic
179C4×C17H(5,𝔽2)11yescharacteristic
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183#553—171no—
184#550D1711yescharacteristic, complemented
185#490—171no—
186#482—171no—
187#480—171no—
188#479—171no—
189#469—171no—
190#466—171no—
191#459—171no—
192#449—171no—
193#449—171no—
194#440—171no—
195#435—171no—
196#417—171no—
197#416—171no—
198C2×C17—41no—
199C2×C17—41no—
200C2×C17—81no—
201C2×C17—81no—
202C2×C17#54611yescharacteristic
203C2×C17#57511yescharacteristic
204C2×C17#57611yescharacteristic
205D17—81no—
206C2×#41C2×D1711yescharacteristic
207C2×#41C2×D1711yescharacteristic
208C2×SD16—171no—
209C2×D8—171no—
210C2×D8—171no—
211C2×M16—171no—
212C22×C8—171no—
213#127—21no—
214#127C2×D1711yescharacteristic
215#126—341no—
216#125—171no—
217#123—341no—
218#121—21no—
219#121—21no—
220#121—171no—
221#121—171no—
222#121—341no—
223#121C2×D1711yescharacteristic
224#121C2×D1711yescharacteristic
225C4×Q8C2×D1711yescharacteristic
226C4×D4—21no—
227C4×D4—171no—
228C4×D4—341no—
229C4×D4C2×D1711yescharacteristic
230#117—171no—
231C2×C4⋊C4—171no—
232#107—171no—
233#107—171no—
234#106—171no—
235C4⋊C8—171no—
236#103—171no—
237#103—171no—
238#103—171no—
239#102—171no—
240#102—171no—
241#102—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246C22⋊C8—171no—
247C22⋊C8—171no—
248#97—171no—
249C17#330911yescharacteristic, complemented, Sylow
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—171no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4C22⋊D1711yescharacteristic
265C2×D4C22×D1711yescharacteristic
266C2×D4C22×D1711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273SD16—342no—
274D8—681no—
275D8—681no—
276M16—681no—
277C2×C8—171no—
278C2×C8—171no—
279C2×C8—171no—
280C2×C8—171no—
281C2×C8—681no—
282C4⋊C4—21no—
283C4⋊C4—171no—
284C4⋊C4—171no—
285C4⋊C4—171no—
286C4⋊C4—341no—
287C4⋊C4—341no—
288C4⋊C4—341no—
289C4⋊C4C22⋊D1711yescharacteristic
290C4⋊C4C22×D1711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4C22⋊D1711yescharacteristic
298C22⋊C4C22⋊D1711yescharacteristic
299C42—21no—
300C42—341no—
301C42C22×D1711yescharacteristic
302C23—21no—
303C23—21no—
304C23—341no—
305C23C2×C22⋊D1711yescharacteristic
306C23C2×C22⋊D1711yescharacteristic
307Q8—21no—
308Q8#6325412yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—341no—
321D4—341no—
322D4—682no—
323D4—1361no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4C2×C22⋊D1711yescharacteristic
338C2×C4C2×C22⋊D1711yescharacteristic
339C2×C4C2×C22⋊D1711yescharacteristic
340C2×C4C2×C22⋊D1711yescharacteristic
341C2×C4C23×D1711yescharacteristic
342C8—341no—
343C8—341no—
344C8—341no—
345C8—341no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22C22×C22⋊D1711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—681no—
369C4—681no—
370C4—681no—
371C4C2×#6325411yescharacteristic
372C4#1058769111yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—1361no—
378C2#1176345311yescharacteristic
379C2#1176360711yescharacteristic
380C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#550C2
socleC22×C17C22×D4
hypercenter#550D17
hypocenterC17#3309
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #11763505C2semidirect
2. #11763173C2semidirect
3. #11763130C2semidirect
4. #11763120C2semidirect
5. #11763116C2semidirect
6. #11763060C2semidirect
7. #11763043C2semidirect
8. #11763001C2semidirect
9. #11762954C2semidirect
10. #11762954C2semidirect
11. #11762907C2semidirect
12. #11762881C2semidirect
13. #11762810C2semidirect
14. #11762805C2semidirect
15. C17×#550C2semidirect
184. #550D17semidirect
249. C17#3309semidirect

Maximal subgroups

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

subgroupindex
#117635052
#117631732
#117631302
#117631202
#117631162
#117630602
#117630432
#117630012
#117629542
#117629072
#117628812
#117628102
#117628052
C17×#5502
#330917