Finite Group Databasebeta

← #14540784

Subgroups — #14540784

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#11763480C211yesmaximal, characteristic, complemented
2#11763175C211yesmaximal, characteristic, complemented
3#11763133C211yesmaximal, characteristic, complemented
4#11763122C211yesmaximal, characteristic, complemented
5#11763113C211yesmaximal, characteristic, complemented
6#11763060C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic, complemented
8#11763004C211yesmaximal, characteristic, complemented
9#11762952C211yesmaximal, characteristic, complemented
10#11762952C211yesmaximal, characteristic, complemented
11#11762909C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762808C211yesmaximal, characteristic, complemented
14#11762803C211yesmaximal, characteristic, complemented
15C17×#553C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C17×#126—21no—
19C17×#125C2211yescharacteristic
20C17×#123—21no—
21C17×#121—21no—
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic
24C4×D4×C17—21no—
25C4×D4×C17C2211yescharacteristic
26C4×D4×C17C2211yescharacteristic
27C17×#117C2211yescharacteristic
28C2×C4⋊C4×C17C2211yescharacteristic
29#10587626—21no—
30C2×#63229C2211yescharacteristic
31C2×#63228C2211yescharacteristic
32C2×#63228C2211yescharacteristic
33#10587616C2211yescharacteristic
34#10587616C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587579—21no—
38#10587570—21no—
39#10587570—21no—
40#10587560C2211yescharacteristic
41C4×Dic34C2211yescharacteristic
42#10587503C2211yescharacteristic
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic
45#10587481C2211yescharacteristic
46#10587481C2211yescharacteristic
47#10587481C2211yescharacteristic
48#10587480C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59C2×D4×C17—21no—
60C2×D4×C17—21no—
61C2×D4×C17D411yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71#63252—22no—
72#63252—22no—
73#63252—41no—
74#63252—41no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×Dic34C2311yescharacteristic
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17D411yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17D411yescharacteristic
94C22⋊C4×C17D411yescharacteristic
95C42×C17—21no—
96C42×C17C2311yescharacteristic
97#63229—22no—
98#63228—41no—
99#63228—41no—
100#63227—21no—
101#63227—21no—
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116D4×C17—21no—
117D4×C17—21no—
118D4×C17—41no—
119D4×C17—42no—
120D4×C17—81no—
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—21no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2411yescharacteristic
135C22⋊D17—42no—
136C22⋊D17—42no—
137C22⋊D17—81no—
138C22⋊D17—81no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141D68—21no—
142D68—21no—
143D68—21no—
144D68—21no—
145D68—42no—
146D68—81no—
147C4×D17—42no—
148C4×D17—42no—
149Dic34—21no—
150Dic34#4112yes—
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×C17C22×D411yescharacteristic
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17C2×#4111yescharacteristic
180C4×C17H(5,𝔽2)11yescharacteristic
181Dic17—22no—
182Dic17—41no—
183#553D1711yescharacteristic, complemented
184#550—171no—
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#54611yescharacteristic
202C2×C17#57511yescharacteristic
203C2×C17#57611yescharacteristic
204D17—81no—
205D17—81no—
206C2×#41—171no—
207C2×#41—171no—
208C2×SD16—171no—
209C2×D8—171no—
210C2×D8—171no—
211C2×M16—171no—
212C22×C8—171no—
213#127—171no—
214#127—341no—
215#126—21no—
216#125C2×D1711yescharacteristic
217#123—21no—
218#121—21no—
219#121—171no—
220#121—171no—
221#121—341no—
222#121—341no—
223#121C2×D1711yescharacteristic
224#121C2×D1711yescharacteristic
225C4×Q8—171no—
226C4×D4—21no—
227C4×D4—341no—
228C4×D4C2×D1711yescharacteristic
229C4×D4C2×D1711yescharacteristic
230#117C2×D1711yescharacteristic
231C2×C4⋊C4C2×D1711yescharacteristic
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—342no—
251#41—342no—
252#41—681no—
253#41—681no—
254C2×Q8—171no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—171no—
258C2×D4—171no—
259C2×D4—341no—
260C2×D4—341no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—341no—
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—21no—
284C4⋊C4—21no—
285C4⋊C4—171no—
286C4⋊C4—341no—
287C4⋊C4C22⋊D1711yescharacteristic
288C4⋊C4C22×D1711yescharacteristic
289C4⋊C4C22×D1711yescharacteristic
290C4⋊C4C22×D1711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4C22⋊D1711yescharacteristic
298C22⋊C4C22⋊D1711yescharacteristic
299C42—21no—
300C42—341no—
301C42C22×D1711yescharacteristic
302C23—21no—
303C23—341no—
304C23—341no—
305C23C2×C22⋊D1711yescharacteristic
306C23C2×C22⋊D1711yescharacteristic
307Q8—172no—
308Q8—341no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—81no—
314D4—341no—
315D4—341no—
316D4—341no—
317D4—341no—
318D4—682no—
319D4—682no—
320D4—682no—
321D4—1361no—
322D4—1361no—
323D4—1361no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—682no—
336C2×C4—682no—
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—681no—
356C22—681no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22C22×C22⋊D1711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—342no—
370C4—681no—
371C4#41×D1711yescharacteristic
372C4#1058769611yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—1361no—
377C2—1361no—
378C2#1176344711yescharacteristic
379C2#1176360711yescharacteristic
380C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#553C2
socleC22×C17C22×D4
hypercenter#553D17
hypocenterC17#3309
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #11763480C2semidirect
2. #11763175C2semidirect
3. #11763133C2semidirect
4. #11763122C2semidirect
5. #11763113C2semidirect
6. #11763060C2semidirect
7. #11763043C2semidirect
8. #11763004C2semidirect
9. #11762952C2semidirect
10. #11762952C2semidirect
11. #11762909C2semidirect
12. #11762881C2semidirect
13. #11762808C2semidirect
14. #11762803C2semidirect
15. C17×#553C2semidirect
183. #553D17semidirect
249. C17#3309semidirect

Maximal subgroups

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

subgroupindex
#117634802
#117631752
#117631332
#117631222
#117631132
#117630602
#117630432
#117630042
#117629522
#117629092
#117628812
#117628082
#117628032
C17×#5532
#330917