Finite Group Databasebeta

← #14540686

Subgroups — #14540686

Order 2176.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763505C211yesmaximal, characteristic, complemented
2#11763163C211yesmaximal, characteristic, complemented
3#11763147C211yesmaximal, characteristic, complemented
4#11763125C211yesmaximal, characteristic, complemented
5#11763116C211yesmaximal, characteristic, complemented
6#11763043C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763021C211yesmaximal, characteristic, complemented
9#11762954C211yesmaximal, characteristic, complemented
10#11762948C211yesmaximal, characteristic, complemented
11C4×#63230C211yesmaximal, characteristic, complemented
12D4×#3622C211yesmaximal, characteristic, complemented
13#11762805C211yesmaximal, characteristic, complemented
14C2×#10587506C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C17×#124—21no—
19C17×#124C2211yescharacteristic
20C17×C22⋊Q8—21no—
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121—21no—
23C17×#121C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×#63230C2211yescharacteristic
28C2×#63230C2211yescharacteristic
29#10587625—21no—
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic
33#10587612C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C22×#3622C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587584—21no—
38#10587569—21no—
39#10587567—21no—
40C4×D68C2211yescharacteristic
41#10587556C2211yescharacteristic
42#10587506C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587506C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587479C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58C4×#3622C2211yescharacteristic
59#41×C17—22no—
60#41×C17—41no—
61C2×Q8×C17—21no—
62C2×Q8×C17—22no—
63C2×Q8×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×Q8×C17C2311yescharacteristic
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17C2311yescharacteristic
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C2×C22⋊D17—21no—
75C2×C22⋊D17—21no—
76C22×Dic17—21no—
77C2×D68C2311yescharacteristic
78C2×C4×D17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17D411yescharacteristic
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
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#63230—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—
105C2×#3622—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115Q8×C17—21no—
116Q8×C17—21no—
117Q8×C17—21no—
118Q8×C17—42no—
119Q8×C17#4112yes—
120D4×C17—22no—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—42no—
124D4×C17—42no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2411yescharacteristic
139C22⋊D17—42no—
140C22⋊D17—81no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—81no—
145D68—21no—
146D68—21no—
147C4×D17—81no—
148#3622—21no—
149#3622—21no—
150#3622—21no—
151#3622—21no—
152#3301—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17#13512yes—
162C22×C17C22×D411yescharacteristic
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—22no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17C2×#4111yescharacteristic
174C4×C17H(5,𝔽2)11yescharacteristic
175Dic17—41no—
176Dic17—41no—
177Dic17—41no—
178#553—171no—
179#548D1711yescharacteristic, complemented
180#488—171no—
181#485—171no—
182#481—171no—
183#479—171no—
184#466—171no—
185#465—171no—
186#463—171no—
187#449—171no—
188#448—171no—
189C4×SD16—171no—
190C8×D4—171no—
191#416—171no—
192C2×#103—171no—
193C2×C17—22no—
194C2×C17—41no—
195C2×C17—81no—
196C2×C17#54611yescharacteristic
197C2×C17C2×#13511yescharacteristic
198C2×C17#57711yescharacteristic
199D17—81no—
200C2×#41C2×D1711yescharacteristic
201C22×Q8C2×D1711yescharacteristic
202C2×SD16—171no—
203C2×SD16—171no—
204C2×D8—171no—
205C22×C8—171no—
206C22×C8—171no—
207#126—341no—
208#125—171no—
209#124—21no—
210#124C2×D1711yescharacteristic
211#123—341no—
212C22⋊Q8—21no—
213C22⋊Q8C2×D1711yescharacteristic
214#121—21no—
215#121—171no—
216#121—171no—
217#121—341no—
218#121C2×D1711yescharacteristic
219C4×Q8C2×D1711yescharacteristic
220C4×D4—21no—
221C4×D4—171no—
222C4×D4—341no—
223C4×D4C2×D1711yescharacteristic
224#117—171no—
225C2×C4⋊C4—171no—
226#107—171no—
227#106—171no—
228#106—171no—
229C4⋊C8—171no—
230#103—171no—
231#103—171no—
232#103—171no—
233#103—171no—
234#103—171no—
235#103—171no—
236#102—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240C22⋊C8—171no—
241C22⋊C8—171no—
242C4×C8—171no—
243C17#330111yescharacteristic, complemented, Sylow
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D1711yescharacteristic
250C2×Q8C22×D1711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—171no—
254C2×D4—341no—
255C2×D4—341no—
256C2×D4C22⋊D1711yescharacteristic
257C2×D4C22×D1711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4C22×D1711yescharacteristic
263C22×C4C22×D1711yescharacteristic
264SD16—342no—
265SD16—681no—
266D8—681no—
267C2×C8—171no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—171no—
271C2×C8—681no—
272C2×C8—681no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—171no—
276C4⋊C4—171no—
277C4⋊C4—171no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4C22⋊D1711yescharacteristic
282C4⋊C4C22×D1711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4C22⋊D1711yescharacteristic
292C22⋊C4C22⋊D1711yescharacteristic
293C42—21no—
294C42—341no—
295C42C22×D1711yescharacteristic
296C23—21no—
297C23—341no—
298C23C2×C22⋊D1711yescharacteristic
299C23C2×C22⋊D1711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#6325412yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—341no—
311D4—341no—
312D4—682no—
313D4—1361no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—1361no—
327C2×C4—1361no—
328C2×C4C2×C22⋊D1711yescharacteristic
329C2×C4C2×C22⋊D1711yescharacteristic
330C2×C4C2×C22⋊D1711yescharacteristic
331C2×C4C2×C22⋊D1711yescharacteristic
332C2×C4C23×D1711yescharacteristic
333C8—341no—
334C8—341no—
335C8—341no—
336C8—341no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—681no—
346C22—681no—
347C22—681no—
348C22#1058763612yes—
349C22C22×C22⋊D1711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—681no—
358C4—681no—
359C4—681no—
360C4C2×#6325411yescharacteristic
361C4#1058769111yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—1361no—
366C2#1176345311yescharacteristic
367C2C2×#1058763611yescharacteristic
368C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#548C2
socleC22×C17C22×D4
hypercenter#548D17
hypocenterC17#3301
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763505C2semidirect
2. #11763163C2semidirect
3. #11763147C2semidirect
4. #11763125C2semidirect
5. #11763116C2semidirect
6. #11763043C2semidirect
7. #11763040C2semidirect
8. #11763021C2semidirect
9. #11762954C2semidirect
10. #11762948C2semidirect
11. C4×#63230C2semidirect
12. D4×#3622C2semidirect
13. #11762805C2semidirect
14. C2×#10587506C2semidirect
15. C17×#548C2semidirect
179. #548D17semidirect
243. C17#3301semidirect

Maximal subgroups

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

subgroupindex
#117635052
#117631632
#117631472
#117631252
#117631162
#117630432
#117630402
#117630212
#117629542
#117629482
C4×#632302
D4×#36222
#117628052
C2×#105875062
C17×#5482
#330117