Finite Group Databasebeta

← #14541850

Subgroups — #14541850

Order 2176.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763544C211yesmaximal, characteristic, complemented
2#11763201C211yesmaximal, characteristic, complemented
3#11763199C211yesmaximal, characteristic
4#11763191C211yesmaximal, characteristic, complemented
5#11763031C211yesmaximal, characteristic, complemented
6#11763028C211yesmaximal, characteristic, complemented
7#11763024C211yesmaximal, characteristic, complemented
8#11763021C211yesmaximal, characteristic, complemented
9#11763011C211yesmaximal, characteristic, complemented
10#11763009C211yesmaximal, characteristic, complemented
11#11762940C211yesmaximal, characteristic, complemented
12#11762919C211yesmaximal, characteristic, complemented
13#11762914C211yesmaximal, characteristic, complemented
14C4×#63230C211yesmaximal, characteristic, complemented
15C17×#556C211yesmaximal, characteristic, complemented
16C17×C4⋊Q8C2211yescharacteristic
17C17×#126—21no—
18C17×#126—21no—
19C17×#124C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C17×C22⋊Q8—21no—
22C4×Q8×C17C2211yescharacteristic
23C4×Q8×C17C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17C2211yescharacteristic
26C17×#117—21no—
27#10587633—21no—
28C2×#63231C2211yescharacteristic
29C2×#63231C2211yescharacteristic
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32C2×#63230C2211yescharacteristic
33C2×#63230C2211yescharacteristic
34C2×#63229C2211yescharacteristic
35C2×#63228C2211yescharacteristic
36C2×#63228C2211yescharacteristic
37#10587580—21no—
38#10587580—21no—
39#10587579—21no—
40#10587561C2211yescharacteristic
41#10587561C2211yescharacteristic
42#10587560C2211yescharacteristic
43C4×D68C2211yescharacteristic
44C4×D68C2211yescharacteristic
45C4×Dic34C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587481C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587479C2211yescharacteristic
53C4⋊#3622C2211yescharacteristic
54C4⋊#3622C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic
56#10587474C2211yescharacteristic
57#10587474C2211yescharacteristic
58C4×#3622C2211yescharacteristic
59C2×Q8×C17D411yescharacteristic
60C2×Q8×C17C2311yescharacteristic
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C2×D68—21no—
65C2×D68—21no—
66C2×D68C2311yescharacteristic
67C2×D68C2311yescharacteristic
68C2×C4×D17—21no—
69C2×C4×D17—21no—
70C2×Dic34C2311yescharacteristic
71C4⋊C4×C17—21no—
72C4⋊C4×C17—21no—
73C4⋊C4×C17—21no—
74C4⋊C4×C17—21no—
75C4⋊C4×C17D411yescharacteristic
76C4⋊C4×C17D411yescharacteristic
77C4⋊C4×C17D411yescharacteristic
78C4⋊C4×C17C2311yescharacteristic
79C22⋊C4×C17—21no—
80C22⋊C4×C17—21no—
81C22⋊C4×C17—21no—
82C22⋊C4×C17—21no—
83C22⋊C4×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17C2311yescharacteristic
87C42×C17C2311yescharacteristic
88C42×C17C2311yescharacteristic
89#63231—22no—
90#63231—41no—
91#63230—22no—
92#63230—41no—
93#63230—41no—
94#63230—41no—
95#63229—41no—
96#63228—22no—
97#63228—41no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104C4⋊Dic17C2311yescharacteristic
105#63225—21no—
106C4×Dic17—21no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C22×D17—21no—
113C22×D17—21no—
114Q8×C17—21no—
115Q8×C17—21no—
116Q8×C17—21no—
117Q8×C17—21no—
118Q8×C17—41no—
119D4×C17—21no—
120D4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—42no—
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C2×Dic17—21no—
133C2×Dic17—21no—
134D68—21no—
135D68—21no—
136D68—21no—
137D68—21no—
138D68—42no—
139D68—81no—
140C4×D17—81no—
141C4×D17—81no—
142Dic34—21no—
143Dic34—21no—
144#3622—21no—
145#3622—21no—
146#3622—21no—
147#3622#4112yes—
148#3401—171nomaximal, Sylow
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17C22×D411yescharacteristic
153C2×D17—41no—
154C2×D17—41no—
155C2×D17—41no—
156C2×D17—41no—
157C2×D17—41no—
158C2×D17—41no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17C2×#4111yescharacteristic
169C4×C17H(5,𝔽2)11yescharacteristic
170Dic17—41no—
171Dic17—41no—
172#559—171no—
173#556D1711yescharacteristic, complemented
174#497—171no—
175#496—171no—
176#495—171no—
177#464—171no—
178#464—171no—
179#463—171no—
180#463—171no—
181#461—171no—
182#460—171no—
183#446—171no—
184#442—171no—
185#441—171no—
186C4×SD16—171no—
187C2×C17—81no—
188C2×C17#55011yescharacteristic
189C2×C17#57511yescharacteristic
190C2×C17#57711yescharacteristic
191D17—81no—
192D17—81no—
193C2×Q16—171no—
194C2×Q16—171no—
195C2×SD16—171no—
196C2×SD16—171no—
197C2×SD16—171no—
198C2×SD16—171no—
199C2×SD16—171no—
200C2×D8—171no—
201C2×D8—171no—
202C4⋊Q8C2×D1711yescharacteristic
203#127—171no—
204#126—21no—
205#126—21no—
206#124—171no—
207#124—171no—
208#124—341no—
209#124C2×D1711yescharacteristic
210#124C2×D1711yescharacteristic
211#123—341no—
212#123—341no—
213C22⋊Q8—21no—
214C4×Q8—171no—
215C4×Q8C2×D1711yescharacteristic
216C4×Q8C2×D1711yescharacteristic
217C4×Q8C2×D1711yescharacteristic
218C4×D4—171no—
219C4×D4—171no—
220C4×D4—341no—
221C4×D4C2×D1711yescharacteristic
222#117—21no—
223#106—171no—
224C4⋊C8—171no—
225C4⋊C8—171no—
226C4⋊C8—171no—
227#103—171no—
228#103—171no—
229#103—171no—
230#102—171no—
231#102—171no—
232#102—171no—
233#97—171no—
234#97—171no—
235C4×C8—171no—
236C17#340111yescharacteristic, complemented, Sylow
237C2×Q8—171no—
238C2×Q8C22⋊D1711yescharacteristic
239C2×Q8C22×D1711yescharacteristic
240C2×Q8C22×D1711yescharacteristic
241C2×D4—171no—
242C2×D4—171no—
243C2×D4—341no—
244C2×D4—341no—
245C2×D4C22×D1711yescharacteristic
246C22×C4—21no—
247C22×C4—341no—
248C22×C4—341no—
249Q16—342no—
250Q16—681no—
251SD16—342no—
252SD16—681no—
253SD16—681no—
254SD16—681no—
255SD16—681no—
256D8—342no—
257D8—681no—
258C2×C8—171no—
259C2×C8—171no—
260C2×C8—171no—
261C2×C8—171no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—171no—
267C4⋊C4—341no—
268C4⋊C4C22⋊D1711yescharacteristic
269C4⋊C4C22⋊D1711yescharacteristic
270C4⋊C4C22⋊D1711yescharacteristic
271C4⋊C4C22×D1711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—341no—
278C22⋊C4—341no—
279C22⋊C4—341no—
280C22⋊C4—341no—
281C22⋊C4—341no—
282C22⋊C4—341no—
283C42—21no—
284C42—21no—
285C42—341no—
286C42C22×D1711yescharacteristic
287C42C22×D1711yescharacteristic
288C42C22×D1711yescharacteristic
289C23—21no—
290C23—341no—
291C23—341no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—341no—
298Q8—341no—
299D4—21no—
300D4—21no—
301D4—341no—
302D4—341no—
303D4—341no—
304D4—341no—
305D4—682no—
306D4—1361no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—1361no—
314C2×C4—1361no—
315C2×C4C2×C22⋊D1711yescharacteristic
316C2×C4C2×C22⋊D1711yescharacteristic
317C2×C4C2×C22⋊D1711yescharacteristic
318C2×C4C2×C22⋊D1711yescharacteristic
319C2×C4C2×C22⋊D1711yescharacteristic
320C2×C4C2×C22⋊D1711yescharacteristic
321C2×C4C23×D1711yescharacteristic
322C8—172no—
323C8—341no—
324C8—341no—
325C8—341no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—681no—
330C22—681no—
331C22—681no—
332C22—681no—
333C22—681no—
334C22—681no—
335C22C22×C22⋊D1711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—681no—
346C4—681no—
347C4#41×D1711yescharacteristic
348C4#1058769611yescharacteristic
349C2—81no—
350C2—1361no—
351C2—1361no—
352C2#1176347811yescharacteristic
353C2#1176360711yescharacteristic
354C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#556C2
socleC22×C17C22×D4
prime radical#11763199C2
hypercenter#556D17
hypocenterC17#3401
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #11763544C2semidirect
2. #11763201C2semidirect
4. #11763191C2semidirect
5. #11763031C2semidirect
6. #11763028C2semidirect
7. #11763024C2semidirect
8. #11763021C2semidirect
9. #11763011C2semidirect
10. #11763009C2semidirect
11. #11762940C2semidirect
12. #11762919C2semidirect
13. #11762914C2semidirect
14. C4×#63230C2semidirect
15. C17×#556C2semidirect
173. #556D17semidirect
236. C17#3401semidirect

Maximal subgroups

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

subgroupindex
#117635442
#117632012
#117631992
#117631912
#117630312
#117630282
#117630242
#117630212
#117630112
#117630092
#117629402
#117629192
#117629142
C4×#632302
C17×#5562
#340117