Finite Group Databasebeta

← #14541220

Subgroups — #14541220

Order 2176.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763494C211yesmaximal, characteristic, complemented
2#11763460C211yesmaximal, characteristic
3#11763217C211yesmaximal, characteristic, complemented
4#11763146C211yesmaximal, characteristic, complemented
5#11763139C211yesmaximal, characteristic, complemented
6#11763136C211yesmaximal, characteristic, complemented
7#11763122C211yesmaximal, characteristic, complemented
8#11763112C211yesmaximal, characteristic, complemented
9#11763103C211yesmaximal, characteristic, complemented
10#11763056C211yesmaximal, characteristic, complemented
11#11763038C211yesmaximal, characteristic, complemented
12#11762908C211yesmaximal, characteristic, complemented
13#11762880C211yesmaximal, characteristic, complemented
14#11762850C211yesmaximal, characteristic, complemented
15C2×C17×#107C211yesmaximal, characteristic, complemented
16C2×#63252—21no—
17C22×C8×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20#10587616C2211yescharacteristic
21#10587615—21no—
22#10587615—21no—
23C4×C22⋊D17C2211yescharacteristic
24#10587612—21no—
25#10587610C2211yescharacteristic
26C2×#63219C2211yescharacteristic
27C2×#63218C2211yescharacteristic
28#10587584—21no—
29#10587583C2211yescharacteristic
30#10587582—21no—
31#10587581C2211yescharacteristic
32#10587580—21no—
33#10587579C2211yescharacteristic
34#10587578C2211yescharacteristic
35C4⋊C4×D17C2211yescharacteristic
36#10587576C2211yescharacteristic
37#10587575—21no—
38#10587574C2211yescharacteristic
39#10587573C2211yescharacteristic
40C17×#107C2211yescharacteristic
41C17×#107C2211yescharacteristic
42C17×#107C2211yescharacteristic
43C17×#107C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic
45#10587492C2211yescharacteristic
46#10587491C2211yescharacteristic
47#10587488C2211yescharacteristic
48#10587487C2211yescharacteristic
49#10587486C2211yescharacteristic
50#10587485C2211yescharacteristic
51#10587481C2211yescharacteristic
52#10587481C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587480C2211yescharacteristic
55#10587479C2211yescharacteristic
56#10587479C2211yescharacteristic
57#10587478C2211yescharacteristic
58#10587478C2211yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17—21no—
63C2×C22⋊D17D411yescharacteristic
64#63252—44no—
65C2×D68—21no—
66C2×D68C2311yescharacteristic
67C2×C4×D17—21no—
68C2×C4×D17—21no—
69C2×C4×D17C2311yescharacteristic
70C2×Dic34—21no—
71C2×Dic34C2311yescharacteristic
72C2×C8×C17—22no—
73C2×C8×C17C2311yescharacteristic
74C2×C8×C17C2311yescharacteristic
75C4⋊C4×C17—21no—
76C4⋊C4×C17—21no—
77C4⋊C4×C17C2311yescharacteristic
78C4⋊C4×C17C2311yescharacteristic
79C4⋊C4×C17C2311yescharacteristic
80C4⋊C4×C17C2311yescharacteristic
81C22⋊Dic17—21no—
82C22⋊Dic17D411yescharacteristic
83#63227—21no—
84#63227—21no—
85#63227—21no—
86#63227—21no—
87#63227—21no—
88#63227D411yescharacteristic
89C4⋊Dic17—21no—
90C4⋊Dic17—21no—
91C4⋊Dic17C2311yescharacteristic
92C4⋊Dic17C2311yescharacteristic
93#63225—21no—
94#63225—21no—
95#63225—21no—
96#63225—21no—
97#63225—21no—
98#63225D411yescharacteristic
99C4×Dic17—21no—
100C4×Dic17—21no—
101C4×Dic17C2311yescharacteristic
102C2×#3622C2311yescharacteristic
103C2×#3622C2311yescharacteristic
104#63219—22no—
105#63218—22no—
106C23×C17C2×D411yescharacteristic
107C22×D17—21no—
108C22×D17C2×D411yescharacteristic
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—42no—
116C2×C4×C17—42no—
117C2×C4×C17C2×D411yescharacteristic
118C2×C4×C17C2411yescharacteristic
119C22⋊D17—22no—
120C22⋊D17—42no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17C2×D411yescharacteristic
125C2×Dic17C2×D411yescharacteristic
126C2×Dic17C2×D411yescharacteristic
127D68—21no—
128D68—21no—
129D68—42no—
130C4×D17—21no—
131C4×D17—21no—
132C4×D17—42no—
133C4×D17—42no—
134Dic34—21no—
135Dic34—21no—
136Dic34—42no—
137C8×C17—21no—
138C8×C17#4112yescomplemented
139#3622—21no—
140#3622—21no—
141#3343—171nomaximal, Sylow
142C22×C17—21no—
143C22×C17—21no—
144C22×C17AGL(1,ℤ/8)12yes—
145C22×C17C22×D411yescharacteristic
146C2×D17—21no—
147C2×D17—21no—
148C2×D17—21no—
149C2×D17—41no—
150C2×D17—41no—
151C2×D17—41no—
152C4×C17—21no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17C2×#4111yescharacteristic
158C4×C172−1+411yescharacteristic
159Dic17—21no—
160Dic17—21no—
161Dic17—21no—
162Dic17—41no—
163Dic17—41no—
164Dic17—41no—
165#552—171no—
166#547—171no—
167#501—171no—
168#484—171no—
169#483—171no—
170#482—171no—
171#480—171no—
172#478—171no—
173#477—171no—
174#468—171no—
175#465—171no—
176#440—171no—
177#435—171no—
178#428—171no—
179C2×#107D1711yescharacteristic, complemented
180C2×C17—22no—
181C2×C17#54711yescharacteristic
182C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C17#57811yescharacteristic
184D17—41no—
185D17—81no—
186C2×#41—341no—
187C2×SD16—171no—
188C2×M16—171no—
189C22×C8C2×D1711yescharacteristic
190C4⋊Q8—171no—
191#126—341no—
192#125—171no—
193#125—341no—
194#123—341no—
195#123—341no—
196#123—341no—
197C22⋊Q8—171no—
198C22⋊Q8—171no—
199C22⋊Q8—341no—
200#121—171no—
201#121—171no—
202C4×Q8—171no—
203C4×D4—171no—
204C4×D4—171no—
205#117—171no—
206#117—341no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4C2×D1711yescharacteristic
209C2×C4⋊C4C2×D1711yescharacteristic
210#107—171no—
211#107—171no—
212#107C2×D1711yescharacteristic
213#107C2×D1711yescharacteristic
214#107C2×D1711yescharacteristic
215#107C2×D1711yescharacteristic
216#106—171no—
217#106—171no—
218#106—171no—
219C4⋊C8—171no—
220#103—171no—
221#103—171no—
222#103—171no—
223#102—171no—
224#102—171no—
225#102—171no—
226C22⋊C8—171no—
227C22⋊C8—171no—
228#97—171no—
229C17#334311yescharacteristic, complemented, Sylow
230#41—684no—
231C2×Q8—171no—
232C2×Q8—341no—
233C2×D4—171no—
234C2×D4—171no—
235C2×D4—341no—
236C2×D4—341no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—171no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4C22×D1711yescharacteristic
243SD16—342no—
244M16—342no—
245C2×C8—22no—
246C2×C8—171no—
247C2×C8—171no—
248C2×C8C22×D1711yescharacteristic
249C2×C8C22×D1711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—171no—
253C4⋊C4—171no—
254C4⋊C4—171no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4—341no—
258C4⋊C4—341no—
259C4⋊C4—341no—
260C4⋊C4—341no—
261C4⋊C4—341no—
262C4⋊C4C22×D1711yescharacteristic
263C4⋊C4C22×D1711yescharacteristic
264C4⋊C4C22×D1711yescharacteristic
265C4⋊C4C22×D1711yescharacteristic
266C22⋊C4—171no—
267C22⋊C4—171no—
268C22⋊C4—341no—
269C22⋊C4—341no—
270C22⋊C4—341no—
271C22⋊C4—341no—
272C22⋊C4—341no—
273C22⋊C4—341no—
274C42—171no—
275C42—341no—
276C42—341no—
277C23—171no—
278C23—341no—
279C23D4×D1711yescharacteristic
280Q8—341no—
281Q8—341no—
282Q8—682no—
283D4—342no—
284D4—341no—
285D4—341no—
286D4—682no—
287D4—682no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—171no—
297C2×C4—171no—
298C2×C4—171no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4D4×D1711yescharacteristic
307C2×C4C23×D1711yescharacteristic
308C8—21no—
309C8—341no—
310C8—341no—
311C8#6325212yescomplemented
312C22—21no—
313C22—21no—
314C22—341no—
315C22—341no—
316C22—341no—
317C22—681no—
318C22—681no—
319C22—681no—
320C22#1058759712yes—
321C22C2×D4×D1711yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—341no—
328C4—341no—
329C4—341no—
330C4—681no—
331C4—681no—
332C4—681no—
333C4C2×#6325211yescharacteristic
334C4#1058769411yescharacteristic
335C2—22no—
336C2—681no—
337C2—1361no—
338C2#1176346011yescharacteristic
339C2C2×#1058759711yescharacteristic
340C2#1176362211yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#107C2
socleC22×C17C22×D4
prime radical#11763460C2
hypercenterC2×#107D17
hypocenterC17#3343
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. #11763494C2semidirect
3. #11763217C2semidirect
4. #11763146C2semidirect
5. #11763139C2semidirect
6. #11763136C2semidirect
7. #11763122C2semidirect
8. #11763112C2semidirect
9. #11763103C2semidirect
10. #11763056C2semidirect
11. #11763038C2semidirect
12. #11762908C2semidirect
13. #11762880C2semidirect
14. #11762850C2semidirect
15. C2×C17×#107C2semidirect
138. C8×C17#41semidirect
179. C2×#107D17semidirect
229. C17#3343semidirect
311. C8#63252semidirect

Maximal subgroups

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

subgroupindex
#117634942
#117634602
#117632172
#117631462
#117631392
#117631362
#117631222
#117631122
#117631032
#117630562
#117630382
#117629082
#117628802
#117628502
C2×C17×#1072
#334317