Finite Group Databasebeta

← #14541265

Subgroups — #14541265

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763514C211yesmaximal, characteristic, complemented
2#11763507C211yesmaximal, characteristic
3#11763217C211yesmaximal, characteristic, complemented
4#11763152C211yesmaximal, characteristic, complemented
5#11763144C211yesmaximal, characteristic, complemented
6#11763127C211yesmaximal, characteristic, complemented
7#11763124C211yesmaximal, characteristic, complemented
8#11763115C211yesmaximal, characteristic, complemented
9#11763088C211yesmaximal, 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
16C22×C8×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19#10587616—21no—
20#10587616C2211yescharacteristic
21C4×C22⋊D17—21no—
22C4×C22⋊D17—21no—
23C4×C22⋊D17C2211yescharacteristic
24#10587610—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
64C2×D68—21no—
65C2×D68C2311yescharacteristic
66C2×C4×D17—21no—
67C2×C4×D17—21no—
68C2×C4×D17C2311yescharacteristic
69C2×Dic34—21no—
70C2×Dic34C2311yescharacteristic
71C2×C8×C17—22no—
72C2×C8×C17C2311yescharacteristic
73C2×C8×C17C2311yescharacteristic
74C4⋊C4×C17—21no—
75C4⋊C4×C17—21no—
76C4⋊C4×C17C2311yescharacteristic
77C4⋊C4×C17C2311yescharacteristic
78C4⋊C4×C17C2311yescharacteristic
79C4⋊C4×C17C2311yescharacteristic
80C22⋊Dic17—21no—
81C22⋊Dic17D411yescharacteristic
82#63227—21no—
83#63227—21no—
84#63227—21no—
85#63227—21no—
86#63227—21no—
87#63227D411yescharacteristic
88C4⋊Dic17—21no—
89C4⋊Dic17—21no—
90C4⋊Dic17C2311yescharacteristic
91C4⋊Dic17C2311yescharacteristic
92#63225—21no—
93#63225—21no—
94#63225—21no—
95#63225—21no—
96#63225—21no—
97#63225D411yescharacteristic
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17C2311yescharacteristic
101C2×#3622C2311yescharacteristic
102C2×#3622C2311yescharacteristic
103#63219—22no—
104#63218—22no—
105C23×C17C2×D411yescharacteristic
106C22×D17—21no—
107C22×D17C2×D411yescharacteristic
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—81no—
115C2×C4×C17—81no—
116C2×C4×C17C2×D411yescharacteristic
117C2×C4×C17C2411yescharacteristic
118C22⋊D17—22no—
119C22⋊D17—42no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17C2×D411yescharacteristic
124C2×Dic17C2×D411yescharacteristic
125C2×Dic17C2×D411yescharacteristic
126D68—21no—
127D68—21no—
128D68—81no—
129C4×D17—21no—
130C4×D17—21no—
131C4×D17—81no—
132C4×D17—81no—
133Dic34—21no—
134Dic34—21no—
135Dic34—81no—
136C8×C17—21no—
137C8×C17#4112yes—
138#3622—21no—
139#3622—21no—
140#3346—171nomaximal, Sylow
141C22×C17—21no—
142C22×C17—21no—
143C22×C17#13712yes—
144C22×C17C22×D411yescharacteristic
145C2×D17—21no—
146C2×D17—21no—
147C2×D17—21no—
148C2×D17—41no—
149C2×D17—41no—
150C2×D17—41no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17C2×#4111yescharacteristic
157C4×C172−1+411yescharacteristic
158Dic17—21no—
159Dic17—21no—
160Dic17—21no—
161Dic17—41no—
162Dic17—41no—
163Dic17—41no—
164#554—171no—
165#553—171no—
166#501—171no—
167#485—171no—
168#484—171no—
169#481—171no—
170#480—171no—
171#479—171no—
172#474—171no—
173#468—171no—
174#465—171no—
175#440—171no—
176#435—171no—
177#428—171no—
178C2×#107D1711yescharacteristic, complemented
179C2×C17—22no—
180C2×C17#54711yescharacteristic
181C2×C17C2×#13711yescharacteristic
182C2×C17#57611yescharacteristic
183D17—41no—
184D17—81no—
185C2×SD16—171no—
186C2×M16—171no—
187C22×C8C2×D1711yescharacteristic
188C4⋊Q8—171no—
189#126—341no—
190#125—171no—
191#125—341no—
192#123—341no—
193C22⋊Q8—171no—
194C22⋊Q8—171no—
195C22⋊Q8—341no—
196C22⋊Q8—341no—
197#121—171no—
198#121—171no—
199#121—341no—
200C4×Q8—171no—
201C4×D4—171no—
202C4×D4—171no—
203C4×D4—341no—
204C4×D4—341no—
205#117—171no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4C2×D1711yescharacteristic
208C2×C4⋊C4C2×D1711yescharacteristic
209#107—171no—
210#107—171no—
211#107C2×D1711yescharacteristic
212#107C2×D1711yescharacteristic
213#107C2×D1711yescharacteristic
214#107C2×D1711yescharacteristic
215#106—171no—
216#106—171no—
217#106—171no—
218C4⋊C8—171no—
219#103—171no—
220#103—171no—
221#103—171no—
222#102—171no—
223#102—171no—
224#102—171no—
225C22⋊C8—171no—
226C22⋊C8—171no—
227#97—171no—
228C17#334611yescharacteristic, complemented, Sylow
229C2×Q8—171no—
230C2×Q8—341no—
231C2×D4—171no—
232C2×D4—171no—
233C2×D4—341no—
234C2×D4—341no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—171no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4C22×D1711yescharacteristic
241SD16—342no—
242M16—342no—
243C2×C8—22no—
244C2×C8—171no—
245C2×C8—171no—
246C2×C8C22×D1711yescharacteristic
247C2×C8C22×D1711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—171no—
251C4⋊C4—171no—
252C4⋊C4—171no—
253C4⋊C4—341no—
254C4⋊C4—341no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4—341no—
258C4⋊C4—341no—
259C4⋊C4—341no—
260C4⋊C4C22×D1711yescharacteristic
261C4⋊C4C22×D1711yescharacteristic
262C4⋊C4C22×D1711yescharacteristic
263C4⋊C4C22×D1711yescharacteristic
264C22⋊C4—171no—
265C22⋊C4—171no—
266C22⋊C4—341no—
267C22⋊C4—341no—
268C22⋊C4—341no—
269C22⋊C4—341no—
270C22⋊C4—341no—
271C22⋊C4—341no—
272C42—171no—
273C42—341no—
274C42—341no—
275C23—171no—
276C23—341no—
277C23D4×D1711yescharacteristic
278Q8—341no—
279Q8—341no—
280Q8—1361no—
281D4—342no—
282D4—341no—
283D4—341no—
284D4—682no—
285D4—1361no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—171no—
295C2×C4—171no—
296C2×C4—171no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—1361no—
303C2×C4—1361no—
304C2×C4D4×D1711yescharacteristic
305C2×C4C23×D1711yescharacteristic
306C8—21no—
307C8—341no—
308C8—341no—
309C8#6325212yes—
310C22—21no—
311C22—21no—
312C22—341no—
313C22—341no—
314C22—341no—
315C22—681no—
316C22—681no—
317C22—681no—
318C22#1058760412yes—
319C22C2×D4×D1711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—341no—
326C4—341no—
327C4—341no—
328C4—681no—
329C4—681no—
330C4—681no—
331C4C2×#6325211yescharacteristic
332C4#1058769411yescharacteristic
333C2—22no—
334C2—681no—
335C2—1361no—
336C2#1176346011yescharacteristic
337C2C2×#1058760411yescharacteristic
338C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#107C2
socleC22×C17C22×D4
prime radical#11763507C2
hypercenterC2×#107D17
hypocenterC17#3346
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. #11763514C2semidirect
3. #11763217C2semidirect
4. #11763152C2semidirect
5. #11763144C2semidirect
6. #11763127C2semidirect
7. #11763124C2semidirect
8. #11763115C2semidirect
9. #11763088C2semidirect
10. #11763056C2semidirect
11. #11763038C2semidirect
12. #11762908C2semidirect
13. #11762880C2semidirect
14. #11762850C2semidirect
15. C2×C17×#107C2semidirect
178. C2×#107D17semidirect
228. C17#3346semidirect

Maximal subgroups

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

subgroupindex
#117635142
#117635072
#117632172
#117631522
#117631442
#117631272
#117631242
#117631152
#117630882
#117630562
#117630382
#117629082
#117628802
#117628502
C2×C17×#1072
#334617