Finite Group Databasebeta

← #14529849

Subgroups — #14529849

Order 2176.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763607C211yesmaximal, characteristic, complemented
2#11763320C211yesmaximal, characteristic, complemented
3#11762827C211yesmaximal, characteristic, complemented
4#11762826C211yesmaximal, characteristic, complemented
5#11762773C211yesmaximal, characteristic, complemented
6#11762565C211yesmaximal, characteristic, complemented
7C17×#525C211yesmaximal, characteristic, complemented
8C2×#41×C17C2211yescharacteristic
9C23×C4×C17C2211yescharacteristic
10C2×#63252C2211yescharacteristic
11C17×#123—21no—
12C17×C22⋊Q8—21no—
13C17×#121—21no—
14C17×#120—21no—
15C4×D4×C17—22no—
16C17×#117C2211yescharacteristic
17#10587639—21no—
18#10587637—21no—
19#10587636—22no—
20#10587635—21no—
21#10587628—21no—
22#10587618—21no—
23#10587616—21no—
24#10587615—21no—
25C4×C22⋊D17—22no—
26#10587612C2211yescharacteristic
27#10587610—21no—
28C2×#63223C2211yescharacteristic
29C2×#63223C2211yescharacteristic
30#10587508—22no—
31C22⋊#3622—22no—
32#10587476—22no—
33C24×C17D411yescharacteristic
34#41×C17—21no—
35#41×C17—21no—
36#41×C17—21no—
37#41×C17—21no—
38C2×Q8×C17D411yescharacteristic, complemented
39C2×D4×C17—22no—
40C2×D4×C17—21no—
41C2×D4×C17D411yescharacteristic, complemented
42C22×C4×C17—21no—
43C22×C4×C17—21no—
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17D411yescharacteristic
50C22×C4×C17C2311yescharacteristic
51C2×C22⋊D17—22no—
52C2×C22⋊D17—21no—
53#63252—21no—
54#63252—21no—
55#63252—21no—
56#63252—21no—
57C2×D68D411yescharacteristic, complemented
58C2×C4×D17—21no—
59C2×Dic34D411yescharacteristic, complemented
60C4⋊C4×C17—22no—
61C4⋊C4×C17—21no—
62C22⋊C4×C17—22no—
63C22⋊C4×C17—22no—
64C22⋊C4×C17—21no—
65C42×C17D412yescomplemented
66C22⋊Dic17—22no—
67C22⋊Dic17—21no—
68#63231—22no—
69#63230—22no—
70#63229—22no—
71#63228—22no—
72#63227—22no—
73C4⋊Dic17—21no—
74#63225—22no—
75C4×Dic17D412yescomplemented
76#63223—21no—
77#63223—21no—
78#63223—21no—
79#63223—21no—
80C2×#3622—21no—
81C2×#3622D412yescomplemented
82C23×C17—21no—
83C23×C17—21no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—42no—
87C23×C17C2×D411yescharacteristic
88C22×D17—21no—
89Q8×C17—21no—
90Q8×C17—21no—
91D4×C17—21no—
92D4×C17—21no—
93D4×C17—44no—
94D4×C17—41no—
95D4×C17—41no—
96C2×C4×C17—22no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17C2×D411yescharacteristic
108C2×C4×C17C2×D411yescharacteristic
109C2×C4×C17C2×D411yescharacteristic
110C2×C4×C17C2×D411yescharacteristic
111C2×C4×C17C2×D411yescharacteristic
112C2×C4×C17#4111yescharacteristic
113C22⋊D17—44no—
114C22⋊D17—41no—
115C22⋊D17—41no—
116C2×Dic17—21no—
117C2×Dic17—22no—
118D68—21no—
119D68—21no—
120C4×D17—41no—
121C4×D17—41no—
122Dic34—21no—
123Dic34—21no—
124#3622—22no—
125#3622—41no—
126#2010—171nomaximal, Sylow
127C22×C17—22no—
128C22×C17—22no—
129C22×C17—22no—
130C22×C17—41no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—42no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17#12011yescharacteristic
138C22×C17#12011yescharacteristic
139C22×C17#12111yescharacteristic
140C2×D17—42no—
141C2×D17—41no—
142C4×C17—42no—
143C4×C17—41no—
144C4×C17—41no—
145C4×C17#12011yescharacteristic
146C4×C17#12111yescharacteristic
147C4×C17#12111yescharacteristic
148C4×C17#12711yescharacteristic
149Dic17—42no—
150Dic17—41no—
151#575—171no—
152#525—171no—
153#525D1711yescharacteristic, complemented
154#421—171no—
155#421—171no—
156#407—171no—
157H(3,ℤ/4)—171no—
158C2×C17—21no—
159C2×C17—21no—
160C2×C17—21no—
161C2×C17—42no—
162C2×C17—81no—
163C2×C17#39211yescharacteristic
164D17—81no—
165C2×#41—171no—
166C2×#41C2×D1711yescharacteristic
167C23×C4C2×D1711yescharacteristic
168#137—341no—
169#137—341no—
170AGL(1,ℤ/8)—341no—
171AGL(1,ℤ/8)—341no—
172#135—342no—
173C2×M16—171no—
174C2×M16—171no—
175#123—21no—
176#123—341no—
177C22⋊Q8—21no—
178C22⋊Q8—341no—
179#121—21no—
180#121—341no—
181#120—21no—
182#120—341no—
183C4×D4—22no—
184C4×D4—342no—
185#117—171no—
186#117C2×D1711yescharacteristic
187#104—342no—
188#104—342no—
189C22⋊C8—342no—
190C17#201011yescharacteristic, complemented, Sylow
191C24C22⋊D1711yescharacteristic
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—341no—
197#41—341no—
198#41—341no—
199#41—341no—
200C2×Q8—171no—
201C2×Q8C22⋊D1711yescharacteristic, complemented
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—171no—
205C2×D4—342no—
206C2×D4—341no—
207C2×D4C22⋊D1711yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—341no—
216C22×C4C22⋊D1711yescharacteristic
217C22×C4C22×D1711yescharacteristic
218Q16—342no—
219SD16—342no—
220SD16—342no—
221D8—342no—
222M16—341no—
223M16—341no—
224M16—341no—
225M16—341no—
226C2×C8—172no—
227C2×C8—341no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—342no—
231C4⋊C4—341no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—21no—
235C22⋊C4—342no—
236C22⋊C4—342no—
237C22⋊C4—341no—
238C42—172no—
239C42C22⋊D1712yescomplemented
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—42no—
245C23—341no—
246C23C2×C22⋊D1711yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—341no—
250Q8—341no—
251D4—21no—
252D4—21no—
253D4—44no—
254D4—41no—
255D4—41no—
256D4—341no—
257D4—341no—
258D4—684no—
259D4—681no—
260D4—681no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—342no—
273C2×C4—341no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4D4×D1711yescharacteristic
277C2×C4D4×D1711yescharacteristic
278C2×C4D4×D1711yescharacteristic
279C2×C4#6325411yescharacteristic
280C2×C4C2×C22⋊D1711yescharacteristic
281C2×C4C2×C22⋊D1711yescharacteristic
282C8—342no—
283C8—681no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—682no—
295C22—681no—
296C22#1058762311yescharacteristic
297C22#1058762511yescharacteristic
298C22#1058763911yescharacteristic
299C4—42no—
300C4—41no—
301C4—41no—
302C4—682no—
303C4—681no—
304C4#1058762311yescharacteristic
305C4#1058762411yescharacteristic
306C4#1058762511yescharacteristic
307C4#1058762611yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—42no—
312C2—81no—
313C2—1361no—
314C2#1176271711yescharacteristic
subgroupG/H
centerC4#10587626
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#525C2
socleC2×C17#392
hypercenter#525D17
hypocenterC17#2010
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #11763607C2semidirect
2. #11763320C2semidirect
3. #11762827C2semidirect
4. #11762826C2semidirect
5. #11762773C2semidirect
6. #11762565C2semidirect
7. C17×#525C2semidirect
38. C2×Q8×C17D4semidirect
41. C2×D4×C17D4semidirect
57. C2×D68D4semidirect
59. C2×Dic34D4semidirect
65. C42×C17D4semidirect
75. C4×Dic17D4semidirect
81. C2×#3622D4semidirect
153. #525D17semidirect
190. C17#2010semidirect
201. C2×Q8C22⋊D17semidirect
207. C2×D4C22⋊D17semidirect
239. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
#117636072
#117633202
#117628272
#117628262
#117627732
#117625652
C17×#5252
#201017