Finite Group Databasebeta

← #14532437

Subgroups — #14532437

Order 2176.

852 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×Q8×D17C211yesmaximal, characteristic, complemented
2C2×#10587571C212yesmaximal, complemented
3C2×#10587570C211yesmaximal, characteristic
4#11762722C211yesmaximal, characteristic, complemented
5#11762721C211yesmaximal, characteristic, complemented
6#11762720C211yesmaximal, characteristic, complemented
7#11762713C212yesmaximal, complemented
8#11762687C212yesmaximal, complemented
9#11762685C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C2×Q8×D17—24no—
16C2×Q8×D17—28no—
17C22×D68C2211yescharacteristic
18C22×C4×D17C2212yes—
19C22×C4×D17C2211yescharacteristic
20C22×Dic34C2211yescharacteristic
21C22×Dic34C2212yes—
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×#63227C2211yescharacteristic
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63227C2212yes—
29C2×#63225C2211yescharacteristic
30C2×C4×Dic17C2212yes—
31#10587571—28no—
32#10587570—24no—
33C17×#95C2212yes—
34C17×#95C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2211yescharacteristic
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C23×D17C2311yescharacteristic
42C2×Q8×C17—22no—
43C2×Q8×C17—24no—
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2312yes—
48C2×C22⋊D17—22no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—22no—
51C2×C22⋊D17—24no—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C22×Dic17C2311yescharacteristic
56Q8×D17—232no—
57C2×D68—22no—
58C2×D68—24no—
59C2×C4×D17—24no—
60C2×C4×D17—24no—
61C2×C4×D17—24no—
62C2×C4×D17—24no—
63C2×C4×D17D414yes—
64C2×Dic34—22no—
65C2×Dic34—24no—
66C2×Dic34—24no—
67C2×Dic34—24no—
68C2×Dic34D418yescomplemented
69C22⋊C4×C17—22no—
70C22⋊C4×C17—24no—
71C22⋊Dic17—24no—
72#63227—24no—
73#63227—22no—
74#63227—24no—
75#63227—24no—
76#63225—22no—
77C4×Dic17—24no—
78C23×C17—22no—
79C23×C17—21no—
80C23×C17—22no—
81C23×C17—21no—
82C23×C17—21no—
83C23×C17C2411yescharacteristic
84C22×D17—22no—
85C22×D17—21no—
86C22×D17—21no—
87C22×D17—21no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17C2×D412yes—
93C22×D17#4114yes—
94Q8×C17—28no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17C2×D412yes—
104C2×C4×C17C2×D414yes—
105C22⋊D17—44no—
106C22⋊D17—44no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17C2×D414yes—
115C2×Dic17C2×D412yes—
116C2×Dic17C2×D414yes—
117D68—44no—
118C4×D17—216no—
119C4×D17—28no—
120Dic34—28no—
121Dic34—216no—
122#2436—171nomaximal, Sylow
123C22×C17—41no—
124C22×C17—41no—
125C22×C17—41no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17C22×D411yescharacteristic
129C22×C17C22×D412yes—
130C22×C17C2×#4112yes—
131C22×C172−1+411yescharacteristic
132C22×C172−1+411yescharacteristic
133C2×D17—22no—
134C2×D17—22no—
135C2×D17—24no—
136C2×D17—24no—
137C2×D17—41no—
138C2×D17—41no—
139C2×D17—41no—
140C2×D17—42no—
141C2×D17—42no—
142C2×D17#12414yescomplemented
143C4×C17—22no—
144C4×C17—24no—
145C4×C17—41no—
146C4×C17—42no—
147Dic17—24no—
148Dic17—41no—
149Dic17—42no—
150Dic17#12014yes—
151C23×Q8—171no—
152C2×#124—172no—
153C2×#121—171no—
154#393—171no—
155#393—171no—
156#393—171no—
157#393D1711yescharacteristic, complemented
158#391—172no—
159#386—172no—
160#386—172no—
161C4×C22⋊C4—171no—
162C2×C17—81no—
163C2×C17C2×#12011yescharacteristic
164C2×C17C2×#12411yescharacteristic
165C2×C17#53611yescharacteristic
166C2×C17#54812yes—
167C2×C17#54812yes—
168D17—24no—
169D17—81no—
170C22×Q8—172no—
171C22×Q8—171no—
172C22×Q8—344no—
173C22×Q8—348no—
174C22×Q8C2×D1711yescharacteristic
175C22×D4—171no—
176C22×D4—171no—
177C22×D4—171no—
178C23×C4—171no—
179C23×C4—172no—
180#124—348no—
181#121—344no—
182C2×C4⋊C4—171no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—171no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4—172no—
187C2×C22⋊C4—172no—
188C2×C22⋊C4C2×D1711yescharacteristic
189C2×C22⋊C4C2×D1712yes—
190C2×C42—172no—
191#95—171no—
192#95—172no—
193#95—172no—
194#95—172no—
195#95C2×D1711yescharacteristic
196#95C2×D1712yes—
197C17#243611yescharacteristic, complemented, Sylow
198C24—171no—
199C24—171no—
200C24C22×D1711yescharacteristic
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—178no—
204C2×Q8—344no—
205C2×Q8—344no—
206C2×Q8—344no—
207C2×Q8—3432no—
208C2×Q8—342no—
209C2×D4—342no—
210C2×D4—344no—
211C2×D4—342no—
212C2×D4—344no—
213C2×D4—342no—
214C2×D4—344no—
215C22×C4—171no—
216C22×C4—172no—
217C22×C4—174no—
218C22×C4—171no—
219C22×C4—172no—
220C22×C4—344no—
221C22×C4—344no—
222C22×C4—344no—
223C22×C4—344no—
224C22×C4C22×D1711yescharacteristic
225C22×C4C22×D1712yes—
226C22×C4C22×D1711yescharacteristic
227C22×C4C22×D1712yes—
228C4⋊C4—342no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—344no—
232C22⋊C4—344no—
233C22⋊C4—344no—
234C22⋊C4—342no—
235C22⋊C4—344no—
236C42—344no—
237C23—22no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—21no—
242C23—172no—
243C23—174no—
244C23—342no—
245C23—341no—
246C23—341no—
247C23—342no—
248C23—341no—
249C23—341no—
250C23—342no—
251C23—341no—
252C23C23×D1711yescharacteristic
253Q8—28no—
254Q8—3416no—
255Q8—348no—
256D4—684no—
257D4—684no—
258D4—684no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—174no—
268C2×C4—172no—
269C2×C4—174no—
270C2×C4—348no—
271C2×C4—3416no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—341no—
276C2×C4—342no—
277C2×C4—342no—
278C2×C4—342no—
279C2×C4D4×D1712yes—
280C2×C4D4×D1714yes—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—174no—
287C22—344no—
288C22—344no—
289C22—342no—
290C22—342no—
291C22—682no—
292C22—682no—
293C22—681no—
294C22—681no—
295C22—681no—
296C22C2×D4×D1712yes—
297C22C2×D4×D1711yescharacteristic
298C22#1058769411yescharacteristic
299C22#41×D1712yes—
300C22#1058769711yescharacteristic
301C4—22no—
302C4—24no—
303C4—42no—
304C4—41no—
305C4—174no—
306C4—344no—
307C4—682no—
308C4—681no—
309C2—81no—
310C2—344no—
311C2—1361no—
312C2#120×D1711yescharacteristic
313C2#124×D1711yescharacteristic
314C2#1176338211yescharacteristic
315C2#1176346712yes—
316C2#1176346912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC2×#10587570C2
hypercenter#393D17
hypocenterC17#2436
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D17C2semidirect
2. C2×#10587571C2semidirect
4. #11762722C2semidirect
5. #11762721C2semidirect
6. #11762720C2semidirect
7. #11762713C2semidirect
8. #11762687C2semidirect
9. #11762685C2semidirect
10. #11762656C2semidirect
11. C17×#393C2semidirect
68. C2×Dic34D4semidirect
142. C2×D17#124semidirect
157. #393D17semidirect
197. C17#2436semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D172
C2×#105875712
C2×#105875702
#117627222
#117627212
#117627202
#117627132
#117626872
#117626852
#117626562
C17×#3932
#243617