Finite Group Databasebeta

← #14532120

Subgroups — #14532120

Order 2176.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#63256C211yesmaximal, characteristic
2C2×#10587580C212yesmaximal, complemented
3C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
4#11762728C211yesmaximal, characteristic, complemented
5#11762727C211yesmaximal, characteristic, complemented
6#11762726C211yesmaximal, characteristic, complemented
7#11762687C212yesmaximal, complemented
8#11762683C212yesmaximal, complemented
9#11762680C212yesmaximal, complemented
10#11762671C211yesmaximal, characteristic, complemented
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C2×#63256—28no—
14C2×#63256—24no—
15C22×D68C2211yescharacteristic
16C22×D68C2212yes—
17C22×C4×D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C22×C4×D17C2212yes—
20C22×C4×D17C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2212yes—
31C2×#63225C2211yescharacteristic
32#10587580—28no—
33C4⋊C4×D17—24no—
34C17×#95C2211yescharacteristic
35C17×#95C2212yes—
36#10587502C2211yescharacteristic
37#10587502C2212yes—
38#10587502C2212yes—
39C23×D17C2312yes—
40C23×D17C2311yescharacteristic
41C2×Q8×C17—24no—
42C2×Q8×C17—22no—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52#63256—232no—
53C2×D68—24no—
54C2×D68—24no—
55C2×D68—24no—
56C2×D68—22no—
57C2×D68D418yes—
58C2×C4×D17—24no—
59C2×C4×D17—24no—
60C2×C4×D17—22no—
61C2×C4×D17—24no—
62C2×C4×D17—22no—
63C2×C4×D17—24no—
64C2×C4×D17—22no—
65C2×C4×D17—22no—
66C2×C4×D17—24no—
67C2×C4×D17D414yes—
68C4⋊C4×C17—22no—
69C4⋊C4×C17—24no—
70#63227—24no—
71#63227—24no—
72#63227—24no—
73#63227—24no—
74#63227—24no—
75C4⋊Dic17—22no—
76#63225—22no—
77#63225—24no—
78#63225—22no—
79C23×C17C2411yescharacteristic
80C22×D17—21no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17C2×D414yes—
90C22×D17C2×D412yes—
91C22×D17C2×D414yes—
92Q8×C17—28no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17C2×D414yes—
104C2×C4×C17C2×D412yes—
105C2×C4×C17#4112yescomplemented
106C2×Dic17—22no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17C2×D412yes—
115C2×Dic17#4112yescomplemented
116D68—28no—
117D68—216no—
118C4×D17—28no—
119C4×D17—216no—
120C4×D17—28no—
121C4×D17—44no—
122#2406—171nomaximal, Sylow
123C22×C17C22×D411yescharacteristic
124C22×C17C22×D412yes—
125C22×C17C2×#4111yescharacteristic
126C22×C17C2×#4111yescharacteristic
127C22×C172−1+412yes—
128C2×D17—22no—
129C2×D17—22no—
130C2×D17—24no—
131C2×D17—24no—
132C2×D17—24no—
133C2×D17—24no—
134C2×D17—24no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17#12014yes—
140C4×C17—24no—
141C4×C17—22no—
142C4×C17—22no—
143C4×C17—41no—
144C4×C17—42no—
145Dic17—22no—
146Dic17—22no—
147Dic17—41no—
148Dic17—42no—
149C22×#41—171no—
150C2×#123—172no—
151C22×C4⋊C4—171no—
152#395—171no—
153#395D1711yescharacteristic, complemented
154#394—171no—
155#394—171no—
156#386—172no—
157#385—172no—
158#385—172no—
159#381—171no—
160C2×C17C2×#12011yescharacteristic
161C2×C17#52511yescharacteristic
162C2×C17#53611yescharacteristic
163C2×C17#54712yes—
164C2×C17#54712yes—
165D17—24no—
166D17—44no—
167C2×#41—344no—
168C2×#41—348no—
169C22×Q8C2×D1711yescharacteristic
170C22×D4—172no—
171C22×D4—171no—
172C23×C4—171no—
173C23×C4—171no—
174C23×C4—171no—
175C23×C4—172no—
176#123—348no—
177C2×C4⋊C4—171no—
178C2×C4⋊C4—171no—
179C2×C4⋊C4—171no—
180C2×C4⋊C4—172no—
181C2×C4⋊C4—344no—
182C2×C4⋊C4C2×D1711yescharacteristic
183C2×C4⋊C4C2×D1712yes—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4—172no—
187C2×C22⋊C4—172no—
188C2×C22⋊C4—172no—
189#95—172no—
190#95—171no—
191#95—172no—
192#95C2×D1711yescharacteristic
193#95C2×D1712yes—
194C17#240611yescharacteristic, complemented, Sylow
195C24—171no—
196C24—172no—
197#41—3432no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—178no—
201C2×D4—344no—
202C2×D4—344no—
203C2×D4—344no—
204C2×D4—342no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—174no—
208C22×C4—172no—
209C22×C4—171no—
210C22×C4—342no—
211C22×C4—344no—
212C22×C4—342no—
213C22×C4—344no—
214C22×C4—344no—
215C22×C4—342no—
216C22×C4—344no—
217C22×C4—344no—
218C22×C4—342no—
219C22×C4C22×D1712yes—
220C22×C4C22×D1711yescharacteristic
221C22×C4C22×D1711yescharacteristic
222C22×C4C22×D1712yes—
223C22×C4C22×D1711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—342no—
227C4⋊C4—342no—
228C4⋊C4—342no—
229C4⋊C4—344no—
230C22⋊C4—344no—
231C22⋊C4—344no—
232C22⋊C4—344no—
233C22⋊C4—344no—
234C22⋊C4—344no—
235C23—174no—
236C23—172no—
237C23—174no—
238C23—341no—
239C23—342no—
240C23—341no—
241C23—342no—
242C23—342no—
243C23—342no—
244C23—342no—
245C23—342no—
246C23—342no—
247C23C23×D1711yescharacteristic
248Q8—28no—
249D4—3416no—
250D4—348no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—172no—
262C2×C4—172no—
263C2×C4—348no—
264C2×C4—3416no—
265C2×C4—348no—
266C2×C4—341no—
267C2×C4—342no—
268C2×C4—341no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4—341no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—684no—
275C2×C4D4×D1712yes—
276C2×C4D4×D1714yes—
277C2×C4#6325612yescomplemented
278C22—174no—
279C22—344no—
280C22—344no—
281C22—342no—
282C22—342no—
283C22—344no—
284C22—344no—
285C22—344no—
286C22—682no—
287C22—682no—
288C22—682no—
289C22—682no—
290C22C2×D4×D1712yes—
291C22C2×D4×D1711yescharacteristic
292C22C2×#6325611yescharacteristic
293C22#1058769412yes—
294C22#41×D1711yescharacteristic
295C4—22no—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—342no—
301C4—342no—
302C4—682no—
303C4—681no—
304C2—344no—
305C2—684no—
306C2#120×D1711yescharacteristic
307C2#1176332211yescharacteristic
308C2#1176338411yescharacteristic
309C2#1176346112yes—
310C2#1176346212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC22×#63256C2
hypercenter#395D17
hypocenterC17#2406
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#10587580C2semidirect
3. C2×C4⋊C4×D17C2semidirect
4. #11762728C2semidirect
5. #11762727C2semidirect
6. #11762726C2semidirect
7. #11762687C2semidirect
8. #11762683C2semidirect
9. #11762680C2semidirect
10. #11762671C2semidirect
11. C17×#395C2semidirect
105. C2×C4×C17#41semidirect
115. C2×Dic17#41semidirect
153. #395D17semidirect
194. C17#2406semidirect
277. C2×C4#63256semidirect

Maximal subgroups

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

subgroupindex
C22×#632562
C2×#105875802
C2×C4⋊C4×D172
#117627282
#117627272
#117627262
#117626872
#117626832
#117626802
#117626712
C17×#3952
#240617