Finite Group Databasebeta

← #14534997

Subgroups — #14534997

Order 2176.

932 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#10587625C212yesmaximal, complemented
2C2×#10587623C211yesmaximal, characteristic, complemented
3C2×#10587623C212yesmaximal, complemented
4#11762723C212yesmaximal, complemented
5#11762717C212yesmaximal, complemented
6#11762717C212yesmaximal, complemented
7#11762700C212yesmaximal, complemented
8#11762689C211yesmaximal, characteristic, complemented
9C2×C17×#127C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×D4×C17C2212yes—
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yescomplemented
14C22×D4×C17C2211yescharacteristic, complemented
15C22×C22⋊D17C2212yes—
16C22×C22⋊D17C2212yes—
17C22×C22⋊D17C2212yes—
18C23×Dic17C2211yescharacteristic, complemented
19C17×#127—24no—
20C2×C42×C17C2211yescharacteristic, complemented
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2212yescomplemented
24C2×C22⋊Dic17C2211yescharacteristic, complemented
25C2×C22⋊Dic17C2212yescomplemented
26#10587625—28no—
27#10587623—28no—
28#10587623—24no—
29C2×#63227C2212yes—
30C2×#63227C2212yes—
31C2×#63227C2212yes—
32C2×#63225C2212yescomplemented
33#10587502C2212yescomplemented
34#10587502C2212yescomplemented
35C24×C17C2312yes—
36C24×C17C2311yescharacteristic
37C24×C17C2311yescharacteristic
38C23×D17—22no—
39C23×D17—24no—
40C23×D17—41no—
41C23×D17—42no—
42C23×D17—41no—
43C23×D17C2312yes—
44C2×D4×C17—24no—
45C2×D4×C17—24no—
46C2×D4×C17—22no—
47C2×D4×C17—24no—
48C2×D4×C17—24no—
49C2×D4×C17—41no—
50C2×D4×C17—41no—
51C2×D4×C17—42no—
52C2×D4×C17—42no—
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17D414yescomplemented
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2312yes—
58C2×C22⋊D17—24no—
59C2×C22⋊D17—24no—
60C2×C22⋊D17—24no—
61C2×C22⋊D17—24no—
62C2×C22⋊D17—24no—
63C2×C22⋊D17—42no—
64C2×C22⋊D17—42no—
65C2×C22⋊D17—42no—
66C2×C22⋊D17—42no—
67C22×Dic17—24no—
68C22×Dic17—41no—
69C22×Dic17C2312yes—
70C22×Dic17C2312yes—
71C22×Dic17C2312yes—
72C42×C17—22no—
73C22⋊Dic17—22no—
74C22⋊Dic17—24no—
75C22⋊Dic17—24no—
76C22⋊Dic17—41no—
77C22⋊Dic17—42no—
78#63227—24no—
79#63227—24no—
80#63227D418yescomplemented
81#63225—24no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—22no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—22no—
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17C2×D412yes—
98C23×C17C2×D412yes—
99C23×C17C2411yescharacteristic
100C22×D17—22no—
101C22×D17—22no—
102C22×D17—22no—
103C22×D17—22no—
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—44no—
107C22×D17—48no—
108C22×D17—48no—
109C22×D17—44no—
110C22×D17—44no—
111C22×D17C2×D414yes—
112D4×C17—44no—
113D4×C17—48no—
114D4×C17—48no—
115D4×C17—44no—
116C2×C4×C17—22no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—22no—
120C2×C4×C17C2×D414yes—
121C2×C4×C17C2×D412yes—
122C22⋊D17—48no—
123C22⋊D17—48no—
124C22⋊D17—48no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—22no—
131C2×Dic17—22no—
132C2×Dic17—22no—
133C2×Dic17—44no—
134C2×Dic17C2×D414yes—
135#2669—171nomaximal, Sylow
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—24no—
139C22×C17—24no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17C22×D412yes—
154C22×C17C22×D411yescharacteristic
155C22×C17H(5,𝔽2)11yescharacteristic
156C22×C17H(5,𝔽2)11yescharacteristic
157C22×C17H(5,𝔽2)12yes—
158C2×D17—24no—
159C2×D17—24no—
160C2×D17—24no—
161C2×D17—416no—
162C2×D17—42no—
163C2×D17—42no—
164C2×D17—42no—
165C2×D17—42no—
166C4×C17—22no—
167C4×C17—24no—
168Dic17—24no—
169Dic17—42no—
170Dic17—42no—
171C2×#127D1711yescharacteristic, complemented
172C2×#121—172no—
173C2×#120—172no—
174C2×#120—171no—
175#394—172no—
176#392—172no—
177#392—172no—
178#388—172no—
179#386—171no—
180C2×C17—42no—
181C2×C17—42no—
182C2×C17—82no—
183C2×C17#53411yescharacteristic
184C2×C17#53512yes—
185C2×C17D4212yes—
186C2×C17D4211yescharacteristic
187C2×C17#56011yescharacteristic
188D17—44no—
189C25—171no—
190C22×D4—172no—
191C22×D4—172no—
192C22×D4—172no—
193C22×D4C2×D1712yes—
194C22×D4C2×D1711yescharacteristic
195C22×D4C2×D1712yescomplemented
196C22×D4C2×D1711yescharacteristic, complemented
197C23×C4—171no—
198#127—24no—
199#121—348no—
200#120—348no—
201#120—344no—
202C2×C4⋊C4—172no—
203C2×C22⋊C4—172no—
204C2×C22⋊C4—172no—
205C2×C22⋊C4—171no—
206C2×C22⋊C4—172no—
207C2×C22⋊C4—172no—
208C2×C22⋊C4—172no—
209C2×C22⋊C4—171no—
210C2×C22⋊C4—172no—
211C2×C42C2×D1711yescharacteristic, complemented
212#95—172no—
213#95—172no—
214C17#266911yescharacteristic, complemented, Sylow
215C24—172no—
216C24—344no—
217C24—342no—
218C24—682no—
219C24—681no—
220C24—681no—
221C24C22×D1712yes—
222C24C22×D1711yescharacteristic
223C24C22×D1711yescharacteristic
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—344no—
236C2×D4—344no—
237C2×D4—344no—
238C2×D4—344no—
239C2×D4—344no—
240C2×D4—682no—
241C2×D4—682no—
242C2×D4—682no—
243C2×D4—682no—
244C2×D4C22⋊D1714yescomplemented
245C22×C4—172no—
246C22×C4—172no—
247C22×C4—172no—
248C22×C4—344no—
249C22×C4—681no—
250C22×C4C22×D1712yes—
251C22×C4C22×D1711yescharacteristic
252C4⋊C4—344no—
253C22⋊C4—178no—
254C22⋊C4—344no—
255C22⋊C4—344no—
256C22⋊C4—342no—
257C22⋊C4—344no—
258C22⋊C4—344no—
259C22⋊C4—681no—
260C22⋊C4—682no—
261C42—22no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—174no—
278C23—342no—
279C23—342no—
280C23—342no—
281C23—342no—
282C23—342no—
283C23—342no—
284C23—688no—
285C23—684no—
286C23—684no—
287C23—688no—
288C23—684no—
289C23C2×C22⋊D1712yes—
290C23C2×C22⋊D1712yes—
291C23C23×D1711yescharacteristic
292D4—44no—
293D4—48no—
294D4—48no—
295D4—44no—
296D4—688no—
297D4—688no—
298D4—688no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—174no—
304C2×C4—342no—
305C2×C4—342no—
306C2×C4—342no—
307C2×C4—342no—
308C2×C4—342no—
309C2×C4—342no—
310C2×C4—342no—
311C2×C4—342no—
312C2×C4—684no—
313C2×C4D4×D1714yes—
314C2×C4C2×C22⋊D1712yes—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—344no—
333C22—344no—
334C22—344no—
335C22—6816no—
336C22—682no—
337C22—682no—
338C22—682no—
339C22—682no—
340C22C2×D4×D1712yes—
341C22#1058769111yescharacteristic
342C22#1058769112yes—
343C22#1058769111yescharacteristic
344C22C22×C22⋊D1711yescharacteristic
345C4—22no—
346C4—24no—
347C4—344no—
348C4—682no—
349C4—682no—
350C2—42no—
351C2—42no—
352C2—82no—
353C2—684no—
354C2#1176336811yescharacteristic
355C2#1176337212yes—
356C2D4×C22⋊D1712yes—
357C2#1176344411yescharacteristic
358C2#1176354611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#127C2
hypercenterC2×#127D17
hypocenterC17#2669
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#10587625C2semidirect
2. C2×#10587623C2semidirect
3. C2×#10587623C2semidirect
4. #11762723C2semidirect
5. #11762717C2semidirect
6. #11762717C2semidirect
7. #11762700C2semidirect
8. #11762689C2semidirect
9. C2×C17×#127C2semidirect
13. C22×D4×C17C22semidirect
14. C22×D4×C17C22semidirect
18. C23×Dic17C22semidirect
20. C2×C42×C17C22semidirect
23. C2×C22⋊Dic17C22semidirect
24. C2×C22⋊Dic17C22semidirect
25. C2×C22⋊Dic17C22semidirect
32. C2×#63225C22semidirect
33. #10587502C22semidirect
34. #10587502C22semidirect
55. C2×D4×C17D4semidirect
80. #63227D4semidirect
171. C2×#127D17semidirect
195. C22×D4C2×D17semidirect
196. C22×D4C2×D17semidirect
211. C2×C42C2×D17semidirect
214. C17#2669semidirect
244. C2×D4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876252
C2×#105876232
#117627232
#117627172
#117627002
#117626892
C2×C17×#1272
#266917