Finite Group Databasebeta

← #14533714

Subgroups — #14533714

Order 2176.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#10587615C211yesmaximal, characteristic, complemented
2C2×#10587625C211yesmaximal, characteristic, complemented
3C2×#10587568C211yesmaximal, characteristic, complemented
4C2×#10587623C211yesmaximal, characteristic, complemented
5C22×#63227C211yesmaximal, characteristic, complemented
6#11762744C212yesmaximal, complemented
7#11762718C212yesmaximal, complemented
8#11762710C212yesmaximal, complemented
9#11762709C212yesmaximal, complemented
10C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
11C2×C17×#123C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C22×C22⋊D17C2212yes—
17C22×C22⋊D17C2211yescharacteristic, complemented
18C23×Dic17C2211yescharacteristic
19C22×D68C2212yes—
20C17×#123—24no—
21C2×C4⋊C4×C17C2212yescomplemented
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2212yescomplemented
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic, complemented
26#10587625—24no—
27#10587623—24no—
28#10587615—24no—
29C2×#63227—24no—
30C2×#63227C2212yes—
31C2×#63227C2212yes—
32C2×#63227C2212yes—
33C2×#63227C2212yes—
34C2×#63227C2211yescharacteristic, complemented
35C2×#63227C2212yescomplemented
36C2×#63225C2212yescomplemented
37C2×#63225C2211yescharacteristic, complemented
38C2×C4×Dic17C2212yescomplemented
39#10587568—24no—
40#10587502C2212yescomplemented
41C24×C17C2311yescharacteristic
42C24×C17C2311yescharacteristic
43C23×D17—22no—
44C23×D17—24no—
45C23×D17—22no—
46C23×D17—22no—
47C23×D17—42no—
48C23×D17C2311yescharacteristic
49C23×D17C2312yes—
50C2×D4×C17—22no—
51C2×D4×C17—22no—
52C2×D4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17C2312yes—
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2312yes—
59C2×C22⋊D17—22no—
60C2×C22⋊D17—22no—
61C2×C22⋊D17—22no—
62C2×C22⋊D17—24no—
63C2×C22⋊D17—24no—
64C2×C22⋊D17—22no—
65C2×C22⋊D17—42no—
66C2×C22⋊D17—42no—
67C22×Dic17—42no—
68C22×Dic17D414yescomplemented
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2312yes—
73C2×D68—24no—
74C2×D68—24no—
75C2×D68—42no—
76C4⋊C4×C17—24no—
77C22⋊C4×C17—24no—
78C22⋊C4×C17D414yescomplemented
79C22⋊Dic17—22no—
80C22⋊Dic17D414yescomplemented
81#63227—24no—
82#63227—24no—
83#63227—24no—
84#63227—24no—
85#63227—24no—
86#63227—22no—
87#63225—24no—
88#63225—22no—
89C4×Dic17—42no—
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17C2×D412yes—
98C23×C17C2×D412yes—
99C23×C17C2×D412yes—
100C23×C17C2411yescharacteristic
101C22×D17—28no—
102C22×D17—22no—
103C22×D17—22no—
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—22no—
107C22×D17—22no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—22no—
112C22×D17—22no—
113C22×D17—48no—
114C22×D17—48no—
115C22×D17—44no—
116C22×D17—44no—
117C22×D17#4114yes—
118D4×C17—44no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—44no—
128C2×C4×C17C2×D414yes—
129C22⋊D17—48no—
130C22⋊D17—44no—
131C22⋊D17—44no—
132C2×Dic17—22no—
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—22no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—44no—
140C2×Dic17C2×D412yes—
141C2×Dic17C2×D414yes—
142C2×Dic17C2×D412yes—
143D68—48no—
144#2539—171nomaximal, Sylow
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—24no—
148C22×C17—24no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17#12714yes—
155C22×C17C22×D411yescharacteristic
156C22×C17C22×D411yescharacteristic
157C22×C17C22×D411yescharacteristic
158C22×C17C2×#4112yes—
159C22×C17H(5,𝔽2)12yes—
160C2×D17—24no—
161C2×D17—24no—
162C2×D17—24no—
163C2×D17—416no—
164C2×D17—42no—
165C2×D17—42no—
166C2×D17—42no—
167C2×D17—42no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—42no—
172C2×D17—42no—
173C4×C17—24no—
174C4×C17—42no—
175C4×C17—41no—
176Dic17—22no—
177Dic17—22no—
178Dic17—24no—
179Dic17—41no—
180C2×#123—171no—
181C2×#123D1711yescharacteristic, complemented
182C2×#121—171no—
183C2×#120—171no—
184C2×#120—171no—
185C22×C22⋊C4—171no—
186#399—172no—
187#392—172no—
188#390—172no—
189#390—172no—
190C4×C22⋊C4—171no—
191C2×C17—24no—
192C2×C17—81no—
193C2×C17#52511yescharacteristic
194C2×C17C2×#12711yescharacteristic
195C2×C17#53411yescharacteristic
196C2×C17#54612yes—
197C2×C17#54612yes—
198D17—44no—
199D17—81no—
200C25—171no—
201C22×D4—171no—
202C22×D4—172no—
203C22×D4—171no—
204C22×D4—172no—
205C22×D4C2×D1711yescharacteristic
206C23×C4—171no—
207C23×C4C2×D1711yescharacteristic
208#123—24no—
209#123—344no—
210#121—344no—
211#120—344no—
212#120—344no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—172no—
215C2×C4⋊C4C2×D1712yescomplemented
216C2×C22⋊C4—171no—
217C2×C22⋊C4—171no—
218C2×C22⋊C4—172no—
219C2×C22⋊C4—172no—
220C2×C22⋊C4—172no—
221C2×C22⋊C4—172no—
222C2×C22⋊C4—171no—
223C2×C22⋊C4—172no—
224C2×C22⋊C4—344no—
225C2×C22⋊C4C2×D1711yescharacteristic
226C2×C22⋊C4C2×D1712yescomplemented
227C2×C42—172no—
228#95—172no—
229C17#253911yescharacteristic, complemented, Sylow
230C24—172no—
231C24—171no—
232C24—344no—
233C24—342no—
234C24—342no—
235C24—342no—
236C24—682no—
237C24C22×D1711yescharacteristic
238C24C22×D1711yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—344no—
243C2×D4—344no—
244C2×D4—342no—
245C2×D4—344no—
246C2×D4—342no—
247C2×D4—342no—
248C2×D4—342no—
249C2×D4—344no—
250C2×D4—682no—
251C2×D4—682no—
252C2×D4—682no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—171no—
257C22×C4—172no—
258C22×C4—171no—
259C22×C4—174no—
260C22×C4—171no—
261C22×C4—682no—
262C22×C4C22×D1712yes—
263C22×C4C22×D1711yescharacteristic
264C22×C4C22×D1712yes—
265C4⋊C4—24no—
266C4⋊C4—342no—
267C4⋊C4—344no—
268C22⋊C4—24no—
269C22⋊C4—174no—
270C22⋊C4—344no—
271C22⋊C4—344no—
272C22⋊C4—344no—
273C22⋊C4—342no—
274C22⋊C4—344no—
275C22⋊C4—342no—
276C22⋊C4—344no—
277C22⋊C4C22⋊D1714yescomplemented
278C42—682no—
279C23—22no—
280C23—22no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—22no—
285C23—21no—
286C23—174no—
287C23—348no—
288C23—342no—
289C23—342no—
290C23—342no—
291C23—342no—
292C23—342no—
293C23—342no—
294C23—341no—
295C23—341no—
296C23—342no—
297C23—342no—
298C23—341no—
299C23—688no—
300C23—688no—
301C23—684no—
302C23—684no—
303C23D4×D1712yes—
304C23D4×D1712yes—
305C23C2×C22⋊D1712yes—
306C23C23×D1711yescharacteristic
307D4—44no—
308D4—688no—
309D4—688no—
310D4—684no—
311D4—684no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—174no—
322C2×C4—172no—
323C2×C4—172no—
324C2×C4—342no—
325C2×C4—342no—
326C2×C4—342no—
327C2×C4—342no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—684no—
332C2×C4C2×C22⋊D1714yes—
333C22—22no—
334C22—22no—
335C22—24no—
336C22—24no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—344no—
343C22—344no—
344C22—344no—
345C22—6816no—
346C22—682no—
347C22—682no—
348C22—682no—
349C22—682no—
350C22—681no—
351C22—681no—
352C22—681no—
353C22—682no—
354C22—682no—
355C22#1058762614yes—
356C22C2×D4×D1711yescharacteristic
357C22C2×D4×D1711yescharacteristic
358C22#41×D1712yes—
359C22#1058769612yes—
360C22C22×C22⋊D1711yescharacteristic
361C4—24no—
362C4—42no—
363C4—41no—
364C4—342no—
365C4—342no—
366C4—344no—
367C4—681no—
368C2—24no—
369C2—81no—
370C2—684no—
371C2—1361no—
372C2#1176331711yescharacteristic
373C2C2×#1058762611yescharacteristic
374C2#1176337011yescharacteristic
375C2#1176344712yes—
376C2#1176345112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
hypercenterC2×#123D17
hypocenterC17#2539
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
2. C2×#10587625C2semidirect
3. C2×#10587568C2semidirect
4. C2×#10587623C2semidirect
5. C22×#63227C2semidirect
6. #11762744C2semidirect
7. #11762718C2semidirect
8. #11762710C2semidirect
9. #11762709C2semidirect
10. C22⋊C4×Dic17C2semidirect
11. C2×C17×#123C2semidirect
17. C22×C22⋊D17C22semidirect
21. C2×C4⋊C4×C17C22semidirect
23. C2×C22⋊C4×C17C22semidirect
25. C2×C22⋊Dic17C22semidirect
34. C2×#63227C22semidirect
35. C2×#63227C22semidirect
36. C2×#63225C22semidirect
37. C2×#63225C22semidirect
38. C2×C4×Dic17C22semidirect
40. #10587502C22semidirect
68. C22×Dic17D4semidirect
78. C22⋊C4×C17D4semidirect
80. C22⋊Dic17D4semidirect
181. C2×#123D17semidirect
215. C2×C4⋊C4C2×D17semidirect
226. C2×C22⋊C4C2×D17semidirect
229. C17#2539semidirect
277. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876252
C2×#105875682
C2×#105876232
C22×#632272
#117627442
#117627182
#117627102
#117627092
C22⋊C4×Dic172
C2×C17×#1232
#253917