Finite Group Databasebeta

← #14534616

Subgroups — #14534616

Order 2176.

472 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587575C211yesmaximal, characteristic, complemented
2C2×#10587578C211yesmaximal, characteristic
3#11762743C211yesmaximal, characteristic, complemented
4#11762742C211yesmaximal, characteristic, complemented
5#11762741C211yesmaximal, characteristic, complemented
6#11762733C211yesmaximal, characteristic
7#11762731C211yesmaximal, characteristic
8#11762731C211yesmaximal, characteristic
9#11762731C211yesmaximal, characteristic
10#11762706C211yesmaximal, characteristic, complemented
11#11762704C211yesmaximal, characteristic, complemented
12#11762678C211yesmaximal, characteristic, complemented
13#11762668C211yesmaximal, characteristic
14#11762668C211yesmaximal, characteristic
15C17×#398C211yesmaximal, characteristic, complemented
16C22×C4×D17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×#63227C2211yescharacteristic
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×C4⋊Dic17C2211yescharacteristic
28C2×C4⋊Dic17C2211yescharacteristic
29C2×C4⋊Dic17C2211yescharacteristic
30C2×C4⋊Dic17C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×C4×Dic17C2211yescharacteristic
42C2×C4×Dic17C2211yescharacteristic
43#10587578—24no—
44#10587575—24no—
45C17×#95C2211yescharacteristic
46C17×#95C2211yescharacteristic
47C17×#95C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C23×D17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17Q814yes—
71C4⋊C4×C17—22no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75#63227—22no—
76#63227—22no—
77#63227—41no—
78#63227—41no—
79#63227—41no—
80#63227—41no—
81C4⋊Dic17—22no—
82C4⋊Dic17—22no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85#63225—22no—
86#63225—22no—
87#63225—22no—
88#63225—22no—
89#63225—22no—
90#63225—22no—
91#63225—41no—
92#63225—41no—
93#63225—41no—
94#63225—41no—
95C4×Dic17—22no—
96C4×Dic17D414yes—
97C23×C17C2411yescharacteristic
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17C2×Q812yes—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17C2×D412yes—
125C2×C4×C17C2×Q812yes—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17C2×D412yes—
144C2×Dic17C2×D412yes—
145C2×Dic17C2×Q812yes—
146C2×Dic17#4112yes—
147C4×D17—28no—
148#2629—171nomaximal, Sylow
149C22×C17C22×D411yescharacteristic
150C22×C17C22×Q811yescharacteristic
151C22×C17C2×#4111yescharacteristic
152C22×C17H(5,𝔽2)11yescharacteristic
153C22×C17H(5,𝔽2)11yescharacteristic
154C22×C172−1+411yescharacteristic
155C22×C172−1+411yescharacteristic
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170Dic17—22no—
171Dic17—41no—
172Dic17—41no—
173Dic17—41no—
174Dic17—41no—
175Dic17—41no—
176Dic17C22⋊Q814yes—
177C2×#125—171no—
178C2×#117—171no—
179#398—171no—
180#398—171no—
181#398—171no—
182#398D1711yescharacteristic, complemented
183#396—171no—
184#396—171no—
185#396—171no—
186#396—171no—
187#389—171no—
188#389—171no—
189#384—171no—
190#380—171no—
191#380—171no—
192C2×C17C2×C22⋊Q811yescharacteristic
193C2×C17#53711yescharacteristic
194C2×C17#53711yescharacteristic
195C2×C17#54011yescharacteristic
196C2×C17#54111yescharacteristic
197C2×C17#54411yescharacteristic
198C2×C17#54411yescharacteristic
199D17—42no—
200C23×C4—171no—
201#125—344no—
202#117—344no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—171no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4C2×D1711yescharacteristic
218C2×C4⋊C4C2×D1711yescharacteristic
219C2×C4⋊C4C2×D1711yescharacteristic
220C2×C4⋊C4C2×D1711yescharacteristic
221C2×C22⋊C4—171no—
222C2×C22⋊C4—171no—
223C2×C22⋊C4—171no—
224C2×C22⋊C4—171no—
225C2×C22⋊C4—171no—
226C2×C22⋊C4—171no—
227C2×C42—171no—
228C2×C42—171no—
229#95—171no—
230#95—171no—
231#95—171no—
232#95—171no—
233#95—171no—
234#95C2×D1711yescharacteristic
235#95C2×D1711yescharacteristic
236#95C2×D1711yescharacteristic
237C17#262911yescharacteristic, complemented, Sylow
238C24—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—174no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—171no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4C22×D1711yescharacteristic
250C22×C4C22×D1711yescharacteristic
251C22×C4C22×D1711yescharacteristic
252C22×C4C22×D1711yescharacteristic
253C22×C4C22×D1711yescharacteristic
254C22×C4C22×D1711yescharacteristic
255C22×C4C22×D1711yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—342no—
261C4⋊C4—342no—
262C4⋊C4—342no—
263C4⋊C4—342no—
264C4⋊C4—342no—
265C4⋊C4—342no—
266C4⋊C4—342no—
267C4⋊C4—342no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C22⋊C4—342no—
275C22⋊C4—342no—
276C22⋊C4—681no—
277C22⋊C4—681no—
278C22⋊C4—681no—
279C22⋊C4—681no—
280C42—174no—
281C42—342no—
282C23—172no—
283C23—341no—
284C23—341no—
285C23—341no—
286C23—341no—
287C23—341no—
288C23—341no—
289C23C23×D1711yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—172no—
310C2×C4—172no—
311C2×C4—172no—
312C2×C4—172no—
313C2×C4—348no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4D4×D1712yes—
332C2×C4Q8×D1712yes—
333C22—342no—
334C22—342no—
335C22—342no—
336C22—681no—
337C22—681no—
338C22—681no—
339C22—681no—
340C22C2×D4×D1711yescharacteristic
341C22#1058769111yescharacteristic
342C22C2×Q8×D1711yescharacteristic
343C22#1058769411yescharacteristic
344C22#41×D1711yescharacteristic
345C22#1058769611yescharacteristic
346C22#1058769711yescharacteristic
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—174no—
355C4—342no—
356C4—681no—
357C4—681no—
358C4—681no—
359C4—681no—
360C4—681no—
361C2—682no—
362C2C22⋊Q8×D1711yescharacteristic
363C2#1176338611yescharacteristic
364C2#1176338911yescharacteristic
365C2#1176341111yescharacteristic
366C2#1176341811yescharacteristic
367C2#1176343611yescharacteristic
368C2#1176343911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC23×D17C23
hypercenter#398D17
hypocenterC17#2629
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#10587575C2semidirect
3. #11762743C2semidirect
4. #11762742C2semidirect
5. #11762741C2semidirect
10. #11762706C2semidirect
11. #11762704C2semidirect
12. #11762678C2semidirect
15. C17×#398C2semidirect
182. #398D17semidirect
237. C17#2629semidirect

Maximal subgroups

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

subgroupindex
C2×#105875752
C2×#105875782
#117627432
#117627422
#117627412
#117627332
#117627312
#117627062
#117627042
#117626782
#117626682
C17×#3982
#262917