Finite Group Databasebeta

← #22595362

Subgroups — #22595362

Order 9088.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556108C211yesmaximal, characteristic, complemented
2C2×#14556105C211yesmaximal, characteristic, complemented
3#17985934C211yesmaximal, characteristic, complemented
4#17985932C211yesmaximal, characteristic, complemented
5#17985930C211yesmaximal, characteristic, complemented
6#17985922C211yesmaximal, characteristic, complemented
7#17985921C211yesmaximal, characteristic, complemented
8#17985903C211yesmaximal, characteristic, complemented
9#17985899C211yesmaximal, characteristic
10#17985898C211yesmaximal, characteristic, complemented
11#17985876C211yesmaximal, characteristic
12#17985875C211yesmaximal, characteristic
13#17985874C211yesmaximal, characteristic, complemented
14#17985873C211yesmaximal, characteristic, complemented
15#399×C71C211yesmaximal, characteristic, complemented
16C22×Dic142C2211yescharacteristic
17C22×C22⋊D71C2211yescharacteristic
18C23×Dic71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20#14556108—24no—
21#14556105—24no—
22C2×C4⋊Dic71C2211yescharacteristic
23C2×C4⋊Dic71C2211yescharacteristic
24C2×C4⋊Dic71C2211yescharacteristic
25C2×C4⋊Dic71C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×Dic71⋊C4C2211yescharacteristic
30C2×Dic71⋊C4C2211yescharacteristic
31C2×C22⋊Dic71C2211yescharacteristic
32C2×C22⋊Dic71C2211yescharacteristic
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×(C2×D71)⋊C4C2211yescharacteristic
36C2×(C2×D71)⋊C4C2211yescharacteristic
37C2×(C2×D71)⋊C4C2211yescharacteristic
38C2×C4×Dic71C2211yescharacteristic
39C71⋊#95C2211yescharacteristic
40C71⋊#95C2211yescharacteristic
41C71⋊#95C2211yescharacteristic
42C71⋊#95C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C71⋊#95C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C2×C4⋊C4×C71C2211yescharacteristic
47C2×C4⋊C4×C71C2211yescharacteristic
48C2×C4⋊C4×C71C2211yescharacteristic
49C2×C22⋊C4×C71C2211yescharacteristic
50C2×C22⋊C4×C71C2211yescharacteristic
51C2×C22⋊C4×C71C2211yescharacteristic
52#95×C71C2211yescharacteristic
53C23×D71C2311yescharacteristic
54C2×Dic142—41no—
55C2×Dic142—41no—
56C2×Dic142D414yescomplemented
57C2×C22⋊D71—41no—
58C2×C22⋊D71—41no—
59C2×C22⋊D71Q814yescomplemented
60C22×Dic71—22no—
61C22×Dic71—41no—
62C22×Dic71—41no—
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2311yescharacteristic
65C22×Dic71C2311yescharacteristic
66C22×Dic71C2311yescharacteristic
67C22×Dic71C2311yescharacteristic
68C22×Dic71C2311yescharacteristic
69C22×Dic71C2311yescharacteristic
70C2×C4×D71—22no—
71C2×C4×D71—41no—
72C2×C4×D71—41no—
73C4⋊Dic71—22no—
74C4⋊Dic71—22no—
75C4⋊Dic71—22no—
76C4⋊Dic71—41no—
77Dic71⋊C4—22no—
78Dic71⋊C4—22no—
79Dic71⋊C4—41no—
80Dic71⋊C4—41no—
81Dic71⋊C4—41no—
82C22⋊Dic71—22no—
83C22⋊Dic71—41no—
84(C2×D71)⋊C4—22no—
85(C2×D71)⋊C4—22no—
86(C2×D71)⋊C4—22no—
87(C2×D71)⋊C4—41no—
88(C2×D71)⋊C4—41no—
89C4×Dic71—41no—
90C24×C71C2311yescharacteristic
91C22×C4×C71C2311yescharacteristic
92C22×C4×C71C2311yescharacteristic
93C22×C4×C71C2311yescharacteristic
94C22×C4×C71C2311yescharacteristic
95C22×C4×C71C2311yescharacteristic
96C22×C4×C71C2311yescharacteristic
97C4⋊C4×C71—22no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71—41no—
100C22⋊C4×C71—22no—
101C22⋊C4×C71—22no—
102C22⋊C4×C71—41no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71C2×Q812yes—
110C23×C71C2411yescharacteristic
111C22×D71—21no—
112C22×D71—21no—
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—21no—
116C22×D71—21no—
117C22×D71C2×Q812yes—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71C2×D412yes—
135C2×C4×C71#4112yes—
136C22⋊D71—44no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—21no—
152C2×Dic71—21no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—44no—
156C2×Dic71C2×D412yes—
157C2×Dic71C2×D412yes—
158C2×Dic71C2×Q812yes—
159C4×D71—44no—
160Dic142—44no—
161C22×C71—22no—
162C22×C71—22no—
163C22×C71—22no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71C22×D411yescharacteristic
169C22×C71C22×Q811yescharacteristic
170C22×C71C2×#4111yescharacteristic
171C22×C71H(5,𝔽2)11yescharacteristic
172C22×C71H(5,𝔽2)11yescharacteristic
173C22×C71H(5,𝔽2)11yescharacteristic
174C22×C712−1+411yescharacteristic
175C2×D71—22no—
176C2×D71—22no—
177C2×D71—22no—
178C2×D71—41no—
179C2×D71—41no—
180C2×D71—41no—
181C2×D71—41no—
182C4×C71—22no—
183C4×C71—22no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188Dic71—22no—
189Dic71—22no—
190Dic71—22no—
191Dic71—41no—
192Dic71—41no—
193Dic71—41no—
194Dic71—41no—
195C2×C71—42no—
196C2×C71#53411yescharacteristic
197C2×C71#53911yescharacteristic
198C2×C71#54311yescharacteristic
199C2×C71D4×Q811yescharacteristic
200C2×C71#55011yescharacteristic
201C2×C71#55411yescharacteristic
202C2×C71#56211yescharacteristic
203D71—42no—
204#2732—711nomaximal, Sylow
205C71#273211yescharacteristic, complemented, Sylow
206C2×C22⋊Q8—711no—
207C2×C22⋊Q8—711no—
208#399—711no—
209#399D7111yescharacteristic, complemented
210#398—711no—
211#398—711no—
212#396—711no—
213#396—711no—
214#391—711no—
215#390—711no—
216#389—711no—
217#386—711no—
218#386—711no—
219#385—711no—
220#385—711no—
221C22×Q8—711no—
222C22×D4—711no—
223C23×C4—711no—
224C23×C4—711no—
225C22⋊Q8—1424no—
226C22⋊Q8—1424no—
227C2×C4⋊C4—711no—
228C2×C4⋊C4—711no—
229C2×C4⋊C4—711no—
230C2×C4⋊C4—711no—
231C2×C4⋊C4—711no—
232C2×C4⋊C4—711no—
233C2×C4⋊C4—711no—
234C2×C4⋊C4—711no—
235C2×C4⋊C4—711no—
236C2×C4⋊C4C2×D7111yescharacteristic
237C2×C4⋊C4C2×D7111yescharacteristic
238C2×C4⋊C4C2×D7111yescharacteristic
239C2×C22⋊C4—711no—
240C2×C22⋊C4—711no—
241C2×C22⋊C4—711no—
242C2×C22⋊C4—711no—
243C2×C22⋊C4—711no—
244C2×C22⋊C4—711no—
245C2×C22⋊C4—711no—
246C2×C22⋊C4C2×D7111yescharacteristic
247C2×C22⋊C4C2×D7111yescharacteristic
248C2×C22⋊C4C2×D7111yescharacteristic
249C2×C42—711no—
250#95—711no—
251#95—711no—
252#95—711no—
253#95—711no—
254#95—711no—
255#95—711no—
256#95—711no—
257#95C2×D7111yescharacteristic
258C24—711no—
259C24C22×D7111yescharacteristic
260C2×Q8—2841no—
261C2×Q8—2841no—
262C2×Q8—714no—
263C2×D4—2841no—
264C2×D4—2841no—
265C2×D4—714no—
266C22×C4—2841no—
267C22×C4—2841no—
268C22×C4—2841no—
269C22×C4—2841no—
270C22×C4—711no—
271C22×C4—711no—
272C22×C4—711no—
273C22×C4—711no—
274C22×C4—711no—
275C22×C4—711no—
276C22×C4—711no—
277C22×C4—1422no—
278C22×C4—1422no—
279C22×C4C22×D7111yescharacteristic
280C22×C4C22×D7111yescharacteristic
281C22×C4C22×D7111yescharacteristic
282C22×C4C22×D7111yescharacteristic
283C22×C4C22×D7111yescharacteristic
284C22×C4C22×D7111yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—2841no—
289C4⋊C4—2841no—
290C4⋊C4—2841no—
291C4⋊C4—2841no—
292C4⋊C4—1422no—
293C4⋊C4—1422no—
294C4⋊C4—1422no—
295C4⋊C4—1422no—
296C4⋊C4—1422no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—2841no—
301C22⋊C4—2841no—
302C22⋊C4—2841no—
303C22⋊C4—1422no—
304C22⋊C4—1422no—
305C22⋊C4—1422no—
306C22⋊C4—1422no—
307C42—2841no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—712no—
315C23—1421no—
316C23—1421no—
317C23—1421no—
318C23—1421no—
319C23—1421no—
320C23—1421no—
321C23Q8×D7112yes—
322C23C23×D7111yescharacteristic
323Q8—2844no—
324D4—2844no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—2844no—
342C2×C4—2844no—
343C2×C4—712no—
344C2×C4—712no—
345C2×C4—712no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4—1421no—
349C2×C4—1421no—
350C2×C4—1421no—
351C2×C4—1421no—
352C2×C4—1421no—
353C2×C4—1421no—
354C2×C4—1421no—
355C2×C4—1421no—
356C2×C4—1421no—
357C2×C4—1421no—
358C2×C4—1421no—
359C2×C4—1421no—
360C2×C4—1421no—
361C2×C4—1421no—
362C2×C4—1421no—
363C2×C4—1421no—
364C2×C4D4×D7112yes—
365C2×C4#1176586412yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—2841no—
374C22—2841no—
375C22—2841no—
376C22—2841no—
377C22—1422no—
378C22—1422no—
379C22—1422no—
380C22C2×D4×D7111yescharacteristic
381C22C2×Q8×D7111yescharacteristic
382C22C2×#1176586411yescharacteristic
383C22#1455617311yescharacteristic
384C22#1455617411yescharacteristic
385C22#1455617411yescharacteristic
386C22#1455617511yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—2841no—
394C4—2841no—
395C4—2841no—
396C4—2841no—
397C4—1422no—
398C4—1422no—
399C4—1422no—
400C2—42no—
401C2—2842no—
402C2#1798656011yescharacteristic
403C2#1798659311yescharacteristic
404C2#1798662411yescharacteristic
405C2#1798666211yescharacteristic
406C2#1798666911yescharacteristic
407C2#1798670711yescharacteristic
408C2#1798674611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2732
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14556108C2semidirect
2. C2×#14556105C2semidirect
3. #17985934C2semidirect
4. #17985932C2semidirect
5. #17985930C2semidirect
6. #17985922C2semidirect
7. #17985921C2semidirect
8. #17985903C2semidirect
10. #17985898C2semidirect
13. #17985874C2semidirect
14. #17985873C2semidirect
15. #399×C71C2semidirect
56. C2×Dic142D4semidirect
59. C2×C22⋊D71Q8semidirect
205. C71#2732semidirect
209. #399D71semidirect