Finite Group Databasebeta

← #22591184

Subgroups — #22591184

Order 9088.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556085C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D71C211yesmaximal, characteristic
3C2×C22⋊C4×D71C211yesmaximal, characteristic
4#17985893C211yesmaximal, characteristic, complemented
5#17985892C211yesmaximal, characteristic, complemented
6#17985887C211yesmaximal, characteristic, complemented
7#17985873C211yesmaximal, characteristic, complemented
8#17985870C211yesmaximal, characteristic, complemented
9#17985861C211yesmaximal, characteristic, complemented
10#17985856C211yesmaximal, characteristic, complemented
11#17985854C211yesmaximal, characteristic
12C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
13#17985846C211yesmaximal, characteristic, complemented
14#17985846C211yesmaximal, characteristic, complemented
15#388×C71C211yesmaximal, characteristic, complemented
16C24×D71C2211yescharacteristic
17C22×C4×D71C2211yescharacteristic
18C22×C4×D71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20C22×C4×D71C2211yescharacteristic
21#14556085—24no—
22C2×C4⋊Dic71C2211yescharacteristic, complemented
23C2×Dic71⋊C4C2211yescharacteristic, complemented
24C2×Dic71⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic71C2211yescharacteristic
26C2×C22⋊Dic71C2211yescharacteristic
27C2×C22⋊Dic71C2211yescharacteristic
28C2×(C2×D71)⋊C4C2211yescharacteristic
29C2×(C2×D71)⋊C4C2211yescharacteristic
30C2×(C2×D71)⋊C4C2211yescharacteristic
31C2×(C2×D71)⋊C4C2211yescharacteristic
32C2×(C2×D71)⋊C4C2211yescharacteristic
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×(C2×D71)⋊C4C2211yescharacteristic
36C22⋊C4×D71—24no—
37C22⋊C4×D71C418yescomplemented
38C2×C4×Dic71C2211yescharacteristic, complemented
39C2×C4×Dic71C2211yescharacteristic, complemented
40C2×C4×Dic71C2211yescharacteristic, complemented
41C71⋊#95C2211yescharacteristic, complemented
42C71⋊#95C2211yescharacteristic, complemented
43C71⋊#95C2211yescharacteristic, complemented
44C71⋊#95C2211yescharacteristic, complemented
45C71⋊#95C2211yescharacteristic, complemented
46C71⋊#95C2211yescharacteristic, complemented
47C2×C4⋊C4×C71C2211yescharacteristic, complemented
48C2×C22⋊C4×C71C2211yescharacteristic
49C2×C22⋊C4×C71C2211yescharacteristic
50C2×C22⋊C4×C71C2211yescharacteristic
51C2×C42×C71C2211yescharacteristic, complemented
52#95×C71C2211yescharacteristic, complemented
53#95×C71C2211yescharacteristic, complemented
54C23×D71—22no—
55C23×D71—22no—
56C23×D71—22no—
57C23×D71—22no—
58C23×D71—41no—
59C23×D71—41no—
60C23×D71C2×C414yes—
61C23×D71C2311yescharacteristic
62C23×D71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2311yescharacteristic
65C22×Dic71C2311yescharacteristic
66C22×Dic71C2311yescharacteristic
67C22×Dic71C2311yescharacteristic
68C22×Dic71C2311yescharacteristic
69C2×C4×D71—22no—
70C2×C4×D71—22no—
71C2×C4×D71—22no—
72C2×C4×D71—22no—
73C2×C4×D71—22no—
74C2×C4×D71—22no—
75C2×C4×D71—41no—
76C2×C4×D71—41no—
77C2×C4×D71—41no—
78C2×C4×D71—41no—
79C2×C4×D71C2×C414yescomplemented
80C2×C4×D71C2×C414yescomplemented
81C4⋊Dic71—41no—
82Dic71⋊C4—22no—
83Dic71⋊C4—22no—
84C22⋊Dic71—22no—
85C22⋊Dic71—41no—
86C22⋊Dic71C2×C414yescomplemented
87(C2×D71)⋊C4—22no—
88(C2×D71)⋊C4—22no—
89(C2×D71)⋊C4—22no—
90(C2×D71)⋊C4—22no—
91(C2×D71)⋊C4—22no—
92(C2×D71)⋊C4C2×C414yes—
93(C2×D71)⋊C4C2×C414yes—
94(C2×D71)⋊C4D414yes—
95C4×Dic71—22no—
96C4×Dic71—41no—
97C4×Dic71—41no—
98C24×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C22×C4×C71C2311yescharacteristic
101C22×C4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104C22×C4×C71C2311yescharacteristic
105C4⋊C4×C71—41no—
106C22⋊C4×C71—22no—
107C22⋊C4×C71—41no—
108C22⋊C4×C71C2×C414yescomplemented
109C42×C71—22no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71C22×C412yes—
117C23×C71C2411yescharacteristic
118C22×D71—21no—
119C22×D71—21no—
120C22×D71—21no—
121C22×D71—21no—
122C22×D71—21no—
123C22×D71—21no—
124C22×D71—21no—
125C22×D71—21no—
126C22×D71—28no—
127C22×D71—28no—
128C22×D71—44no—
129C22×D71—44no—
130C22×D71—44no—
131C22×D71—44no—
132C22×D71—44no—
133C22×D71C22×C412yes—
134C22×D71C22×C412yes—
135C22×D71C2×D412yes—
136C22×D71#4112yes—
137C22×D71#4112yes—
138C22×D71#4112yes—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—21no—
153C2×C4×C71—21no—
154C2×C4×C71C22×C412yes—
155C2×C4×C71C22×C412yes—
156C2×C4×C71C2×D412yes—
157C2×Dic71—21no—
158C2×Dic71—21no—
159C2×Dic71—21no—
160C2×Dic71—21no—
161C2×Dic71—21no—
162C2×Dic71—21no—
163C2×Dic71—21no—
164C2×Dic71—21no—
165C2×Dic71—21no—
166C2×Dic71—21no—
167C2×Dic71—21no—
168C2×Dic71—21no—
169C2×Dic71—21no—
170C2×Dic71—21no—
171C2×Dic71—21no—
172C2×Dic71C22×C412yes—
173C2×Dic71C22×C412yes—
174C2×Dic71C2×D412yes—
175C4×D71—28no—
176C4×D71—44no—
177C4×D71—44no—
178C4×D71—44no—
179C22×C71—22no—
180C22×C71—22no—
181C22×C71—22no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71—41no—
185C22×C71—41no—
186C22×C71C23×C411yescharacteristic
187C22×C71C22×D411yescharacteristic
188C22×C71C2×#4111yescharacteristic
189C22×C71C2×#4111yescharacteristic
190C22×C71C2×#4111yescharacteristic
191C22×C71H(5,𝔽2)11yescharacteristic
192C22×C71H(5,𝔽2)11yescharacteristic
193C2×D71—22no—
194C2×D71—22no—
195C2×D71—22no—
196C2×D71—22no—
197C2×D71—22no—
198C2×D71—22no—
199C2×D71—22no—
200C2×D71—22no—
201C2×D71—22no—
202C2×D71—22no—
203C2×D71—22no—
204C2×D71—22no—
205C2×D71—416no—
206C2×D71#11714yescomplemented
207C2×D71C4×D414yes—
208C4×C71—22no—
209C4×C71—22no—
210C4×C71—22no—
211C4×C71—41no—
212C4×C71—41no—
213C4×C71—41no—
214Dic71—22no—
215Dic71—22no—
216Dic71—22no—
217Dic71—41no—
218Dic71—41no—
219Dic71—41no—
220C2×C71—42no—
221C2×C71C2×#11711yescharacteristic
222C2×C71C2×C4×D411yescharacteristic
223C2×C71#51811yescharacteristic
224C2×C71#54611yescharacteristic
225C2×C71#54611yescharacteristic
226C2×C71#55111yescharacteristic
227C2×C71#55111yescharacteristic
228D71—24no—
229D71—24no—
230#2342—711nomaximal, Sylow
231C71#234211yescharacteristic, complemented, Sylow
232C2×#117—711no—
233C22×C22⋊C4—711no—
234C22×C22⋊C4—711no—
235#388—711no—
236#388—711no—
237#388—711no—
238#388D7111yescharacteristic, complemented
239#385—711no—
240#385—711no—
241#381—711no—
242#380—711no—
243#379—711no—
244C4×C22⋊C4—711no—
245C4×C22⋊C4—711no—
246C4×C22⋊C4—711no—
247C25—711no—
248C23×C4—711no—
249C23×C4—711no—
250C23×C4—711no—
251C23×C4—711no—
252#117—1424no—
253C2×C4⋊C4—711no—
254C2×C4⋊C4—711no—
255C2×C4⋊C4—711no—
256C2×C4⋊C4C2×D7111yescharacteristic, complemented
257C2×C22⋊C4—711no—
258C2×C22⋊C4—711no—
259C2×C22⋊C4—711no—
260C2×C22⋊C4—711no—
261C2×C22⋊C4—711no—
262C2×C22⋊C4—718no—
263C2×C22⋊C4—711no—
264C2×C22⋊C4—711no—
265C2×C22⋊C4—711no—
266C2×C22⋊C4—711no—
267C2×C22⋊C4—711no—
268C2×C22⋊C4—711no—
269C2×C22⋊C4—1424no—
270C2×C22⋊C4C2×D7111yescharacteristic
271C2×C22⋊C4C2×D7111yescharacteristic
272C2×C22⋊C4C2×D7111yescharacteristic
273C2×C42—711no—
274C2×C42—711no—
275C2×C42—711no—
276C2×C42C2×D7111yescharacteristic, complemented
277#95—711no—
278#95—711no—
279#95—711no—
280#95—711no—
281#95—711no—
282#95—711no—
283#95C2×D7111yescharacteristic, complemented
284#95C2×D7111yescharacteristic, complemented
285C24—2841no—
286C24—2841no—
287C24—714no—
288C24—711no—
289C24—711no—
290C24—1422no—
291C24—1422no—
292C24—1422no—
293C24—1422no—
294C24C22×D7111yescharacteristic
295C22×C4—2841no—
296C22×C4—2841no—
297C22×C4—2841no—
298C22×C4—2841no—
299C22×C4—714no—
300C22×C4—711no—
301C22×C4—711no—
302C22×C4—711no—
303C22×C4—714no—
304C22×C4—711no—
305C22×C4—711no—
306C22×C4—711no—
307C22×C4—1422no—
308C22×C4—1422no—
309C22×C4—1422no—
310C22×C4—1422no—
311C22×C4—1422no—
312C22×C4—1422no—
313C22×C4C22×D7111yescharacteristic
314C22×C4C22×D7111yescharacteristic
315C22×C4C22×D7111yescharacteristic
316C22×C4C22×D7111yescharacteristic
317C22×C4C22×D7111yescharacteristic
318C22×C4C22×D7111yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—2841no—
321C4⋊C4—1422no—
322C4⋊C4—1422no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—2841no—
326C22⋊C4—714no—
327C22⋊C4—714no—
328C22⋊C4—714no—
329C22⋊C4—714no—
330C22⋊C4—1422no—
331C22⋊C4—1422no—
332C22⋊C4—1422no—
333C22⋊C4—1422no—
334C22⋊C4—1422no—
335C22⋊C4—1422no—
336C22⋊C4C4×D7114yescomplemented
337C42—22no—
338C42—2841no—
339C42—2841no—
340C42—1422no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2844no—
348C23—2844no—
349C23—2844no—
350C23—2844no—
351C23—2844no—
352C23—712no—
353C23—712no—
354C23—712no—
355C23—712no—
356C23—712no—
357C23—712no—
358C23—1428no—
359C23—1428no—
360C23—1421no—
361C23—1421no—
362C23—1421no—
363C23—1421no—
364C23—1421no—
365C23—1421no—
366C23—1421no—
367C23—1421no—
368C23C2×C4×D7112yes—
369C23C23×D7111yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—2844no—
386C2×C4—2844no—
387C2×C4—2844no—
388C2×C4—712no—
389C2×C4—712no—
390C2×C4—712no—
391C2×C4—1428no—
392C2×C4—1421no—
393C2×C4—1421no—
394C2×C4—1421no—
395C2×C4—1421no—
396C2×C4—1421no—
397C2×C4—1421no—
398C2×C4—1421no—
399C2×C4—1421no—
400C2×C4—1421no—
401C2×C4—1421no—
402C2×C4—1421no—
403C2×C4—1421no—
404C2×C4—1421no—
405C2×C4—1421no—
406C2×C4—1421no—
407C2×C4C2×C4×D7112yes—
408C2×C4C2×C4×D7112yes—
409C2×C4D4×D7112yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—28416no—
418C22—714no—
419C22—714no—
420C22—1422no—
421C22—1422no—
422C22—1422no—
423C22—1422no—
424C22—1422no—
425C22—1422no—
426C22—1422no—
427C22—1422no—
428C22—1422no—
429C22—1422no—
430C22—1422no—
431C22—1422no—
432C22C22×C4×D7111yescharacteristic
433C22C2×D4×D7111yescharacteristic
434C22#41×D7111yescharacteristic
435C22#41×D7111yescharacteristic
436C22#41×D7111yescharacteristic
437C22#1455617311yescharacteristic
438C22#1455617411yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—2841no—
446C4—2841no—
447C4—2841no—
448C4—1422no—
449C4—1422no—
450C4—1422no—
451C2—42no—
452C2—1424no—
453C2—1424no—
454C2#117×D7111yescharacteristic
455C2C4×D4×D7111yescharacteristic
456C2#1798647111yescharacteristic
457C2#1798664111yescharacteristic
458C2#1798664611yescharacteristic
459C2#1798667411yescharacteristic
460C2#1798667911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutatorC22×C71C23×C4
Fitting#388×C71C2
socleC23×C71C24
hypercenter#388D71
hypocenterC71#2342
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#14556085C2semidirect
4. #17985893C2semidirect
5. #17985892C2semidirect
6. #17985887C2semidirect
7. #17985873C2semidirect
8. #17985870C2semidirect
9. #17985861C2semidirect
10. #17985856C2semidirect
12. C4×(C2×D71)⋊C4C2semidirect
13. #17985846C2semidirect
14. #17985846C2semidirect
15. #388×C71C2semidirect
22. C2×C4⋊Dic71C22semidirect
23. C2×Dic71⋊C4C22semidirect
24. C2×Dic71⋊C4C22semidirect
37. C22⋊C4×D71C4semidirect
38. C2×C4×Dic71C22semidirect
39. C2×C4×Dic71C22semidirect
40. C2×C4×Dic71C22semidirect
41. C71⋊#95C22semidirect
42. C71⋊#95C22semidirect
43. C71⋊#95C22semidirect
44. C71⋊#95C22semidirect
45. C71⋊#95C22semidirect
46. C71⋊#95C22semidirect
47. C2×C4⋊C4×C71C22semidirect
51. C2×C42×C71C22semidirect
52. #95×C71C22semidirect
53. #95×C71C22semidirect
79. C2×C4×D71C2×C4semidirect
80. C2×C4×D71C2×C4semidirect
86. C22⋊Dic71C2×C4semidirect
108. C22⋊C4×C71C2×C4semidirect
206. C2×D71#117semidirect
231. C71#2342semidirect
238. #388D71semidirect
256. C2×C4⋊C4C2×D71semidirect
276. C2×C42C2×D71semidirect
283. #95C2×D71semidirect
284. #95C2×D71semidirect
336. C22⋊C4C4×D71semidirect