Finite Group Databasebeta

← #22591032

Subgroups — #22591032

Order 9088.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556101C211yesmaximal, characteristic
2C2×C4×C22⋊D71C211yesmaximal, characteristic, complemented
3C2×#14556086C211yesmaximal, characteristic, complemented
4#17985900C211yesmaximal, characteristic, complemented
5#17985899C211yesmaximal, characteristic, complemented
6#17985899C211yesmaximal, characteristic, complemented
7#17985894C211yesmaximal, characteristic, complemented
8#17985892C211yesmaximal, characteristic, complemented
9#17985891C211yesmaximal, characteristic, complemented
10#17985878C211yesmaximal, characteristic, complemented
11#17985877C211yesmaximal, characteristic, complemented
12#17985870C211yesmaximal, characteristic, complemented
13#17985858C211yesmaximal, characteristic, complemented
14#17985850C211yesmaximal, characteristic, complemented
15#385×C71C211yesmaximal, characteristic, complemented
16C22×D284C2211yescharacteristic
17C22×C22⋊D71C2211yescharacteristic
18C22×C22⋊D71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20C22×C4×D71C2211yescharacteristic
21#14556101C418yescomplemented
22C4×C22⋊D71—24no—
23#14556086—24no—
24C2×C4⋊Dic71C2211yescharacteristic
25C2×Dic71⋊C4C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×C22⋊Dic71C2211yescharacteristic
30C2×C22⋊Dic71C2211yescharacteristic
31C2×(C2×D71)⋊C4C2211yescharacteristic
32C2×(C2×D71)⋊C4C2211yescharacteristic
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×(C2×D71)⋊C4C2211yescharacteristic
36C2×(C2×D71)⋊C4C2211yescharacteristic
37C2×(C2×D71)⋊C4C2211yescharacteristic
38C2×(C2×D71)⋊C4C2211yescharacteristic
39C2×C4×Dic71C2211yescharacteristic
40C2×C4×Dic71C2211yescharacteristic
41C2×C4×Dic71C2211yescharacteristic
42C2×C4×Dic71C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C71⋊#95C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C71⋊#95C2211yescharacteristic
47C23×C4×C71C2211yescharacteristic
48C2×C4⋊C4×C71C2211yescharacteristic
49C2×C4⋊C4×C71C2211yescharacteristic
50C2×C22⋊C4×C71C2211yescharacteristic
51C2×C22⋊C4×C71C2211yescharacteristic
52#95×C71C2211yescharacteristic
53#95×C71C2211yescharacteristic
54C23×D71C2311yescharacteristic
55C23×D71C2311yescharacteristic
56C2×D284—41no—
57C2×D284—41no—
58C2×D284C2×C414yescomplemented
59C2×C22⋊D71—22no—
60C2×C22⋊D71—22no—
61C2×C22⋊D71—41no—
62C2×C22⋊D71—41no—
63C2×C22⋊D71C2×C414yes—
64C2×C22⋊D71C2×C414yescomplemented
65C22×Dic71C2311yescharacteristic
66C22×Dic71C2311yescharacteristic
67C22×Dic71C2311yescharacteristic
68C22×Dic71C2311yescharacteristic
69C22×Dic71C2311yescharacteristic
70C22×Dic71C2311yescharacteristic
71C2×C4×D71—22no—
72C2×C4×D71—22no—
73C2×C4×D71—22no—
74C2×C4×D71—41no—
75C2×C4×D71—41no—
76C2×C4×D71C2×C414yescomplemented
77C4⋊Dic71—22no—
78Dic71⋊C4—22no—
79Dic71⋊C4—22no—
80Dic71⋊C4—41no—
81Dic71⋊C4C2×C414yescomplemented
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—
89(C2×D71)⋊C4—41no—
90(C2×D71)⋊C4—41no—
91(C2×D71)⋊C4C2×C414yescomplemented
92C4×Dic71—22no—
93C4×Dic71—41no—
94C4×Dic71—41no—
95C4×Dic71D414yescomplemented
96C24×C71C2311yescharacteristic
97C22×C4×C71—22no—
98C22×C4×C71—41no—
99C22×C4×C71—41no—
100C22×C4×C71C2311yescharacteristic
101C22×C4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104C22×C4×C71C2311yescharacteristic
105C22×C4×C71C2311yescharacteristic
106C4⋊C4×C71—22no—
107C4⋊C4×C71—41no—
108C22⋊C4×C71—41no—
109C22⋊C4×C71C2×C414yescomplemented
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—21no—
127C22×D71—21no—
128C22×D71—21no—
129C22×D71—21no—
130C22×D71C22×C412yes—
131C22×D71C22×C412yes—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
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—44no—
148C2×C4×C71C22×C412yes—
149C2×C4×C71C22×C412yes—
150C2×C4×C71C2×D412yes—
151C22⋊D71—28no—
152C22⋊D71—44no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—21no—
156C2×Dic71—21no—
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×Dic71C22×C412yes—
167C2×Dic71C22×C412yes—
168C2×Dic71C2×D412yes—
169C2×Dic71C2×D412yes—
170C2×Dic71#4112yes—
171D284—44no—
172C4×D71—28no—
173C4×D71—44no—
174C22×C71—22no—
175C22×C71—22no—
176C22×C71—22no—
177C22×C71—41no—
178C22×C71—41no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71C23×C411yescharacteristic
182C22×C71C22×D411yescharacteristic
183C22×C71C2×#4111yescharacteristic
184C22×C71H(5,𝔽2)11yescharacteristic
185C22×C71H(5,𝔽2)11yescharacteristic
186C22×C71H(5,𝔽2)11yescharacteristic
187C22×C712−1+411yescharacteristic
188C2×D71—22no—
189C2×D71—22no—
190C2×D71—22no—
191C2×D71—22no—
192C2×D71—22no—
193C2×D71—22no—
194C2×D71—41no—
195C2×D71—41no—
196C2×D71—41no—
197C2×D71—41no—
198C2×D71—41no—
199C2×D71—41no—
200C2×D71—41no—
201C2×D71—41no—
202C4×C71—22no—
203C4×C71—22no—
204C4×C71—22no—
205C4×C71—41no—
206C4×C71—41no—
207C4×C71—41no—
208Dic71—22no—
209Dic71—22no—
210Dic71—41no—
211Dic71—41no—
212Dic71—41no—
213Dic71C4×D414yes—
214C2×C71—42no—
215C2×C71C2×C4×D411yescharacteristic
216C2×C71#51811yescharacteristic
217C2×C71#52011yescharacteristic
218C2×C71#53511yescharacteristic
219C2×C71#53711yescharacteristic
220C2×C71#54011yescharacteristic
221C2×C71#54211yescharacteristic
222D71—42no—
223D71—42no—
224#2324—711nomaximal, Sylow
225C71#232411yescharacteristic, complemented, Sylow
226C2×#121—711no—
227C2×C4×D4—711no—
228C2×#117—711no—
229#390—711no—
230#390—711no—
231#390—711no—
232#389—711no—
233#388—711no—
234#388—711no—
235#386—711no—
236#386—711no—
237#385—711no—
238#385D7111yescharacteristic, complemented
239#380—711no—
240C4×C4⋊C4—711no—
241C22×D4—711no—
242C22×D4—711no—
243C22×D4—711no—
244C23×C4—711no—
245C23×C4—711no—
246C23×C4C2×D7111yescharacteristic
247#121—718no—
248C4×D4—1424no—
249#117—1424no—
250C2×C4⋊C4—711no—
251C2×C4⋊C4—711no—
252C2×C4⋊C4—711no—
253C2×C4⋊C4—711no—
254C2×C4⋊C4—711no—
255C2×C4⋊C4C2×D7111yescharacteristic
256C2×C4⋊C4C2×D7111yescharacteristic
257C2×C22⋊C4—711no—
258C2×C22⋊C4—711no—
259C2×C22⋊C4—711no—
260C2×C22⋊C4—711no—
261C2×C22⋊C4—711no—
262C2×C22⋊C4—711no—
263C2×C22⋊C4—711no—
264C2×C22⋊C4—711no—
265C2×C22⋊C4—711no—
266C2×C22⋊C4—711no—
267C2×C22⋊C4C2×D7111yescharacteristic
268C2×C22⋊C4C2×D7111yescharacteristic
269C2×C42—711no—
270C2×C42—711no—
271C2×C42—711no—
272C2×C42—711no—
273#95—711no—
274#95—711no—
275#95—711no—
276#95—711no—
277#95C2×D7111yescharacteristic
278#95C2×D7111yescharacteristic
279C24—711no—
280C24—711no—
281C24C22×D7111yescharacteristic
282C2×D4—2841no—
283C2×D4—2841no—
284C2×D4—2841no—
285C2×D4—2841no—
286C2×D4—714no—
287C2×D4—714no—
288C2×D4—714no—
289C2×D4—1422no—
290C2×D4—1422no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—2841no—
295C22×C4—2841no—
296C22×C4—711no—
297C22×C4—711no—
298C22×C4—711no—
299C22×C4—714no—
300C22×C4—711no—
301C22×C4—711no—
302C22×C4—711no—
303C22×C4—1422no—
304C22×C4—1422no—
305C22×C4—1422no—
306C22×C4C22×D7111yescharacteristic
307C22×C4C22×D7111yescharacteristic
308C22×C4C22×D7111yescharacteristic
309C22×C4C22×D7111yescharacteristic
310C22×C4C22×D7111yescharacteristic
311C22×C4C22×D7111yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—2841no—
315C4⋊C4—714no—
316C4⋊C4—1422no—
317C4⋊C4—1422no—
318C4⋊C4—1422no—
319C22⋊C4—41no—
320C22⋊C4—2841no—
321C22⋊C4—2841no—
322C22⋊C4—2841no—
323C22⋊C4—2841no—
324C22⋊C4—2841no—
325C22⋊C4—714no—
326C22⋊C4—1422no—
327C22⋊C4—1422no—
328C22⋊C4—1422no—
329C22⋊C4—1422no—
330C22⋊C4C4×D7114yescomplemented
331C42—2841no—
332C42—2841no—
333C42—714no—
334C42—1422no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—712no—
342C23—712no—
343C23—1421no—
344C23—1421no—
345C23—1421no—
346C23—1421no—
347C23—1421no—
348C23—1421no—
349C23—1421no—
350C23—1421no—
351C23—1421no—
352C23—1421no—
353C23—1421no—
354C23—1421no—
355C23C2×C4×D7112yes—
356C23C23×D7111yescharacteristic
357D4—2844no—
358D4—2844no—
359D4—1428no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—2844no—
377C2×C4—712no—
378C2×C4—712no—
379C2×C4—712no—
380C2×C4—712no—
381C2×C4—712no—
382C2×C4—1428no—
383C2×C4—1421no—
384C2×C4—1421no—
385C2×C4—1421no—
386C2×C4—1421no—
387C2×C4—1421no—
388C2×C4—1421no—
389C2×C4—1421no—
390C2×C4—1421no—
391C2×C4—1421no—
392C2×C4—1421no—
393C2×C4—1421no—
394C2×C4—1421no—
395C2×C4—1421no—
396C2×C4C2×C4×D7112yes—
397C2×C4C2×C4×D7112yes—
398C2×C4D4×D7112yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2841no—
407C22—2841no—
408C22—2841no—
409C22—2841no—
410C22—2841no—
411C22—2841no—
412C22—2841no—
413C22—2841no—
414C22—1422no—
415C22—1422no—
416C22—1422no—
417C22—1422no—
418C22—1422no—
419C22—1422no—
420C22C22×C4×D7111yescharacteristic
421C22C2×D4×D7111yescharacteristic
422C22#41×D7111yescharacteristic
423C22#1455617311yescharacteristic
424C22#1455617311yescharacteristic
425C22#1455617411yescharacteristic
426C22#1455617611yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—2841no—
434C4—2841no—
435C4—2841no—
436C4—714no—
437C4—1422no—
438C4—1422no—
439C2—42no—
440C2—2842no—
441C2—2842no—
442C2C4×D4×D7111yescharacteristic
443C2#1798647111yescharacteristic
444C2#1798648011yescharacteristic
445C2#1798656411yescharacteristic
446C2#1798657911yescharacteristic
447C2#1798660111yescharacteristic
448C2#1798661911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutatorC22×C71C23×C4
Fitting#385×C71C2
socleC23×C71C24
hypercenter#385D71
hypocenterC71#2324
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D71C2semidirect
3. C2×#14556086C2semidirect
4. #17985900C2semidirect
5. #17985899C2semidirect
6. #17985899C2semidirect
7. #17985894C2semidirect
8. #17985892C2semidirect
9. #17985891C2semidirect
10. #17985878C2semidirect
11. #17985877C2semidirect
12. #17985870C2semidirect
13. #17985858C2semidirect
14. #17985850C2semidirect
15. #385×C71C2semidirect
21. #14556101C4semidirect
58. C2×D284C2×C4semidirect
64. C2×C22⋊D71C2×C4semidirect
76. C2×C4×D71C2×C4semidirect
81. Dic71⋊C4C2×C4semidirect
91. (C2×D71)⋊C4C2×C4semidirect
95. C4×Dic71D4semidirect
109. C22⋊C4×C71C2×C4semidirect
225. C71#2324semidirect
238. #385D71semidirect
330. C22⋊C4C4×D71semidirect