Finite Group Databasebeta

← #22601993

Subgroups — #22601993

Order 9088.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556175C211yesmaximal, characteristic, complemented
2C2×#41×D71C211yesmaximal, characteristic, complemented
3#17986669C213yesmaximal, complemented
4#17986668C211yesmaximal, characteristic, complemented
5#17986667C213yesmaximal, complemented
6#17986661C211yesmaximal, characteristic, complemented
7#17986658C213yesmaximal, complemented
8#17986656C211yesmaximal, characteristic, complemented
9#17986655C213yesmaximal, complemented
10#17986565C213yesmaximal, complemented
11#17986562C213yesmaximal, complemented
12#17986509C213yesmaximal, complemented
13#17986508C213yesmaximal, complemented
14#17986475C211yesmaximal, characteristic, complemented
15#550×C71C211yesmaximal, characteristic, complemented
16#14556175—28no—
17C2×#11765866C2211yescharacteristic
18C2×#11765865C2213yes—
19C2×#11765865C2213yes—
20C2×#11765864C2213yes—
21C2×#11765864C2213yes—
22C2×#11765864C2213yescomplemented
23#41×D71—28no—
24C22×Dic142C2213yescomplemented
25C2×Q8×D71C2211yescharacteristic
26C2×Q8×D71C2211yescharacteristic, complemented
27C2×D4×D71C2213yes—
28C22×C4×D71C2213yes—
29#14556124C2213yescomplemented
30#14556124C2216yescomplemented
31#14556116C2213yescomplemented
32#14556115C2213yescomplemented
33#14556114C2216yescomplemented
34#14556113C2211yescharacteristic
35#14556110C2213yes—
36#14556108C2213yescomplemented
37#14556107C2213yescomplemented
38#14556106C2213yescomplemented
39#14556105C2213yescomplemented
40#14556103C2213yescomplemented
41#14556102C2213yescomplemented
42#14556101C2213yescomplemented
43#14556100C2216yescomplemented
44#14556100C2216yescomplemented
45#14556100C2213yescomplemented
46#14556099C2213yescomplemented
47#14556099C2213yescomplemented
48#14556098C2213yes—
49#14556096C2213yes—
50#14556096C2216yes—
51Q8×Dic71C2211yescharacteristic, complemented
52C4×Dic142C2213yescomplemented
53#14556093C2213yescomplemented
54C4×D284C2213yescomplemented
55D4×Dic71C2216yescomplemented
56C4×C22⋊D71C2213yescomplemented
57C4×C22⋊D71C2213yescomplemented
58(C22⋊D71)⋊C4C2216yescomplemented
59#14556086C2213yes—
60#14556085C2213yes—
61C2×#41×C71C2211yescharacteristic
62C2×#41×C71C2211yescharacteristic, complemented
63#127×C71C2213yescomplemented
64#121×C71C2213yescomplemented
65#121×C71C2213yescomplemented
66C4×Q8×C71C2211yescharacteristic, complemented
67C4×D4×C71C2213yescomplemented
68C23×D71C2313yes—
69#11765866—24no—
70#11765865—212no—
71#11765865—212no—
72#11765864—212no—
73#11765864—212no—
74#11765864—212no—
75C2×Dic142—212no—
76C2×Dic142C2313yes—
77C2×Dic142C2313yes—
78C2×Dic142C2313yes—
79Q8×D71—24no—
80Q8×D71D418yescomplemented
81C2×D284C2313yes—
82C2×D284C2311yescharacteristic
83C2×D284C2313yes—
84C2×C22⋊D71C2316yes—
85C2×C22⋊D71C2313yes—
86C2×C22⋊D71C2313yes—
87C2×C22⋊D71C2316yes—
88C2×C22⋊D71C2313yes—
89C2×C22⋊D71C2313yes—
90D4×D71—212no—
91C22×Dic71C2313yes—
92C22×Dic71C2316yes—
93C2×C4×D71—212no—
94C2×C4×D71C2311yescharacteristic
95C2×C4×D71C2313yes—
96C2×C4×D71C2313yes—
97C2×C4×D71C2313yes—
98C2×C4×D71C2313yes—
99C4⋊Dic71C2313yes—
100C4⋊Dic71C2313yes—
101Dic71⋊C4C2313yes—
102Dic71⋊C4C2313yes—
103Dic71⋊C4C2313yes—
104Dic71⋊C4C2311yescharacteristic
105C22⋊Dic71C2313yes—
106C22⋊Dic71C2313yes—
107C22⋊Dic71C2316yes—
108(C2×D71)⋊C4C2311yescharacteristic
109(C2×D71)⋊C4C2313yes—
110(C2×D71)⋊C4C2316yes—
111(C2×D71)⋊C4C2311yescharacteristic
112(C2×D71)⋊C4C2313yes—
113(C2×D71)⋊C4C2311yescharacteristic
114(C2×D71)⋊C4C2313yes—
115C4×Dic71C2313yes—
116C4×Dic71C2316yes—
117#41×C71—24no—
118#41×C71—24no—
119C2×Q8×C71C2311yescharacteristic
120C2×D4×C71C2313yes—
121C2×D4×C71C2313yes—
122C2×D4×C71C2313yes—
123C2×D4×C71C2316yes—
124C22×C4×C71C2313yes—
125C22×C4×C71C2313yes—
126C4⋊C4×C71C2311yescharacteristic
127C4⋊C4×C71C2313yes—
128C22⋊C4×C71C2313yes—
129C22⋊C4×C71C2313yes—
130C42×C71C2313yes—
131C23×C71C2413yes—
132C23×C71C2413yes—
133C22×D71—26no—
134C22×D71—43no—
135C22×D71—43no—
136C22×D71C2413yes—
137C22×D71C2411yescharacteristic
138C22×D71C2411yescharacteristic
139Q8×C71C2×D414yes—
140D4×C71—26no—
141D4×C71—26no—
142D4×C71—46no—
143D4×C71—43no—
144C2×C4×C71—26no—
145C2×C4×C71—26no—
146C2×C4×C71C2411yescharacteristic
147C2×C4×C71C2413yes—
148C2×C4×C71C2411yescharacteristic
149C2×C4×C71C2411yescharacteristic
150C2×C4×C71C2413yes—
151C22⋊D71—26no—
152C22⋊D71—212no—
153C22⋊D71—26no—
154C22⋊D71—43no—
155C22⋊D71—43no—
156C22⋊D71—46no—
157C2×Dic71—26no—
158C2×Dic71—212no—
159C2×Dic71C2411yescharacteristic
160C2×Dic71C2413yes—
161C2×Dic71C2413yes—
162C2×Dic71C2411yescharacteristic
163C2×Dic71C2413yes—
164D284—22no—
165D284—26no—
166D284—43no—
167C4×D71—22no—
168C4×D71—26no—
169C4×D71—26no—
170C4×D71—26no—
171C4×D71C2×D4112yes—
172Dic142—26no—
173Dic142—26no—
174Dic142C2×D4112yes—
175C22×C71—23no—
176C22×C71—23no—
177C22×C71—23no—
178C22×C71—23no—
179C22×C71—23no—
180C22×C71—23no—
181C22×C71C2511yescharacteristic
182C2×D71—21no—
183C2×D71—21no—
184C2×D71—21no—
185C2×D71—23no—
186C2×D71—21no—
187C2×D71—23no—
188C2×D71—21no—
189C2×D71—23no—
190C2×D71—412no—
191C2×D71C22×D412yes—
192C4×C71—21no—
193C4×C71—21no—
194C4×C71—23no—
195C4×C71—21no—
196C4×C71C22×D416yes—
197Dic71—23no—
198Dic71—21no—
199Dic71—21no—
200Dic71—23no—
201Dic71C22×D416yes—
202C2×C71—43no—
203C2×C71—43no—
204C2×C71C23×D411yescharacteristic
205C2×C71#58511yescharacteristic
206C2×C71#58511yescharacteristic
207D71—22no—
208D71—41no—
209D71—43no—
210#3487—711nomaximal, Sylow
211C71#348711yescharacteristic, complemented, Sylow
212C2×2−1+4—711no—
213C22×#41—711no—
214#550—711no—
215#550—713no—
216#550—713no—
217#550D7111yescharacteristic, complemented
218#548—711no—
219#548—713no—
220#548—711no—
221#548—713no—
222#535—713no—
223#535—713no—
224#525—713no—
225#525—713no—
226#519—711no—
2272−1+4—1428no—
228C2×#41—711no—
229C2×#41—713no—
230C2×#41—713no—
231C2×#41—713no—
232C2×#41—713no—
233C2×#41—713no—
234C2×#41—1428no—
235C2×#41C2×D7111yescharacteristic
236C2×#41C2×D7111yescharacteristic, complemented
237C22×Q8—711no—
238C22×Q8—711no—
239C22×Q8—713no—
240C22×D4—713no—
241C23×C4—713no—
242#127—713no—
243#127—716no—
244#127C2×D7113yescomplemented
245#124—713no—
246#124—716no—
247#124—713no—
248#123—711no—
249#123—713no—
250C22⋊Q8—713no—
251C22⋊Q8—713no—
252C22⋊Q8—713no—
253C22⋊Q8—713no—
254#121—713no—
255#121—713no—
256#121—713no—
257#121—716no—
258#121—716no—
259#121—713no—
260#121—713no—
261#121—713no—
262#121C2×D7113yescomplemented
263#121C2×D7113yescomplemented
264#120—716no—
265#120—713no—
266#120—713no—
267C4×Q8—711no—
268C4×Q8—713no—
269C4×Q8—713no—
270C4×Q8C2×D7111yescharacteristic, complemented
271C4×D4—716no—
272C4×D4—713no—
273C4×D4—713no—
274C4×D4—716no—
275C4×D4—713no—
276C4×D4C2×D7113yescomplemented
277#117—713no—
278#117—713no—
279C24—713no—
280#41—24no—
281#41—24no—
282#41—14212no—
283#41—14212no—
284#41—14212no—
285#41—14212no—
286#41—14212no—
287#41—1424no—
288C2×Q8—718no—
289C2×Q8—713no—
290C2×Q8—713no—
291C2×Q8—713no—
292C2×Q8—14212no—
293C2×Q8—1424no—
294C2×Q8C22×D7111yescharacteristic
295C2×D4—716no—
296C2×D4—713no—
297C2×D4—713no—
298C2×D4—713no—
299C2×D4—716no—
300C2×D4—713no—
301C2×D4—713no—
302C2×D4—711no—
303C2×D4—713no—
304C2×D4—14212no—
305C2×D4C22×D7113yes—
306C2×D4C22×D7113yes—
307C2×D4C22×D7116yes—
308C2×D4C22×D7113yes—
309C22×C4—713no—
310C22×C4—716no—
311C22×C4—713no—
312C22×C4—713no—
313C22×C4—713no—
314C22×C4—713no—
315C22×C4—711no—
316C22×C4—14212no—
317C22×C4C22×D7113yes—
318C22×C4C22×D7113yes—
319C4⋊C4—713no—
320C4⋊C4—713no—
321C4⋊C4—713no—
322C4⋊C4—711no—
323C4⋊C4—713no—
324C4⋊C4—713no—
325C4⋊C4C22×D7113yes—
326C4⋊C4C22×D7111yescharacteristic
327C22⋊C4—713no—
328C22⋊C4—713no—
329C22⋊C4—716no—
330C22⋊C4—713no—
331C22⋊C4—711no—
332C22⋊C4—711no—
333C22⋊C4—711no—
334C22⋊C4—713no—
335C22⋊C4—716no—
336C22⋊C4—713no—
337C22⋊C4C22×D7113yes—
338C22⋊C4C22×D7113yes—
339C42—716no—
340C42—713no—
341C42C22×D7113yes—
342C23—2843no—
343C23—2843no—
344C23—711no—
345C23—713no—
346C23—711no—
347C23—1426no—
348C23C23×D7113yes—
349C23C23×D7113yes—
350Q8—7112no—
351Q8—1426no—
352Q8—1426no—
353Q8D4×D7114yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—2843no—
359D4—2843no—
360D4—2846no—
361D4—2843no—
362D4—1422no—
363D4—14212no—
364D4—1426no—
365D4—1426no—
366D4—1426no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—713no—
370C2×C4—7112no—
371C2×C4—711no—
372C2×C4—713no—
373C2×C4—713no—
374C2×C4—711no—
375C2×C4—1426no—
376C2×C4—1422no—
377C2×C4—1426no—
378C2×C4—14212no—
379C2×C4—1426no—
380C2×C4—1426no—
381C2×C4C23×D7113yes—
382C2×C4C23×D7111yescharacteristic
383C2×C4C23×D7111yescharacteristic
384C2×C4C23×D7111yescharacteristic
385C2×C4C23×D7113yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—28412no—
393C22—712no—
394C22—1421no—
395C22—1423no—
396C22—1421no—
397C22—1423no—
398C22—1421no—
399C22—1423no—
400C22—1421no—
401C22—1421no—
402C22C24×D7111yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—716no—
408C4—1421no—
409C4—1423no—
410C4—1421no—
411C4—1423no—
412C4C2×D4×D7116yes—
413C2—43no—
414C2—43no—
415C2—2843no—
416C2—2841no—
417C2—1422no—
418C2C22×D4×D7111yescharacteristic
419C2#1798683411yescharacteristic
420C2#1798683511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#550×C71C2
hypercenter#550D71
hypocenterC71#3487
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#14556175C2semidirect
2. C2×#41×D71C2semidirect
3. #17986669C2semidirect
4. #17986668C2semidirect
5. #17986667C2semidirect
6. #17986661C2semidirect
7. #17986658C2semidirect
8. #17986656C2semidirect
9. #17986655C2semidirect
10. #17986565C2semidirect
11. #17986562C2semidirect
12. #17986509C2semidirect
13. #17986508C2semidirect
14. #17986475C2semidirect
15. #550×C71C2semidirect
22. C2×#11765864C22semidirect
24. C22×Dic142C22semidirect
26. C2×Q8×D71C22semidirect
29. #14556124C22semidirect
30. #14556124C22semidirect
31. #14556116C22semidirect
32. #14556115C22semidirect
33. #14556114C22semidirect
36. #14556108C22semidirect
37. #14556107C22semidirect
38. #14556106C22semidirect
39. #14556105C22semidirect
40. #14556103C22semidirect
41. #14556102C22semidirect
42. #14556101C22semidirect
43. #14556100C22semidirect
44. #14556100C22semidirect
45. #14556100C22semidirect
46. #14556099C22semidirect
47. #14556099C22semidirect
51. Q8×Dic71C22semidirect
52. C4×Dic142C22semidirect
53. #14556093C22semidirect
54. C4×D284C22semidirect
55. D4×Dic71C22semidirect
56. C4×C22⋊D71C22semidirect
57. C4×C22⋊D71C22semidirect
58. (C22⋊D71)⋊C4C22semidirect
62. C2×#41×C71C22semidirect
63. #127×C71C22semidirect
64. #121×C71C22semidirect
65. #121×C71C22semidirect
66. C4×Q8×C71C22semidirect
67. C4×D4×C71C22semidirect
80. Q8×D71D4semidirect
211. C71#3487semidirect
217. #550D71semidirect
236. C2×#41C2×D71semidirect
244. #127C2×D71semidirect
262. #121C2×D71semidirect
263. #121C2×D71semidirect
270. C4×Q8C2×D71semidirect
276. C4×D4C2×D71semidirect