Finite Group Databasebeta

← #11743782

Subgroups — #11743782

Order 896.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×#41C211yesmaximal, characteristic, complemented
2C2×#6551C211yesmaximal, characteristic, complemented
3C7×#548C211yesmaximal, characteristic, complemented
4#90522C211yesmaximal, characteristic, complemented
5Q8×C22⋊D7C211yesmaximal, characteristic, complemented
6#90367C213yesmaximal, complemented
7#90366C213yesmaximal, complemented
8#90322C213yesmaximal, complemented
9D7×C22⋊Q8C213yesmaximal, complemented
10#90303C213yesmaximal, complemented
11#90299C213yesmaximal, complemented
12Q8×D28C211yesmaximal, characteristic, complemented
13#90265C211yesmaximal, characteristic, complemented
14#90242C213yesmaximal, complemented
15#90238C213yesmaximal, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic, complemented
18D7×#41—28no—
19#6551—28no—
20C2×#1161C2211yescharacteristic
21C2×#1161C2213yescomplemented
22C2×Q8×D7C2211yescharacteristic
23C2×Q8×D7C2211yescharacteristic, complemented
24C2×Q8×D7C2213yescomplemented
25C2×#1159C2213yes—
26C2×D4×D7C2213yes—
27C2×#1157C2213yes—
28C2×#1157C2213yescomplemented
29C2×#1157C2213yescomplemented
30C22×C4×D7C2213yes—
31C7×#124C2213yescomplemented
32C7×C22⋊Q8C2213yescomplemented
33C7×#121C2213yescomplemented
34C4×C7×Q8C2211yescharacteristic, complemented
35C4×C7×D4C2213yescomplemented
36#6509C2213yes—
37Q8×Dic7C2211yescharacteristic, complemented
38#6507C2213yescomplemented
39#6502C2213yescomplemented
40#6501C2216yescomplemented
41#6501C2213yescomplemented
42#6500C2213yes—
43#6499C2213yescomplemented
44C4×C22⋊D7C2213yes—
45C4×C22⋊D7C2213yescomplemented
46#6487C2213yescomplemented
47#6487C2213yescomplemented
48#6460C2213yes—
49#6460C2213yescomplemented
50#6459C2213yes—
51#6457C2213yescomplemented
52#6456C2213yescomplemented
53D7×C4⋊C4C2216yescomplemented
54D7×C4⋊C4C2213yescomplemented
55#6447C2211yescharacteristic
56#6447C2216yescomplemented
57#6446C2213yescomplemented
58#6446C2216yescomplemented
59#6446C2216yescomplemented
60#6445C2213yes—
61D7×C22⋊C4C2216yes—
62#6441C2213yescomplemented
63#6441C2216yescomplemented
64C4×D28C2213yes—
65C4×D28C2213yescomplemented
66#6435C2213yescomplemented
67#6432C2213yescomplemented
68#3489—71nomaximal, Sylow
69C23×D7C2313yes—
70C7×#41—24no—
71C2×C7×Q8—24no—
72C2×C7×Q8C2313yes—
73C2×C7×Q8C2311yescharacteristic
74C2×C7×D4C2313yes—
75C2×C7×D4C2313yes—
76C22×C4×C7C2313yes—
77C22×C4×C7C2313yes—
78C2×C22⋊D7C2316yes—
79C2×C22⋊D7C2313yes—
80C2×C22⋊D7C2311yescharacteristic
81C2×C22⋊D7C2311yescharacteristic
82C2×C22⋊D7C2313yes—
83C22×Dic7C2313yes—
84#1161—212no—
85#1161D418yescomplemented
86Q8×D7—24no—
87Q8×D7—24no—
88Q8×D7—212no—
89#1159—212no—
90D4×D7—212no—
91#1157—212no—
92#1157—212no—
93#1157—212no—
94C2×D28C2311yescharacteristic
95C2×D28C2313yes—
96C2×D28C2313yes—
97C2×C4×D7—212no—
98C2×C4×D7C2313yes—
99C2×C4×D7C2313yes—
100C2×C4×D7C2316yes—
101C2×C4×D7C2313yes—
102C2×C4×D7C2311yescharacteristic
103C2×C4×D7C2316yes—
104C2×C4×D7C2313yes—
105C2×Dic14C2313yes—
106C2×Dic14C2313yes—
107C2×Dic14C2313yes—
108C7×C4⋊C4C2313yes—
109C7×C4⋊C4C2313yes—
110C7×C22⋊C4C2316yes—
111C7×C22⋊C4C2313yes—
112C7×C22⋊C4C2311yescharacteristic
113C42×C7C2313yes—
114C22⋊Dic7C2316yes—
115C22⋊Dic7C2313yes—
116C22⋊Dic7C2313yes—
117(C2×D7)⋊C4C2313yes—
118(C2×D7)⋊C4C2313yes—
119(C2×D7)⋊C4C2313yes—
120(C2×D7)⋊C4C2313yes—
121(C2×D7)⋊C4C2313yes—
122(C2×D7)⋊C4C2311yescharacteristic
123C4⋊Dic7C2313yes—
124C4⋊Dic7C2316yes—
125C4⋊Dic7C2313yes—
126Dic7⋊C4C2311yescharacteristic
127Dic7⋊C4C2313yes—
128Dic7⋊C4C2313yes—
129Dic7⋊C4C2316yes—
130Dic7⋊C4C2313yes—
131C4×Dic7C2313yes—
132C2×2−1+4—71no—
133C22×#41—71no—
134D4×Q8—71no—
135D4×Q8—71no—
136#548—71no—
137#548D711yescharacteristic, complemented
138#547—73no—
139#547—73no—
140#546—73no—
141#546—73no—
142#537—73no—
143#536—73no—
144#525—73no—
145C2×C22⋊Q8—73no—
146#520—71no—
147C23×C7C2413yes—
148C23×C7C2411yescharacteristic
149C22×D7—26no—
150C22×D7—26no—
151C22×D7—43no—
152C22×D7C2411yescharacteristic
153C22×D7C2413yes—
154C22×D7C2411yescharacteristic
155C7×Q8—26no—
156C7×Q8C2×D414yes—
157C7×D4—26no—
158C7×D4—43no—
159C2×C4×C7—26no—
160C2×C4×C7—26no—
161C2×C4×C7C2413yes—
162C2×C4×C7C2411yescharacteristic
163C2×C4×C7C2411yescharacteristic
164C2×C4×C7C2413yes—
165C2×C4×C7C2413yes—
166C22⋊D7—26no—
167C22⋊D7—212no—
168C22⋊D7—22no—
169C22⋊D7—22no—
170C22⋊D7—43no—
171C2×Dic7—26no—
172C2×Dic7C2411yescharacteristic
173C2×Dic7C2413yes—
174C2×Dic7C2413yes—
175C2×Dic7C2413yes—
176C2×Dic7C2411yescharacteristic
177D28—26no—
178D28—22no—
179D28C2×D4112yes—
180C4×D7—26no—
181C4×D7—212no—
182C4×D7—26no—
183C4×D7—26no—
184C4×D7—22no—
185C4×D7—46no—
186C4×D7C2×D4112yes—
187Dic14—26no—
188Dic14—26no—
189Dic14—26no—
1902−1+4—148no—
191C2×#41—71no—
192C2×#41—73no—
193C2×#41—73no—
194C2×#41—73no—
195C2×#41—73no—
196C2×#41—73no—
197C2×#41—148no—
198C2×#41C2×D711yescharacteristic
199C22×Q8—73no—
200C22×Q8—71no—
201C22×Q8—71no—
202C22×Q8C2×D711yescharacteristic, complemented
203C22×D4—73no—
204C23×C4—73no—
205C4⋊Q8—73no—
206C4⋊Q8—73no—
207#124—73no—
208#124C2×D713yescomplemented
209#123—73no—
210#123—73no—
211#123—76no—
212#123—76no—
213C22⋊Q8—73no—
214C22⋊Q8—73no—
215C22⋊Q8—73no—
216C22⋊Q8—73no—
217C22⋊Q8—73no—
218C22⋊Q8—73no—
219C22⋊Q8—76no—
220C22⋊Q8—73no—
221C22⋊Q8C2×D713yescomplemented
222#121—73no—
223#121—76no—
224#121—73no—
225#121—71no—
226#121—76no—
227#121C2×D713yescomplemented
228#120—73no—
229#120—73no—
230C4×Q8—71no—
231C4×Q8C2×D711yescharacteristic, complemented
232C4×D4—73no—
233C4×D4—73no—
234C4×D4—73no—
235C4×D4—73no—
236C4×D4—73no—
237C4×D4C2×D713yescomplemented
238#117—73no—
239C2×C4⋊C4—73no—
240C2×C4⋊C4—76no—
241C2×C22⋊C4—76no—
242C22×C7—23no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—23no—
246C22×C7—23no—
247C22×C7—21no—
248C22×C7C2511yescharacteristic
249C2×D7—21no—
250C2×D7—23no—
251C2×D7—23no—
252C2×D7—21no—
253C2×D7—21no—
254C2×D7—21no—
255C2×D7—21no—
256C2×D7—412no—
257C2×D7C22×D416yes—
258C2×D72−1+412yescomplemented
259C4×C7—23no—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—23no—
263C4×C7C22×D416yes—
264Dic7—23no—
265Dic7—23no—
266Dic7—21no—
267Dic7—23no—
268Dic7C22×D412yes—
269C24—73no—
270#41—24no—
271#41—78no—
272#41—1412no—
273#41—1412no—
274#41—1412no—
275#41—1412no—
276#41—1412no—
277C2×Q8—24no—
278C2×Q8—73no—
279C2×Q8—73no—
280C2×Q8—73no—
281C2×Q8—1412no—
282C2×Q8—144no—
283C2×Q8—144no—
284C2×Q8C22×D713yes—
285C2×Q8C22×D711yescharacteristic
286C2×D4—71no—
287C2×D4—73no—
288C2×D4—71no—
289C2×D4—76no—
290C2×D4—73no—
291C2×D4—73no—
292C2×D4—71no—
293C2×D4—73no—
294C2×D4—1412no—
295C2×D4C22×D713yes—
296C2×D4C22×D713yes—
297C22×C4—73no—
298C22×C4—73no—
299C22×C4—76no—
300C22×C4—73no—
301C22×C4—73no—
302C22×C4—73no—
303C22×C4—71no—
304C22×C4—76no—
305C22×C4—1412no—
306C22×C4C22×D713yes—
307C22×C4C22×D713yes—
308C4⋊C4—73no—
309C4⋊C4—73no—
310C4⋊C4—76no—
311C4⋊C4—71no—
312C4⋊C4—73no—
313C4⋊C4—73no—
314C4⋊C4—76no—
315C4⋊C4—73no—
316C4⋊C4C22×D713yes—
317C4⋊C4C22×D713yes—
318C22⋊C4—76no—
319C22⋊C4—73no—
320C22⋊C4—73no—
321C22⋊C4—73no—
322C22⋊C4—73no—
323C22⋊C4—73no—
324C22⋊C4—73no—
325C22⋊C4—71no—
326C22⋊C4—73no—
327C22⋊C4C22×D716yes—
328C22⋊C4C22×D711yescharacteristic
329C22⋊C4C22×D713yes—
330C42—73no—
331C42C22×D713yes—
332C2×C7—43no—
333C2×C7—41no—
334C2×C7C23×D411yescharacteristic
335C2×C7C2×2−1+411yescharacteristic
336C2×C7#58511yescharacteristic
337D7—22no—
338D7—26no—
339D7—41no—
340C23—71no—
341C23—73no—
342C23—71no—
343C23—146no—
344C23—146no—
345C23—283no—
346C23C23×D713yes—
347C23C23×D711yescharacteristic
348Q8—26no—
349Q8—146no—
350Q8—146no—
351Q8—146no—
352Q8D4×D714yes—
353D4—26no—
354D4—43no—
355D4—712no—
356D4—142no—
357D4—146no—
358D4—142no—
359D4—142no—
360D4—1412no—
361D4—146no—
362D4—283no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—71no—
366C2×C4—712no—
367C2×C4—71no—
368C2×C4—73no—
369C2×C4—73no—
370C2×C4—73no—
371C2×C4—142no—
372C2×C4—146no—
373C2×C4—1412no—
374C2×C4—146no—
375C2×C4—146no—
376C2×C4—146no—
377C2×C4—286no—
378C2×C4C23×D711yescharacteristic
379C2×C4C23×D713yes—
380C2×C4C23×D713yes—
381C2×C4C23×D711yescharacteristic
382C2×C4C23×D713yes—
383C7#348911yescharacteristic, complemented, Sylow
384C22—21no—
385C22—23no—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—76no—
391C22—72no—
392C22—141no—
393C22—143no—
394C22—141no—
395C22—143no—
396C22—141no—
397C22—141no—
398C22—141no—
399C22—2812no—
400C22C24×D711yescharacteristic
401C4—21no—
402C4—23no—
403C4—23no—
404C4—21no—
405C4—72no—
406C4—143no—
407C4—143no—
408C4—143no—
409C4—141no—
410C4C2×D4×D716yes—
411C2—43no—
412C2—41no—
413C2—142no—
414C2—146no—
415C2—281no—
416C2C22×D4×D711yescharacteristic
417C2#9061611yescharacteristic
418C2D7×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#548C2
hypercenter#548D7
hypocenterC7#3489
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×D7×#41C2semidirect
2. C2×#6551C2semidirect
3. C7×#548C2semidirect
4. #90522C2semidirect
5. Q8×C22⋊D7C2semidirect
6. #90367C2semidirect
7. #90366C2semidirect
8. #90322C2semidirect
9. D7×C22⋊Q8C2semidirect
10. #90303C2semidirect
11. #90299C2semidirect
12. Q8×D28C2semidirect
13. #90265C2semidirect
14. #90242C2semidirect
15. #90238C2semidirect
17. C22×C7×Q8C22semidirect
21. C2×#1161C22semidirect
23. C2×Q8×D7C22semidirect
24. C2×Q8×D7C22semidirect
28. C2×#1157C22semidirect
29. C2×#1157C22semidirect
31. C7×#124C22semidirect
32. C7×C22⋊Q8C22semidirect
33. C7×#121C22semidirect
34. C4×C7×Q8C22semidirect
35. C4×C7×D4C22semidirect
37. Q8×Dic7C22semidirect
38. #6507C22semidirect
39. #6502C22semidirect
40. #6501C22semidirect
41. #6501C22semidirect
43. #6499C22semidirect
45. C4×C22⋊D7C22semidirect
46. #6487C22semidirect
47. #6487C22semidirect
49. #6460C22semidirect
51. #6457C22semidirect
52. #6456C22semidirect
53. D7×C4⋊C4C22semidirect
54. D7×C4⋊C4C22semidirect
56. #6447C22semidirect
57. #6446C22semidirect
58. #6446C22semidirect
59. #6446C22semidirect
62. #6441C22semidirect
63. #6441C22semidirect
65. C4×D28C22semidirect
66. #6435C22semidirect
67. #6432C22semidirect
85. #1161D4semidirect
137. #548D7semidirect
202. C22×Q8C2×D7semidirect
208. #124C2×D7semidirect
221. C22⋊Q8C2×D7semidirect
227. #121C2×D7semidirect
231. C4×Q8C2×D7semidirect
237. C4×D4C2×D7semidirect
258. C2×D72−1+4semidirect
383. C7#3489semidirect

Maximal subgroups

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

subgroupindex
C2×D7×#412
C2×#65512
C7×#5482
#905222
Q8×C22⋊D72
#903672
#903662
#903222
D7×C22⋊Q82
#903032
#902992
Q8×D282
#902652
#902422
#902382
#34897