Finite Group Databasebeta

← #11736828

Subgroups — #11736828

Order 896.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90225C211yesmaximal, characteristic, complemented
2#90212C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89618C211yesmaximal, characteristic, complemented
5#89616C211yesmaximal, characteristic, complemented
6#89616C211yesmaximal, characteristic, complemented
7#89614C211yesmaximal, characteristic, complemented
8#89509C211yesmaximal, characteristic, complemented
9#89504C211yesmaximal, characteristic
10#89503C211yesmaximal, characteristic, complemented
11#89495C211yesmaximal, characteristic, complemented
12#89483C211yesmaximal, characteristic, complemented
13#89482C211yesmaximal, characteristic, complemented
14C4×D56C211yesmaximal, characteristic, complemented
15C4×#1132C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C7×#117C2211yescharacteristic
18#6493C2211yescharacteristic
19#6489C2211yescharacteristic
20#6487C2211yescharacteristic
21C2×D56C2211yescharacteristic
22C2×D56C2211yescharacteristic
23C2×#1132C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25#6461—21no—
26#6459—21no—
27#6456—21no—
28#6456—21no—
29#6448—21no—
30#6446—21no—
31D7×C22⋊C4—21no—
32D7×C22⋊C4—21no—
33#6438C2211yescharacteristic, complemented
34C4×D28C2211yescharacteristic
35C4×D28C2211yescharacteristic
36C4×D28C2211yescharacteristic
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic, complemented
39#6432C2211yescharacteristic, complemented
40C4×Dic14C2211yescharacteristic, complemented
41C7×C4⋊C8C2211yescharacteristic, complemented
42C7×C4⋊C8C2211yescharacteristic, complemented
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45C7×#97C2211yescharacteristic, complemented
46C4×C7×C8C2211yescharacteristic, complemented
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic, complemented
52#6395C2211yescharacteristic, complemented
53#6392C2211yescharacteristic, complemented
54#6392C2211yescharacteristic, complemented
55#6391C2211yescharacteristic, complemented
56#6391C2211yescharacteristic, complemented
57#6390C2211yescharacteristic, complemented
58#6390C2211yescharacteristic, complemented
59#3118—71nomaximal, Sylow
60C23×D7—21no—
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×D28—21no—
64C2×D28—22no—
65C2×D28—22no—
66C2×D28C2311yescharacteristic
67C2×D28C2311yescharacteristic
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4D411yescharacteristic
81C7×C4⋊C4D411yescharacteristic
82C7×C22⋊C4D411yescharacteristic
83C7×C22⋊C4D411yescharacteristic
84C42×C7C2311yescharacteristic
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105D56—22no—
106D56—22no—
107#1132—22no—
108#1132—22no—
109#542—71no—
110#518—71no—
111#485—71no—
112#482—71no—
113#476—71no—
114#474—71no—
115#463—71no—
116#463—71no—
117#450—71no—
118#447—71no—
119#442—71no—
120#440—71no—
121C4×SD16—71no—
122C4×D8—71no—
123#433D711yescharacteristic, complemented
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—24no—
128C22×D7—21no—
129C22×D7—81no—
130C22×D7—81no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2411yescharacteristic
139C22⋊D7—81no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145D28—24no—
146D28—21no—
147D28—21no—
148D28—21no—
149D28—21no—
150D28#4112yes—
151D28#4112yes—
152C4×D7—81no—
153C4×D7—81no—
154C4×D7—81no—
155C4×D7—81no—
156C4×D7—81no—
157Dic14—21no—
158Dic14—21no—
159Dic14—81no—
160C7×C8—21no—
161C7×C8—21no—
162C7×C8—21no—
163C7×C8—21no—
164C22×D4—71no—
165C2×SD16—71no—
166C2×SD16—71no—
167C2×D8—71no—
168C2×D8—71no—
169C4⋊Q8—71no—
170#126—141no—
171#124—71no—
172#124—141no—
173#123—141no—
174C22⋊Q8—71no—
175C22⋊Q8—141no—
176#121—71no—
177C4×Q8—71no—
178C4×D4—71no—
179C4×D4—71no—
180C4×D4—71no—
181C4×D4—71no—
182C4×D4—71no—
183C4×D4—141no—
184C4×D4—141no—
185#117—71no—
186#117C2×D711yescharacteristic
187C2×C22⋊C4—141no—
188C2×C22⋊C4—141no—
189#107—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8C2×D711yescharacteristic, complemented
194C4⋊C8C2×D711yescharacteristic, complemented
195#103—71no—
196#103—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203C22⋊C8C2×D711yescharacteristic
204C22⋊C8C2×D711yescharacteristic
205#97C2×D711yescharacteristic, complemented
206C4×C8C2×D711yescharacteristic, complemented
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7C22×D411yescharacteristic
211C2×D7—22no—
212C2×D7—22no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—84no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7C2×#4111yescharacteristic
227C4×C7C2×#4111yescharacteristic
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C24—141no—
234C2×Q8—71no—
235C2×Q8—141no—
236C2×D4—71no—
237C2×D4—71no—
238C2×D4—71no—
239C2×D4—142no—
240C2×D4—142no—
241C2×D4—141no—
242C2×D4—141no—
243C22×C4—141no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4C22×D711yescharacteristic
249SD16—142no—
250SD16—142no—
251D8—142no—
252D8—142no—
253C2×C8C22×D711yescharacteristic
254C2×C8C22×D711yescharacteristic
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4D2811yescharacteristic
265C4⋊C4D2811yescharacteristic
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4D2811yescharacteristic
277C22⋊C4D2811yescharacteristic
278C42—141no—
279C42—141no—
280C42C22×D711yescharacteristic
281C42C22×D711yescharacteristic
282C2×C7—41no—
283C2×C7#52511yescharacteristic
284C2×C7#57611yescharacteristic
285C2×C7#57711yescharacteristic
286D7—42no—
287D7—42no—
288D7—81no—
289C23—144no—
290C23—141no—
291C23—141no—
292C23—141no—
293C23—561no—
294C23—561no—
295C23C2×D2811yescharacteristic
296Q8—141no—
297Q8—141no—
298Q8—561no—
299D4—72no—
300D4—72no—
301D4—144no—
302D4—141no—
303D4—141no—
304D4—141no—
305D4—141no—
306D4—561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—561no—
315C2×C4—561no—
316C2×C4—561no—
317C2×C4—561no—
318C2×C4—561no—
319C2×C4C2×D2811yescharacteristic
320C2×C4C2×D2811yescharacteristic
321C2×C4C2×D2811yescharacteristic
322C2×C4C2×D2811yescharacteristic
323C2×C4C2×D2811yescharacteristic
324C2×C4C23×D711yescharacteristic
325C8—21no—
326C8—21no—
327C8—21no—
328C8—21no—
329C7#311811yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22—21no—
333C22—142no—
334C22—142no—
335C22—281no—
336C22—281no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22—281no—
342C22—564no—
343C22C22×D2811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4—281no—
354C4D7×#4111yescharacteristic
355C4D7×#4111yescharacteristic
356C2—41no—
357C2—282no—
358C2—282no—
359C2—561no—
360C2#9021511yescharacteristic
361C2#9044211yescharacteristic
362C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
prime radical#89504C2
hypercenter#433D7
hypocenterC7#3118
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #90225C2semidirect
2. #90212C2semidirect
3. C7×#433C2semidirect
4. #89618C2semidirect
5. #89616C2semidirect
6. #89616C2semidirect
7. #89614C2semidirect
8. #89509C2semidirect
10. #89503C2semidirect
11. #89495C2semidirect
12. #89483C2semidirect
13. #89482C2semidirect
14. C4×D56C2semidirect
15. C4×#1132C2semidirect
33. #6438C22semidirect
38. C4×D28C22semidirect
39. #6432C22semidirect
40. C4×Dic14C22semidirect
41. C7×C4⋊C8C22semidirect
42. C7×C4⋊C8C22semidirect
45. C7×#97C22semidirect
46. C4×C7×C8C22semidirect
51. #6395C22semidirect
52. #6395C22semidirect
53. #6392C22semidirect
54. #6392C22semidirect
55. #6391C22semidirect
56. #6391C22semidirect
57. #6390C22semidirect
58. #6390C22semidirect
123. #433D7semidirect
193. C4⋊C8C2×D7semidirect
194. C4⋊C8C2×D7semidirect
205. #97C2×D7semidirect
206. C4×C8C2×D7semidirect
329. C7#3118semidirect

Maximal subgroups

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

subgroupindex
#902252
#902122
C7×#4332
#896182
#896162
#896142
#895092
#895042
#895032
#894952
#894832
#894822
C4×D562
C4×#11322
#31187