Finite Group Databasebeta

← #11736916

Subgroups — #11736916

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90256C211yesmaximal, characteristic, complemented
2#90246C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89907C211yesmaximal, characteristic, complemented
5#89893C211yesmaximal, characteristic, complemented
6#89887C211yesmaximal, characteristic, complemented
7C2×#6391C211yesmaximal, characteristic, complemented
8#89619C211yesmaximal, characteristic, complemented
9#89614C211yesmaximal, characteristic, complemented
10#89509C211yesmaximal, characteristic, complemented
11#89508C211yesmaximal, characteristic, complemented
12#89507C211yesmaximal, characteristic, complemented
13#89507C211yesmaximal, characteristic, complemented
14#89483C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×C7×M16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6502—21no—
21#6493C2211yescharacteristic
22#6493C2211yescharacteristic
23#6493C2211yescharacteristic
24#6493C2211yescharacteristic
25#6489C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×D56C2211yescharacteristic
30#6461—21no—
31#6460—21no—
32#6449—21no—
33#6449—21no—
34#6446—21no—
35(C22⋊D7)⋊C4—21no—
36C4×D28C2211yescharacteristic
37C4×D28C2211yescharacteristic
38#6433C2211yescharacteristic
39#6432C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42C7×C22⋊C8C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6391C2211yescharacteristic
56#6391C2211yescharacteristic
57#6391C2211yescharacteristic
58#6391C2211yescharacteristic
59#3345—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68C22×Dic7—21no—
69C22×Dic7—21no—
70#1159—44no—
71C2×D28C2311yescharacteristic
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×Dic14—21no—
76C7×M16—22no—
77C2×C7×C8—22no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4D411yescharacteristic
83C7×C22⋊C4D411yescharacteristic
84C7×C22⋊C4D411yescharacteristic
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87(C2×D7)⋊C4—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—
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106D56—22no—
107#553—71no—
108#547—71no—
109#501—71no—
110#482—71no—
111#481—71no—
112#481—71no—
113#480—71no—
114#478—71no—
115#476—71no—
116#469—71no—
117#466—71no—
118#442—71no—
119#435D711yescharacteristic, complemented
120#428—71no—
121C2×#106—71no—
122C23×C7C2×D411yescharacteristic
123C23×C7C2×D411yescharacteristic
124C22×D7—21no—
125C22×D7—21no—
126C7×D4—22no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—42no—
137C22⋊D7—42no—
138C22⋊D7—42no—
139C22⋊D7—81no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—81no—
149D28—21no—
150D28—21no—
151D28—21no—
152D28—21no—
153C4×D7—42no—
154C4×D7—81no—
155Dic14—42no—
156C7×C8—21no—
157C7×C8—21no—
158C7×C8—21no—
159C7×C8#4112yescomplemented
160C2×#41—141no—
161C2×D8—71no—
162C2×M16C2×D711yescharacteristic
163C22×C8C2×D711yescharacteristic
164C4⋊Q8—71no—
165#126—141no—
166#125—71no—
167#123—141no—
168#123—141no—
169#123—141no—
170C22⋊Q8—141no—
171#121—71no—
172#121—71no—
173#121—71no—
174#121—71no—
175#121—141no—
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—141no—
179C4×D4C2×D711yescharacteristic
180#117—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184#107—71no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190#106—71no—
191#106—71no—
192#106—71no—
193C4⋊C8C2×D711yescharacteristic
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8C2×D711yescharacteristic
201C22⋊C8C2×D711yescharacteristic
202#97C2×D711yescharacteristic
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7AGL(1,ℤ/8)12yes—
209C22×C7C22×D411yescharacteristic
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C4×C7—21no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7C2×#4111yescharacteristic
221C4×C72−1+411yescharacteristic
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228#41—284no—
229C2×Q8—141no—
230C2×D4—71no—
231C2×D4—71no—
232C2×D4—141no—
233C2×D4—141no—
234C2×D4—141no—
235C2×D4—141no—
236C2×D4D2811yescharacteristic
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244D8—142no—
245M16—22no—
246C2×C8—22no—
247C2×C8C22×D711yescharacteristic
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C2×C8C22×D711yescharacteristic
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4D2811yescharacteristic
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4D2811yescharacteristic
272C22⋊C4D2811yescharacteristic
273C42—141no—
274C42C22×D711yescharacteristic
275C2×C7—22no—
276C2×C7—41no—
277C2×C7#54711yescharacteristic
278C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
279C2×C7#57711yescharacteristic
280D7—81no—
281D7—81no—
282C23—141no—
283C23—141no—
284C23C2×D2811yescharacteristic
285C23C2×D2811yescharacteristic
286Q8—282no—
287D4—22no—
288D4—141no—
289D4—141no—
290D4—141no—
291D4—141no—
292D4—282no—
293D4—282no—
294D4—282no—
295D4—561no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—282no—
307C2×C4—282no—
308C2×C4—282no—
309C2×C4—561no—
310C2×C4—561no—
311C2×C4C2×D2811yescharacteristic
312C2×C4C2×D2811yescharacteristic
313C2×C4C2×D2811yescharacteristic
314C2×C4C2×D2811yescharacteristic
315C2×C4C23×D711yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8#115912yescomplemented
320C7#334511yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22#647112yes—
333C22C22×D2811yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4—281no—
343C4—281no—
344C4C2×#115911yescharacteristic
345C4#655411yescharacteristic
346C2—22no—
347C2—41no—
348C2—561no—
349C2—561no—
350C2#9024611yescharacteristic
351C2C2×#647111yescharacteristic
352C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
hypercenter#435D7
hypocenterC7#3345
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #90256C2semidirect
2. #90246C2semidirect
3. C7×#435C2semidirect
4. #89907C2semidirect
5. #89893C2semidirect
6. #89887C2semidirect
7. C2×#6391C2semidirect
8. #89619C2semidirect
9. #89614C2semidirect
10. #89509C2semidirect
11. #89508C2semidirect
12. #89507C2semidirect
13. #89507C2semidirect
14. #89483C2semidirect
15. #89474C2semidirect
119. #435D7semidirect
159. C7×C8#41semidirect
319. C8#1159semidirect
320. C7#3345semidirect

Maximal subgroups

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

subgroupindex
#902562
#902462
C7×#4352
#899072
#898932
#898872
C2×#63912
#896192
#896142
#895092
#895082
#895072
#894832
#894742
#33457