Finite Group Databasebeta

← #11738739

Subgroups — #11738739

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#90309C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89810C211yesmaximal, characteristic, complemented
5C2×#6382C211yesmaximal, characteristic, complemented
6#89663C211yesmaximal, characteristic, complemented
7#89657C211yesmaximal, characteristic, complemented
8#89653C211yesmaximal, characteristic, complemented
9#89648C211yesmaximal, characteristic, complemented
10#89557C211yesmaximal, characteristic, complemented
11#89554C211yesmaximal, characteristic, complemented
12#89549C211yesmaximal, characteristic, complemented
13#89533C211yesmaximal, characteristic, complemented
14#89507C211yesmaximal, characteristic, complemented
15#89502C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C7×#121C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19#6502—21no—
20#6501C2211yescharacteristic
21D4×Dic7—21no—
22D4×Dic7C2211yescharacteristic
23C2×#1141C2211yescharacteristic
24#6493C2211yescharacteristic
25#6492—21no—
26C4×C22⋊D7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C22×C7⋊C8C2211yescharacteristic
29C2×#1131C2211yescharacteristic
30#6459—21no—
31#6458C2211yescharacteristic
32#6457—21no—
33#6456C2211yescharacteristic
34#6455C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6453C2211yescharacteristic
37#6451C2211yescharacteristic
38#6449—21no—
39#6442—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6391C2211yescharacteristic
50#6391C2211yescharacteristic
51#6389C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3345—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68C22×Dic7—21no—
69#1157—44no—
70C2×D28—21no—
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7D411yescharacteristic
86#1141—22no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4D411yescharacteristic
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8—22no—
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#1131—22no—
107#553—71no—
108#547—71no—
109#501—71no—
110#482—71no—
111#481—71no—
112#481—71no—
113#480D711yescharacteristic, complemented
114#478—71no—
115#476—71no—
116#469—71no—
117#466—71no—
118#442—71no—
119#435—71no—
120#428—71no—
121C2×#106—71no—
122C23×C7—21no—
123C23×C7C2×D411yescharacteristic
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—42no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—22no—
138C22⋊D7—42no—
139C22⋊D7—81no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—81no—
144C2×Dic7—81no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148D28—21no—
149D28—21no—
150D28—42no—
151C4×D7—21no—
152C4×D7—21no—
153C4×D7—42no—
154C4×D7—42no—
155Dic14—42no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8#4112yescomplemented
160C2×#41—141no—
161C2×D8—71no—
162C2×M16—71no—
163C22×C8—71no—
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—141no—
175#121C2×D711yescharacteristic
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—71no—
179C4×D4—141no—
180#117—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4C2×D711yescharacteristic
184#107—71no—
185#107C2×D711yescharacteristic
186#107C2×D711yescharacteristic
187#106—71no—
188#106—71no—
189#106—71no—
190#106—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102C2×D711yescharacteristic
199#102C2×D711yescharacteristic
200C22⋊C8—71no—
201C22⋊C8C2×D711yescharacteristic
202#97—71no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7AGL(1,ℤ/8)12yes—
209C22×C7C22×D411yescharacteristic
210C2×D7—21no—
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C72−1+411yescharacteristic
222Dic7—21no—
223Dic7—21no—
224Dic7—21no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228#41—284no—
229C2×Q8—141no—
230C2×D4—21no—
231C2×D4—71no—
232C2×D4—71no—
233C2×D4—141no—
234C2×D4—141no—
235C2×D4—141no—
236C2×D4C22×D711yescharacteristic
237C22×C4—21no—
238C22×C4—71no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244D8—142no—
245M16—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8—142no—
249C2×C8C22×D711yescharacteristic
250C2×C8C22×D711yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4C22×D711yescharacteristic
262C4⋊C4C22×D711yescharacteristic
263C4⋊C4C22×D711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—71no—
266C22⋊C4—71no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C42—71no—
274C42—141no—
275C2×C7—22no—
276C2×C7—81no—
277C2×C7#54711yescharacteristic
278C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
279C2×C7#57711yescharacteristic
280D7—41no—
281D7—81no—
282C23—21no—
283C23—71no—
284C23—141no—
285C23D4×D711yescharacteristic
286Q8—282no—
287D4—21no—
288D4—21no—
289D4—42no—
290D4—142no—
291D4—141no—
292D4—141no—
293D4—282no—
294D4—282no—
295D4—561no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—71no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4—561no—
313C2×C4—561no—
314C2×C4D4×D711yescharacteristic
315C2×C4C23×D711yescharacteristic
316C8—21no—
317C8—21no—
318C8—72no—
319C8—141no—
320C7#334511yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—141no—
327C22—141no—
328C22—141no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22#647412yes—
333C22C2×D4×D711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—141no—
339C4—141no—
340C4—141no—
341C4—281no—
342C4—281no—
343C4—281no—
344C4D7×#4111yescharacteristic
345C4#655411yescharacteristic
346C2—22no—
347C2—81no—
348C2—281no—
349C2—561no—
350C2#9034611yescharacteristic
351C2C2×#647411yescharacteristic
352C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
hypercenter#480D7
hypocenterC7#3345
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #90309C2semidirect
2. #90198C2semidirect
3. C7×#480C2semidirect
4. #89810C2semidirect
5. C2×#6382C2semidirect
6. #89663C2semidirect
7. #89657C2semidirect
8. #89653C2semidirect
9. #89648C2semidirect
10. #89557C2semidirect
11. #89554C2semidirect
12. #89549C2semidirect
13. #89533C2semidirect
14. #89507C2semidirect
15. #89502C2semidirect
113. #480D7semidirect
159. C7⋊C8#41semidirect
320. C7#3345semidirect

Maximal subgroups

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

subgroupindex
#903092
#901982
C7×#4802
#898102
C2×#63822
#896632
#896572
#896532
#896482
#895572
#895542
#895492
#895332
#895072
#895022
#33457