Finite Group Databasebeta

← #11736914

Subgroups — #11736914

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90246C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89893C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7C2×#6392C211yesmaximal, characteristic, complemented
8#89622C211yesmaximal, characteristic, complemented
9#89619C211yesmaximal, characteristic, complemented
10#89509C211yesmaximal, characteristic, complemented
11#89508C211yesmaximal, characteristic, complemented
12#89494C211yesmaximal, characteristic, complemented
13#89493C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×C7×M16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6498—21no—
21#6493C2211yescharacteristic
22#6493C2211yescharacteristic
23#6489C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×#1132C2211yescharacteristic
30#6460—21no—
31#6452—21no—
32#6449—21no—
33#6449—21no—
34#6442—21no—
35#6440—21no—
36C4×D28C2211yescharacteristic
37#6433C2211yescharacteristic
38#6432C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42C7×C22⋊C8C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6392C2211yescharacteristic
48#6392C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6391C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3343—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68#1159—44no—
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×Dic14—21no—
72C2×Dic14C2311yescharacteristic
73C7×M16—22no—
74C2×C7×C8—22no—
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4D411yescharacteristic
80C7×C22⋊C4D411yescharacteristic
81C7×C22⋊C4D411yescharacteristic
82C42×C7C2311yescharacteristic
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106#1132—22no—
107#552—71no—
108#547—71no—
109#501—71no—
110#484—71no—
111#483—71no—
112#482—71no—
113#480—71no—
114#478—71no—
115#477—71no—
116#468—71no—
117#465—71no—
118#440—71no—
119#435D711yescharacteristic, complemented
120#428—71no—
121C2×#107—71no—
122C23×C7C2×D411yescharacteristic
123C23×C7C2×D411yescharacteristic
124C22×D7—21no—
125C7×D4—22no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C22⋊D7—42no—
136C22⋊D7—42no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—42no—
147D28—21no—
148D28—21no—
149C4×D7—42no—
150Dic14—21no—
151Dic14—21no—
152Dic14—42no—
153C7×C8—21no—
154C7×C8—21no—
155C7×C8—21no—
156C7×C8#4112yescomplemented
157C2×#41—141no—
158C2×SD16—71no—
159C2×M16C2×D711yescharacteristic
160C22×C8C2×D711yescharacteristic
161C4⋊Q8—71no—
162#126—141no—
163#125—71no—
164#125—141no—
165#123—141no—
166#123—141no—
167#123—141no—
168C22⋊Q8—71no—
169C22⋊Q8—71no—
170C22⋊Q8—141no—
171#121—71no—
172#121—71no—
173C4×Q8—71no—
174C4×D4—71no—
175C4×D4C2×D711yescharacteristic
176#117—71no—
177#117—141no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8C2×D711yescharacteristic
191#103—71no—
192#103—71no—
193#103—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197C22⋊C8C2×D711yescharacteristic
198C22⋊C8C2×D711yescharacteristic
199#97C2×D711yescharacteristic
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7AGL(1,ℤ/8)12yes—
206C22×C7C22×D411yescharacteristic
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7C2×#4111yescharacteristic
215C4×C72−1+411yescharacteristic
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223#41—284no—
224C2×Q8—71no—
225C2×Q8—141no—
226C2×D4—71no—
227C2×D4—141no—
228C2×D4—141no—
229C2×D4D2811yescharacteristic
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4—141no—
234C22×C4C22×D711yescharacteristic
235C22×C4C22×D711yescharacteristic
236SD16—142no—
237M16—22no—
238C2×C8—22no—
239C2×C8C22×D711yescharacteristic
240C2×C8C22×D711yescharacteristic
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4D2811yescharacteristic
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4D2811yescharacteristic
266C22⋊C4D2811yescharacteristic
267C42—141no—
268C42—141no—
269C42C22×D711yescharacteristic
270C2×C7—22no—
271C2×C7—41no—
272C2×C7#54711yescharacteristic
273C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C7#57811yescharacteristic
275D7—81no—
276C23—141no—
277C23C2×D2811yescharacteristic
278C23C2×D2811yescharacteristic
279Q8—141no—
280Q8—141no—
281Q8—282no—
282D4—22no—
283D4—141no—
284D4—141no—
285D4—282no—
286D4—282no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4—282no—
302C2×C4C2×D2811yescharacteristic
303C2×C4C2×D2811yescharacteristic
304C2×C4C2×D2811yescharacteristic
305C2×C4C2×D2811yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8#115912yescomplemented
311C7#334311yescharacteristic, complemented, Sylow
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—281no—
318C22—281no—
319C22—281no—
320C22#647112yes—
321C22C22×D2811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4—281no—
331C4—281no—
332C4—281no—
333C4C2×#115911yescharacteristic
334C4#655411yescharacteristic
335C2—22no—
336C2—41no—
337C2—561no—
338C2#9024611yescharacteristic
339C2C2×#647111yescharacteristic
340C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radical#90246C2
hypercenter#435D7
hypocenterC7#3343
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
2. #90232C2semidirect
3. C7×#435C2semidirect
4. #89906C2semidirect
5. #89893C2semidirect
6. #89886C2semidirect
7. C2×#6392C2semidirect
8. #89622C2semidirect
9. #89619C2semidirect
10. #89509C2semidirect
11. #89508C2semidirect
12. #89494C2semidirect
13. #89493C2semidirect
14. #89482C2semidirect
15. #89474C2semidirect
119. #435D7semidirect
156. C7×C8#41semidirect
310. C8#1159semidirect
311. C7#3343semidirect

Maximal subgroups

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

subgroupindex
#902462
#902322
C7×#4352
#899062
#898932
#898862
C2×#63922
#896222
#896192
#895092
#895082
#894942
#894932
#894822
#894742
#33437