Finite Group Databasebeta

← #11736680

Subgroups — #11736680

Order 896.

386 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#90223C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic, complemented
3C7×#428C211yesmaximal, characteristic, complemented
4#89907C211yesmaximal, characteristic, complemented
5#89900C211yesmaximal, characteristic, complemented
6#89775C211yesmaximal, characteristic, complemented
7#89740C211yesmaximal, characteristic, complemented
8#89662C211yesmaximal, characteristic, complemented
9#89657C211yesmaximal, characteristic, complemented
10#89653C211yesmaximal, characteristic, complemented
11#89644C211yesmaximal, characteristic, complemented
12#89642C211yesmaximal, characteristic, complemented
13#89637C211yesmaximal, characteristic, complemented
14#89635C211yesmaximal, characteristic, complemented
15#89627C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C2×C7×M16C2211yescharacteristic
18C7×#117C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20#6493C2211yescharacteristic
21#6493C2211yescharacteristic
22#6492—21no—
23C4×C22⋊D7C2211yescharacteristic
24C2×D56C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26#6459—21no—
27#6458C2211yescharacteristic
28#6458C2211yescharacteristic
29#6457—21no—
30#6456C2211yescharacteristic
31#6456C2211yescharacteristic
32#6455C2211yescharacteristic
33D7×C4⋊C4C2211yescharacteristic
34#6451C2211yescharacteristic
35#6451C2211yescharacteristic
36#6448—21no—
37#6447—21no—
38#6438—21no—
39#6435—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6391C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50C8×Dic7C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3354—71nomaximal, Sylow
60C22×C4×C7—21no—
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7D411yescharacteristic
65#1157—44no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28C2311yescharacteristic
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×Dic14—21no—
76C7×M16—22no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C42×C7—21no—
86C22⋊Dic7D411yescharacteristic
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—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105D56—22no—
106#1131—22no—
107#555—71no—
108#547—71no—
109#498—71no—
110#482—71no—
111#482—71no—
112#481—71no—
113#481—71no—
114#476—71no—
115#474—71no—
116#469—71no—
117#469—71no—
118C4×D8—71no—
119#436—71no—
120#428—71no—
121#428D711yescharacteristic, complemented
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C22⋊D7—22no—
136C22⋊D7—42no—
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—21no—
144D28—21no—
145D28—21no—
146D28—21no—
147D28—42no—
148D28—81no—
149C4×D7—21no—
150C4×D7—21no—
151C4×D7—42no—
152C4×D7—42no—
153C4×D7—42no—
154Dic14—42no—
155Dic14—81no—
156C7×C8—21no—
157C7×C8#4112yescomplemented
158C7⋊C8—21no—
159C7⋊C8—21no—
160C2×#41—141no—
161C2×D8—71no—
162C2×M16—71no—
163C2×M16C2×D711yescharacteristic
164C4⋊Q8—71no—
165C4⋊Q8—71no—
166#124—141no—
167#124—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—71no—
179#117—71no—
180#117—141no—
181#117C2×D711yescharacteristic
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#106C2×D711yescharacteristic
192#106C2×D711yescharacteristic
193C4⋊C8—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8—71no—
201C22⋊C8—71no—
202C4×C8—71no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7C22×D411yescharacteristic
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—21no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7C2×#4111yescharacteristic
222C4×C72−1+411yescharacteristic
223Dic7—21no—
224Dic7—21no—
225Dic7—41no—
226Dic7—41no—
227Dic7AGL(1,ℤ/8)12yes—
228#41—284no—
229C2×Q8—141no—
230C2×Q8—141no—
231C2×D4—71no—
232C2×D4—71no—
233C2×D4—71no—
234C2×D4—141no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C22×C4—21no—
239C22×C4—71no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244D8—142no—
245M16—22no—
246M16—142no—
247C2×C8—71no—
248C2×C8—71no—
249C2×C8C22×D711yescharacteristic
250C2×C8C22×D711yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4C22×D711yescharacteristic
259C4⋊C4C22×D711yescharacteristic
260C4⋊C4C22×D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—71no—
264C22⋊C4—71no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C42—21no—
273C42—71no—
274C42—141no—
275C2×C7—41no—
276C2×C7#54711yescharacteristic
277C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
278C2×C7#57711yescharacteristic
279D7—41no—
280D7—81no—
281D7—81no—
282C23—71no—
283C23—141no—
284C23—141no—
285C23D4×D711yescharacteristic
286Q8—282no—
287Q8—561no—
288D4—142no—
289D4—141no—
290D4—141no—
291D4—141no—
292D4—141no—
293D4—282no—
294D4—282no—
295D4—561no—
296D4—561no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—71no—
305C2×C4—71no—
306C2×C4—71no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—282no—
312C2×C4—282no—
313C2×C4—282no—
314C2×C4D4×D711yescharacteristic
315C2×C4C23×D711yescharacteristic
316C8—21no—
317C8—141no—
318C8—141no—
319C8#115712yescomplemented
320C7#335411yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—141no—
325C22—141no—
326C22—141no—
327C22—281no—
328C22—281no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22—281no—
333C22C2×D4×D711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—72no—
340C4—141no—
341C4—141no—
342C4—281no—
343C4—281no—
344C4C2×#115711yescharacteristic
345C4#655111yescharacteristic
346C2—41no—
347C2—281no—
348C2—561no—
349C2—561no—
350C2#9019811yescharacteristic
351C2D7×AGL(1,ℤ/8)11yescharacteristic
352C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
hypercenter#428D7
hypocenterC7#3354
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #90223C2semidirect
2. #90198C2semidirect
3. C7×#428C2semidirect
4. #89907C2semidirect
5. #89900C2semidirect
6. #89775C2semidirect
7. #89740C2semidirect
8. #89662C2semidirect
9. #89657C2semidirect
10. #89653C2semidirect
11. #89644C2semidirect
12. #89642C2semidirect
13. #89637C2semidirect
14. #89635C2semidirect
15. #89627C2semidirect
121. #428D7semidirect
157. C7×C8#41semidirect
319. C8#1157semidirect
320. C7#3354semidirect

Maximal subgroups

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

subgroupindex
#902232
#901982
C7×#4282
#899072
#899002
#897752
#897402
#896622
#896572
#896532
#896442
#896422
#896372
#896352
#896272
#33547