Finite Group Databasebeta

← #11739437

Subgroups — #11739437

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#90417C211yesmaximal, characteristic, complemented
2#90386C211yesmaximal, characteristic, complemented
3C7×#501C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89838C211yesmaximal, characteristic, complemented
6#89657C211yesmaximal, characteristic, complemented
7#89655C211yesmaximal, characteristic, complemented
8#89654C211yesmaximal, characteristic, complemented
9#89653C211yesmaximal, characteristic, complemented
10#89637C211yesmaximal, characteristic, complemented
11#89635C211yesmaximal, characteristic, complemented
12#89635C211yesmaximal, characteristic, complemented
13D7×#106C211yesmaximal, characteristic, complemented
14#89483C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic, complemented
16C2×#1161—21no—
17C7×C4⋊Q8C2211yescharacteristic
18C7×#125C2211yescharacteristic
19#6509—21no—
20C2×D56C2211yescharacteristic
21C2×#1131C2211yescharacteristic
22C2×C8×D7C2211yescharacteristic
23#6461—21no—
24#6458—21no—
25#6458C2211yescharacteristic
26#6458C2211yescharacteristic
27#6458C2211yescharacteristic
28#6458C2211yescharacteristic
29#6457—21no—
30#6457—21no—
31#6457—21no—
32#6456—21no—
33#6455C2211yescharacteristic
34D7×C4⋊C4C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36D7×C4⋊C4C2211yescharacteristic
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C4×D28C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6391C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3345—71nomaximal, Sylow
60C2×C7×Q8—21no—
61#1161—44no—
62C2×D28—21no—
63C2×D28—21no—
64C2×D28—21no—
65C2×D28—21no—
66C2×D28D411yescharacteristic
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×C4×D7C2311yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C42×C7C2311yescharacteristic
86(C2×D7)⋊C4—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—
93(C2×D7)⋊C4D411yescharacteristic
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7D411yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104D56—22no—
105#1131—22no—
106C8×D7—22no—
107#553—71no—
108#547—71no—
109#501D711yescharacteristic, complemented
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#435—71no—
120#428—71no—
121C2×#106—71no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7C2×D411yescharacteristic
125C22×D7C2×D411yescharacteristic
126C7×Q8—42no—
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×C7C2411yescharacteristic
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138D28—22no—
139D28—21no—
140D28—21no—
141D28—21no—
142D28—21no—
143D28—42no—
144D28—42no—
145D28—42no—
146D28—81no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—21no—
150C4×D7—21no—
151C4×D7—42no—
152C4×D7—42no—
153C4×D7—42no—
154C4×D7—81no—
155C4×D7—81no—
156C7×C8—21no—
157C7×C8#4112yescomplemented
158C7⋊C8—21no—
159C7⋊C8—21no—
160C2×#41—141no—
161C2×D8—71no—
162C2×M16—71no—
163C22×C8—71no—
164C4⋊Q8C2×D711yescharacteristic
165#126—141no—
166#125C2×D711yescharacteristic
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—71no—
179C4×D4—141no—
180#117—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
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—
202#97C2×D711yescharacteristic
203C22×C7C22×D411yescharacteristic
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7AGL(1,ℤ/8)12yes—
216C4×C7—21no—
217C4×C7—21no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C72−1+411yescharacteristic
224Dic7—21no—
225Dic7—21no—
226Dic7—41no—
227Dic7—41no—
228#41—284no—
229C2×Q8—21no—
230C2×D4—71no—
231C2×D4—71no—
232C2×D4—71no—
233C2×D4—141no—
234C2×D4—141no—
235C2×D4—141no—
236C2×D4—141no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4—141no—
244D8—142no—
245M16—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8—142no—
249C2×C8C22×D711yescharacteristic
250C2×C8C22×D711yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4C22×D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C4⋊C4C22×D711yescharacteristic
263C4⋊C4C22×D711yescharacteristic
264C22⋊C4—71no—
265C22⋊C4—71no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C42—141no—
274C42C22×D711yescharacteristic
275C2×C7#54711yescharacteristic
276C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
277C2×C7#57711yescharacteristic
278D7—22no—
279D7—41no—
280D7—81no—
281D7—81no—
282C23—71no—
283C23—71no—
284C23—141no—
285C23—141no—
286Q8—42no—
287D4—142no—
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—71no—
301C2×C4—71no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—282no—
309C2×C4—282no—
310C2×C4—282no—
311C2×C4—561no—
312C2×C4—561no—
313C2×C4D4×D711yescharacteristic
314C2×C4D4×D711yescharacteristic
315C2×C4C23×D711yescharacteristic
316C8—21no—
317C8—141no—
318C8—141no—
319C8#116112yescomplemented
320C7#334511yescharacteristic, complemented, Sylow
321C22—72no—
322C22—141no—
323C22—141no—
324C22—141no—
325C22—141no—
326C22—141no—
327C22—281no—
328C22—281no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22—281no—
333C22C2×D4×D711yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—141no—
341C4—141no—
342C4—281no—
343C4—281no—
344C4C2×#116111yescharacteristic
345C4#655111yescharacteristic
346C2—142no—
347C2—281no—
348C2—561no—
349C2—561no—
350C2#9041711yescharacteristic
351C2D7×AGL(1,ℤ/8)11yescharacteristic
352C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
hypercenter#501D7
hypocenterC7#3345
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #90417C2semidirect
2. #90386C2semidirect
3. C7×#501C2semidirect
4. #89856C2semidirect
5. #89838C2semidirect
6. #89657C2semidirect
7. #89655C2semidirect
8. #89654C2semidirect
9. #89653C2semidirect
10. #89637C2semidirect
11. #89635C2semidirect
12. #89635C2semidirect
13. D7×#106C2semidirect
14. #89483C2semidirect
15. #89478C2semidirect
109. #501D7semidirect
157. C7×C8#41semidirect
319. C8#1161semidirect
320. C7#3345semidirect

Maximal subgroups

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

subgroupindex
#904172
#903862
C7×#5012
#898562
#898382
#896572
#896552
#896542
#896532
#896372
#896352
D7×#1062
#894832
#894782
#33457