Finite Group Databasebeta

← #11737328

Subgroups — #11737328

Order 896.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90254C211yesmaximal, characteristic, complemented
2#90245C211yesmaximal, characteristic, complemented
3C7×#442C211yesmaximal, characteristic, complemented
4#89930C211yesmaximal, characteristic, complemented
5#89928C211yesmaximal, characteristic, complemented
6#89788C211yesmaximal, characteristic, complemented
7#89779C211yesmaximal, characteristic, complemented
8#89639C211yesmaximal, characteristic, complemented
9#89638C211yesmaximal, characteristic, complemented
10#89552C211yesmaximal, characteristic, complemented
11#89547C211yesmaximal, characteristic, complemented
12#89544C211yesmaximal, characteristic, complemented
13#89542C211yesmaximal, characteristic, complemented
14#89486C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×C7×D8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic, complemented
21#6501C2211yescharacteristic
22#6501C2211yescharacteristic, complemented
23C2×#1142C2211yescharacteristic, complemented
24C2×#1141C2211yescharacteristic
25#6493—21no—
26#6493—21no—
27C2×(C2×D7)⋊C4—21no—
28#6489—21no—
29C2×#1131C2211yescharacteristic
30C2×C8×D7C2211yescharacteristic, complemented
31#6460C2211yescharacteristic
32#6458C2211yescharacteristic
33#6455C2211yescharacteristic
34#6455C2211yescharacteristic, complemented
35#6449—21no—
36#6448—21no—
37#6448—21no—
38#6445—21no—
39#6438C2211yescharacteristic
40C4×D28C2211yescharacteristic
41#6432C2211yescharacteristic, complemented
42C7×#106C2211yescharacteristic, complemented
43C7×#102C2211yescharacteristic
44C7×#102C2211yescharacteristic, complemented
45C7×#97C2211yescharacteristic, complemented
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic, complemented
52#6390C2211yescharacteristic, complemented
53#6390C2211yescharacteristic, complemented
54#6384C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6381C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58C4⋊(C7⋊C8)C2211yescharacteristic, complemented
59#3319—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70#1159—22no—
71#1159—41no—
72D4×D7—22no—
73D4×D7—41no—
74C2×D28—21no—
75C2×D28D411yescharacteristic
76C2×D28C2311yescharacteristic
77C2×C4×D7C2311yescharacteristic
78C2×C4×D7C2311yescharacteristic
79C2×Dic14—21no—
80C2×Dic14C2311yescharacteristic
81C7×D8—22no—
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91#1142—41no—
92#1141—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4D411yescharacteristic
100(C2×D7)⋊C4D411yescharacteristic
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7D411yescharacteristic
104C4⋊Dic7C2311yescharacteristic
105C4⋊Dic7C2311yescharacteristic
106C4×Dic7—21no—
107C4×Dic7—21no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1131—41no—
111C8×D7—41no—
112#555—71no—
113#546—71no—
114#489—71no—
115#485—71no—
116#482—71no—
117#474—71no—
118#468—71no—
119#466—71no—
120#463—71no—
121#449—71no—
122#447—71no—
123#442D711yescharacteristic, complemented
124#435—71no—
125#418—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7—21no—
129C22×D7—21no—
130C22×D7—42no—
131C22×D7—42no—
132C22×D7—81no—
133C22×D7C2×D411yescharacteristic
134C22×D7C2×D411yescharacteristic
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4#4112yes—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7—81no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—42no—
147C22⋊D7—42no—
148C22⋊D7—81no—
149C22⋊D7—81no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—42no—
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156D28—21no—
157D28—21no—
158D28—41no—
159D28—81no—
160C4×D7—21no—
161C4×D7—21no—
162C4×D7—21no—
163C4×D7—21no—
164Dic14—21no—
165Dic14—21no—
166Dic14—81no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C22×D4—71no—
173C2×SD16—71no—
174C2×D8—71no—
175C2×D8C2×D711yescharacteristic
176C2×M16—71no—
177C22×C8—71no—
178C4⋊Q8—71no—
179#124—71no—
180#124—141no—
181#124—141no—
182#123—141no—
183C22⋊Q8—71no—
184#121—71no—
185#121—71no—
186#121—71no—
187#121—141no—
188#121—141no—
189#120—141no—
190C4×D4—71no—
191C4×D4C2×D711yescharacteristic
192C4×D4C2×D711yescharacteristic, complemented
193#117—71no—
194#117—71no—
195#117—141no—
196C2×C22⋊C4—141no—
197#107—71no—
198#107—71no—
199#106C2×D711yescharacteristic, complemented
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102—71no—
209#102C2×D711yescharacteristic
210#102C2×D711yescharacteristic, complemented
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97C2×D711yescharacteristic, complemented
214C22×C7—22no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—21no—
224C2×D7—21no—
225C2×D7—21no—
226C2×D7—21no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—84no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7C2×#4111yescharacteristic
236C4×C7H(5,𝔽2)11yescharacteristic
237Dic7—21no—
238Dic7—21no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243#41—142no—
244#41—281no—
245C2×Q8—71no—
246C2×Q8—141no—
247C2×D4—71no—
248C2×D4—71no—
249C2×D4—141no—
250C2×D4—141no—
251C2×D4—142no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—281no—
256C2×D4C22×D711yescharacteristic
257C2×D4C22×D711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—141no—
263SD16—281no—
264D8—22no—
265D8—281no—
266M16—281no—
267C2×C8—71no—
268C2×C8—71no—
269C2×C8—281no—
270C2×C8C22×D711yescharacteristic
271C2×C8C22×D711yescharacteristic
272C4⋊C4—71no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4C22×D711yescharacteristic
278C4⋊C4C22×D711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—71no—
282C22⋊C4—71no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C42—141no—
292C42—141no—
293C42C22×D711yescharacteristic
294C2×C7—42no—
295C2×C7—81no—
296C2×C7#54611yescharacteristic
297C2×C7#57511yescharacteristic
298C2×C7#57611yescharacteristic
299D7—41no—
300D7—41no—
301D7—81no—
302C23—21no—
303C23—21no—
304C23—71no—
305C23—71no—
306C23—141no—
307C23—282no—
308C23—282no—
309C23—561no—
310Q8—141no—
311Q8—141no—
312Q8—561no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—141no—
317D4—141no—
318D4—281no—
319D4—282no—
320D4—282no—
321D4—561no—
322D4—561no—
323D4—561no—
324D4#115712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—71no—
329C2×C4—71no—
330C2×C4—81no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—282no—
339C2×C4D4×D711yescharacteristic
340C2×C4D4×D711yescharacteristic
341C2×C4C23×D711yescharacteristic
342C8—21no—
343C8—21no—
344C8—141no—
345C8—141no—
346C7#331911yescharacteristic, complemented, Sylow
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—141no—
354C22—141no—
355C22—141no—
356C22—141no—
357C22—141no—
358C22—141no—
359C22—281no—
360C22—281no—
361C22—281no—
362C22—564no—
363C22C2×D4×D711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—141no—
369C4—141no—
370C4—281no—
371C4—281no—
372C4—281no—
373C4C2×#115711yescharacteristic
374C4#655311yescharacteristic
375C2—42no—
376C2—81no—
377C2—281no—
378C2—281no—
379C2—561no—
380C2#9024111yescharacteristic
381C2#9046411yescharacteristic
382C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#442C2
socleC22×C7C22×D4
hypercenter#442D7
hypocenterC7#3319
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #90254C2semidirect
2. #90245C2semidirect
3. C7×#442C2semidirect
4. #89930C2semidirect
5. #89928C2semidirect
6. #89788C2semidirect
7. #89779C2semidirect
8. #89639C2semidirect
9. #89638C2semidirect
10. #89552C2semidirect
11. #89547C2semidirect
12. #89544C2semidirect
13. #89542C2semidirect
14. #89486C2semidirect
15. #89478C2semidirect
20. C4×C7×D4C22semidirect
22. #6501C22semidirect
23. C2×#1142C22semidirect
30. C2×C8×D7C22semidirect
34. #6455C22semidirect
41. #6432C22semidirect
42. C7×#106C22semidirect
44. C7×#102C22semidirect
45. C7×#97C22semidirect
51. (C2×D7)⋊C8C22semidirect
52. #6390C22semidirect
53. #6390C22semidirect
54. #6384C22semidirect
56. #6381C22semidirect
57. #6381C22semidirect
58. C4⋊(C7⋊C8)C22semidirect
123. #442D7semidirect
192. C4×D4C2×D7semidirect
199. #106C2×D7semidirect
210. #102C2×D7semidirect
213. #97C2×D7semidirect
346. C7#3319semidirect

Maximal subgroups

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

subgroupindex
#902542
#902452
C7×#4422
#899302
#899282
#897882
#897792
#896392
#896382
#895522
#895472
#895442
#895422
#894862
#894782
#33197