Finite Group Databasebeta

← #11736998

Subgroups — #11736998

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#90256C211yesmaximal, characteristic, complemented
2#90241C211yesmaximal, characteristic, complemented
3C7×#436C211yesmaximal, characteristic, complemented
4#89907C211yesmaximal, characteristic, complemented
5#89906C211yesmaximal, characteristic, complemented
6#89905C211yesmaximal, characteristic, complemented
7#89903C211yesmaximal, characteristic, complemented
8#89619C211yesmaximal, characteristic, complemented
9#89616C211yesmaximal, characteristic, complemented
10#89508C211yesmaximal, characteristic, complemented
11#89506C211yesmaximal, characteristic, complemented
12#89504C211yesmaximal, characteristic, complemented
13#89495C211yesmaximal, characteristic, complemented
14#89469C211yesmaximal, characteristic, complemented
15C4×D56C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C2×C7×M16C2211yescharacteristic, complemented
20C4×C7×D4C2211yescharacteristic
21#6502—21no—
22#6500—21no—
23#6493C2211yescharacteristic
24#6493C2211yescharacteristic
25#6493C2211yescharacteristic, complemented
26#6489C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic, complemented
28#6487C2211yescharacteristic
29C2×D56C2211yescharacteristic
30C2×D56C2211yescharacteristic
31C2×#1132C2211yescharacteristic, complemented
32#6461—21no—
33#6457—21no—
34#6447—21no—
35#6446—21no—
36(C22⋊D7)⋊C4—21no—
37D7×C22⋊C4—21no—
38#6438C2211yescharacteristic
39C4×D28C2211yescharacteristic
40C4×D28C2211yescharacteristic, complemented
41#6433C2211yescharacteristic, complemented
42C7×C4⋊C8C2211yescharacteristic, complemented
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic, complemented
45C4×C7×C8C2211yescharacteristic, complemented
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52#6395C2211yescharacteristic, complemented
53#6392C2211yescharacteristic, complemented
54#6392C2211yescharacteristic, complemented
55#6391C2211yescharacteristic, complemented
56#6390C2211yescharacteristic, complemented
57#6390C2211yescharacteristic, complemented
58#6390C2211yescharacteristic, complemented
59#3322—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4D411yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C22×Dic7—21no—
69#1157—22no—
70#1157—41no—
71C2×D28—22no—
72C2×D28—21no—
73C2×D28—41no—
74C2×D28C2311yescharacteristic
75C2×D28C2311yescharacteristic
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×Dic14C2311yescharacteristic
81C7×M16—41no—
82C7×M16—41no—
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C2×C7×C8C2311yescharacteristic
87C7×C4⋊C4D411yescharacteristic
88C7×C22⋊C4D411yescharacteristic
89C7×C22⋊C4D411yescharacteristic
90C42×C7C2311yescharacteristic
91C22⋊Dic7—21no—
92C22⋊Dic7—21no—
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)⋊C4—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104C4⋊Dic7C2311yescharacteristic
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107Dic7⋊C4—21no—
108C4×Dic7—21no—
109D56—22no—
110D56—41no—
111#1132—41no—
112#553—71no—
113#546—71no—
114#488—71no—
115#485—71no—
116#480—71no—
117#478—71no—
118#469—71no—
119#468—71no—
120#463—71no—
121#449—71no—
122#447—71no—
123C4×D8—71no—
124#436D711yescharacteristic, complemented
125#418—71no—
126#417—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—42no—
133C22×D7—42no—
134C22×D7—81no—
135C7×D4—41no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2411yescharacteristic
145C22⋊D7—42no—
146C22⋊D7—42no—
147C22⋊D7—81no—
148C22⋊D7—81no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—81no—
155D28—21no—
156D28—21no—
157D28—21no—
158D28—21no—
159D28—21no—
160D28—42no—
161D28#4112yes—
162C4×D7—42no—
163C4×D7—81no—
164C4×D7—81no—
165Dic14—21no—
166Dic14—21no—
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×D8—71no—
176C2×M16C2×D711yescharacteristic
177C2×M16C2×D711yescharacteristic, complemented
178#126—141no—
179#125—71no—
180#124—71no—
181#123—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×D4—71no—
192C4×D4—141no—
193C4×D4C2×D711yescharacteristic
194#117—71no—
195C2×C4⋊C4—71no—
196C2×C22⋊C4—141no—
197#107—71no—
198#107—71no—
199#106—71no—
200C4⋊C8C2×D711yescharacteristic, complemented
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#102—71no—
210#102—71no—
211C22⋊C8C2×D711yescharacteristic
212C22⋊C8C2×D711yescharacteristic, complemented
213C4×C8C2×D711yescharacteristic, complemented
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7C22×D411yescharacteristic
221C2×D7—22no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—84no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7C2×#4111yescharacteristic
236C4×C7H(5,𝔽2)11yescharacteristic
237Dic7—41no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243#41—142no—
244#41—281no—
245C2×Q8—71no—
246C2×D4—71no—
247C2×D4—71no—
248C2×D4—71no—
249C2×D4—142no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—281no—
256C2×D4D2811yescharacteristic
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263SD16—281no—
264D8—142no—
265D8—281no—
266M16—41no—
267M16—41no—
268C2×C8C22×D711yescharacteristic
269C2×C8C22×D711yescharacteristic
270C2×C8C22×D711yescharacteristic
271C2×C8C22×D711yescharacteristic
272C4⋊C4—71no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—71no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4D2811yescharacteristic
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4D2811yescharacteristic
291C22⋊C4D2811yescharacteristic
292C42—141no—
293C42C22×D711yescharacteristic
294C2×C7—41no—
295C2×C7—41no—
296C2×C7#54611yescharacteristic
297C2×C7#57511yescharacteristic
298C2×C7#57611yescharacteristic
299D7—42no—
300D7—81no—
301D7—81no—
302C23—141no—
303C23—141no—
304C23—141no—
305C23—282no—
306C23—282no—
307C23—561no—
308C23C2×D2811yescharacteristic
309C23C2×D2811yescharacteristic
310Q8—141no—
311Q8—141no—
312D4—41no—
313D4—72no—
314D4—141no—
315D4—141no—
316D4—141no—
317D4—141no—
318D4—141no—
319D4—282no—
320D4—282no—
321D4—282no—
322D4—561no—
323D4—561no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—282no—
334C2×C4—561no—
335C2×C4—561no—
336C2×C4—561no—
337C2×C4C2×D2811yescharacteristic
338C2×C4C2×D2811yescharacteristic
339C2×C4C2×D2811yescharacteristic
340C2×C4C2×D2811yescharacteristic
341C2×C4C23×D711yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C7#332211yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—142no—
354C22—281no—
355C22—281no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22—281no—
362C22—564no—
363C22C22×D2811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—281no—
369C4—281no—
370C4—281no—
371C4—281no—
372C4—281no—
373C4#654811yescharacteristic
374C4D7×#4111yescharacteristic
375C2—41no—
376C2—41no—
377C2—282no—
378C2—561no—
379C2—561no—
380C2#9024511yescharacteristic
381C2#9043911yescharacteristic
382C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#436C2
socleC22×C7C22×D4
hypercenter#436D7
hypocenterC7#3322
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #90256C2semidirect
2. #90241C2semidirect
3. C7×#436C2semidirect
4. #89907C2semidirect
5. #89906C2semidirect
6. #89905C2semidirect
7. #89903C2semidirect
8. #89619C2semidirect
9. #89616C2semidirect
10. #89508C2semidirect
11. #89506C2semidirect
12. #89504C2semidirect
13. #89495C2semidirect
14. #89469C2semidirect
15. C4×D56C2semidirect
19. C2×C7×M16C22semidirect
25. #6493C22semidirect
27. C2×C4⋊Dic7C22semidirect
31. C2×#1132C22semidirect
40. C4×D28C22semidirect
41. #6433C22semidirect
42. C7×C4⋊C8C22semidirect
44. C7×C22⋊C8C22semidirect
45. C4×C7×C8C22semidirect
52. #6395C22semidirect
53. #6392C22semidirect
54. #6392C22semidirect
55. #6391C22semidirect
56. #6390C22semidirect
57. #6390C22semidirect
58. #6390C22semidirect
124. #436D7semidirect
177. C2×M16C2×D7semidirect
200. C4⋊C8C2×D7semidirect
212. C22⋊C8C2×D7semidirect
213. C4×C8C2×D7semidirect
346. C7#3322semidirect

Maximal subgroups

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

subgroupindex
#902562
#902412
C7×#4362
#899072
#899062
#899052
#899032
#896192
#896162
#895082
#895062
#895042
#894952
#894692
C4×D562
#33227