Finite Group Databasebeta

← #11737607

Subgroups — #11737607

Order 896.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90497C211yesmaximal, characteristic
2#90332C211yesmaximal, characteristic, complemented
3C7×#450C211yesmaximal, characteristic, complemented
4#89987C211yesmaximal, characteristic, complemented
5#89945C211yesmaximal, characteristic, complemented
6#89943C211yesmaximal, characteristic, complemented
7#89938C211yesmaximal, characteristic, complemented
8#89936C211yesmaximal, characteristic, complemented
9#89824C211yesmaximal, characteristic, complemented
10#89556C211yesmaximal, characteristic, complemented
11#89544C211yesmaximal, characteristic, complemented
12#89537C211yesmaximal, characteristic, complemented
13#89533C211yesmaximal, characteristic, complemented
14#89500C211yesmaximal, characteristic, complemented
15#89495C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21#6516—21no—
22C2×C22⋊Dic7—21no—
23#6509C2211yescharacteristic, complemented
24Q8×Dic7C2211yescharacteristic, complemented
25C2×#1144C2211yescharacteristic, complemented
26#6502—21no—
27#6501C2211yescharacteristic
28#6499C2211yescharacteristic
29#6498—21no—
30D4×Dic7C2211yescharacteristic
31C4×C22⋊D7C2211yescharacteristic
32#6489C2211yescharacteristic
33#6487C2211yescharacteristic
34C2×#1136C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic, complemented
36#6461—21no—
37#6455C2211yescharacteristic, complemented
38#6451C2211yescharacteristic, complemented
39#6450—21no—
40#6448—21no—
41#6442—21no—
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic, complemented
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic, complemented
51#6392C2211yescharacteristic, complemented
52#6391C2211yescharacteristic, complemented
53#6390C2211yescharacteristic, complemented
54#6390C2211yescharacteristic, complemented
55#6389C2211yescharacteristic, complemented
56Dic7⋊C8C2211yescharacteristic, complemented
57#6384C2211yescharacteristic, complemented
58#6382C2211yescharacteristic, complemented
59#3311—71nomaximal, Sylow
60C24×C7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—22no—
63C2×C7×D4—21no—
64C2×C7×D4—41no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7D411yescharacteristic
71C22×Dic7—21no—
72C22×Dic7—21no—
73#1159—22no—
74#1159—41no—
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14C2311yescharacteristic
78C7×SD16—22no—
79C7×SD16—41no—
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7D411yescharacteristic
92#1144—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108#1136—41no—
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C8×D7—41no—
112#556—71no—
113#546—71no—
114#489—71no—
115#485—71no—
116#482—71no—
117#476—71no—
118#470—71no—
119#465—71no—
120#464—71no—
121#452—71no—
122#450D711yescharacteristic, complemented
123#440—71no—
124#435—71no—
125#418—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—42no—
130C23×C7—42no—
131C23×C7—81no—
132C23×C7C2×D411yescharacteristic
133C22×D7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—21no—
139C7×D4—42no—
140C7×D4#4112yes—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—41no—
148C22⋊D7—42no—
149C22⋊D7—81no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—42no—
155C2×Dic7—81no—
156C2×Dic7C2×D411yescharacteristic
157C2×Dic7C2×D411yescharacteristic
158C2×Dic7C2×D411yescharacteristic
159C4×D7—21no—
160C4×D7—21no—
161Dic14—21no—
162Dic14—21no—
163Dic14—42no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×D4C2×D711yescharacteristic
170C2×Q16—71no—
171C2×SD16C2×D711yescharacteristic
172C2×SD16C2×D711yescharacteristic
173C2×M16—71no—
174C22×C8—71no—
175C4⋊Q8—71no—
176#126—141no—
177#126—141no—
178#124—71no—
179#124—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8C2×D711yescharacteristic
184#121—71no—
185#121—141no—
186#120—141no—
187C4×Q8—71no—
188C4×Q8—141no—
189C4×D4—71no—
190C4×D4—71no—
191#117—71no—
192#117—71no—
193C2×C22⋊C4—141no—
194#107—71no—
195#106—71no—
196#106—71no—
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102C2×D711yescharacteristic
207#102C2×D711yescharacteristic
208C22⋊C8—71no—
209C22⋊C8C2×D711yescharacteristic
210#97—71no—
211C22×C7—22no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—84no—
221C22×C7C22×D411yescharacteristic
222C2×D7—21no—
223C2×D7—21no—
224C2×D7—21no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7C2×#4111yescharacteristic
229C4×C7H(5,𝔽2)11yescharacteristic
230Dic7—21no—
231Dic7—21no—
232Dic7—21no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237C24—21no—
238#41—142no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8—141no—
242C2×Q8C22×D711yescharacteristic
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—71no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4C22×D711yescharacteristic
250C2×D4C22×D711yescharacteristic
251C22×C4—71no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255Q16—281no—
256SD16—22no—
257SD16—41no—
258M16—281no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—281no—
262C2×C8C22×D711yescharacteristic
263C2×C8C22×D711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—71no—
276C22⋊C4—71no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C42—71no—
286C42—141no—
287C42—141no—
288C42—141no—
289C2×C7—41no—
290C2×C7—42no—
291C2×C7—81no—
292C2×C7#54611yescharacteristic
293C2×C7#57511yescharacteristic
294C2×C7#57711yescharacteristic
295D7—41no—
296C23—21no—
297C23—21no—
298C23—42no—
299C23—42no—
300C23—71no—
301C23—81no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—141no—
306Q8—141no—
307Q8—282no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—281no—
313D4—282no—
314D4—561no—
315D4#115912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—71no—
321C2×C4—71no—
322C2×C4—71no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—282no—
330C2×C4—561no—
331C2×C4D4×D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—21no—
334C8—21no—
335C8—141no—
336C8—141no—
337C7#331111yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—141no—
349C22—141no—
350C22—141no—
351C22C2×D4×D711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—141no—
356C4—141no—
357C4—141no—
358C4—281no—
359C4—281no—
360C4—281no—
361C4—281no—
362C4C2×#115911yescharacteristic
363C4#654811yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—281no—
368C2#9028211yescharacteristic
369C2#9045311yescharacteristic
370C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#450C2
socleC22×C7C22×D4
prime radical#90497C2
hypercenter#450D7
hypocenterC7#3311
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #90332C2semidirect
3. C7×#450C2semidirect
4. #89987C2semidirect
5. #89945C2semidirect
6. #89943C2semidirect
7. #89938C2semidirect
8. #89936C2semidirect
9. #89824C2semidirect
10. #89556C2semidirect
11. #89544C2semidirect
12. #89537C2semidirect
13. #89533C2semidirect
14. #89500C2semidirect
15. #89495C2semidirect
23. #6509C22semidirect
24. Q8×Dic7C22semidirect
25. C2×#1144C22semidirect
35. C2×C8×D7C22semidirect
37. #6455C22semidirect
38. #6451C22semidirect
45. #6409C22semidirect
50. (C2×D7)⋊C8C22semidirect
51. #6392C22semidirect
52. #6391C22semidirect
53. #6390C22semidirect
54. #6390C22semidirect
55. #6389C22semidirect
56. Dic7⋊C8C22semidirect
57. #6384C22semidirect
58. #6382C22semidirect
122. #450D7semidirect
337. C7#3311semidirect

Maximal subgroups

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

subgroupindex
#904972
#903322
C7×#4502
#899872
#899452
#899432
#899382
#899362
#898242
#895562
#895442
#895372
#895332
#895002
#894952
#33117