Finite Group Databasebeta

← #11736343

Subgroups — #11736343

Order 896.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90520C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic, complemented
3C7×#417C211yesmaximal, characteristic, complemented
4#90011C211yesmaximal, characteristic, complemented
5#89906C211yesmaximal, characteristic, complemented
6#89899C211yesmaximal, characteristic, complemented
7#89739C211yesmaximal, characteristic, complemented
8#89585C211yesmaximal, characteristic, complemented
9#89581C211yesmaximal, characteristic, complemented
10#89570C211yesmaximal, characteristic, complemented
11#89567C211yesmaximal, characteristic, complemented
12#89548C211yesmaximal, characteristic, complemented
13D7×#102C211yesmaximal, characteristic, complemented
14#89531C211yesmaximal, characteristic, complemented
15#89529C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic, complemented
19C2×C7×C4⋊C4C2211yescharacteristic, complemented
20#6510C2211yescharacteristic
21#6509C2211yescharacteristic
22C2×#1144C2211yescharacteristic, complemented
23#6502—21no—
24#6500—21no—
25D4×Dic7C2211yescharacteristic
26C2×#1142C2211yescharacteristic
27#6493C2211yescharacteristic
28#6492—21no—
29C4×C22⋊D7—21no—
30C4×C22⋊D7C2211yescharacteristic
31C2×(C2×D7)⋊C4—21no—
32#6487—21no—
33#6487C2211yescharacteristic, complemented
34C2×#1132C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic
36#6460C2211yescharacteristic
37#6459—21no—
38D7×C4⋊C4C2211yescharacteristic
39#6452—21no—
40#6451C2211yescharacteristic, complemented
41#6450C2211yescharacteristic, complemented
42C7×#103C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic, complemented
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic, complemented
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic, complemented
52#6390C2211yescharacteristic, complemented
53#6389C2211yescharacteristic, complemented
54Dic7⋊C8C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3314—71nomaximal, Sylow
60C23×D7—21no—
61C7×#41—22no—
62C7×#41—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7D411yescharacteristic
72C22×Dic7—21no—
73D4×D7—22no—
74D4×D7—41no—
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C2×Dic14—21no—
79C2×Dic14C2311yescharacteristic
80C7×M16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7D411yescharacteristic
89#1144—41no—
90#1142—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4D411yescharacteristic
97C4⋊Dic7—21no—
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4D411yescharacteristic
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1132—41no—
111C8×D7—41no—
112#554—71no—
113#546—71no—
114#489—71no—
115#484—71no—
116#480—71no—
117#474—71no—
118#468—71no—
119#467—71no—
120#464—71no—
121#452—71no—
122#450—71no—
123#440—71no—
124#435—71no—
125#417D711yescharacteristic, complemented
126C2×#102—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—42no—
131C22×D7—42no—
132C22×D7—81no—
133C22×D7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×D4—21no—
137C7×D4—42no—
138C7×D4#4112yes—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—42no—
145C2×C4×C7—81no—
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C22⋊D7—22no—
149C22⋊D7—42no—
150C22⋊D7—81no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—81no—
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157C2×Dic7C2×D411yescharacteristic
158D28—21no—
159D28—21no—
160C4×D7—21no—
161C4×D7—21no—
162C4×D7—81no—
163Dic14—21no—
164Dic14—21no—
165Dic14—81no—
166C7×C8—21no—
167C7×C8—21no—
168C7⋊C8—21no—
169C7⋊C8—21no—
170C2×#41C2×D711yescharacteristic
171C22×D4—71no—
172C2×Q16—71no—
173C2×SD16—71no—
174C2×SD16—71no—
175C2×M16C2×D711yescharacteristic, complemented
176C22×C8—71no—
177C4⋊Q8—71no—
178#125—141no—
179#124—71no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8—71no—
184C22⋊Q8—141no—
185C22⋊Q8—141no—
186#121—71no—
187#121—141no—
188#120—141no—
189C4×Q8—71no—
190C4×D4—71no—
191C4×D4—71no—
192C4×D4—141no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4C2×D711yescharacteristic, complemented
195C2×C22⋊C4—141no—
196#107—71no—
197#107—71no—
198#107—71no—
199C4⋊C8—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic, complemented
203#103C2×D711yescharacteristic, complemented
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102C2×D711yescharacteristic
209#102C2×D711yescharacteristic
210C22⋊C8—71no—
211C22⋊C8—71no—
212#97—71no—
213C22×C7—22no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×D411yescharacteristic
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—84no—
226C2×D7#13712yes—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—21no—
234Dic7—21no—
235Dic7—21no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239C24—141no—
240#41—22no—
241#41—41no—
242C2×Q8—71no—
243C2×Q8—141no—
244C2×Q8C22×D711yescharacteristic
245C2×D4—21no—
246C2×D4—71no—
247C2×D4—71no—
248C2×D4—141no—
249C2×D4—142no—
250C2×D4—141no—
251C2×D4—281no—
252C2×D4C22×D711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—71no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4C22×D711yescharacteristic
259Q16—281no—
260SD16—142no—
261SD16—281no—
262M16—41no—
263C2×C8—71no—
264C2×C8—71no—
265C2×C8—281no—
266C2×C8C22×D711yescharacteristic
267C2×C8C22×D711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—71no—
270C4⋊C4—71no—
271C4⋊C4—71no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4C22×D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C22⋊C4—71no—
281C22⋊C4—71no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C42—71no—
290C42—141no—
291C2×C7—42no—
292C2×C7—41no—
293C2×C7#54611yescharacteristic
294C2×C7C2×#13711yescharacteristic
295C2×C7#57611yescharacteristic
296D7—22no—
297D7—81no—
298C23—21no—
299C23—71no—
300C23—141no—
301C23—282no—
302C23—282no—
303C23—561no—
304C23D4×D711yescharacteristic
305Q8—21no—
306Q8—21no—
307Q8—141no—
308Q8—141no—
309Q8—561no—
310D4—21no—
311D4—42no—
312D4—142no—
313D4—141no—
314D4—141no—
315D4—282no—
316D4—561no—
317D4#115712yes—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—71no—
325C2×C4—71no—
326C2×C4—71no—
327C2×C4—81no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—561no—
334C2×C4—561no—
335C2×C4D4×D711yescharacteristic
336C2×C4C23×D711yescharacteristic
337C8—21no—
338C8—21no—
339C8—141no—
340C8—141no—
341C7#331411yescharacteristic, complemented, Sylow
342C22—21no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—72no—
349C22—141no—
350C22—141no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22—564no—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—141no—
361C4—141no—
362C4—141no—
363C4—281no—
364C4—281no—
365C4—281no—
366C4C2×#115711yescharacteristic
367C4#654811yescharacteristic
368C2—41no—
369C2—42no—
370C2—142no—
371C2—561no—
372C2#9018111yescharacteristic
373C2#9046511yescharacteristic
374C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
prime radical#90520C2
hypercenter#417D7
hypocenterC7#3314
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #90202C2semidirect
3. C7×#417C2semidirect
4. #90011C2semidirect
5. #89906C2semidirect
6. #89899C2semidirect
7. #89739C2semidirect
8. #89585C2semidirect
9. #89581C2semidirect
10. #89570C2semidirect
11. #89567C2semidirect
12. #89548C2semidirect
13. D7×#102C2semidirect
14. #89531C2semidirect
15. #89529C2semidirect
18. C2×C7×M16C22semidirect
19. C2×C7×C4⋊C4C22semidirect
22. C2×#1144C22semidirect
33. #6487C22semidirect
40. #6451C22semidirect
41. #6450C22semidirect
42. C7×#103C22semidirect
43. C7×#103C22semidirect
46. #6409C22semidirect
48. C22⋊(C7⋊C8)C22semidirect
51. #6392C22semidirect
52. #6390C22semidirect
53. #6389C22semidirect
54. Dic7⋊C8C22semidirect
57. #6381C22semidirect
58. #6381C22semidirect
125. #417D7semidirect
175. C2×M16C2×D7semidirect
194. C2×C4⋊C4C2×D7semidirect
202. #103C2×D7semidirect
203. #103C2×D7semidirect
341. C7#3314semidirect

Maximal subgroups

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

subgroupindex
#905202
#902022
C7×#4172
#900112
#899062
#898992
#897392
#895852
#895812
#895702
#895672
#895482
D7×#1022
#895312
#895292
#33147