Finite Group Databasebeta

← #11737987

Subgroups — #11737987

Order 896.

444 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#124C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic, complemented
3C7×#463C211yesmaximal, characteristic, complemented
4#89948C211yesmaximal, characteristic, complemented
5#89942C211yesmaximal, characteristic, complemented
6#89928C211yesmaximal, characteristic, complemented
7#89926C211yesmaximal, characteristic, complemented
8#89829C211yesmaximal, characteristic, complemented
9#89788C211yesmaximal, characteristic, complemented
10#89616C211yesmaximal, characteristic, complemented
11#89607C211yesmaximal, characteristic, complemented
12#89578C211yesmaximal, characteristic, complemented
13#89567C211yesmaximal, characteristic, complemented
14#89545C211yesmaximal, characteristic, complemented
15#89537C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×#1157—21no—
19C2×C7×SD16C2211yescharacteristic, complemented
20C2×C7×D8C2211yescharacteristic
21C7×#124C2211yescharacteristic
22C4×C7×D4C2211yescharacteristic, complemented
23#6510C2211yescharacteristic
24C2×#1143C2211yescharacteristic, complemented
25#6503C2211yescharacteristic, complemented
26#6501C2211yescharacteristic
27#6499C2211yescharacteristic
28C2×#1142C2211yescharacteristic, complemented
29C2×#1141C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31C4×C22⋊D7—21no—
32C2×D56C2211yescharacteristic
33C2×#1132C2211yescharacteristic, complemented
34#6458C2211yescharacteristic
35#6451C2211yescharacteristic, complemented
36#6448—21no—
37#6448—21no—
38#6448—21no—
39#6447—21no—
40D7×C22⋊C4—21no—
41D7×C22⋊C4—21no—
42#6438C2211yescharacteristic
43C4×D28C2211yescharacteristic, complemented
44C42×D7C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic, complemented
46C7×#103C2211yescharacteristic, complemented
47C7×#102C2211yescharacteristic
48#6409C2211yescharacteristic, complemented
49#6406C2211yescharacteristic
50#6395C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic
53#6390C2211yescharacteristic, complemented
54#6389C2211yescharacteristic, complemented
55#6389C2211yescharacteristic, complemented
56#6384C2211yescharacteristic, complemented
57#6383C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic, complemented
59#3213—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69Q8×D7—41no—
70Q8×D7—41no—
71D4×D7—41no—
72D4×D7—41no—
73#1157—44no—
74C2×D28—21no—
75C2×D28C2311yescharacteristic
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7D411yescharacteristic
80C2×C4×D7D411yescharacteristic
81C2×C4×D7C2311yescharacteristic
82C2×Dic14—21no—
83C2×Dic14—21no—
84C2×Dic14C2311yescharacteristic
85C7×SD16—41no—
86C7×D8—41no—
87C2×C7×C8C2311yescharacteristic
88C2×C7×C8C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
93C42×C7C2311yescharacteristic
94C22⋊Dic7—21no—
95C22⋊Dic7—21no—
96C22⋊Dic7—21no—
97#1143—41no—
98#1142—41no—
99#1141—41no—
100#1141—41no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107C4⋊Dic7C2311yescharacteristic
108Dic7⋊C4—21no—
109C4×Dic7D411yescharacteristic
110C4×Dic7D411yescharacteristic
111C4×Dic7C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113C2×C7⋊C8C2311yescharacteristic
114D56—41no—
115#1132—41no—
116#532—71no—
117C2×#124—71no—
118#496—71no—
119#496—71no—
120#489—71no—
121#489—71no—
122#463—71no—
123#463—71no—
124#463—71no—
125#463D711yescharacteristic, complemented
126#448—71no—
127#448—71no—
128#447—71no—
129#447—71no—
130#432—71no—
131C23×C7—21no—
132C23×C7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—42no—
136C22×D7—42no—
137C22×D7—42no—
138C22×D7C2×D411yescharacteristic
139C7×Q8—21no—
140C7×Q8—21no—
141C7×D4—21no—
142C7×D4—21no—
143C7×D4—21no—
144C7×D4—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—42no—
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2411yescharacteristic
151C22⋊D7—42no—
152C22⋊D7—42no—
153C22⋊D7—42no—
154C22⋊D7—42no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7C2×D411yescharacteristic
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160D28—21no—
161D28—21no—
162D28—21no—
163D28—21no—
164D28—42no—
165C4×D7—22no—
166C4×D7—21no—
167C4×D7—21no—
168C4×D7—41no—
169C4×D7—42no—
170C4×D7—42no—
171Dic14—21no—
172Dic14—21no—
173Dic14—42no—
174Dic14—42no—
175C7×C8—21no—
176C7×C8—21no—
177C7⋊C8—21no—
178C7⋊C8—21no—
179C2×#41—141no—
180C22×Q8—71no—
181C22×D4—71no—
182C2×SD16—71no—
183C2×SD16—71no—
184C2×SD16—71no—
185C2×SD16C2×D711yescharacteristic, complemented
186C2×D8—71no—
187C2×D8—71no—
188C2×D8—71no—
189C2×D8C2×D711yescharacteristic
190C4⋊Q8—71no—
191#127—71no—
192#124—71no—
193#124—71no—
194#124—71no—
195#124—141no—
196#124—141no—
197#124—141no—
198#124C2×D711yescharacteristic
199#121—71no—
200#121—71no—
201#121—141no—
202C4×D4—71no—
203C4×D4—141no—
204C4×D4C2×D711yescharacteristic, complemented
205C2×C22⋊C4—141no—
206C2×C22⋊C4—141no—
207C2×C42—71no—
208C4⋊C8—71no—
209C4⋊C8C2×D711yescharacteristic, complemented
210#103—71no—
211#103—71no—
212#103—71no—
213#103C2×D711yescharacteristic, complemented
214#102—71no—
215#102—71no—
216#102—71no—
217#102C2×D711yescharacteristic
218C22⋊C8—71no—
219C22⋊C8—71no—
220#97—71no—
221#97—71no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7C22×D411yescharacteristic
229C2×D7—21no—
230C2×D7—21no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—84no—
238C2×D7AGL(1,ℤ/8)12yes—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7H(5,𝔽2)11yescharacteristic
244C4×C7H(5,𝔽2)11yescharacteristic
245Dic7—21no—
246Dic7—21no—
247Dic7—21no—
248Dic7—41no—
249Dic7—41no—
250C24—141no—
251#41—284no—
252C2×Q8—71no—
253C2×Q8—141no—
254C2×Q8—141no—
255C2×Q8—281no—
256C2×Q8—281no—
257C2×Q8C22×D711yescharacteristic
258C2×D4—71no—
259C2×D4—71no—
260C2×D4—141no—
261C2×D4—141no—
262C2×D4—141no—
263C2×D4—141no—
264C2×D4—141no—
265C2×D4—281no—
266C2×D4—281no—
267C2×D4C22×D711yescharacteristic
268C2×D4C22×D711yescharacteristic
269C22×C4—21no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—141no—
274C22×C4—141no—
275SD16—41no—
276SD16—281no—
277SD16—281no—
278SD16—281no—
279D8—41no—
280D8—281no—
281D8—281no—
282D8—281no—
283C2×C8—71no—
284C2×C8—71no—
285C2×C8C22×D711yescharacteristic
286C2×C8C22×D711yescharacteristic
287C4⋊C4—71no—
288C4⋊C4—141no—
289C4⋊C4C22×D711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C22⋊C4—141no—
297C22⋊C4—141no—
298C22⋊C4—141no—
299C22⋊C4—141no—
300C22⋊C4—141no—
301C22⋊C4—141no—
302C42—71no—
303C42—71no—
304C42—71no—
305C42C22×D711yescharacteristic
306C2×C7—81no—
307C2×C7—81no—
308C2×C7#53511yescharacteristic
309C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
310C2×C7#57511yescharacteristic
311D7—22no—
312D7—81no—
313D7—81no—
314C23—21no—
315C23—21no—
316C23—71no—
317C23—141no—
318C23—141no—
319C23—282no—
320C23—282no—
321C23—282no—
322Q8—21no—
323Q8—21no—
324Q8—141no—
325Q8—141no—
326Q8—282no—
327Q8—282no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—141no—
333D4—141no—
334D4—141no—
335D4—141no—
336D4—282no—
337D4—282no—
338D4—282no—
339D4—282no—
340D4—282no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—71no—
345C2×C4—71no—
346C2×C4—71no—
347C2×C4—142no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—281no—
353C2×C4—282no—
354C2×C4—282no—
355C2×C4D4×D711yescharacteristic
356C2×C4D4×D711yescharacteristic
357C2×C4C23×D711yescharacteristic
358C8—21no—
359C8—21no—
360C8—141no—
361C8—141no—
362C7#321311yescharacteristic, complemented, Sylow
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—72no—
370C22—141no—
371C22—141no—
372C22—281no—
373C22—281no—
374C22—281no—
375C22—281no—
376C22—281no—
377C22—281no—
378C22—564no—
379C22C2×D4×D711yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—141no—
385C4—141no—
386C4—141no—
387C4—281no—
388C4—281no—
389C4#654811yescharacteristic
390C4#655311yescharacteristic
391C2—81no—
392C2—81no—
393C2—142no—
394C2—561no—
395C2—561no—
396C2#9029811yescharacteristic
397C2#9045911yescharacteristic
398C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#463C2
socleC22×C7C22×D4
hypercenter#463D7
hypocenterC7#3213
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. D7×#124C2semidirect
2. #90239C2semidirect
3. C7×#463C2semidirect
4. #89948C2semidirect
5. #89942C2semidirect
6. #89928C2semidirect
7. #89926C2semidirect
8. #89829C2semidirect
9. #89788C2semidirect
10. #89616C2semidirect
11. #89607C2semidirect
12. #89578C2semidirect
13. #89567C2semidirect
14. #89545C2semidirect
15. #89537C2semidirect
19. C2×C7×SD16C22semidirect
22. C4×C7×D4C22semidirect
24. C2×#1143C22semidirect
25. #6503C22semidirect
28. C2×#1142C22semidirect
33. C2×#1132C22semidirect
35. #6451C22semidirect
43. C4×D28C22semidirect
45. C7×C4⋊C8C22semidirect
46. C7×#103C22semidirect
48. #6409C22semidirect
53. #6390C22semidirect
54. #6389C22semidirect
55. #6389C22semidirect
56. #6384C22semidirect
58. C4⋊(C7⋊C8)C22semidirect
125. #463D7semidirect
185. C2×SD16C2×D7semidirect
204. C4×D4C2×D7semidirect
209. C4⋊C8C2×D7semidirect
213. #103C2×D7semidirect
362. C7#3213semidirect

Maximal subgroups

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

subgroupindex
D7×#1242
#902392
C7×#4632
#899482
#899422
#899282
#899262
#898292
#897882
#896162
#896072
#895782
#895672
#895452
#895372
#32137