Finite Group Databasebeta

← #11738216

Subgroups — #11738216

Order 896.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6481C211yesmaximal, characteristic, complemented
2C22×#1132C211yesmaximal, characteristic, complemented
3#90323C211yesmaximal, characteristic, complemented
4#90321C211yesmaximal, characteristic
5C7×#467C211yesmaximal, characteristic, complemented
6#89962C211yesmaximal, characteristic, complemented
7#89883C211yesmaximal, characteristic, complemented
8#89822C211yesmaximal, characteristic, complemented
9#89818C211yesmaximal, characteristic, complemented
10#89661C211yesmaximal, characteristic, complemented
11#89658C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic, complemented
13#89582C211yesmaximal, characteristic, complemented
14#89578C211yesmaximal, characteristic, complemented
15#89570C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic, complemented
18C22×D28C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×Q16C2211yescharacteristic, complemented
21C22×C7×C8C2211yescharacteristic
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24#6510C2211yescharacteristic
25#6507C2211yescharacteristic, complemented
26C2×#1144C2211yescharacteristic, complemented
27C2×#1144C2211yescharacteristic, complemented
28C2×#1143C2211yescharacteristic
29C2×#1143C2211yescharacteristic
30C4×C22⋊D7C2211yescharacteristic
31#6481—24no—
32C2×#1132—24no—
33C2×#1132C2211yescharacteristic
34C2×#1132C2211yescharacteristic
35C2×#1132C2211yescharacteristic
36C2×#1132C2211yescharacteristic
37C2×#1131C2211yescharacteristic, complemented
38#6460C2211yescharacteristic, complemented
39#6459—21no—
40#6458—21no—
41#6458C2211yescharacteristic
42#6457—21no—
43#6456C2211yescharacteristic
44#6451—21no—
45#6450C2211yescharacteristic, complemented
46#6445—21no—
47(C22⋊D7)⋊C4—21no—
48D7×C22⋊C4—21no—
49#6441—21no—
50C7×#107C2211yescharacteristic, complemented
51C7×#103C2211yescharacteristic, complemented
52C7×#103C2211yescharacteristic, complemented
53C22⋊(C7⋊C8)C2211yescharacteristic
54(C2×D7)⋊C8C2211yescharacteristic, complemented
55#6389C2211yescharacteristic, complemented
56Dic7⋊C8C2211yescharacteristic, complemented
57#6384C2211yescharacteristic, complemented
58#6384C2211yescharacteristic, complemented
59#6383C2211yescharacteristic
60#6383C2211yescharacteristic
61#3266—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68#1161—22no—
69#1161—41no—
70Q8×D7—22no—
71Q8×D7—41no—
72C2×D28—22no—
73C2×D28—21no—
74C2×D28—21no—
75C2×D28—21no—
76C2×D28—41no—
77C2×D28C2311yescharacteristic
78C2×D28C2311yescharacteristic
79C2×C4×D7—21no—
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×C4×D7C2311yescharacteristic
83C2×Dic14—22no—
84C2×Dic14—21no—
85C2×Dic14—21no—
86C2×Dic14—41no—
87C2×Dic14C2311yescharacteristic
88C2×Dic14C2311yescharacteristic
89C7×Q16—22no—
90C2×C7×C8—22no—
91C2×C7×C8C2311yescharacteristic
92C2×C7×C8C2311yescharacteristic
93C7×C4⋊C4—21no—
94C7×C4⋊C4—21no—
95C7×C4⋊C4C2311yescharacteristic
96C7×C4⋊C4C2311yescharacteristic
97C7×C22⋊C4—21no—
98C7×C22⋊C4—21no—
99C22⋊Dic7D411yescharacteristic
100#1144—22no—
101#1144—41no—
102#1143—22no—
103#1143—41no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4D411yescharacteristic
110C4⋊Dic7—21no—
111Dic7⋊C4—21no—
112Dic7⋊C4—21no—
113Dic7⋊C4—21no—
114Dic7⋊C4D411yescharacteristic
115C4×Dic7—21no—
116C4×Dic7C2311yescharacteristic
117C2×C7⋊C8C2311yescharacteristic
118C2×C7⋊C8C2311yescharacteristic
119#1132—22no—
120#1132—22no—
121#1132—22no—
122#1132D414yescomplemented
123#1131—22no—
124C2×#137—71no—
125C22×SD16—71no—
126D4×Q8—71no—
127#546—71no—
128#497—71no—
129#468—71no—
130#467D711yescharacteristic, complemented
131#464—71no—
132#461—71no—
133#452—71no—
134#451—71no—
135#450—71no—
136#448—71no—
137#440—71no—
138#435—71no—
139C23×C7C2×D411yescharacteristic
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—42no—
143C22×D7—42no—
144C22×D7—81no—
145C22×D7C2×D411yescharacteristic
146C7×Q8—21no—
147C7×Q8—21no—
148C7×Q8—21no—
149C7×Q8—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7C2×D411yescharacteristic
157C2×C4×C7C2411yescharacteristic
158C22⋊D7—22no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—42no—
162C2×Dic7C2×D411yescharacteristic
163C2×Dic7C2×D411yescharacteristic
164C2×Dic7C2×D411yescharacteristic
165D28—21no—
166D28—21no—
167D28—21no—
168D28—42no—
169D28—42no—
170D28—81no—
171D28C2×D412yes—
172C4×D7—21no—
173C4×D7—21no—
174C4×D7—42no—
175C4×D7—42no—
176C4×D7—81no—
177Dic14—21no—
178Dic14—21no—
179Dic14—21no—
180Dic14—42no—
181Dic14—42no—
182Dic14C2×D412yes—
183C7×C8—21no—
184C7×C8C2×D412yes—
185C7⋊C8—21no—
186C7⋊C8—21no—
187C2×#41—71no—
188C22×Q8—71no—
189C22×Q8—71no—
190C22×D4—71no—
191#137—144no—
192C2×Q16—71no—
193C2×Q16—71no—
194C2×Q16C2×D711yescharacteristic, complemented
195C2×SD16—71no—
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×SD16—144no—
202C2×M16—71no—
203C22×C8C2×D711yescharacteristic
204C4⋊Q8—71no—
205C4⋊Q8—141no—
206#124—71no—
207#123—141no—
208C22⋊Q8—71no—
209C22⋊Q8—141no—
210C22⋊Q8—141no—
211C22⋊Q8C2×D711yescharacteristic
212C22⋊Q8C2×D711yescharacteristic
213#121—71no—
214#121—141no—
215#120—141no—
216C4×Q8—71no—
217C4×D4—71no—
218C4×D4—71no—
219C4×D4—141no—
220C2×C22⋊C4—141no—
221#107C2×D711yescharacteristic, complemented
222C4⋊C8—71no—
223#103—71no—
224#103—71no—
225#103C2×D711yescharacteristic, complemented
226#103C2×D711yescharacteristic, complemented
227#102—71no—
228#102—71no—
229C22⋊C8—71no—
230C22⋊C8—71no—
231#97—71no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7#13712yescomplemented
235C22×C7C22×D411yescharacteristic
236C2×D7—21no—
237C2×D7—22no—
238C2×D7—21no—
239C2×D7—21no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—84no—
246C4×C7—21no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7—41no—
251C4×C7C22×D411yescharacteristic
252C4×C7H(5,𝔽2)11yescharacteristic
253Dic7—22no—
254Dic7—21no—
255Dic7—21no—
256Dic7—21no—
257Dic7—41no—
258C24—141no—
259#41—142no—
260#41—281no—
261C2×Q8—71no—
262C2×Q8—71no—
263C2×Q8—141no—
264C2×Q8—142no—
265C2×Q8—141no—
266C2×Q8—142no—
267C2×Q8—281no—
268C2×Q8—281no—
269C2×Q8C22×D711yescharacteristic
270C2×Q8C22×D711yescharacteristic
271C2×D4—71no—
272C2×D4—71no—
273C2×D4—71no—
274C2×D4—142no—
275C2×D4—141no—
276C2×D4—141no—
277C2×D4—141no—
278C2×D4—281no—
279C22×C4—71no—
280C22×C4—141no—
281C22×C4—141no—
282C22×C4—141no—
283C22×C4—141no—
284C22×C4C22×D711yescharacteristic
285Q16—22no—
286Q16—142no—
287Q16—281no—
288SD16—74no—
289SD16—142no—
290SD16—142no—
291SD16—142no—
292SD16—142no—
293SD16—281no—
294M16—142no—
295C2×C8—22no—
296C2×C8—71no—
297C2×C8—71no—
298C2×C8C22×D711yescharacteristic
299C2×C8C22×D711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—71no—
303C4⋊C4—141no—
304C4⋊C4—141no—
305C4⋊C4—141no—
306C4⋊C4—141no—
307C4⋊C4C22×D711yescharacteristic
308C4⋊C4C22×D711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—71no—
312C22⋊C4—71no—
313C22⋊C4—141no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C42—71no—
319C42—141no—
320C2×C7—22no—
321C2×C7D4211yescharacteristic
322C2×C7C2×#13711yescharacteristic
323C2×C7#57711yescharacteristic
324D7—41no—
325D7—42no—
326D7—81no—
327C23—71no—
328C23—141no—
329C23—141no—
330C23—282no—
331C23—282no—
332C23—561no—
333C23D4×D711yescharacteristic
334Q8—21no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—72no—
339Q8—141no—
340Q8—141no—
341Q8—141no—
342Q8—282no—
343Q8—282no—
344D4—72no—
345D4—142no—
346D4—141no—
347D4—141no—
348D4—141no—
349D4—282no—
350D4—282no—
351D4—561no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—71no—
359C2×C4—71no—
360C2×C4—71no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—282no—
366C2×C4—282no—
367C2×C4—282no—
368C2×C4—561no—
369C2×C4D4×D711yescharacteristic
370C2×C4C23×D711yescharacteristic
371C8—21no—
372C8—141no—
373C8—141no—
374C8D4×D712yes—
375C7#326611yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—141no—
379C22—141no—
380C22—141no—
381C22—142no—
382C22—281no—
383C22—281no—
384C22—281no—
385C22—281no—
386C22—281no—
387C22—564no—
388C22#648112yescomplemented
389C22C2×D4×D711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—141no—
396C4—141no—
397C4—141no—
398C4—142no—
399C4—281no—
400C4C2×D4×D711yescharacteristic
401C4#655311yescharacteristic
402C2—22no—
403C2—281no—
404C2—282no—
405C2—561no—
406C2#9030011yescharacteristic
407C2C2×#648111yescharacteristic
408C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#467C2
socleC22×C7C22×D4
prime radical#90321C2
hypercenter#467D7
hypocenterC7#3266
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#6481C2semidirect
2. C22×#1132C2semidirect
3. #90323C2semidirect
5. C7×#467C2semidirect
6. #89962C2semidirect
7. #89883C2semidirect
8. #89822C2semidirect
9. #89818C2semidirect
10. #89661C2semidirect
11. #89658C2semidirect
12. #89590C2semidirect
13. #89582C2semidirect
14. #89578C2semidirect
15. #89570C2semidirect
17. C2×Q8×D7C22semidirect
20. C2×C7×Q16C22semidirect
25. #6507C22semidirect
26. C2×#1144C22semidirect
27. C2×#1144C22semidirect
37. C2×#1131C22semidirect
38. #6460C22semidirect
45. #6450C22semidirect
50. C7×#107C22semidirect
51. C7×#103C22semidirect
52. C7×#103C22semidirect
54. (C2×D7)⋊C8C22semidirect
55. #6389C22semidirect
56. Dic7⋊C8C22semidirect
57. #6384C22semidirect
58. #6384C22semidirect
122. #1132D4semidirect
130. #467D7semidirect
194. C2×Q16C2×D7semidirect
221. #107C2×D7semidirect
225. #103C2×D7semidirect
226. #103C2×D7semidirect
234. C22×C7#137semidirect
375. C7#3266semidirect
388. C22#6481semidirect

Maximal subgroups

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

subgroupindex
C2×#64812
C22×#11322
#903232
#903212
C7×#4672
#899622
#898832
#898222
#898182
#896612
#896582
#895902
#895822
#895782
#895702
#32667