Finite Group Databasebeta

← #11738366

Subgroups — #11738366

Order 896.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6482C211yesmaximal, characteristic, complemented
2C2×#6471C211yesmaximal, characteristic, complemented
3#90322C211yesmaximal, characteristic, complemented
4#90321C211yesmaximal, characteristic, complemented
5C7×#470C211yesmaximal, characteristic, complemented
6#89962C211yesmaximal, characteristic, complemented
7#89899C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89818C211yesmaximal, characteristic, complemented
10#89641C211yesmaximal, characteristic, complemented
11#89636C211yesmaximal, characteristic, complemented
12#89591C211yesmaximal, characteristic, complemented
13#89590C211yesmaximal, characteristic, complemented
14#89583C211yesmaximal, characteristic, complemented
15#89582C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1161C2211yescharacteristic, complemented
18C2×#1157C2211yescharacteristic
19C22×D28C2211yescharacteristic
20C2×C7×Q16C2211yescharacteristic, complemented
21C2×C7×M16C2211yescharacteristic
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24#6510C2211yescharacteristic
25#6507C2211yescharacteristic, complemented
26C2×#1144C2211yescharacteristic, complemented
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1143C2211yescharacteristic, complemented
30C4×C22⋊D7C2211yescharacteristic
31#6482—24no—
32#6471—24no—
33C2×D56C2211yescharacteristic
34C2×D56C2211yescharacteristic
35C2×#1132C2211yescharacteristic
36C2×#1132C2211yescharacteristic
37C2×C8×D7C2211yescharacteristic, complemented
38#6460C2211yescharacteristic, complemented
39#6458—21no—
40#6458C2211yescharacteristic
41#6457—21no—
42#6457—21no—
43#6456C2211yescharacteristic
44#6456C2211yescharacteristic, complemented
45D7×C4⋊C4—21no—
46#6447—21no—
47#6446—21no—
48#6445—21no—
49D7×C22⋊C4—21no—
50C7×#106C2211yescharacteristic, 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#6383C2211yescharacteristic
59#6383C2211yescharacteristic
60#6383C2211yescharacteristic, complemented
61#3267—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×C22⋊D7D411yescharacteristic
68#1161—22no—
69#1161—22no—
70#1161—41no—
71#1161—41no—
72#1157—22no—
73#1157—41no—
74C2×D28—22no—
75C2×D28—21no—
76C2×D28—21no—
77C2×D28—21no—
78C2×D28—21no—
79C2×D28—41no—
80C2×D28C2311yescharacteristic
81C2×D28C2311yescharacteristic
82C2×D28C2311yescharacteristic
83C2×C4×D7—21no—
84C2×C4×D7—21no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7C2311yescharacteristic
89C2×Dic14C2311yescharacteristic
90C7×Q16—22no—
91C7×M16—22no—
92C2×C7×C8C2311yescharacteristic
93C2×C7×C8C2311yescharacteristic
94C7×C4⋊C4—21no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4C2311yescharacteristic
97C7×C4⋊C4C2311yescharacteristic
98C7×C22⋊C4—21no—
99C7×C22⋊C4—21no—
100C22⋊Dic7D411yescharacteristic
101#1144—41no—
102#1143—22no—
103#1143—22no—
104#1143—41no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4—21no—
110(C2×D7)⋊C4—21no—
111(C2×D7)⋊C4D411yescharacteristic
112C4⋊Dic7—21no—
113Dic7⋊C4—21no—
114Dic7⋊C4—21no—
115Dic7⋊C4D411yescharacteristic
116C4×Dic7C2311yescharacteristic
117C2×C7⋊C8C2311yescharacteristic
118C2×C7⋊C8C2311yescharacteristic
119D56—22no—
120D56D414yescomplemented
121#1132—22no—
122#1132—22no—
123C8×D7—22no—
124C2×AGL(1,ℤ/8)—71no—
125C2×#135—71no—
126#547—71no—
127#546—71no—
128#497—71no—
129#470D711yescharacteristic, complemented
130#465—71no—
131#463—71no—
132#461—71no—
133#452—71no—
134#452—71no—
135#450—71no—
136#449—71no—
137#442—71no—
138#435—71no—
139C23×C7C2×D411yescharacteristic
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—42no—
144C22×D7—42no—
145C22×D7—81no—
146C22×D7C2×D411yescharacteristic
147C7×Q8—21no—
148C7×Q8—21no—
149C7×Q8—21no—
150C7×Q8—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7C2×D411yescharacteristic
158C2×C4×C7C2411yescharacteristic
159C22⋊D7—41no—
160C22⋊D7—42no—
161C2×Dic7—21no—
162C2×Dic7C2×D411yescharacteristic
163C2×Dic7C2×D411yescharacteristic
164C2×Dic7C2×D411yescharacteristic
165D28—21no—
166D28—21no—
167D28—21no—
168D28—21no—
169D28—42no—
170D28—42no—
171D28—42no—
172D28—81no—
173D28C2×D412yes—
174D28C2×D412yes—
175C4×D7—21no—
176C4×D7—21no—
177C4×D7—42no—
178C4×D7—42no—
179C4×D7—42no—
180C4×D7—42no—
181C4×D7—81no—
182Dic14—21no—
183Dic14—21no—
184C7×C8—21no—
185C7×C8C2×D412yes—
186C7⋊C8—21no—
187C7⋊C8—21no—
188C2×#41—71no—
189C2×#41—71no—
190C2×#41—71no—
191C22×D4—71no—
192AGL(1,ℤ/8)—144no—
193#135—144no—
194C2×Q16—71no—
195C2×Q16C2×D711yescharacteristic, complemented
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×D8—71no—
202C2×D8—71no—
203C2×M16C2×D711yescharacteristic
204C22×C8—71no—
205C4⋊Q8—71no—
206#124—71no—
207#123—141no—
208#123—141no—
209#123—141no—
210C22⋊Q8—71no—
211C22⋊Q8C2×D711yescharacteristic
212C22⋊Q8C2×D711yescharacteristic
213#121—71no—
214#121—141no—
215#121—141no—
216#120—141no—
217C4×D4—71no—
218C4×D4—71no—
219C4×D4—71no—
220C2×C4⋊C4—141no—
221C2×C22⋊C4—141no—
222#106C2×D711yescharacteristic, complemented
223C4⋊C8—71no—
224#103—71no—
225#103C2×D711yescharacteristic, complemented
226#103C2×D711yescharacteristic, complemented
227#102—71no—
228#102—71no—
229#102—71no—
230C22⋊C8—71no—
231C22⋊C8—71no—
232#97—71no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7C22×D411yescharacteristic
237C2×D7—21no—
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—21no—
241C2×D7—21no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—84no—
250C4×C7—21no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7—41no—
255C4×C7C22×D411yescharacteristic
256C4×C7H(5,𝔽2)11yescharacteristic
257Dic7—21no—
258Dic7—21no—
259Dic7—21no—
260Dic7—41no—
261C24—141no—
262#41—142no—
263#41—142no—
264#41—142no—
265#41—281no—
266#41—281no—
267#41—281no—
268C2×Q8—71no—
269C2×Q8C22×D711yescharacteristic
270C2×Q8C22×D711yescharacteristic
271C2×D4—71no—
272C2×D4—71no—
273C2×D4—71no—
274C2×D4—71no—
275C2×D4—142no—
276C2×D4—141no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—281no—
282C22×C4—71no—
283C22×C4—141no—
284C22×C4—141no—
285C22×C4—141no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4C22×D711yescharacteristic
289Q16—22no—
290Q16—281no—
291SD16—142no—
292SD16—142no—
293SD16—142no—
294SD16—142no—
295SD16—281no—
296D8—74no—
297D8—142no—
298M16—22no—
299C2×C8—71no—
300C2×C8—71no—
301C2×C8—142no—
302C2×C8C22×D711yescharacteristic
303C2×C8C22×D711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—71no—
307C4⋊C4—141no—
308C4⋊C4—141no—
309C4⋊C4—141no—
310C4⋊C4C22×D711yescharacteristic
311C4⋊C4C22×D711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—71no—
315C22⋊C4—71no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C22⋊C4—141no—
319C22⋊C4—141no—
320C22⋊C4—141no—
321C22⋊C4—141no—
322C42—71no—
323C2×C7—41no—
324C2×C7D4211yescharacteristic
325C2×C7#57511yescharacteristic
326C2×C7#57711yescharacteristic
327D7—41no—
328D7—42no—
329D7—42no—
330D7—81no—
331C23—71no—
332C23—141no—
333C23—141no—
334C23—141no—
335C23—282no—
336C23—282no—
337C23—561no—
338C23D4×D711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—141no—
344Q8—141no—
345D4—72no—
346D4—72no—
347D4—141no—
348D4—141no—
349D4—141no—
350D4—141no—
351D4—282no—
352D4—282no—
353D4—282no—
354D4—281no—
355D4—282no—
356D4—561no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—71no—
364C2×C4—71no—
365C2×C4—71no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—282no—
370C2×C4—282no—
371C2×C4—282no—
372C2×C4—282no—
373C2×C4—561no—
374C2×C4D4×D711yescharacteristic
375C2×C4C23×D711yescharacteristic
376C8—21no—
377C8—141no—
378C8—141no—
379C8D4×D712yes—
380C7#326711yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—141no—
385C22—141no—
386C22—141no—
387C22—142no—
388C22—142no—
389C22—281no—
390C22—281no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—564no—
397C22C2×D4×D711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—141no—
404C4—141no—
405C4—141no—
406C4—281no—
407C4C2×D4×D711yescharacteristic
408C4#655311yescharacteristic
409C2—41no—
410C2—281no—
411C2—282no—
412C2—282no—
413C2—561no—
414C2#9030011yescharacteristic
415C2#9046811yescharacteristic
416C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#470C2
socleC22×C7C22×D4
hypercenter#470D7
hypocenterC7#3267
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#6482C2semidirect
2. C2×#6471C2semidirect
3. #90322C2semidirect
4. #90321C2semidirect
5. C7×#470C2semidirect
6. #89962C2semidirect
7. #89899C2semidirect
8. #89819C2semidirect
9. #89818C2semidirect
10. #89641C2semidirect
11. #89636C2semidirect
12. #89591C2semidirect
13. #89590C2semidirect
14. #89583C2semidirect
15. #89582C2semidirect
17. C2×#1161C22semidirect
20. C2×C7×Q16C22semidirect
25. #6507C22semidirect
26. C2×#1144C22semidirect
29. C2×#1143C22semidirect
37. C2×C8×D7C22semidirect
38. #6460C22semidirect
44. #6456C22semidirect
50. C7×#106C22semidirect
51. C7×#103C22semidirect
52. C7×#103C22semidirect
54. (C2×D7)⋊C8C22semidirect
55. #6389C22semidirect
56. Dic7⋊C8C22semidirect
57. #6384C22semidirect
60. #6383C22semidirect
120. D56D4semidirect
129. #470D7semidirect
195. C2×Q16C2×D7semidirect
222. #106C2×D7semidirect
225. #103C2×D7semidirect
226. #103C2×D7semidirect
380. C7#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#64822
C2×#64712
#903222
#903212
C7×#4702
#899622
#898992
#898192
#898182
#896412
#896362
#895912
#895902
#895832
#895822
#32677