Finite Group Databasebeta

← #11737713

Subgroups — #11737713

Order 896.

498 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
1#90522C211yesmaximal, characteristic, complemented
2C2×#6512C211yesmaximal, characteristic, complemented
3C2×#6481C211yesmaximal, characteristic, complemented
4#90321C211yesmaximal, characteristic, complemented
5C7×#452C211yesmaximal, characteristic, complemented
6#89957C211yesmaximal, characteristic, complemented
7#89944C211yesmaximal, characteristic, complemented
8#89940C211yesmaximal, characteristic, complemented
9#89824C211yesmaximal, characteristic, complemented
10#89578C211yesmaximal, characteristic, complemented
11#89560C211yesmaximal, characteristic, complemented
12#89559C211yesmaximal, characteristic, complemented
13#89554C211yesmaximal, characteristic, complemented
14#89503C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic, complemented
19C22×D28C2211yescharacteristic
20C2×C7×Q16C2211yescharacteristic, complemented
21C2×C7×SD16C2211yescharacteristic, complemented
22C7×C22⋊Q8C2211yescharacteristic
23#6512—24no—
24#6510C2211yescharacteristic
25Q8×Dic7C2211yescharacteristic, complemented
26C2×#1144C2211yescharacteristic, complemented
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29#6501—21no—
30#6501C2211yescharacteristic, complemented
31#6499—21no—
32#6499C2211yescharacteristic, complemented
33C2×#1141C2211yescharacteristic
34C2×#1141C2211yescharacteristic
35C4×C22⋊D7—21no—
36C4×C22⋊D7C2211yescharacteristic
37#6487—21no—
38#6487C2211yescharacteristic
39C2×#1136C2211yescharacteristic
40#6481—24no—
41C2×#1132C2211yescharacteristic
42C2×#1132C2211yescharacteristic
43C2×#1131C2211yescharacteristic, complemented
44#6458—21no—
45#6458C2211yescharacteristic
46#6457—21no—
47#6456C2211yescharacteristic
48#6445—21no—
49D7×C22⋊C4—21no—
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic, complemented
52C7×C22⋊C8C2211yescharacteristic
53#6409C2211yescharacteristic, complemented
54#6395C2211yescharacteristic
55#6395C2211yescharacteristic
56(C2×D7)⋊C8C2211yescharacteristic, complemented
57Dic7⋊C8C2211yescharacteristic, complemented
58C8×Dic7C2211yescharacteristic, complemented
59#6384C2211yescharacteristic, complemented
60#6382C2211yescharacteristic, complemented
61#3275—71nomaximal, Sylow
62C23×D7—21no—
63C7×#41—22no—
64C7×#41—41no—
65C2×C7×Q8C2311yescharacteristic
66C2×C7×Q8C2311yescharacteristic
67C2×C7×D4—21no—
68C2×C7×D4C2311yescharacteristic
69C22×C4×C7—21no—
70C22×C4×C7C2311yescharacteristic
71C2×C22⋊D7—21no—
72C2×C22⋊D7D411yescharacteristic
73#1161—22no—
74#1161—41no—
75Q8×D7—22no—
76Q8×D7—41no—
77C2×D28—22no—
78C2×D28—21no—
79C2×D28—21no—
80C2×D28—21no—
81C2×D28—41no—
82C2×D28C2311yescharacteristic
83C2×D28C2311yescharacteristic
84C2×C4×D7—21no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7C2311yescharacteristic
88C2×Dic14—21no—
89C2×Dic14C2311yescharacteristic
90C7×Q16—22no—
91C7×SD16—41no—
92C2×C7×C8C2311yescharacteristic
93C2×C7×C8C2311yescharacteristic
94C7×C4⋊C4—21no—
95C7×C4⋊C4C2311yescharacteristic
96C7×C22⋊C4—21no—
97C22⋊Dic7—21no—
98C22⋊Dic7—21no—
99C22⋊Dic7—21no—
100C22⋊Dic7—21no—
101C22⋊Dic7D411yescharacteristic
102#1144—22no—
103#1143—22no—
104#1143D414yescomplemented
105#1141—22no—
106#1141—22no—
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—
113C4⋊Dic7C2311yescharacteristic
114Dic7⋊C4—21no—
115Dic7⋊C4D411yescharacteristic
116C4×Dic7—21no—
117C4×Dic7C2311yescharacteristic
118#1136—22no—
119C2×C7⋊C8C2311yescharacteristic
120C2×C7⋊C8C2311yescharacteristic
121#1132—22no—
122#1132—41no—
123#1131—22no—
124C2×#137—71no—
125C2×AGL(1,ℤ/8)—71no—
126#548—71no—
127#546—71no—
128#495—71no—
129#470—71no—
130#469—71no—
131#464—71no—
132#463—71no—
133#452D711yescharacteristic, complemented
134#450—71no—
135#449—71no—
136#448—71no—
137C4×SD16—71no—
138#436—71no—
139C23×C7—21no—
140C23×C7C2×D411yescharacteristic
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×Q8C2×D412yes—
151C7×D4—21no—
152C7×D4—21no—
153C7×D4—42no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—42no—
160C2×C4×C7C2×D411yescharacteristic
161C2×C4×C7C2411yescharacteristic
162C22⋊D7—41no—
163C22⋊D7—42no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7C2×D411yescharacteristic
167C2×Dic7C2×D411yescharacteristic
168C2×Dic7C2×D411yescharacteristic
169D28—21no—
170D28—21no—
171D28—21no—
172D28—42no—
173D28—42no—
174D28—81no—
175D28C2×D412yes—
176C4×D7—21no—
177C4×D7—21no—
178C4×D7—42no—
179C4×D7—42no—
180C4×D7—81no—
181Dic14—21no—
182Dic14—21no—
183Dic14—42no—
184C7×C8—21no—
185C7×C8—21no—
186C7⋊C8—21no—
187C7⋊C8C2×D412yes—
188C2×#41—71no—
189C2×#41C2×D711yescharacteristic
190C22×Q8—71no—
191C22×D4—71no—
192#137—144no—
193AGL(1,ℤ/8)—144no—
194C2×Q16—71no—
195C2×Q16C2×D711yescharacteristic, complemented
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16C2×D711yescharacteristic, complemented
201C2×D8—71no—
202C2×D8—71no—
203C2×M16—71no—
204C2×M16—71no—
205#124—71no—
206#124—71no—
207#124—141no—
208#123—141no—
209C22⋊Q8—71no—
210C22⋊Q8—141no—
211C22⋊Q8C2×D711yescharacteristic
212#121—71no—
213#121—71no—
214#121—141no—
215#121—141no—
216#120—141no—
217C4×Q8—71no—
218C4×D4—71no—
219C4×D4—71no—
220C4×D4—141no—
221C2×C22⋊C4—141no—
222#106—71no—
223C4⋊C8—71no—
224#103—71no—
225#103—71no—
226#103C2×D711yescharacteristic, complemented
227#102—71no—
228#102—71no—
229#102C2×D711yescharacteristic, complemented
230C22⋊C8—71no—
231C22⋊C8C2×D711yescharacteristic
232C4×C8—71no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7C22×D411yescharacteristic
240C2×D7—21no—
241C2×D7—22no—
242C2×D7—21no—
243C2×D7—21no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—84no—
250C4×C7—22no—
251C4×C7—21no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7C22×D411yescharacteristic
255C4×C7H(5,𝔽2)11yescharacteristic
256Dic7—21no—
257Dic7—21no—
258Dic7—21no—
259Dic7—41no—
260Dic7—41no—
261C24—141no—
262#41—22no—
263#41—41no—
264#41—142no—
265#41—281no—
266C2×Q8—71no—
267C2×Q8—141no—
268C2×Q8—142no—
269C2×Q8—281no—
270C2×Q8C22×D711yescharacteristic
271C2×Q8C22×D711yescharacteristic
272C2×D4—21no—
273C2×D4—71no—
274C2×D4—71no—
275C2×D4—71no—
276C2×D4—142no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—281no—
282C2×D4C22×D711yescharacteristic
283C22×C4—21no—
284C22×C4—71no—
285C22×C4—141no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4C22×D711yescharacteristic
289Q16—22no—
290Q16—142no—
291SD16—41no—
292SD16—74no—
293SD16—142no—
294SD16—142no—
295SD16—281no—
296D8—142no—
297D8—142no—
298M16—142no—
299M16—142no—
300C2×C8—71no—
301C2×C8—71no—
302C2×C8C22×D711yescharacteristic
303C2×C8C22×D711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—71no—
306C4⋊C4—71no—
307C4⋊C4—141no—
308C4⋊C4—141no—
309C4⋊C4C22×D711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—71no—
312C22⋊C4—71no—
313C22⋊C4—141no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C22⋊C4—141no—
319C22⋊C4—141no—
320C22⋊C4—141no—
321C42—71no—
322C42—141no—
323C2×C7—41no—
324C2×C7—81no—
325C2×C7D4211yescharacteristic
326C2×C7#57511yescharacteristic
327C2×C7#57711yescharacteristic
328D7—41no—
329D7—42no—
330D7—81no—
331C23—21no—
332C23—71no—
333C23—141no—
334C23—141no—
335C23—282no—
336C23—282no—
337C23—561no—
338C23D4×D711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—141no—
343Q8—141no—
344Q8—282no—
345Q8D4×D712yes—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—72no—
350D4—141no—
351D4—141no—
352D4—141no—
353D4—282no—
354D4—282no—
355D4—281no—
356D4—282no—
357D4—561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—71no—
365C2×C4—71no—
366C2×C4—71no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—282no—
372C2×C4—282no—
373C2×C4—561no—
374C2×C4D4×D711yescharacteristic
375C2×C4C23×D711yescharacteristic
376C8—21no—
377C8—21no—
378C8—72no—
379C8—141no—
380C7#327511yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—141no—
388C22—141no—
389C22—141no—
390C22—142no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—564no—
397C22C2×D4×D711yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—141no—
403C4—141no—
404C4—141no—
405C4—281no—
406C4—281no—
407C4C2×D4×D711yescharacteristic
408C4#654811yescharacteristic
409C2—41no—
410C2—81no—
411C2—281no—
412C2—282no—
413C2—561no—
414C2#9028111yescharacteristic
415C2#9046111yescharacteristic
416C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
hypercenter#452D7
hypocenterC7#3275
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #90522C2semidirect
2. C2×#6512C2semidirect
3. C2×#6481C2semidirect
4. #90321C2semidirect
5. C7×#452C2semidirect
6. #89957C2semidirect
7. #89944C2semidirect
8. #89940C2semidirect
9. #89824C2semidirect
10. #89578C2semidirect
11. #89560C2semidirect
12. #89559C2semidirect
13. #89554C2semidirect
14. #89503C2semidirect
15. #89501C2semidirect
18. C2×Q8×D7C22semidirect
20. C2×C7×Q16C22semidirect
21. C2×C7×SD16C22semidirect
25. Q8×Dic7C22semidirect
26. C2×#1144C22semidirect
30. #6501C22semidirect
32. #6499C22semidirect
43. C2×#1131C22semidirect
50. C7×#103C22semidirect
51. C7×#102C22semidirect
53. #6409C22semidirect
56. (C2×D7)⋊C8C22semidirect
57. Dic7⋊C8C22semidirect
58. C8×Dic7C22semidirect
59. #6384C22semidirect
60. #6382C22semidirect
104. #1143D4semidirect
133. #452D7semidirect
195. C2×Q16C2×D7semidirect
200. C2×SD16C2×D7semidirect
226. #103C2×D7semidirect
229. #102C2×D7semidirect
380. C7#3275semidirect

Maximal subgroups

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

subgroupindex
#905222
C2×#65122
C2×#64812
#903212
C7×#4522
#899572
#899442
#899402
#898242
#895782
#895602
#895592
#895542
#895032
#895012
#32757