Finite Group Databasebeta

← #11737544

Subgroups — #11737544

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#90520C211yesmaximal, characteristic, complemented
2C2×#6514C211yesmaximal, characteristic, complemented
3C2×#6474C211yesmaximal, characteristic, complemented
4#90289C211yesmaximal, characteristic, complemented
5C7×#449C211yesmaximal, characteristic, complemented
6#89945C211yesmaximal, characteristic, complemented
7#89940C211yesmaximal, characteristic, complemented
8#89925C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic, complemented
10#89588C211yesmaximal, characteristic, complemented
11#89570C211yesmaximal, characteristic, complemented
12#89548C211yesmaximal, characteristic, complemented
13#89530C211yesmaximal, characteristic, complemented
14#89510C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×D4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic, complemented
20C2×C7×SD16C2211yescharacteristic, complemented
21C2×C7×D8C2211yescharacteristic
22C7×#121C2211yescharacteristic, complemented
23#6514—24no—
24#6510C2211yescharacteristic
25#6509C2211yescharacteristic
26C2×#1144C2211yescharacteristic, complemented
27C2×#1144C2211yescharacteristic, complemented
28#6502—21no—
29#6502—21no—
30#6500—21no—
31#6499C2211yescharacteristic, complemented
32D4×Dic7—21no—
33D4×Dic7C2211yescharacteristic
34C2×#1142C2211yescharacteristic
35C2×#1142C2211yescharacteristic
36C2×#1141C2211yescharacteristic
37#6493C2211yescharacteristic
38#6492—21no—
39C4×C22⋊D7C2211yescharacteristic
40C2×(C2×D7)⋊C4—21no—
41C2×#1136C2211yescharacteristic, complemented
42#6474—24no—
43C2×#1132C2211yescharacteristic
44C2×#1132C2211yescharacteristic
45C2×#1131C2211yescharacteristic
46#6460C2211yescharacteristic
47#6450C2211yescharacteristic, complemented
48#6448—21no—
49#6441—21no—
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic
52C7×C22⋊C8C2211yescharacteristic, complemented
53#6406C2211yescharacteristic, complemented
54(C2×D7)⋊C8C2211yescharacteristic
55#6390C2211yescharacteristic, complemented
56#6390C2211yescharacteristic, complemented
57Dic7⋊C8C2211yescharacteristic, complemented
58C8×Dic7C2211yescharacteristic, complemented
59#6383C2211yescharacteristic
60#6382C2211yescharacteristic, complemented
61#3275—71nomaximal, Sylow
62C23×D7—21no—
63C7×#41—22no—
64C7×#41—41no—
65C2×C7×Q8C2311yescharacteristic
66C2×C7×D4—21no—
67C2×C7×D4—21no—
68C2×C7×D4C2311yescharacteristic
69C2×C7×D4C2311yescharacteristic
70C22×C4×C7—21no—
71C22×C4×C7C2311yescharacteristic
72C2×C22⋊D7—21no—
73C2×C22⋊D7—21no—
74C2×C22⋊D7—21no—
75C2×C22⋊D7D411yescharacteristic
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79#1159—22no—
80#1159—41no—
81D4×D7—22no—
82D4×D7—41no—
83C2×D28C2311yescharacteristic
84C2×C4×D7C2311yescharacteristic
85C2×Dic14—22no—
86C2×Dic14—21no—
87C2×Dic14—41no—
88C2×Dic14C2311yescharacteristic
89C2×Dic14C2311yescharacteristic
90C7×SD16—41no—
91C7×D8—22no—
92C2×C7×C8C2311yescharacteristic
93C2×C7×C8C2311yescharacteristic
94C7×C4⋊C4C2311yescharacteristic
95C7×C22⋊C4—21no—
96C22⋊Dic7—21no—
97C22⋊Dic7—21no—
98C22⋊Dic7—21no—
99C22⋊Dic7D411yescharacteristic
100#1144—22no—
101#1144—22no—
102#1142—22no—
103#1142D414yescomplemented
104#1141—22no—
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)⋊C4D411yescharacteristic
111C4⋊Dic7—21no—
112C4⋊Dic7C2311yescharacteristic
113Dic7⋊C4—21no—
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#452—71no—
134#450—71no—
135#449D711yescharacteristic, complemented
136#448—71no—
137C4×SD16—71no—
138#436—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7C2×D411yescharacteristic
142C22×D7—21no—
143C22×D7—42no—
144C22×D7—42no—
145C22×D7—81no—
146C22×D7C2×D411yescharacteristic
147C7×Q8—21no—
148C7×Q8—21no—
149C7×D4—21no—
150C7×D4—21no—
151C7×D4—21no—
152C7×D4—42no—
153C7×D4—42no—
154C7×D4C2×D412yes—
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—
164C22⋊D7—42no—
165C22⋊D7—81no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—42no—
170C2×Dic7—42no—
171C2×Dic7—81no—
172C2×Dic7C2×D411yescharacteristic
173C2×Dic7C2×D411yescharacteristic
174C2×Dic7C2×D411yescharacteristic
175D28—21no—
176D28—21no—
177C4×D7—21no—
178C4×D7—21no—
179Dic14—21no—
180Dic14—21no—
181Dic14—21no—
182Dic14—42no—
183Dic14C2×D412yes—
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×Q16—71no—
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16C2×D711yescharacteristic, complemented
201C2×D8—71no—
202C2×D8C2×D711yescharacteristic
203C2×M16—71no—
204C2×M16—71no—
205#124—71no—
206#124—71no—
207#124—141no—
208#123—141no—
209C22⋊Q8—71no—
210C22⋊Q8—71no—
211C22⋊Q8—141no—
212#121—71no—
213#121—141no—
214#121—141no—
215#121C2×D711yescharacteristic, complemented
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
230C22⋊C8—71no—
231C22⋊C8C2×D711yescharacteristic, complemented
232C4×C8—71no—
233C22×C7—21no—
234C22×C7—22no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7C22×D411yescharacteristic
243C2×D7—21no—
244C2×D7—21no—
245C2×D7—21no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—84no—
250C4×C7—21no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7C22×D411yescharacteristic
254C4×C7H(5,𝔽2)11yescharacteristic
255Dic7—22no—
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—71no—
268C2×Q8—142no—
269C2×Q8—141no—
270C2×Q8—281no—
271C2×Q8C22×D711yescharacteristic
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—71no—
275C2×D4—71no—
276C2×D4—141no—
277C2×D4—142no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—281no—
281C2×D4C22×D711yescharacteristic
282C2×D4C22×D711yescharacteristic
283C22×C4—21no—
284C22×C4—71no—
285C22×C4—141no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4C22×D711yescharacteristic
289Q16—142no—
290Q16—142no—
291SD16—41no—
292SD16—74no—
293SD16—142no—
294SD16—142no—
295SD16—281no—
296D8—22no—
297D8—142no—
298M16—142no—
299M16—142no—
300C2×C8—71no—
301C2×C8—71no—
302C2×C8C22×D711yescharacteristic
303C2×C8C22×D711yescharacteristic
304C4⋊C4—71no—
305C4⋊C4—71no—
306C4⋊C4—141no—
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—42no—
324C2×C7—41no—
325C2×C7—81no—
326C2×C7D4211yescharacteristic
327C2×C7#57511yescharacteristic
328C2×C7#57711yescharacteristic
329D7—41no—
330D7—81no—
331C23—21no—
332C23—21no—
333C23—71no—
334C23—141no—
335C23—282no—
336C23—282no—
337C23—561no—
338C23D4×D711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—72no—
342Q8—141no—
343Q8—141no—
344Q8—141no—
345Q8—282no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—141no—
352D4—141no—
353D4—281no—
354D4—282no—
355D4—282no—
356D4—561no—
357D4D4×D712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—71no—
364C2×C4—71no—
365C2×C4—71no—
366C2×C4—141no—
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—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—141no—
391C22—141no—
392C22—141no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—564no—
397C22C2×D4×D711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—141no—
402C4—141no—
403C4—141no—
404C4—142no—
405C4—281no—
406C4—281no—
407C4C2×D4×D711yescharacteristic
408C4#654811yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—281no—
413C2—561no—
414C2#9028111yescharacteristic
415C2#9046111yescharacteristic
416C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#449C2
socleC22×C7C22×D4
hypercenter#449D7
hypocenterC7#3275
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #90520C2semidirect
2. C2×#6514C2semidirect
3. C2×#6474C2semidirect
4. #90289C2semidirect
5. C7×#449C2semidirect
6. #89945C2semidirect
7. #89940C2semidirect
8. #89925C2semidirect
9. #89811C2semidirect
10. #89588C2semidirect
11. #89570C2semidirect
12. #89548C2semidirect
13. #89530C2semidirect
14. #89510C2semidirect
15. #89501C2semidirect
19. C22×Dic14C22semidirect
20. C2×C7×SD16C22semidirect
22. C7×#121C22semidirect
26. C2×#1144C22semidirect
27. C2×#1144C22semidirect
31. #6499C22semidirect
41. C2×#1136C22semidirect
47. #6450C22semidirect
50. C7×#103C22semidirect
52. C7×C22⋊C8C22semidirect
53. #6406C22semidirect
55. #6390C22semidirect
56. #6390C22semidirect
57. Dic7⋊C8C22semidirect
58. C8×Dic7C22semidirect
60. #6382C22semidirect
103. #1142D4semidirect
135. #449D7semidirect
200. C2×SD16C2×D7semidirect
215. #121C2×D7semidirect
226. #103C2×D7semidirect
231. C22⋊C8C2×D7semidirect
380. C7#3275semidirect

Maximal subgroups

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

subgroupindex
#905202
C2×#65142
C2×#64742
#902892
C7×#4492
#899452
#899402
#899252
#898112
#895882
#895702
#895482
#895302
#895102
#895012
#32757