Finite Group Databasebeta

← #11740949

Subgroups — #11740949

Order 896.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#546C211yesmaximal, characteristic, complemented
2C2×#6512C211yesmaximal, characteristic, complemented
3C2×#6495C211yesmaximal, characteristic, complemented
4#90241C211yesmaximal, characteristic, complemented
5#90011C211yesmaximal, characteristic, complemented
6#89989C211yesmaximal, characteristic, complemented
7#89831C211yesmaximal, characteristic, complemented
8#89829C211yesmaximal, characteristic, complemented
9#89821C211yesmaximal, characteristic, complemented
10#89819C211yesmaximal, characteristic, complemented
11#89814C211yesmaximal, characteristic, complemented
12#89808C211yesmaximal, characteristic, complemented
13#89788C211yesmaximal, characteristic, complemented
14C4×#1141C211yesmaximal, characteristic, complemented
15#89784C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic, complemented
18C2×#1157C2211yescharacteristic
19C22×D28C2211yescharacteristic, complemented
20C7×#124C2211yescharacteristic, complemented
21C7×#123—21no—
22C7×C22⋊Q8C2211yescharacteristic, complemented
23C7×#121—21no—
24C7×#121C2211yescharacteristic
25C7×#120—21no—
26C4×C7×D4C2211yescharacteristic
27C4×C7×D4C2211yescharacteristic, complemented
28C2×C7×C22⋊C4—21no—
29#6512—24no—
30C2×#1144C2211yescharacteristic, complemented
31C2×#1143C2211yescharacteristic, complemented
32C2×#1143C2211yescharacteristic, complemented
33#6500—21no—
34C2×#1142C2211yescharacteristic, complemented
35C2×#1142C2211yescharacteristic, complemented
36#6495—24no—
37C2×#1141C2211yescharacteristic
38C2×#1141C2211yescharacteristic
39C2×#1141C2211yescharacteristic
40C2×#1141C2211yescharacteristic
41#6493C2211yescharacteristic
42#6487C2211yescharacteristic, complemented
43C2×#1136C2211yescharacteristic, complemented
44C2×#1136C2211yescharacteristic, complemented
45#6457—21no—
46#6447—21no—
47D7×C22⋊C4—21no—
48#6438C2211yescharacteristic, complemented
49C4×D28C2211yescharacteristic, complemented
50#6409C2211yescharacteristic, complemented
51#6406C2211yescharacteristic, complemented
52#6406C2211yescharacteristic, complemented
53C22⋊(C7⋊C8)C2211yescharacteristic, complemented
54C22⋊(C7⋊C8)C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic, complemented
57#6383C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59C4⋊(C7⋊C8)C2211yescharacteristic, complemented
60C4×C7⋊C8C2211yescharacteristic, complemented
61#3273—71nomaximal, Sylow
62C24×C7—21no—
63C23×D7—21no—
64C7×#41—22no—
65C7×#41—41no—
66C2×C7×Q8C2311yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—21no—
69C2×C7×D4—21no—
70C2×C7×D4—21no—
71C2×C7×D4—41no—
72C2×C7×D4D411yescharacteristic
73C2×C7×D4C2311yescharacteristic
74C2×C7×D4C2311yescharacteristic
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7C2311yescharacteristic
78C22×C4×C7C2311yescharacteristic
79C2×C22⋊D7—21no—
80C2×C22⋊D7—21no—
81#1157—22no—
82#1157—41no—
83C2×D28—22no—
84C2×D28—21no—
85C2×D28—41no—
86C2×D28C2311yescharacteristic
87C2×D28C2311yescharacteristic
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×Dic14C2311yescharacteristic
91C7×C4⋊C4—21no—
92C7×C4⋊C4D411yescharacteristic
93C7×C4⋊C4C2311yescharacteristic
94C7×C22⋊C4—21no—
95C7×C22⋊C4—21no—
96C7×C22⋊C4—21no—
97C7×C22⋊C4—21no—
98C7×C22⋊C4—21no—
99C7×C22⋊C4D411yescharacteristic
100C7×C22⋊C4D411yescharacteristic
101C42×C7C2311yescharacteristic
102C22⋊Dic7—21no—
103#1144—41no—
104#1143—22no—
105#1143—22no—
106#1142—22no—
107#1142—22no—
108#1141—22no—
109#1141—22no—
110#1141—41no—
111#1141D414yescomplemented
112(C2×D7)⋊C4—21no—
113(C2×D7)⋊C4—21no—
114(C2×D7)⋊C4—21no—
115(C2×D7)⋊C4—21no—
116C4⋊Dic7C2311yescharacteristic
117Dic7⋊C4—21no—
118#1136—22no—
119#1136—22no—
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122C2×C7⋊C8C2311yescharacteristic
123C2×C7⋊C8C2311yescharacteristic
124C2×AGL(1,ℤ/8)—71no—
125C2×AGL(1,ℤ/8)—71no—
126#546—71no—
127#546D711yescharacteristic, complemented
128#495—71no—
129#468—71no—
130#468—71no—
131#463—71no—
132#463—71no—
133#452—71no—
134#452—71no—
135#447—71no—
136#447—71no—
137C4×D8—71no—
138#436—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—81no—
144C23×C7C2×D411yescharacteristic
145C23×C7C2×D411yescharacteristic
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—42no—
149C22×D7—42no—
150C22×D7—81no—
151C7×Q8—21no—
152C7×Q8—21no—
153C7×D4—21no—
154C7×D4—21no—
155C7×D4—21no—
156C7×D4—41no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4—81no—
160C7×D4C2×D412yes—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—42no—
168C2×C4×C7—81no—
169C2×C4×C7C2×D411yescharacteristic
170C2×C4×C7C2×D411yescharacteristic
171C2×C4×C7C2×D411yescharacteristic
172C2×C4×C7C2×D411yescharacteristic
173C2×C4×C7C2411yescharacteristic
174C22⋊D7—42no—
175C22⋊D7—81no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178D28—21no—
179D28—21no—
180D28—21no—
181D28—42no—
182D28C2×D412yes—
183C4×D7—42no—
184C4×D7—81no—
185Dic14—21no—
186Dic14—21no—
187C7⋊C8—21no—
188C7⋊C8—21no—
189C7⋊C8—21no—
190C7⋊C8C2×D412yes—
191C2×#41—71no—
192C2×#41C2×D711yescharacteristic
193C22×D4—71no—
194C22×D4C2×D711yescharacteristic, complemented
195AGL(1,ℤ/8)—144no—
196AGL(1,ℤ/8)—144no—
197C2×Q16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×SD16—71no—
202C2×D8—71no—
203C2×D8—71no—
204C2×D8—71no—
205C2×D8—71no—
206C2×M16—71no—
207C2×M16—71no—
208#124—71no—
209#124C2×D711yescharacteristic, complemented
210#123—21no—
211#123—141no—
212C22⋊Q8—71no—
213C22⋊Q8C2×D711yescharacteristic, complemented
214#121—21no—
215#121—71no—
216#121—141no—
217#121C2×D711yescharacteristic
218#120—21no—
219#120—141no—
220C4×D4—71no—
221C4×D4C2×D711yescharacteristic
222C4×D4C2×D711yescharacteristic, complemented
223C2×C22⋊C4—21no—
224C2×C22⋊C4—141no—
225#107—71no—
226C4⋊C8—71no—
227#103—71no—
228#103—71no—
229#102—71no—
230#102—71no—
231#102—71no—
232#102—71no—
233C22⋊C8—71no—
234C22⋊C8—71no—
235C4×C8—71no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—22no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—84no—
249C22×C7C22×D411yescharacteristic
250C2×D7—22no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—84no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7C22×D411yescharacteristic
264C4×C7H(5,𝔽2)11yescharacteristic
265Dic7—41no—
266Dic7—41no—
267C24—21no—
268C24—141no—
269#41—22no—
270#41—41no—
271#41—142no—
272#41—281no—
273C2×Q8—71no—
274C2×Q8C22×D711yescharacteristic
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—71no—
281C2×D4—71no—
282C2×D4—142no—
283C2×D4—141no—
284C2×D4—141no—
285C2×D4—141no—
286C2×D4—281no—
287C2×D4C22⋊D711yescharacteristic
288C2×D4C22×D711yescharacteristic
289C2×D4C22×D711yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—141no—
293C22×C4—141no—
294C22×C4C22×D711yescharacteristic
295C22×C4C22×D711yescharacteristic
296Q16—281no—
297SD16—142no—
298SD16—142no—
299SD16—142no—
300SD16—142no—
301D8—74no—
302D8—142no—
303D8—142no—
304D8—281no—
305M16—142no—
306M16—142no—
307C2×C8—71no—
308C2×C8—71no—
309C2×C8—71no—
310C2×C8—71no—
311C4⋊C4—21no—
312C4⋊C4—71no—
313C4⋊C4—141no—
314C4⋊C4C22⋊D711yescharacteristic
315C4⋊C4C22×D711yescharacteristic
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—141no—
322C22⋊C4—141no—
323C22⋊C4—141no—
324C22⋊C4—141no—
325C22⋊C4—141no—
326C22⋊C4C22⋊D711yescharacteristic
327C22⋊C4C22⋊D711yescharacteristic
328C42C22×D711yescharacteristic
329C2×C7—41no—
330C2×C7—41no—
331C2×C7—42no—
332C2×C7—81no—
333C2×C7D4211yescharacteristic
334C2×C7#57511yescharacteristic
335C2×C7#57611yescharacteristic
336D7—42no—
337D7—81no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—141no—
344C23—141no—
345C23—282no—
346C23—282no—
347C23—561no—
348C23C2×C22⋊D711yescharacteristic
349C23C2×C22⋊D711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—141no—
353Q8—141no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—41no—
358D4—42no—
359D4—42no—
360D4—72no—
361D4—81no—
362D4—141no—
363D4—141no—
364D4—141no—
365D4—282no—
366D4—282no—
367D4—561no—
368D4D4×D712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—282no—
380C2×C4—561no—
381C2×C4C2×C22⋊D711yescharacteristic
382C2×C4C2×C22⋊D711yescharacteristic
383C2×C4C2×C22⋊D711yescharacteristic
384C2×C4C2×C22⋊D711yescharacteristic
385C2×C4C23×D711yescharacteristic
386C8—72no—
387C8—141no—
388C8—141no—
389C8—141no—
390C7#327311yescharacteristic, complemented, Sylow
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—142no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22—281no—
409C22—281no—
410C22—564no—
411C22C22×C22⋊D711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—281no—
419C4—281no—
420C4C2×D4×D711yescharacteristic
421C4#654811yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—282no—
427C2—561no—
428C2D4×C22⋊D711yescharacteristic
429C2#9051311yescharacteristic
430C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#546C2
socleC22×C7C22×D4
hypercenter#546D7
hypocenterC7#3273
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C7×#546C2semidirect
2. C2×#6512C2semidirect
3. C2×#6495C2semidirect
4. #90241C2semidirect
5. #90011C2semidirect
6. #89989C2semidirect
7. #89831C2semidirect
8. #89829C2semidirect
9. #89821C2semidirect
10. #89819C2semidirect
11. #89814C2semidirect
12. #89808C2semidirect
13. #89788C2semidirect
14. C4×#1141C2semidirect
15. #89784C2semidirect
17. C22×C7×D4C22semidirect
19. C22×D28C22semidirect
20. C7×#124C22semidirect
22. C7×C22⋊Q8C22semidirect
27. C4×C7×D4C22semidirect
30. C2×#1144C22semidirect
31. C2×#1143C22semidirect
32. C2×#1143C22semidirect
34. C2×#1142C22semidirect
35. C2×#1142C22semidirect
42. #6487C22semidirect
43. C2×#1136C22semidirect
44. C2×#1136C22semidirect
48. #6438C22semidirect
49. C4×D28C22semidirect
50. #6409C22semidirect
51. #6406C22semidirect
52. #6406C22semidirect
53. C22⋊(C7⋊C8)C22semidirect
54. C22⋊(C7⋊C8)C22semidirect
55. #6384C22semidirect
56. #6383C22semidirect
57. #6383C22semidirect
58. #6381C22semidirect
59. C4⋊(C7⋊C8)C22semidirect
60. C4×C7⋊C8C22semidirect
111. #1141D4semidirect
127. #546D7semidirect
194. C22×D4C2×D7semidirect
209. #124C2×D7semidirect
213. C22⋊Q8C2×D7semidirect
222. C4×D4C2×D7semidirect
390. C7#3273semidirect

Maximal subgroups

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

subgroupindex
C7×#5462
C2×#65122
C2×#64952
#902412
#900112
#899892
#898312
#898292
#898212
#898192
#898142
#898082
#897882
C4×#11412
#897842
#32737