Finite Group Databasebeta

← #11738173

Subgroups — #11738173

Order 896.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6474C211yesmaximal, characteristic, complemented
2C2×#6467C211yesmaximal, characteristic, complemented
3#90303C211yesmaximal, characteristic, complemented
4#90290C211yesmaximal, characteristic, complemented
5C7×#466C211yesmaximal, characteristic, complemented
6#89926C211yesmaximal, characteristic, complemented
7#89883C211yesmaximal, characteristic, complemented
8#89809C211yesmaximal, characteristic, complemented
9#89809C211yesmaximal, characteristic, complemented
10#89661C211yesmaximal, characteristic, complemented
11#89658C211yesmaximal, characteristic, complemented
12#89558C211yesmaximal, characteristic, complemented
13#89555C211yesmaximal, characteristic, complemented
14#89545C211yesmaximal, characteristic, complemented
15#89539C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C2×#1157C2211yescharacteristic, complemented
20C2×C7×D8C2211yescharacteristic
21C22×C7×C8C2211yescharacteristic, complemented
22C7×#121C2211yescharacteristic
23C7×#121C2211yescharacteristic, complemented
24#6503—21no—
25#6503C2211yescharacteristic, complemented
26#6502—21no—
27#6501—21no—
28#6500—21no—
29#6499C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31C2×#1142C2211yescharacteristic, complemented
32C2×#1141C2211yescharacteristic
33C2×#1141C2211yescharacteristic
34C4×C22⋊D7C2211yescharacteristic
35#6474—24no—
36C2×Dic28C2211yescharacteristic, complemented
37#6467—24no—
38C2×D56C2211yescharacteristic
39C2×#1132C2211yescharacteristic
40C2×#1132C2211yescharacteristic, complemented
41C2×#1131C2211yescharacteristic
42#6460C2211yescharacteristic
43#6458C2211yescharacteristic
44#6456C2211yescharacteristic
45#6450C2211yescharacteristic, complemented
46#6447—21no—
47#6446—21no—
48(C22⋊D7)⋊C4—21no—
49D7×C22⋊C4—21no—
50C7×#107C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic
52C7×#102C2211yescharacteristic, complemented
53C22⋊(C7⋊C8)C2211yescharacteristic, complemented
54(C2×D7)⋊C8C2211yescharacteristic
55#6389C2211yescharacteristic, complemented
56Dic7⋊C8C2211yescharacteristic, complemented
57#6384C2211yescharacteristic, complemented
58#6384C2211yescharacteristic, complemented
59#6383C2211yescharacteristic
60#6383C2211yescharacteristic, complemented
61#3268—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C2×C22⋊D7—21no—
73C2×C22⋊D7—21no—
74C2×C22⋊D7D411yescharacteristic
75C22×Dic7—21no—
76#1159—22no—
77#1159—41no—
78D4×D7—22no—
79D4×D7—41no—
80#1157—22no—
81#1157—22no—
82#1157—41no—
83#1157—41no—
84C2×D28—21no—
85C2×D28C2311yescharacteristic
86C2×D28C2311yescharacteristic
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7C2311yescharacteristic
91C2×Dic14C2311yescharacteristic
92C2×Dic14C2311yescharacteristic
93C7×D8—22no—
94C2×C7×C8—22no—
95C2×C7×C8C2311yescharacteristic
96C2×C7×C8C2311yescharacteristic
97C7×C4⋊C4C2311yescharacteristic
98C7×C4⋊C4C2311yescharacteristic
99C7×C22⋊C4—21no—
100C7×C22⋊C4—21no—
101C22⋊Dic7—21no—
102C22⋊Dic7—21no—
103C22⋊Dic7D411yescharacteristic
104#1142—22no—
105#1142—41no—
106#1141—22no—
107#1141—41no—
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⋊C4D411yescharacteristic
115C4×Dic7—21no—
116C4×Dic7C2311yescharacteristic
117C2×C7⋊C8C2311yescharacteristic
118C2×C7⋊C8C2311yescharacteristic
119Dic28—22no—
120D56—22no—
121#1132—22no—
122#1132D414yescomplemented
123#1131—22no—
124C2×AGL(1,ℤ/8)—71no—
125C2×#135—71no—
126#550—71no—
127#546—71no—
128#496—71no—
129#468—71no—
130#466D711yescharacteristic, complemented
131#464—71no—
132#459—71no—
133#452—71no—
134#449—71no—
135#449—71no—
136#447—71no—
137#440—71no—
138#435—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7C2×D411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—42no—
145C22×D7—42no—
146C22×D7—81no—
147C22×D7C2×D411yescharacteristic
148C7×D4—21no—
149C7×D4—21no—
150C7×D4—21no—
151C7×D4—21no—
152C7×D4—81no—
153C7×D4—81no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7C2×D411yescharacteristic
159C2×C4×C7C2411yescharacteristic
160C22⋊D7—41no—
161C22⋊D7—42no—
162C22⋊D7—42no—
163C22⋊D7—42no—
164C22⋊D7—42no—
165C22⋊D7—42no—
166C22⋊D7—81no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—42no—
170C2×Dic7C2×D411yescharacteristic
171C2×Dic7C2×D411yescharacteristic
172C2×Dic7C2×D411yescharacteristic
173D28—21no—
174D28—21no—
175D28—21no—
176D28—81no—
177D28C2×D412yes—
178C4×D7—21no—
179C4×D7—21no—
180C4×D7—42no—
181C4×D7—42no—
182C4×D7—81no—
183Dic14—21no—
184Dic14—21no—
185Dic14—21no—
186Dic14C2×D412yes—
187C7×C8—21no—
188C7×C8C2×D412yes—
189C7⋊C8—21no—
190C7⋊C8—21no—
191C2×#41—71no—
192C2×#41—71no—
193C2×#41—71no—
194C22×D4—71no—
195AGL(1,ℤ/8)—144no—
196#135—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×D8C2×D711yescharacteristic
206C2×M16—71no—
207C22×C8C2×D711yescharacteristic, complemented
208#127—71no—
209#127—141no—
210#124—71no—
211#123—141no—
212C22⋊Q8—71no—
213#121—71no—
214#121—141no—
215#121—141no—
216#121—141no—
217#121C2×D711yescharacteristic
218#121C2×D711yescharacteristic, complemented
219#120—141no—
220C4×Q8—71no—
221C4×D4—71no—
222C4×D4—71no—
223C4×D4—141no—
224C2×C22⋊C4—141no—
225#107C2×D711yescharacteristic, complemented
226C4⋊C8—71no—
227#103—71no—
228#103—71no—
229#102—71no—
230#102—71no—
231#102C2×D711yescharacteristic
232#102C2×D711yescharacteristic, complemented
233C22⋊C8—71no—
234C22⋊C8—71no—
235#97—71no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7C22×D411yescharacteristic
246C2×D7—21no—
247C2×D7—22no—
248C2×D7—21no—
249C2×D7—21no—
250C2×D7—41no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—84no—
256C4×C7—21no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7C22×D411yescharacteristic
260C4×C7H(5,𝔽2)11yescharacteristic
261Dic7—22no—
262Dic7—21no—
263Dic7—21no—
264Dic7—21no—
265Dic7—41no—
266C24—141no—
267#41—142no—
268#41—142no—
269#41—142no—
270#41—281no—
271#41—281no—
272#41—281no—
273C2×Q8—71no—
274C2×Q8—71no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—71no—
278C2×D4—71no—
279C2×D4—71no—
280C2×D4—141no—
281C2×D4—141no—
282C2×D4—141no—
283C2×D4—141no—
284C2×D4—142no—
285C2×D4—141no—
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—281no—
289C2×D4C22×D711yescharacteristic
290C2×D4C22×D711yescharacteristic
291C22×C4—71no—
292C22×C4—141no—
293C22×C4—141no—
294C22×C4—141no—
295C22×C4—141no—
296C22×C4C22×D711yescharacteristic
297Q16—142no—
298SD16—74no—
299SD16—142no—
300SD16—142no—
301SD16—281no—
302D8—22no—
303D8—142no—
304D8—142no—
305D8—281no—
306M16—142no—
307C2×C8—22no—
308C2×C8—71no—
309C2×C8—71no—
310C2×C8C22×D711yescharacteristic
311C2×C8C22×D711yescharacteristic
312C4⋊C4—71no—
313C4⋊C4—141no—
314C4⋊C4—141no—
315C4⋊C4C22×D711yescharacteristic
316C4⋊C4C22×D711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—71no—
320C22⋊C4—71no—
321C22⋊C4—141no—
322C22⋊C4—141no—
323C22⋊C4—141no—
324C22⋊C4—141no—
325C22⋊C4—141no—
326C42—71no—
327C42—141no—
328C2×C7—41no—
329C2×C7—81no—
330C2×C7—81no—
331C2×C7D4211yescharacteristic
332C2×C7#57511yescharacteristic
333C2×C7#57611yescharacteristic
334D7—41no—
335D7—42no—
336D7—81no—
337C23—21no—
338C23—21no—
339C23—71no—
340C23—141no—
341C23—141no—
342C23—282no—
343C23—282no—
344C23—561no—
345C23D4×D711yescharacteristic
346Q8—72no—
347Q8—141no—
348Q8—141no—
349Q8—141no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—72no—
355D4—81no—
356D4—81no—
357D4—141no—
358D4—141no—
359D4—141no—
360D4—281no—
361D4—282no—
362D4—282no—
363D4—282no—
364D4—282no—
365D4—282no—
366D4—561no—
367D4—561no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—71no—
373C2×C4—71no—
374C2×C4—71no—
375C2×C4—141no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4—561no—
383C2×C4D4×D711yescharacteristic
384C2×C4C23×D711yescharacteristic
385C8—21no—
386C8—141no—
387C8—141no—
388C8D4×D712yes—
389C7#326811yescharacteristic, complemented, Sylow
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—141no—
400C22—141no—
401C22—141no—
402C22—142no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22—564no—
409C22C2×D4×D711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—141no—
414C4—141no—
415C4—141no—
416C4—142no—
417C4—281no—
418C4C2×D4×D711yescharacteristic
419C4#655311yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—281no—
424C2—282no—
425C2—561no—
426C2#9030011yescharacteristic
427C2#9044811yescharacteristic
428C2#9045911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#466C2
socleC22×C7C22×D4
hypercenter#466D7
hypocenterC7#3268
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#6474C2semidirect
2. C2×#6467C2semidirect
3. #90303C2semidirect
4. #90290C2semidirect
5. C7×#466C2semidirect
6. #89926C2semidirect
7. #89883C2semidirect
8. #89809C2semidirect
9. #89809C2semidirect
10. #89661C2semidirect
11. #89658C2semidirect
12. #89558C2semidirect
13. #89555C2semidirect
14. #89545C2semidirect
15. #89539C2semidirect
19. C2×#1157C22semidirect
21. C22×C7×C8C22semidirect
23. C7×#121C22semidirect
25. #6503C22semidirect
31. C2×#1142C22semidirect
36. C2×Dic28C22semidirect
40. C2×#1132C22semidirect
45. #6450C22semidirect
50. C7×#107C22semidirect
52. C7×#102C22semidirect
53. C22⋊(C7⋊C8)C22semidirect
55. #6389C22semidirect
56. Dic7⋊C8C22semidirect
57. #6384C22semidirect
58. #6384C22semidirect
60. #6383C22semidirect
122. #1132D4semidirect
130. #466D7semidirect
207. C22×C8C2×D7semidirect
218. #121C2×D7semidirect
225. #107C2×D7semidirect
232. #102C2×D7semidirect
389. C7#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#64742
C2×#64672
#903032
#902902
C7×#4662
#899262
#898832
#898092
#896612
#896582
#895582
#895552
#895452
#895392
#32687