Finite Group Databasebeta

← #11736941

Subgroups — #11736941

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×#6471C211yesmaximal, characteristic, complemented
2C2×#6467C211yesmaximal, characteristic, complemented
3#90242C211yesmaximal, characteristic, complemented
4#90241C211yesmaximal, characteristic, complemented
5C7×#435C211yesmaximal, characteristic, complemented
6#89909C211yesmaximal, characteristic, complemented
7#89886C211yesmaximal, characteristic, complemented
8#89616C211yesmaximal, characteristic, complemented
9#89613C211yesmaximal, characteristic, complemented
10#89506C211yesmaximal, characteristic, complemented
11#89505C211yesmaximal, characteristic, complemented
12#89505C211yesmaximal, characteristic, complemented
13#89503C211yesmaximal, characteristic, complemented
14#89487C211yesmaximal, characteristic, complemented
15#89483C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×#1157C2211yescharacteristic, complemented
19C22×D28C2211yescharacteristic
20C2×C7×M16C2211yescharacteristic
21C22×C7×C8C2211yescharacteristic, complemented
22C4×C7×D4C2211yescharacteristic
23#6501—21no—
24#6500—21no—
25#6493C2211yescharacteristic
26#6487C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic, complemented
29#6471—24no—
30C2×Dic28C2211yescharacteristic, complemented
31C2×Dic28C2211yescharacteristic, complemented
32#6467—24no—
33C2×D56C2211yescharacteristic
34C2×D56C2211yescharacteristic
35C2×#1132C2211yescharacteristic
36C2×#1132C2211yescharacteristic
37C2×#1132C2211yescharacteristic
38C2×#1132C2211yescharacteristic
39C2×#1132C2211yescharacteristic, complemented
40#6457—21no—
41D7×C4⋊C4—21no—
42#6447—21no—
43#6446—21no—
44#6446—21no—
45D7×C22⋊C4—21no—
46#6438C2211yescharacteristic
47C4×D28C2211yescharacteristic
48C4×D28C2211yescharacteristic, complemented
49#6432C2211yescharacteristic, complemented
50C7×C4⋊C8C2211yescharacteristic, complemented
51C7×C22⋊C8C2211yescharacteristic
52C7×C22⋊C8C2211yescharacteristic, complemented
53C7×#97C2211yescharacteristic, complemented
54#6395C2211yescharacteristic
55#6395C2211yescharacteristic
56#6395C2211yescharacteristic, complemented
57#6391C2211yescharacteristic, complemented
58#6390C2211yescharacteristic, complemented
59#6390C2211yescharacteristic, complemented
60#6390C2211yescharacteristic, complemented
61#3267—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×D4D411yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70#1157—22no—
71#1157—22no—
72#1157—22no—
73#1157—41no—
74#1157—41no—
75#1157—41no—
76C2×D28—22no—
77C2×D28—21no—
78C2×D28—41no—
79C2×D28C2311yescharacteristic
80C2×D28C2311yescharacteristic
81C2×D28C2311yescharacteristic
82C2×C4×D7—21no—
83C2×C4×D7—21no—
84C2×C4×D7—21no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×Dic14C2311yescharacteristic
88C2×Dic14C2311yescharacteristic
89C2×Dic14C2311yescharacteristic
90C7×M16—22no—
91C2×C7×C8—22no—
92C2×C7×C8C2311yescharacteristic
93C2×C7×C8C2311yescharacteristic
94C2×C7×C8C2311yescharacteristic
95C2×C7×C8C2311yescharacteristic
96C7×C4⋊C4D411yescharacteristic
97C7×C22⋊C4D411yescharacteristic
98C7×C22⋊C4D411yescharacteristic
99C42×C7C2311yescharacteristic
100C22⋊Dic7—21no—
101C22⋊Dic7—21no—
102C22⋊Dic7—21no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108C4⋊Dic7—21no—
109C4⋊Dic7—21no—
110C4⋊Dic7C2311yescharacteristic
111C4⋊Dic7C2311yescharacteristic
112Dic7⋊C4—21no—
113Dic7⋊C4—21no—
114Dic7⋊C4—21no—
115Dic28—22no—
116Dic28—41no—
117D56—22no—
118D56D414yescomplemented
119#1132—22no—
120#1132—22no—
121#1132—22no—
122#1132—22no—
123#1132—41no—
124C2×AGL(1,ℤ/8)—71no—
125C2×#135—71no—
126#547—71no—
127#546—71no—
128#497—71no—
129#470—71no—
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#435D711yescharacteristic, complemented
139C23×C7C2×D411yescharacteristic
140C23×C7C2×D411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—42no—
145C22×D7—42no—
146C22×D7—81no—
147C7×D4—41no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2×D411yescharacteristic
155C2×C4×C7C2×D411yescharacteristic
156C2×C4×C7C2411yescharacteristic
157C22⋊D7—42no—
158C22⋊D7—42no—
159C22⋊D7—42no—
160C22⋊D7—81no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166D28—21no—
167D28—21no—
168D28—21no—
169D28—21no—
170D28—42no—
171D28C2×D412yes—
172D28C2×D412yes—
173C4×D7—42no—
174C4×D7—42no—
175C4×D7—42no—
176C4×D7—42no—
177C4×D7—81no—
178Dic14—21no—
179Dic14—21no—
180Dic14—21no—
181Dic14—21no—
182Dic14—21no—
183Dic14—21no—
184C7×C8—21no—
185C7×C8—21no—
186C7×C8—21no—
187C7×C8C2×D412yes—
188C2×#41—71no—
189C2×#41—71no—
190C2×#41—71no—
191C22×D4—71no—
192AGL(1,ℤ/8)—144no—
193#135—144no—
194C2×Q16—71no—
195C2×Q16—71no—
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×C8C2×D711yescharacteristic, complemented
205C4⋊Q8—71no—
206#124—71no—
207#123—141no—
208#123—141no—
209#123—141no—
210C22⋊Q8—71no—
211C22⋊Q8—71no—
212C22⋊Q8—71no—
213#121—71no—
214#121—141no—
215#121—141no—
216#120—141no—
217C4×D4—71no—
218C4×D4—71no—
219C4×D4C2×D711yescharacteristic
220C2×C4⋊C4—141no—
221C2×C22⋊C4—141no—
222#106—71no—
223C4⋊C8C2×D711yescharacteristic, complemented
224#103—71no—
225#103—71no—
226#103—71no—
227#102—71no—
228#102—71no—
229#102—71no—
230C22⋊C8C2×D711yescharacteristic
231C22⋊C8C2×D711yescharacteristic, complemented
232#97C2×D711yescharacteristic, complemented
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7C22×D411yescharacteristic
240C2×D7—22no—
241C2×D7—22no—
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—21no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7C22×D411yescharacteristic
255C4×C7H(5,𝔽2)11yescharacteristic
256Dic7—41no—
257Dic7—41no—
258Dic7—41no—
259Dic7—41no—
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×Q8—71no—
270C2×Q8—71no—
271C2×D4—71no—
272C2×D4—71no—
273C2×D4—71no—
274C2×D4—142no—
275C2×D4—141no—
276C2×D4—141no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—281no—
281C2×D4D2811yescharacteristic
282C22×C4—141no—
283C22×C4—141no—
284C22×C4—141no—
285C22×C4—141no—
286C22×C4—141no—
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289Q16—142no—
290Q16—281no—
291SD16—142no—
292SD16—142no—
293SD16—142no—
294SD16—142no—
295SD16—281no—
296D8—74no—
297D8—142no—
298M16—22no—
299C2×C8—22no—
300C2×C8C22×D711yescharacteristic
301C2×C8C22×D711yescharacteristic
302C2×C8C22×D711yescharacteristic
303C2×C8C22×D711yescharacteristic
304C4⋊C4—71no—
305C4⋊C4—71no—
306C4⋊C4—141no—
307C4⋊C4—141no—
308C4⋊C4—141no—
309C4⋊C4—141no—
310C4⋊C4—141no—
311C4⋊C4D2811yescharacteristic
312C22⋊C4—141no—
313C22⋊C4—141no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C22⋊C4—141no—
319C22⋊C4—141no—
320C22⋊C4D2811yescharacteristic
321C22⋊C4D2811yescharacteristic
322C42C22×D711yescharacteristic
323C2×C7—41no—
324C2×C7—41no—
325C2×C7D4211yescharacteristic
326C2×C7#57511yescharacteristic
327C2×C7#57711yescharacteristic
328D7—42no—
329D7—42no—
330D7—81no—
331C23—141no—
332C23—141no—
333C23—141no—
334C23—282no—
335C23—282no—
336C23—561no—
337C23C2×D2811yescharacteristic
338C23C2×D2811yescharacteristic
339Q8—141no—
340Q8—141no—
341Q8—141no—
342Q8—141no—
343Q8—141no—
344Q8—141no—
345D4—41no—
346D4—72no—
347D4—72no—
348D4—141no—
349D4—141no—
350D4—141no—
351D4—141no—
352D4—282no—
353D4—282no—
354D4—282no—
355D4—282no—
356D4—561no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—282no—
367C2×C4—282no—
368C2×C4—282no—
369C2×C4—282no—
370C2×C4—561no—
371C2×C4C2×D2811yescharacteristic
372C2×C4C2×D2811yescharacteristic
373C2×C4C2×D2811yescharacteristic
374C2×C4C2×D2811yescharacteristic
375C2×C4C23×D711yescharacteristic
376C8—21no—
377C8—21no—
378C8—21no—
379C8D4×D712yes—
380C7#326711yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—142no—
388C22—142no—
389C22—281no—
390C22—281no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—564no—
397C22C22×D2811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—281no—
403C4—281no—
404C4—281no—
405C4—281no—
406C4—281no—
407C4C2×D4×D711yescharacteristic
408C4#655311yescharacteristic
409C2—41no—
410C2—41no—
411C2—282no—
412C2—282no—
413C2—561no—
414C2D4×D2811yescharacteristic
415C2#9043911yescharacteristic
416C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
hypercenter#435D7
hypocenterC7#3267
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#6471C2semidirect
2. C2×#6467C2semidirect
3. #90242C2semidirect
4. #90241C2semidirect
5. C7×#435C2semidirect
6. #89909C2semidirect
7. #89886C2semidirect
8. #89616C2semidirect
9. #89613C2semidirect
10. #89506C2semidirect
11. #89505C2semidirect
12. #89505C2semidirect
13. #89503C2semidirect
14. #89487C2semidirect
15. #89483C2semidirect
18. C2×#1157C22semidirect
21. C22×C7×C8C22semidirect
28. #6487C22semidirect
30. C2×Dic28C22semidirect
31. C2×Dic28C22semidirect
39. C2×#1132C22semidirect
48. C4×D28C22semidirect
49. #6432C22semidirect
50. C7×C4⋊C8C22semidirect
52. C7×C22⋊C8C22semidirect
53. C7×#97C22semidirect
56. #6395C22semidirect
57. #6391C22semidirect
58. #6390C22semidirect
59. #6390C22semidirect
60. #6390C22semidirect
118. D56D4semidirect
138. #435D7semidirect
204. C22×C8C2×D7semidirect
223. C4⋊C8C2×D7semidirect
231. C22⋊C8C2×D7semidirect
232. #97C2×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×#64712
C2×#64672
#902422
#902412
C7×#4352
#899092
#898862
#896162
#896132
#895062
#895052
#895032
#894872
#894832
#32677