Finite Group Databasebeta

← #11738799

Subgroups — #11738799

Order 896.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90310C211yesmaximal, characteristic, complemented
2#90212C211yesmaximal, characteristic, complemented
3C7×#482C211yesmaximal, characteristic, complemented
4#89808C211yesmaximal, characteristic, complemented
5#89775C211yesmaximal, characteristic, complemented
6#89663C211yesmaximal, characteristic, complemented
7#89644C211yesmaximal, characteristic, complemented
8#89643C211yesmaximal, characteristic, complemented
9#89641C211yesmaximal, characteristic, complemented
10#89558C211yesmaximal, characteristic, complemented
11#89558C211yesmaximal, characteristic, complemented
12#89557C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89504C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C7×#121C2211yescharacteristic
18C7×#117C2211yescharacteristic
19#6503C2211yescharacteristic, complemented
20#6502—21no—
21#6501—21no—
22D4×Dic7C2211yescharacteristic, complemented
23C2×#1141C2211yescharacteristic
24C2×#1141C2211yescharacteristic
25#6493C2211yescharacteristic
26#6489C2211yescharacteristic
27C2×D56C2211yescharacteristic
28C2×D56C2211yescharacteristic
29#6456C2211yescharacteristic
30#6456C2211yescharacteristic
31#6456C2211yescharacteristic
32#6456C2211yescharacteristic
33#6456C2211yescharacteristic, complemented
34#6453C2211yescharacteristic, complemented
35#6449—21no—
36#6446—21no—
37D7×C22⋊C4—21no—
38D7×C22⋊C4—21no—
39C4×D28—21no—
40C4×D28—21no—
41C7×#107C2211yescharacteristic, complemented
42C7×#106C2211yescharacteristic, complemented
43C7×#102C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic, complemented
45C7×C22⋊C8C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6389C2211yescharacteristic, complemented
50Dic7⋊C8C2211yescharacteristic, complemented
51Dic7⋊C8C2211yescharacteristic, complemented
52C8×Dic7C2211yescharacteristic, complemented
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic, complemented
56#6383C2211yescharacteristic, complemented
57#6382C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3122—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68C2×D28—21no—
69C2×D28—22no—
70C2×D28—22no—
71C2×D28C2311yescharacteristic
72C2×D28C2311yescharacteristic
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C42×C7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7D411yescharacteristic
89C22⋊Dic7D411yescharacteristic
90#1141—22no—
91#1141—22no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98C4⋊Dic7—21no—
99C4⋊Dic7D411yescharacteristic
100C4⋊Dic7D411yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103C4×Dic7C2311yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107D56—22no—
108D56—22no—
109#540—71no—
110#518—71no—
111#482—71no—
112#482D711yescharacteristic, complemented
113#478—71no—
114#478—71no—
115#459—71no—
116#459—71no—
117#447—71no—
118#447—71no—
119#442—71no—
120#442—71no—
121C4×D8—71no—
122C4×D8—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—24no—
129C22×D7—21no—
130C22×D7—81no—
131C22×D7—81no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—81no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—81no—
143C22⋊D7—81no—
144C22⋊D7—81no—
145C2×Dic7—21no—
146C2×Dic7—81no—
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—24no—
152D28—21no—
153D28—21no—
154D28—21no—
155D28—21no—
156D28#4112yes—
157D28#4112yes—
158C4×D7—81no—
159C4×D7—81no—
160C4×D7—81no—
161C4×D7—81no—
162C4×D7—81no—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×D4—71no—
168C2×D8—71no—
169C2×D8—71no—
170C2×D8—71no—
171C2×D8—71no—
172#127—71no—
173#125—71no—
174#123—141no—
175#123—141no—
176#121—71no—
177#121—141no—
178#121—141no—
179#121C2×D711yescharacteristic
180C4×D4—71no—
181C4×D4—71no—
182C4×D4—71no—
183C4×D4—71no—
184C4×D4—71no—
185C4×D4—71no—
186C4×D4—141no—
187C4×D4—141no—
188#117—71no—
189#117C2×D711yescharacteristic
190C2×C22⋊C4—141no—
191C2×C22⋊C4—141no—
192#107—71no—
193#107C2×D711yescharacteristic, complemented
194#106—71no—
195#106C2×D711yescharacteristic, complemented
196C4⋊C8—71no—
197C4⋊C8—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#102C2×D711yescharacteristic, complemented
205#102C2×D711yescharacteristic, complemented
206C22⋊C8—71no—
207C22⋊C8C2×D711yescharacteristic
208#97—71no—
209C4×C8—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7C22×D411yescharacteristic
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—84no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7C2×#4111yescharacteristic
233Dic7—21no—
234Dic7—21no—
235Dic7—21no—
236Dic7—21no—
237Dic7—41no—
238C24—141no—
239C2×D4—21no—
240C2×D4—71no—
241C2×D4—71no—
242C2×D4—71no—
243C2×D4—142no—
244C2×D4—142no—
245C2×D4—141no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4C22×D711yescharacteristic
250C22×C4—141no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4—141no—
256C22×C4C22×D711yescharacteristic
257D8—142no—
258D8—142no—
259D8—142no—
260D8—142no—
261C2×C8—71no—
262C2×C8—71no—
263C2×C8C22×D711yescharacteristic
264C2×C8C22×D711yescharacteristic
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4C22×D711yescharacteristic
271C4⋊C4C22×D711yescharacteristic
272C4⋊C4C22×D711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—71no—
276C22⋊C4—71no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C42—21no—
285C42—71no—
286C42—71no—
287C2×C7—41no—
288C2×C7—81no—
289C2×C7#52511yescharacteristic
290C2×C7#57611yescharacteristic
291C2×C7#57611yescharacteristic
292D7—42no—
293D7—42no—
294D7—81no—
295C23—21no—
296C23—144no—
297C23—141no—
298C23—141no—
299C23—141no—
300C23—561no—
301C23—561no—
302C23D4×D711yescharacteristic
303D4—21no—
304D4—21no—
305D4—72no—
306D4—72no—
307D4—81no—
308D4—144no—
309D4—141no—
310D4—141no—
311D4—141no—
312D4—141no—
313D4—561no—
314D4—561no—
315D4—561no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—71no—
322C2×C4—71no—
323C2×C4—71no—
324C2×C4—71no—
325C2×C4—141no—
326C2×C4—561no—
327C2×C4—561no—
328C2×C4—561no—
329C2×C4—561no—
330C2×C4—561no—
331C2×C4—561no—
332C2×C4D4×D711yescharacteristic
333C2×C4C23×D711yescharacteristic
334C8—21no—
335C8—21no—
336C8—141no—
337C8—141no—
338C7#312211yescharacteristic, complemented, Sylow
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—142no—
346C22—142no—
347C22—281no—
348C22—281no—
349C22—281no—
350C22—281no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22—564no—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—141no—
361C4—141no—
362C4—141no—
363C4—141no—
364C4—281no—
365C4D7×#4111yescharacteristic
366C4D7×#4111yescharacteristic
367C2—41no—
368C2—81no—
369C2—282no—
370C2—282no—
371C2—561no—
372C2#9034711yescharacteristic
373C2#9046011yescharacteristic
374C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#482C2
socleC22×C7C22×D4
hypercenter#482D7
hypocenterC7#3122
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #90310C2semidirect
2. #90212C2semidirect
3. C7×#482C2semidirect
4. #89808C2semidirect
5. #89775C2semidirect
6. #89663C2semidirect
7. #89644C2semidirect
8. #89643C2semidirect
9. #89641C2semidirect
10. #89558C2semidirect
11. #89558C2semidirect
12. #89557C2semidirect
13. #89528C2semidirect
14. #89504C2semidirect
15. #89492C2semidirect
19. #6503C22semidirect
22. D4×Dic7C22semidirect
33. #6456C22semidirect
34. #6453C22semidirect
41. C7×#107C22semidirect
42. C7×#106C22semidirect
43. C7×#102C22semidirect
44. C7×#102C22semidirect
49. #6389C22semidirect
50. Dic7⋊C8C22semidirect
51. Dic7⋊C8C22semidirect
52. C8×Dic7C22semidirect
55. #6383C22semidirect
56. #6383C22semidirect
57. #6382C22semidirect
58. #6381C22semidirect
112. #482D7semidirect
193. #107C2×D7semidirect
195. #106C2×D7semidirect
204. #102C2×D7semidirect
205. #102C2×D7semidirect
338. C7#3122semidirect

Maximal subgroups

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

subgroupindex
#903102
#902122
C7×#4822
#898082
#897752
#896632
#896442
#896432
#896412
#895582
#895572
#895282
#895042
#894922
#31227