Finite Group Databasebeta

← #11737331

Subgroups — #11737331

Order 896.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90252C211yesmaximal, characteristic, complemented
2#90236C211yesmaximal, characteristic, complemented
3C7×#442C211yesmaximal, characteristic, complemented
4#89928C211yesmaximal, characteristic
5#89925C211yesmaximal, characteristic, complemented
6C4×#1142C211yesmaximal, characteristic, complemented
7C4×#1141C211yesmaximal, characteristic, complemented
8#89642C211yesmaximal, characteristic, complemented
9#89639C211yesmaximal, characteristic, complemented
10#89559C211yesmaximal, characteristic, complemented
11#89556C211yesmaximal, characteristic, complemented
12#89548C211yesmaximal, characteristic, complemented
13#89531C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89481C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C2×C7×D8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic, complemented
20#6501C2211yescharacteristic
21#6499C2211yescharacteristic, complemented
22D4×Dic7C2211yescharacteristic
23C2×#1142C2211yescharacteristic
24C2×#1141C2211yescharacteristic
25#6492—21no—
26C4×C22⋊D7—21no—
27C2×(C2×D7)⋊C4—21no—
28#6487—21no—
29C2×#1132C2211yescharacteristic
30#6460C2211yescharacteristic
31#6456C2211yescharacteristic
32#6455C2211yescharacteristic
33#6451C2211yescharacteristic, complemented
34#6448—21no—
35(C22⋊D7)⋊C4—21no—
36D7×C22⋊C4—21no—
37#6442—21no—
38C4×D28C2211yescharacteristic
39#6435C2211yescharacteristic
40C4×Dic14C2211yescharacteristic, complemented
41C7×#106C2211yescharacteristic, complemented
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic, complemented
44C7×#97C2211yescharacteristic, complemented
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic, complemented
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic, complemented
51#6390C2211yescharacteristic, complemented
52Dic7⋊C8C2211yescharacteristic, complemented
53Dic7⋊C8C2211yescharacteristic, complemented
54#6384C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58C4×C7⋊C8C2211yescharacteristic, complemented
59#3118—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68D4×D7—22no—
69D4×D7—22no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C7×D8—22no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C42×C7C2311yescharacteristic
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88#1142—22no—
89#1141—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108#1132—22no—
109#542—71no—
110#518—71no—
111#485—71no—
112#482—71no—
113#476—71no—
114#474—71no—
115#463—71no—
116#463—71no—
117#450—71no—
118#447—71no—
119#442D711yescharacteristic, complemented
120#440—71no—
121C4×SD16—71no—
122C4×D8—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7—21no—
126C22×D7—21no—
127C22×D7—24no—
128C22×D7—81no—
129C22×D7—81no—
130C22×D7C2×D411yescharacteristic
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4#4112yes—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—81no—
138C2×C4×C7—81no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—24no—
143C22⋊D7—81no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—81no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—21no—
152D28#4112yes—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—81no—
156C4×D7—81no—
157Dic14—21no—
158Dic14—21no—
159Dic14—81no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C22×D4—71no—
165C2×SD16—71no—
166C2×SD16—71no—
167C2×D8—71no—
168C2×D8C2×D711yescharacteristic
169C4⋊Q8—71no—
170#126—141no—
171#124—71no—
172#124—141no—
173#123—141no—
174C22⋊Q8—71no—
175C22⋊Q8—141no—
176#121—71no—
177C4×Q8—71no—
178C4×D4—71no—
179C4×D4—71no—
180C4×D4—71no—
181C4×D4—141no—
182C4×D4—141no—
183C4×D4C2×D711yescharacteristic
184C4×D4C2×D711yescharacteristic, complemented
185#117—71no—
186#117—71no—
187C2×C22⋊C4—141no—
188C2×C22⋊C4—141no—
189#107—71no—
190#107—71no—
191#106—71no—
192#106C2×D711yescharacteristic, complemented
193C4⋊C8—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102C2×D711yescharacteristic
202#102C2×D711yescharacteristic, complemented
203C22⋊C8—71no—
204C22⋊C8—71no—
205#97C2×D711yescharacteristic, complemented
206C4×C8—71no—
207C22×C7—22no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C2×D7—21no—
215C2×D7—22no—
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—84no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#4111yescharacteristic
226C4×C7C2×#4111yescharacteristic
227Dic7—21no—
228Dic7—21no—
229Dic7—21no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C24—141no—
234C2×Q8—71no—
235C2×Q8—141no—
236C2×D4—71no—
237C2×D4—141no—
238C2×D4—142no—
239C2×D4—142no—
240C2×D4—141no—
241C2×D4C22×D711yescharacteristic
242C2×D4C22×D711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—71no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249SD16—142no—
250SD16—142no—
251D8—22no—
252D8—142no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4C22×D711yescharacteristic
265C4⋊C4C22×D711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—71no—
269C22⋊C4—71no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—71no—
279C42—141no—
280C42—141no—
281C42C22×D711yescharacteristic
282C2×C7—42no—
283C2×C7—81no—
284C2×C7#52511yescharacteristic
285C2×C7#57611yescharacteristic
286C2×C7#57711yescharacteristic
287D7—41no—
288D7—42no—
289C23—21no—
290C23—21no—
291C23—71no—
292C23—144no—
293C23—141no—
294C23—561no—
295C23—561no—
296Q8—141no—
297Q8—141no—
298Q8—561no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—72no—
303D4—144no—
304D4—141no—
305D4—561no—
306D4#115712yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—71no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4—561no—
322C2×C4D4×D711yescharacteristic
323C2×C4D4×D711yescharacteristic
324C2×C4C23×D711yescharacteristic
325C8—21no—
326C8—21no—
327C8—141no—
328C8—141no—
329C7#311811yescharacteristic, complemented, Sylow
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—141no—
337C22—141no—
338C22—141no—
339C22—142no—
340C22—281no—
341C22—281no—
342C22—564no—
343C22C2×D4×D711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—141no—
349C4—141no—
350C4—141no—
351C4—281no—
352C4—281no—
353C4—281no—
354C4C2×#115711yescharacteristic
355C4D7×#4111yescharacteristic
356C2—42no—
357C2—81no—
358C2—281no—
359C2—282no—
360C2#9023811yescharacteristic
361C2#9046511yescharacteristic
362C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#442C2
socleC22×C7C22×D4
prime radical#89928C2
hypercenter#442D7
hypocenterC7#3118
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #90252C2semidirect
2. #90236C2semidirect
3. C7×#442C2semidirect
5. #89925C2semidirect
6. C4×#1142C2semidirect
7. C4×#1141C2semidirect
8. #89642C2semidirect
9. #89639C2semidirect
10. #89559C2semidirect
11. #89556C2semidirect
12. #89548C2semidirect
13. #89531C2semidirect
14. #89482C2semidirect
15. #89481C2semidirect
19. C4×C7×D4C22semidirect
21. #6499C22semidirect
33. #6451C22semidirect
40. C4×Dic14C22semidirect
41. C7×#106C22semidirect
43. C7×#102C22semidirect
44. C7×#97C22semidirect
46. #6406C22semidirect
50. #6392C22semidirect
51. #6390C22semidirect
52. Dic7⋊C8C22semidirect
53. Dic7⋊C8C22semidirect
54. #6384C22semidirect
56. #6382C22semidirect
57. #6381C22semidirect
58. C4×C7⋊C8C22semidirect
119. #442D7semidirect
184. C4×D4C2×D7semidirect
192. #106C2×D7semidirect
202. #102C2×D7semidirect
205. #97C2×D7semidirect
329. C7#3118semidirect

Maximal subgroups

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

subgroupindex
#902522
#902362
C7×#4422
#899282
#899252
C4×#11422
C4×#11412
#896422
#896392
#895592
#895562
#895482
#895312
#894822
#894812
#31187