Finite Group Databasebeta

← #11738820

Subgroups — #11738820

Order 896.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90298C211yesmaximal, characteristic
2#90214C211yesmaximal, characteristic, complemented
3C7×#482C211yesmaximal, characteristic, complemented
4#89805C211yesmaximal, characteristic, complemented
5#89775C211yesmaximal, characteristic, complemented
6#89654C211yesmaximal, characteristic, complemented
7#89653C211yesmaximal, characteristic, complemented
8#89638C211yesmaximal, characteristic, complemented
9#89632C211yesmaximal, characteristic, complemented
10#89551C211yesmaximal, characteristic, complemented
11#89549C211yesmaximal, characteristic, complemented
12#89542C211yesmaximal, characteristic, complemented
13D7×#102C211yesmaximal, characteristic, complemented
14#89507C211yesmaximal, characteristic, complemented
15#89499C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×#121C2211yescharacteristic
19C7×#117C2211yescharacteristic, complemented
20#6503—21no—
21#6501C2211yescharacteristic
22#6500—21no—
23#6493C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic, complemented
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6458C2211yescharacteristic
28#6455C2211yescharacteristic
29#6455C2211yescharacteristic
30#6455C2211yescharacteristic, complemented
31D7×C4⋊C4C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic, complemented
33#6448—21no—
34#6445—21no—
35(C22⋊D7)⋊C4—21no—
36(C22⋊D7)⋊C4—21no—
37C4×D28—21no—
38C4×Dic14—21no—
39C7×#107C2211yescharacteristic, complemented
40C7×#106C2211yescharacteristic, complemented
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic, complemented
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic, complemented
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic, complemented
50(C2×D7)⋊C8C2211yescharacteristic, complemented
51#6391C2211yescharacteristic, complemented
52#6391C2211yescharacteristic, complemented
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic, complemented
56#6381C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3096—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—
67C2×C22⋊D7D411yescharacteristic
68C2×C22⋊D7D411yescharacteristic
69C22×Dic7—21no—
70D4×D7—22no—
71D4×D7—22no—
72#1157—22no—
73#1157—22no—
74C2×D28—21no—
75C2×D28D411yescharacteristic
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7C2311yescharacteristic
80C2×C4×D7C2311yescharacteristic
81C2×Dic14D411yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C22⋊C4—21no—
88C7×C22⋊C4—21no—
89C42×C7—21no—
90C22⋊Dic7—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#535—71no—
110#520—71no—
111#482—71no—
112#482—71no—
113#482—71no—
114#482D711yescharacteristic, complemented
115#481—71no—
116#481—71no—
117#480—71no—
118#480—71no—
119#428—71no—
120#427—71no—
121#418—71no—
122C2×#102—71no—
123#408—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C22×D7—42no—
128C22×D7—81no—
129C22×D7—81no—
130C22×D7C2×D411yescharacteristic
131C22×D7C2×D411yescharacteristic
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—22no—
143C22⋊D7—22no—
144C22⋊D7—42no—
145C22⋊D7—81no—
146C22⋊D7—81no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—81no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—22no—
154D28—21no—
155D28—21no—
156D28—81no—
157C4×D7—21no—
158C4×D7—21no—
159C4×D7—81no—
160C4×D7—81no—
161C4×D7#4112yes—
162C4×D7#4112yes—
163Dic14—22no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×D4—71no—
170C2×M16—71no—
171C22×C8—71no—
172#127—141no—
173#124—141no—
174#121—71no—
175#121—71no—
176#121—71no—
177#121C2×D711yescharacteristic
178#120—141no—
179#120—141no—
180C4×Q8—141no—
181C4×D4—141no—
182C4×D4—141no—
183C4×D4—141no—
184#117—71no—
185#117—71no—
186#117—71no—
187#117C2×D711yescharacteristic, complemented
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191#107—71no—
192#107—71no—
193#107—71no—
194#107C2×D711yescharacteristic, complemented
195#106—71no—
196#106—71no—
197#106—71no—
198#106C2×D711yescharacteristic, complemented
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102C2×D711yescharacteristic
206#102C2×D711yescharacteristic
207C22⋊C8—71no—
208C22⋊C8—71no—
209C22⋊C8—71no—
210C22⋊C8C2×D711yescharacteristic, complemented
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C2×D7—21no—
219C2×D7—21no—
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—84no—
227C2×D7#12312yes—
228C4×C7—21no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7C2×#4111yescharacteristic
233C4×C7C2×#4111yescharacteristic
234Dic7—21no—
235Dic7—41no—
236Dic7—41no—
237Dic7—41no—
238Dic7#12312yes—
239C24—141no—
240#41—142no—
241#41—142no—
242C2×Q8—71no—
243C2×D4—21no—
244C2×D4—71no—
245C2×D4—71no—
246C2×D4—71no—
247C2×D4—71no—
248C2×D4—141no—
249C2×D4—142no—
250C2×D4—142no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4C22×D711yescharacteristic
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4C22×D711yescharacteristic
261M16—142no—
262C2×C8—71no—
263C2×C8—71no—
264C2×C8—142no—
265C2×C8C22×D711yescharacteristic
266C2×C8C22×D711yescharacteristic
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—71no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22×D711yescharacteristic
274C4⋊C4C22×D711yescharacteristic
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C42—21no—
285C42—141no—
286C42—141no—
287C42—141no—
288C2×C7—41no—
289C2×C7—81no—
290C2×C7C2×#12311yescharacteristic
291C2×C7#57611yescharacteristic
292C2×C7#57711yescharacteristic
293D7—22no—
294D7—41no—
295D7—81no—
296C23—21no—
297C23—71no—
298C23—71no—
299C23—141no—
300C23—282no—
301C23—561no—
302C23—561no—
303C23D4×D711yescharacteristic
304Q8—142no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—142no—
309D4—142no—
310D4—142no—
311D4—141no—
312D4—141no—
313D4—282no—
314D4—561no—
315D4—561no—
316D4—561no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—71no—
323C2×C4—72no—
324C2×C4—72no—
325C2×C4—71no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—561no—
332C2×C4—561no—
333C2×C4—561no—
334C2×C4D4×D711yescharacteristic
335C2×C4C23×D711yescharacteristic
336C8—21no—
337C8—21no—
338C8—141no—
339C8—141no—
340C7#309611yescharacteristic, complemented, Sylow
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—72no—
348C22—141no—
349C22—141no—
350C22—141no—
351C22—141no—
352C22—141no—
353C22—281no—
354C22—281no—
355C22—281no—
356C22—564no—
357C22C2×D4×D711yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—72no—
363C4—141no—
364C4—281no—
365C4—281no—
366C4—281no—
367C4D7×#4111yescharacteristic
368C4D7×#4111yescharacteristic
369C2—41no—
370C2—81no—
371C2—142no—
372C2—281no—
373C2—561no—
374C2D7×#12311yescharacteristic
375C2#9046111yescharacteristic
376C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#482C2
socleC22×C7C22×D4
prime radical#90298C2
hypercenter#482D7
hypocenterC7#3096
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
2. #90214C2semidirect
3. C7×#482C2semidirect
4. #89805C2semidirect
5. #89775C2semidirect
6. #89654C2semidirect
7. #89653C2semidirect
8. #89638C2semidirect
9. #89632C2semidirect
10. #89551C2semidirect
11. #89549C2semidirect
12. #89542C2semidirect
13. D7×#102C2semidirect
14. #89507C2semidirect
15. #89499C2semidirect
19. C7×#117C22semidirect
24. C2×C4⋊Dic7C22semidirect
30. #6455C22semidirect
32. D7×C4⋊C4C22semidirect
39. C7×#107C22semidirect
40. C7×#106C22semidirect
43. C7×C22⋊C8C22semidirect
46. C22⋊(C7⋊C8)C22semidirect
49. (C2×D7)⋊C8C22semidirect
50. (C2×D7)⋊C8C22semidirect
51. #6391C22semidirect
52. #6391C22semidirect
55. #6382C22semidirect
56. #6381C22semidirect
57. #6381C22semidirect
58. #6381C22semidirect
114. #482D7semidirect
187. #117C2×D7semidirect
194. #107C2×D7semidirect
198. #106C2×D7semidirect
210. C22⋊C8C2×D7semidirect
340. C7#3096semidirect

Maximal subgroups

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

subgroupindex
#902982
#902142
C7×#4822
#898052
#897752
#896542
#896532
#896382
#896322
#895512
#895492
#895422
D7×#1022
#895072
#894992
#30967