Finite Group Databasebeta

← #11737973

Subgroups — #11737973

Order 896.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90372C211yesmaximal, characteristic, complemented
2#90233C211yesmaximal, characteristic, complemented
3C7×#463C211yesmaximal, characteristic, complemented
4#89940C211yesmaximal, characteristic, complemented
5#89938C211yesmaximal, characteristic, complemented
6#89926C211yesmaximal, characteristic
7#89925C211yesmaximal, characteristic, complemented
8#89832C211yesmaximal, characteristic, complemented
9#89786C211yesmaximal, characteristic, complemented
10#89613C211yesmaximal, characteristic, complemented
11#89600C211yesmaximal, characteristic, complemented
12#89570C211yesmaximal, characteristic, complemented
13#89561C211yesmaximal, characteristic, complemented
14#89534C211yesmaximal, characteristic, complemented
15#89530C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×D8C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1144C2211yescharacteristic
24#6503C2211yescharacteristic
25#6499C2211yescharacteristic
26#6499C2211yescharacteristic
27D4×Dic7C2211yescharacteristic
28C2×#1142C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6489—21no—
32#6487—21no—
33C2×#1132C2211yescharacteristic
34C2×#1132C2211yescharacteristic
35#6450C2211yescharacteristic
36#6450C2211yescharacteristic
37#6449—21no—
38#6448—21no—
39#6446—21no—
40(C22⋊D7)⋊C4—21no—
41#6442—21no—
42#6442—21no—
43C4×D28C2211yescharacteristic
44#6432C2211yescharacteristic
45C4×Dic14C2211yescharacteristic
46C7×C4⋊C8C2211yescharacteristic
47C7×#103C2211yescharacteristic
48C7×#102C2211yescharacteristic
49#6406C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3401—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C22×Dic7—21no—
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×Dic14D411yescharacteristic
70C2×Dic14C2311yescharacteristic
71C2×Dic14C2311yescharacteristic
72C7×SD16—41no—
73C7×D8—41no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C42×C7C2311yescharacteristic
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86#1144—22no—
87#1144—41no—
88#1142—22no—
89#1142—41no—
90#1141—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7D411yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7C2311yescharacteristic
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1132—41no—
111#1132—41no—
112#559—71no—
113#556—71no—
114#497—71no—
115#496—71no—
116#495—71no—
117#464—71no—
118#464—71no—
119#463—71no—
120#463D711yescharacteristic, complemented
121#461—71no—
122#460—71no—
123#446—71no—
124#442—71no—
125#441—71no—
126C4×SD16—71no—
127C23×C7—21no—
128C23×C7—21no—
129C22×D7—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—21no—
135C7×D4—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—42no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—42no—
143C22⋊D7—81no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—81no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152D28—21no—
153D28—21no—
154C4×D7—81no—
155Dic14—21no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—41no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8#4112yes—
164C2×Q16—71no—
165C2×Q16—71no—
166C2×SD16—71no—
167C2×SD16—71no—
168C2×SD16—71no—
169C2×SD16—71no—
170C2×SD16C2×D711yescharacteristic
171C2×D8—71no—
172C2×D8C2×D711yescharacteristic
173C4⋊Q8—71no—
174#127—71no—
175#126—141no—
176#126—141no—
177#124—71no—
178#124—71no—
179#124—71no—
180#124—141no—
181#124C2×D711yescharacteristic
182#123—141no—
183#123—141no—
184C22⋊Q8—141no—
185C4×Q8—71no—
186C4×Q8—71no—
187C4×Q8—71no—
188C4×Q8—71no—
189C4×D4—71no—
190C4×D4—71no—
191C4×D4—141no—
192C4×D4C2×D711yescharacteristic
193#117—141no—
194#106—71no—
195C4⋊C8—71no—
196C4⋊C8—71no—
197C4⋊C8C2×D711yescharacteristic
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic
201#102—71no—
202#102—71no—
203#102C2×D711yescharacteristic
204#97—71no—
205#97—71no—
206C4×C8—71no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7C2×#4111yescharacteristic
222C4×C7H(5,𝔽2)11yescharacteristic
223Dic7—21no—
224Dic7—21no—
225Dic7—21no—
226Dic7—21no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230C2×Q8—71no—
231C2×Q8—71no—
232C2×Q8—71no—
233C2×Q8C22×D711yescharacteristic
234C2×D4—71no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4C22×D711yescharacteristic
238C2×D4C22×D711yescharacteristic
239C22×C4—21no—
240C22×C4—141no—
241C22×C4—141no—
242Q16—142no—
243Q16—281no—
244SD16—41no—
245SD16—142no—
246SD16—281no—
247SD16—281no—
248SD16—281no—
249D8—41no—
250D8—142no—
251C2×C8—71no—
252C2×C8—71no—
253C2×C8C22×D711yescharacteristic
254C2×C8C22×D711yescharacteristic
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4C22×D711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—71no—
277C42—71no—
278C42—141no—
279C42—141no—
280C42—141no—
281C42C22×D711yescharacteristic
282C2×C7—81no—
283C2×C7—81no—
284C2×C7#55011yescharacteristic
285C2×C7#57511yescharacteristic
286C2×C7#57711yescharacteristic
287D7—81no—
288C23—21no—
289C23—21no—
290C23—141no—
291Q8—21no—
292Q8—21no—
293Q8—141no—
294Q8—141no—
295Q8—141no—
296Q8—141no—
297Q8—281no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—141no—
303D4—141no—
304D4—282no—
305D4—561no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—71no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—561no—
317C2×C4—561no—
318C2×C4D4×D711yescharacteristic
319C2×C4D4×D711yescharacteristic
320C2×C4C23×D711yescharacteristic
321C8—21no—
322C8—21no—
323C8—72no—
324C8—141no—
325C7#340111yescharacteristic, complemented, Sylow
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22C2×D4×D711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—141no—
341C4—141no—
342C4—141no—
343C4—141no—
344C4—281no—
345C4—281no—
346C4—281no—
347C4#654811yescharacteristic
348C4D7×#4111yescharacteristic
349C2—81no—
350C2—81no—
351C2—561no—
352C2#9029011yescharacteristic
353C2#9046111yescharacteristic
354C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#463C2
socleC22×C7C22×D4
prime radical#89926C2
hypercenter#463D7
hypocenterC7#3401
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #90372C2semidirect
2. #90233C2semidirect
3. C7×#463C2semidirect
4. #89940C2semidirect
5. #89938C2semidirect
7. #89925C2semidirect
8. #89832C2semidirect
9. #89786C2semidirect
10. #89613C2semidirect
11. #89600C2semidirect
12. #89570C2semidirect
13. #89561C2semidirect
14. #89534C2semidirect
15. #89530C2semidirect
120. #463D7semidirect
325. C7#3401semidirect

Maximal subgroups

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

subgroupindex
#903722
#902332
C7×#4632
#899402
#899382
#899262
#899252
#898322
#897862
#896132
#896002
#895702
#895612
#895342
#895302
#34017