Finite Group Databasebeta

← #11738240

Subgroups — #11738240

Order 896.

354 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90315C211yesmaximal, characteristic, complemented
2#90291C211yesmaximal, characteristic, complemented
3C7×#468C211yesmaximal, characteristic, complemented
4#89938C211yesmaximal, characteristic, complemented
5#89937C211yesmaximal, characteristic, complemented
6#89906C211yesmaximal, characteristic, complemented
7#89890C211yesmaximal, characteristic, complemented
8#89824C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic
10#89663C211yesmaximal, characteristic, complemented
11#89649C211yesmaximal, characteristic, complemented
12#89570C211yesmaximal, characteristic, complemented
13#89568C211yesmaximal, characteristic, complemented
14#89559C211yesmaximal, characteristic, complemented
15#89538C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23#6499C2211yescharacteristic
24#6498—21no—
25D4×Dic7—21no—
26D4×Dic7C2211yescharacteristic
27C2×#1141C2211yescharacteristic
28#6493C2211yescharacteristic
29#6487C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C2×#1136C2211yescharacteristic
32C2×#1132C2211yescharacteristic
33#6461—21no—
34#6456C2211yescharacteristic
35#6455—21no—
36#6453C2211yescharacteristic
37#6453C2211yescharacteristic
38#6450C2211yescharacteristic
39#6449—21no—
40(C22⋊D7)⋊C4—21no—
41#6442—21no—
42#6440—21no—
43C7×#107C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49#6392C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3256—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66C22×Dic7D411yescharacteristic
67C22×Dic7D411yescharacteristic
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×Dic14C2311yescharacteristic
71C7×SD16—41no—
72C7×M16—41no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84#1144—41no—
85#1141—41no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7D411yescharacteristic
102C4×Dic7D411yescharacteristic
103C4×Dic7C2311yescharacteristic
104C4×Dic7C2311yescharacteristic
105#1136—41no—
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108#1132—41no—
109#529—71no—
110#529—71no—
111#479—71no—
112#479—71no—
113#478—71no—
114#478—71no—
115#470—71no—
116#469—71no—
117#468—71no—
118#468D711yescharacteristic, complemented
119#464—71no—
120#464—71no—
121#463—71no—
122#463—71no—
123#423—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—42no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2411yescharacteristic
139C22⋊D7—42no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—41no—
144C2×Dic7—41no—
145C2×Dic7—42no—
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150D28—21no—
151D28—21no—
152C4×D7—42no—
153Dic14—21no—
154Dic14—21no—
155C7×C8—21no—
156C7×C8—21no—
157C7⋊C8—21no—
158C7⋊C8—21no—
159C2×Q16—71no—
160C2×SD16—71no—
161C2×SD16C2×D711yescharacteristic
162C2×D8—71no—
163C2×M16—71no—
164C2×M16C2×D711yescharacteristic
165#126—141no—
166#126—141no—
167#125—71no—
168#125—71no—
169#124—71no—
170#124—71no—
171#123—141no—
172#123—141no—
173C22⋊Q8—71no—
174C22⋊Q8C2×D711yescharacteristic
175#121—71no—
176#121C2×D711yescharacteristic
177C4×Q8—71no—
178C4×Q8—71no—
179C4×D4—71no—
180C4×D4—71no—
181C4×D4—141no—
182C4×D4—141no—
183#117—141no—
184#117—141no—
185C2×C42—71no—
186#107—71no—
187#107C2×D711yescharacteristic
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192C4⋊C8—71no—
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103C2×D711yescharacteristic
198#102—71no—
199#102—71no—
200#102—71no—
201#102C2×D711yescharacteristic
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7C22×D411yescharacteristic
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7H(5,𝔽2)11yescharacteristic
217C4×C72−1+411yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—71no—
226C2×Q8C22×D711yescharacteristic
227C2×D4—21no—
228C2×D4—71no—
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4C22×D711yescharacteristic
236Q16—281no—
237SD16—41no—
238SD16—281no—
239D8—281no—
240M16—41no—
241M16—281no—
242C2×C8—71no—
243C2×C8—71no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—71no—
269C42—71no—
270C42—71no—
271C42—71no—
272C42—141no—
273C42—141no—
274C2×C7—41no—
275C2×C7—81no—
276C2×C7#53711yescharacteristic
277C2×C7#57511yescharacteristic
278C2×C7#57511yescharacteristic
279D7—81no—
280C23—21no—
281C23—141no—
282C23D4×D711yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—141no—
286Q8—141no—
287D4—21no—
288D4—21no—
289D4—42no—
290D4—141no—
291D4—141no—
292D4—282no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—71no—
299C2×C4—71no—
300C2×C4—71no—
301C2×C4—71no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—282no—
306C2×C4—281no—
307C2×C4—281no—
308C2×C4—282no—
309C2×C4D4×D711yescharacteristic
310C2×C4C23×D711yescharacteristic
311C8—21no—
312C8—21no—
313C8—141no—
314C8—141no—
315C7#325611yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—281no—
323C22—281no—
324C22—281no—
325C22C2×D4×D711yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—141no—
331C4—141no—
332C4—141no—
333C4—141no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4#654811yescharacteristic
338C4#655411yescharacteristic
339C2—41no—
340C2—81no—
341C2—561no—
342C2#9028811yescharacteristic
343C2#9046411yescharacteristic
344C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#468C2
socleC22×C7C22×D4
prime radical#89811C2
hypercenter#468D7
hypocenterC7#3256
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #90315C2semidirect
2. #90291C2semidirect
3. C7×#468C2semidirect
4. #89938C2semidirect
5. #89937C2semidirect
6. #89906C2semidirect
7. #89890C2semidirect
8. #89824C2semidirect
10. #89663C2semidirect
11. #89649C2semidirect
12. #89570C2semidirect
13. #89568C2semidirect
14. #89559C2semidirect
15. #89538C2semidirect
118. #468D7semidirect
315. C7#3256semidirect

Maximal subgroups

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

subgroupindex
#903152
#902912
C7×#4682
#899382
#899372
#899062
#898902
#898242
#898112
#896632
#896492
#895702
#895682
#895592
#895382
#32567