Finite Group Databasebeta

← #11738956

Subgroups — #11738956

Order 896.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90333C211yesmaximal, characteristic
2#90220C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic, complemented
4#89824C211yesmaximal, characteristic, complemented
5#89767C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic, complemented
7#89650C211yesmaximal, characteristic, complemented
8#89633C211yesmaximal, characteristic, complemented
9#89629C211yesmaximal, characteristic, complemented
10#89588C211yesmaximal, characteristic, complemented
11#89581C211yesmaximal, characteristic, complemented
12#89568C211yesmaximal, characteristic, complemented
13#89562C211yesmaximal, characteristic, complemented
14#89501C211yesmaximal, characteristic
15#89494C211yesmaximal, characteristic, complemented
16C7×C22⋊Q8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18#6509C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C4×C22⋊D7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23#6487C2211yescharacteristic
24C2×#1136C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26#6461—21no—
27#6460C2211yescharacteristic
28#6455C2211yescharacteristic
29D7×C4⋊C4C2211yescharacteristic
30#6453—21no—
31#6453C2211yescharacteristic
32#6453C2211yescharacteristic
33#6450C2211yescharacteristic
34#6449—21no—
35#6446—21no—
36#6442—21no—
37#6440—21no—
38#6435—21no—
39#6433—21no—
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6392C2211yescharacteristic
48#6392C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3352—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7D411yescharacteristic
63C22×Dic7—21no—
64C2×C4×D7—21no—
65C2×C4×D7C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×C7×C8C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4C2311yescharacteristic
71C7×C4⋊C4C2311yescharacteristic
72C7×C4⋊C4C2311yescharacteristic
73C7×C22⋊C4—21no—
74C7×C22⋊C4—21no—
75C42×C7—21no—
76C22⋊Dic7—21no—
77C22⋊Dic7D411yescharacteristic
78#1144—22no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4D411yescharacteristic
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102#1136—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106#552—71no—
107#552—71no—
108#499—71no—
109#485—71no—
110#485D711yescharacteristic, complemented
111#484—71no—
112#484—71no—
113#479—71no—
114#479—71no—
115#470—71no—
116#470—71no—
117C4×Q16—71no—
118#436—71no—
119#428—71no—
120#428—71no—
121C23×C7C2×D411yescharacteristic
122C22×D7C2×D411yescharacteristic
123C7×Q8—21no—
124C7×Q8—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C22⋊D7—41no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—42no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C4×D7—21no—
143C4×D7—21no—
144C4×D7—42no—
145Dic14—21no—
146Dic14—21no—
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8#4112yes—
151C2×Q16—71no—
152C2×M16—71no—
153C2×M16—71no—
154#126—141no—
155#126—141no—
156#125—71no—
157#125—71no—
158#125—141no—
159#125—141no—
160#123—141no—
161#123—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164C22⋊Q8—71no—
165C22⋊Q8C2×D711yescharacteristic
166C4×Q8—71no—
167C4×Q8—71no—
168C4×D4—71no—
169#117—71no—
170#117—141no—
171#117—141no—
172#117C2×D711yescharacteristic
173C2×C4⋊C4—71no—
174C2×C4⋊C4—71no—
175#107—71no—
176#107—71no—
177#107—71no—
178#107—71no—
179#107C2×D711yescharacteristic
180#106—71no—
181#106—71no—
182#106—71no—
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
189#103C2×D711yescharacteristic
190#103C2×D711yescharacteristic
191C22⋊C8—71no—
192C22⋊C8C2×D711yescharacteristic
193C4×C8—71no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7C22×D411yescharacteristic
198C2×D7—21no—
199C2×D7—21no—
200C2×D7—21no—
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7C2×#4111yescharacteristic
207C4×C72−1+411yescharacteristic
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215C2×Q8—71no—
216C2×Q8C22×D711yescharacteristic
217C2×D4—71no—
218C22×C4—71no—
219C22×C4—141no—
220C22×C4—141no—
221C22×C4C22×D711yescharacteristic
222Q16—142no—
223M16—142no—
224M16—142no—
225C2×C8—71no—
226C2×C8—71no—
227C2×C8C22×D711yescharacteristic
228C2×C8C22×D711yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4C22×D711yescharacteristic
246C4⋊C4C22×D711yescharacteristic
247C4⋊C4C22×D711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—71no—
251C22⋊C4—71no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C42—21no—
257C42—71no—
258C42—141no—
259C42—141no—
260C42—141no—
261C2×C7—41no—
262C2×C7#54711yescharacteristic
263C2×C7#57511yescharacteristic
264C2×C7#57811yescharacteristic
265D7—41no—
266C23—71no—
267C23D4×D711yescharacteristic
268Q8—21no—
269Q8—21no—
270Q8—141no—
271Q8—141no—
272D4—281no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—282no—
289C2×C4—282no—
290C2×C4D4×D711yescharacteristic
291C2×C4C23×D711yescharacteristic
292C8—21no—
293C8—21no—
294C8—72no—
295C8—141no—
296C7#335211yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—141no—
301C22—141no—
302C22—141no—
303C22C2×D4×D711yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—141no—
310C4—141no—
311C4—141no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4D7×#4111yescharacteristic
317C4#655411yescharacteristic
318C2—41no—
319C2—281no—
320C2#9034611yescharacteristic
321C2#9046211yescharacteristic
322C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#485D7
hypocenterC7#3352
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. C7×#485C2semidirect
4. #89824C2semidirect
5. #89767C2semidirect
6. #89660C2semidirect
7. #89650C2semidirect
8. #89633C2semidirect
9. #89629C2semidirect
10. #89588C2semidirect
11. #89581C2semidirect
12. #89568C2semidirect
13. #89562C2semidirect
15. #89494C2semidirect
110. #485D7semidirect
296. C7#3352semidirect

Maximal subgroups

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

subgroupindex
#903332
#902202
C7×#4852
#898242
#897672
#896602
#896502
#896332
#896292
#895882
#895812
#895682
#895622
#895012
#894942
#33527