Finite Group Databasebeta

← #11738978

Subgroups — #11738978

Order 896.

340 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#90332C211yesmaximal, characteristic
2#90224C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic, complemented
4#89823C211yesmaximal, characteristic, complemented
5#89770C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic, complemented
7#89646C211yesmaximal, characteristic, complemented
8#89632C211yesmaximal, characteristic, complemented
9#89626C211yesmaximal, characteristic, complemented
10#89586C211yesmaximal, characteristic, complemented
11#89585C211yesmaximal, characteristic, complemented
12#89563C211yesmaximal, characteristic, complemented
13#89562C211yesmaximal, characteristic, complemented
14(C22⋊D7)⋊C8C211yesmaximal, characteristic
15#89495C211yesmaximal, characteristic, complemented
16C7×C22⋊Q8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18#6509C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C4×C22⋊D7C2211yescharacteristic
22#6489C2211yescharacteristic
23#6487C2211yescharacteristic
24C22×C7⋊C8C2211yescharacteristic
25C2×C8×D7C2211yescharacteristic
26#6461—21no—
27#6460C2211yescharacteristic
28#6455C2211yescharacteristic
29#6455C2211yescharacteristic
30#6451C2211yescharacteristic
31#6451C2211yescharacteristic
32#6450—21no—
33#6450C2211yescharacteristic
34#6448—21no—
35#6448—21no—
36#6442—21no—
37#6442—21no—
38#6439—21no—
39C4×Dic14—21no—
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6392C2211yescharacteristic
48#6391C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3342—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7D411yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C2×Dic14—21no—
65C2×Dic14—21no—
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⋊Dic7—21no—
78C22⋊Dic7D411yescharacteristic
79#1144—22no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7C2311yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4D411yescharacteristic
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C8×D7—22no—
106#556—71no—
107#556—71no—
108#500—71no—
109#485—71no—
110#485—71no—
111#485—71no—
112#485D711yescharacteristic, complemented
113#475—71no—
114#475—71no—
115#467—71no—
116#467—71no—
117C4×Q16—71no—
118C8×D4—71no—
119#427—71no—
120#427—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—22no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C4×D7—21no—
142C4×D7—21no—
143Dic14—21no—
144Dic14—21no—
145Dic14—42no—
146Dic14—42no—
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8#4112yes—
151C2×Q16—71no—
152C22×C8—71no—
153C22×C8—71no—
154C4⋊Q8—71no—
155C4⋊Q8—71no—
156#126—141no—
157#126—141no—
158#126—141no—
159#126—141no—
160#124—141no—
161#124—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164C22⋊Q8—71no—
165C22⋊Q8C2×D711yescharacteristic
166C4×Q8—71no—
167C4×Q8—71no—
168C4×Q8—141no—
169C4×Q8—141no—
170C4×D4—71no—
171#117—71no—
172#117—71no—
173#117—71no—
174#117C2×D711yescharacteristic
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—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7#13512yes—
215C2×Q8—71no—
216C2×Q8—141no—
217C2×Q8—141no—
218C2×Q8C22×D711yescharacteristic
219C2×D4—71no—
220C22×C4—71no—
221C22×C4C22×D711yescharacteristic
222Q16—142no—
223C2×C8—71no—
224C2×C8—71no—
225C2×C8—142no—
226C2×C8—142no—
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⋊C4C22×D711yescharacteristic
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—71no—
247C22⋊C4—71no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C42—21no—
255C42—71no—
256C42—141no—
257C42—141no—
258C42—141no—
259C42—141no—
260C42—141no—
261C2×C7—41no—
262C2×C7#54711yescharacteristic
263C2×C7C2×#13511yescharacteristic
264C2×C7#57811yescharacteristic
265D7—41no—
266C23—71no—
267C23D4×D711yescharacteristic
268Q8—21no—
269Q8—21no—
270Q8—141no—
271Q8—141no—
272Q8—282no—
273Q8—282no—
274D4—142no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—71no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4D4×D711yescharacteristic
291C2×C4C23×D711yescharacteristic
292C8—21no—
293C8—21no—
294C8—72no—
295C8—141no—
296C7#334211yescharacteristic, 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—72no—
310C4—141no—
311C4—141no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4D7×#4111yescharacteristic
317C4#655411yescharacteristic
318C2—41no—
319C2—281no—
320C2#9034611yescharacteristic
321C2D7×#13511yescharacteristic
322C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#485D7
hypocenterC7#3342
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. #89823C2semidirect
5. #89770C2semidirect
6. #89660C2semidirect
7. #89646C2semidirect
8. #89632C2semidirect
9. #89626C2semidirect
10. #89586C2semidirect
11. #89585C2semidirect
12. #89563C2semidirect
13. #89562C2semidirect
15. #89495C2semidirect
112. #485D7semidirect
296. C7#3342semidirect

Maximal subgroups

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

subgroupindex
#903322
#902242
C7×#4852
#898232
#897702
#896602
#896462
#896322
#896262
#895862
#895852
#895632
#895622
(C22⋊D7)⋊C82
#894952
#33427