Finite Group Databasebeta

← #11738962

Subgroups — #11738962

Order 896.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×C22⋊Q8C211yesmaximal, characteristic
2D7×#117C211yesmaximal, characteristic
3C7×#485C211yesmaximal, characteristic, complemented
4#89817C211yesmaximal, characteristic, complemented
5#89768C211yesmaximal, characteristic, complemented
6#89656C211yesmaximal, characteristic, complemented
7#89653C211yesmaximal, characteristic, complemented
8#89634C211yesmaximal, characteristic, complemented
9#89633C211yesmaximal, characteristic, complemented
10#89581C211yesmaximal, characteristic, complemented
11#89577C211yesmaximal, characteristic, complemented
12#89574C211yesmaximal, characteristic, complemented
13#89574C211yesmaximal, characteristic, complemented
14#89498C211yesmaximal, characteristic
15#89495C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#117C2211yescharacteristic
20#6509C2211yescharacteristic
21#6489C2211yescharacteristic
22#6487C2211yescharacteristic
23C2×#1131C2211yescharacteristic
24C2×#1131C2211yescharacteristic
25#6460C2211yescharacteristic
26#6459—21no—
27#6455C2211yescharacteristic
28#6455C2211yescharacteristic
29D7×C4⋊C4—21no—
30D7×C4⋊C4C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic
33D7×C22⋊C4—21no—
34D7×C22⋊C4—21no—
35#6441—21no—
36#6440—21no—
37#6435—21no—
38C42×D7—21no—
39C7×#107C2211yescharacteristic, complemented
40C7×#106C2211yescharacteristic, complemented
41C7×#103C2211yescharacteristic, complemented
42C7×#103C2211yescharacteristic, complemented
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic, complemented
45#6409C2211yescharacteristic, complemented
46C22⋊(C7⋊C8)C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic, complemented
50#6391C2211yescharacteristic, complemented
51#6390C2211yescharacteristic, complemented
52#6390C2211yescharacteristic, complemented
53#6384C2211yescharacteristic, complemented
54#6384C2211yescharacteristic, complemented
55#6382C2211yescharacteristic, complemented
56#6382C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3095—71nomaximal, Sylow
60C23×D7D411yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7D411yescharacteristic
64Q8×D7—22no—
65Q8×D7—22no—
66C2×C4×D7—22no—
67C2×C4×D7—22no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7D411yescharacteristic
73C2×C4×D7D411yescharacteristic
74C2×C4×D7C2311yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C42×C7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108#1131—22no—
109C2×C22⋊Q8—71no—
110C2×#117—71no—
111#485—71no—
112#485—71no—
113#485—71no—
114#485D711yescharacteristic, complemented
115#484—71no—
116#484—71no—
117#483—71no—
118#483—71no—
119#428—71no—
120#428—71no—
121#419—71no—
122#419—71no—
123#407—71no—
124C23×C7C2×D411yescharacteristic
125C22×D7—22no—
126C22×D7—22no—
127C22×D7—41no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C2×Dic7—22no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C4×D7—22no—
148C4×D7—21no—
149C4×D7—21no—
150C4×D7—22no—
151C4×D7—42no—
152C4×D7—42no—
153C4×D7—42no—
154C4×D7—42no—
155C4×D7#4112yes—
156C4×D7#4112yes—
157Dic14—21no—
158Dic14—21no—
159Dic14—42no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C22×Q8—71no—
165C23×C4—71no—
166C2×M16—71no—
167C2×M16—71no—
168C22⋊Q8—71no—
169C22⋊Q8—71no—
170C22⋊Q8—71no—
171C22⋊Q8—141no—
172C22⋊Q8—141no—
173C22⋊Q8C2×D711yescharacteristic
174#117—71no—
175#117—71no—
176#117—71no—
177#117—141no—
178#117—141no—
179#117C2×D711yescharacteristic
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—141no—
184C2×C22⋊C4—141no—
185C2×C22⋊C4—141no—
186C2×C42—141no—
187#107—71no—
188#107—71no—
189#107—71no—
190#107C2×D711yescharacteristic, complemented
191#106—71no—
192#106—71no—
193#106—71no—
194#106C2×D711yescharacteristic, complemented
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103C2×D711yescharacteristic, complemented
202#103C2×D711yescharacteristic, complemented
203C22⋊C8—71no—
204C22⋊C8—71no—
205C22⋊C8—71no—
206C22⋊C8C2×D711yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—44no—
216C2×D7#12312yes—
217C2×D7#13712yes—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C7C2×#4111yescharacteristic
225Dic7—21no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7#12312yes—
231C24—71no—
232C2×Q8—71no—
233C2×Q8—141no—
234C2×Q8—142no—
235C2×Q8—142no—
236C2×Q8C22×D711yescharacteristic
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—142no—
243C22×C4—142no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4C22×D711yescharacteristic
249M16—142no—
250M16—142no—
251C2×C8—71no—
252C2×C8—71no—
253C2×C8C22×D711yescharacteristic
254C2×C8C22×D711yescharacteristic
255C4⋊C4—21no—
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⋊C4—141no—
265C4⋊C4C22×D711yescharacteristic
266C4⋊C4C22×D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—21no—
277C42—141no—
278C42—141no—
279C42—141no—
280C2×C7—41no—
281C2×C7C2×#12311yescharacteristic
282C2×C7C2×#13711yescharacteristic
283C2×C7#57511yescharacteristic
284D7—22no—
285D7—22no—
286C23—71no—
287C23—71no—
288C23—142no—
289C23—142no—
290C23—281no—
291C23D4×D711yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—141no—
295Q8—141no—
296Q8—282no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—71no—
304C2×C4—72no—
305C2×C4—72no—
306C2×C4—71no—
307C2×C4—142no—
308C2×C4—142no—
309C2×C4—142no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4D4×D711yescharacteristic
321C2×C4C23×D711yescharacteristic
322C8—21no—
323C8—21no—
324C8—141no—
325C8—141no—
326C7#309511yescharacteristic, complemented, Sylow
327C22—21no—
328C22—21no—
329C22—21no—
330C22—72no—
331C22—72no—
332C22—141no—
333C22—141no—
334C22—141no—
335C22—141no—
336C22—284no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—72no—
344C4—141no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4D7×#4111yescharacteristic
350C4D7×#4111yescharacteristic
351C2—41no—
352C2—142no—
353C2—142no—
354C2D7×#12311yescharacteristic
355C2#9045911yescharacteristic
356C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#485C2
socleC22×C7C22×D4
prime radicalC23×D7D4
hypercenter#485D7
hypocenterC7#3095
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. #89817C2semidirect
5. #89768C2semidirect
6. #89656C2semidirect
7. #89653C2semidirect
8. #89634C2semidirect
9. #89633C2semidirect
10. #89581C2semidirect
11. #89577C2semidirect
12. #89574C2semidirect
13. #89574C2semidirect
15. #89495C2semidirect
39. C7×#107C22semidirect
40. C7×#106C22semidirect
41. C7×#103C22semidirect
42. C7×#103C22semidirect
44. #6409C22semidirect
45. #6409C22semidirect
49. #6392C22semidirect
50. #6391C22semidirect
51. #6390C22semidirect
52. #6390C22semidirect
53. #6384C22semidirect
54. #6384C22semidirect
55. #6382C22semidirect
56. #6382C22semidirect
57. #6381C22semidirect
58. #6381C22semidirect
114. #485D7semidirect
190. #107C2×D7semidirect
194. #106C2×D7semidirect
201. #103C2×D7semidirect
202. #103C2×D7semidirect
326. C7#3095semidirect

Maximal subgroups

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

subgroupindex
D7×C22⋊Q82
D7×#1172
C7×#4852
#898172
#897682
#896562
#896532
#896342
#896332
#895812
#895772
#895742
#894982
#894952
#30957