Finite Group Databasebeta

← #11738862

Subgroups — #11738862

Order 896.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90334C211yesmaximal, characteristic, complemented
2#90195C211yesmaximal, characteristic, complemented
3C7×#483C211yesmaximal, characteristic, complemented
4#89819C211yesmaximal, characteristic, complemented
5#89740C211yesmaximal, characteristic, complemented
6#89642C211yesmaximal, characteristic, complemented
7#89641C211yesmaximal, characteristic, complemented
8#89626C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic, complemented
10#89591C211yesmaximal, characteristic, complemented
11#89570C211yesmaximal, characteristic, complemented
12#89561C211yesmaximal, characteristic, complemented
13#89560C211yesmaximal, characteristic, complemented
14#89506C211yesmaximal, characteristic
15#89492C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C7×C22⋊Q8C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19#6510C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6493C2211yescharacteristic
24C4×C22⋊D7—21no—
25C4×C22⋊D7—21no—
26#6489C2211yescharacteristic
27C2×D56C2211yescharacteristic
28C2×#1132C2211yescharacteristic
29#6461—21no—
30#6460—21no—
31#6456C2211yescharacteristic
32#6456C2211yescharacteristic
33#6456C2211yescharacteristic
34#6455—21no—
35D7×C4⋊C4—21no—
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6450C2211yescharacteristic
39#6448—21no—
40#6446—21no—
41C7×#106C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×C22⋊C8C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6395C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3125—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7—21no—
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65#1157—22no—
66#1157—22no—
67C2×D28C2311yescharacteristic
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C22⋊Dic7D411yescharacteristic
84C22⋊Dic7D411yescharacteristic
85#1144—22no—
86#1143—22no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—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⋊Dic7D411yescharacteristic
95C4⋊Dic7D411yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7C2311yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107D56—22no—
108#1132—22no—
109#542—71no—
110#520—71no—
111#483D711yescharacteristic, complemented
112#481—71no—
113#476—71no—
114#475—71no—
115#464—71no—
116#463—71no—
117#452—71no—
118#449—71no—
119#442—71no—
120#441—71no—
121C4×SD16—71no—
122C4×SD16—71no—
123#433—71no—
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C22×D7—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—81no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—24no—
139C22⋊D7—81no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—21no—
147D28—21no—
148D28—21no—
149D28#4112yes—
150C4×D7—24no—
151C4×D7—81no—
152C4×D7—81no—
153C4×D7—81no—
154Dic14—21no—
155Dic14—81no—
156Dic14#4112yes—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C2×#41—71no—
162C2×Q16—71no—
163C2×SD16—71no—
164C2×SD16—71no—
165C2×D8—71no—
166C4⋊Q8—71no—
167#126—141no—
168#124—71no—
169#124—141no—
170#123—141no—
171C22⋊Q8—141no—
172C22⋊Q8C2×D711yescharacteristic
173#121—71no—
174C4×Q8—71no—
175C4×Q8—71no—
176C4×Q8—71no—
177C4×D4—71no—
178C4×D4—71no—
179C4×D4—71no—
180C4×D4—141no—
181C4×D4—141no—
182#117—71no—
183#117—141no—
184C2×C4⋊C4—141no—
185C2×C4⋊C4C2×D711yescharacteristic
186#106—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189#106C2×D711yescharacteristic
190C4⋊C8—71no—
191C4⋊C8—71no—
192#103—71no—
193#103—71no—
194#103C2×D711yescharacteristic
195#103C2×D711yescharacteristic
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8—71no—
201C22⋊C8C2×D711yescharacteristic
202#97—71no—
203C4×C8—71no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7C22×D411yescharacteristic
208C2×D7—22no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C4×C7—21no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C7C2×#4111yescharacteristic
221Dic7—22no—
222Dic7—21no—
223Dic7—21no—
224Dic7—21no—
225Dic7—21no—
226Dic7—41no—
227#41—142no—
228#41—142no—
229C2×Q8—71no—
230C2×Q8—141no—
231C2×Q8C22×D711yescharacteristic
232C2×D4—71no—
233C2×D4—71no—
234C2×D4—141no—
235C2×D4—141no—
236C22×C4—21no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4C22×D711yescharacteristic
242Q16—142no—
243SD16—142no—
244SD16—142no—
245D8—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4C22×D711yescharacteristic
260C4⋊C4C22×D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—71no—
264C22⋊C4—71no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C42—71no—
272C42—71no—
273C42—141no—
274C42—141no—
275C42—141no—
276C2×C7—41no—
277C2×C7#52511yescharacteristic
278C2×C7#57711yescharacteristic
279C2×C7#57711yescharacteristic
280D7—42no—
281D7—81no—
282C23—141no—
283C23—141no—
284C23D4×D711yescharacteristic
285Q8—21no—
286Q8—21no—
287Q8—72no—
288Q8—141no—
289Q8—561no—
290D4—72no—
291D4—144no—
292D4—141no—
293D4—141no—
294D4—141no—
295D4—561no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—71no—
305C2×C4—71no—
306C2×C4—81no—
307C2×C4—144no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—561no—
311C2×C4—561no—
312C2×C4—561no—
313C2×C4D4×D711yescharacteristic
314C2×C4C23×D711yescharacteristic
315C8—21no—
316C8—21no—
317C8—141no—
318C8—141no—
319C7#312511yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—21no—
323C22—142no—
324C22—281no—
325C22—281no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—141no—
336C4—141no—
337C4—141no—
338C4—141no—
339C4—142no—
340C4—281no—
341C4D7×#4111yescharacteristic
342C4D7×#4111yescharacteristic
343C2—41no—
344C2—282no—
345C2—561no—
346C2#9034711yescharacteristic
347C2#9045311yescharacteristic
348C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#483C2
socleC22×C7C22×D4
prime radical#89506C2
hypercenter#483D7
hypocenterC7#3125
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #90334C2semidirect
2. #90195C2semidirect
3. C7×#483C2semidirect
4. #89819C2semidirect
5. #89740C2semidirect
6. #89642C2semidirect
7. #89641C2semidirect
8. #89626C2semidirect
9. #89624C2semidirect
10. #89591C2semidirect
11. #89570C2semidirect
12. #89561C2semidirect
13. #89560C2semidirect
15. #89492C2semidirect
111. #483D7semidirect
319. C7#3125semidirect

Maximal subgroups

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

subgroupindex
#903342
#901952
C7×#4832
#898192
#897402
#896422
#896412
#896262
#896242
#895912
#895702
#895612
#895602
#895062
#894922
#31257