Finite Group Databasebeta

← #11738699

Subgroups — #11738699

Order 896.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90386C211yesmaximal, characteristic
2#90272C211yesmaximal, characteristic, complemented
3C7×#479C211yesmaximal, characteristic, complemented
4#89836C211yesmaximal, characteristic, complemented
5C4×#1143C211yesmaximal, characteristic, complemented
6#89654C211yesmaximal, characteristic, complemented
7#89652C211yesmaximal, characteristic, complemented
8#89638C211yesmaximal, characteristic, complemented
9D7×#106C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89606C211yesmaximal, characteristic, complemented
12#89587C211yesmaximal, characteristic, complemented
13#89581C211yesmaximal, characteristic, complemented
14#89579C211yesmaximal, characteristic, complemented
15#89579C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6509C2211yescharacteristic
19#6509C2211yescharacteristic
20C2×#1143C2211yescharacteristic
21C2×C8×D7C2211yescharacteristic
22C2×C8×D7C2211yescharacteristic
23#6461—21no—
24#6458—21no—
25#6458C2211yescharacteristic
26#6458C2211yescharacteristic
27#6457—21no—
28#6457—21no—
29#6456—21no—
30#6455C2211yescharacteristic
31#6455C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic
33D7×C4⋊C4C2211yescharacteristic
34#6452—21no—
35#6439—21no—
36C4×D28C2211yescharacteristic
37C4×D28C2211yescharacteristic
38#6435—21no—
39#6433C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6383C2211yescharacteristic
53#6382C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3337—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×D28—21no—
62C2×D28—21no—
63C2×D28D411yescharacteristic
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7C2311yescharacteristic
69C2×C4×D7C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C42×C7—21no—
79C42×C7C2311yescharacteristic
80#1143—22no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89(C2×D7)⋊C4D411yescharacteristic
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C8×D7—22no—
105C8×D7—22no—
106#553—71no—
107#552—71no—
108#499—71no—
109#485—71no—
110#484—71no—
111#482—71no—
112#480—71no—
113#479D711yescharacteristic, complemented
114#478—71no—
115#465—71no—
116#465—71no—
117C4×SD16—71no—
118C8×D4—71no—
119#427—71no—
120C2×#106—71no—
121C22×D7—21no—
122C22×D7C2×D411yescharacteristic
123C22×D7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138D28—22no—
139D28—21no—
140D28—21no—
141D28—42no—
142D28—81no—
143C4×D7—21no—
144C4×D7—21no—
145C4×D7—21no—
146C4×D7—21no—
147C4×D7—42no—
148C4×D7—81no—
149C4×D7—81no—
150C7×C8—21no—
151C7×C8—21no—
152C7⋊C8—21no—
153C7⋊C8#4112yes—
154C2×SD16—71no—
155C22×C8—71no—
156C22×C8—71no—
157#126—141no—
158#126—141no—
159#125—71no—
160#125—141no—
161#125C2×D711yescharacteristic
162#123—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166#121—71no—
167#121—71no—
168#121—141no—
169C4×Q8C2×D711yescharacteristic
170C4×D4—71no—
171C4×D4—71no—
172C4×D4—141no—
173#117—71no—
174#117—71no—
175#117—141no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107C2×D711yescharacteristic
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186#106C2×D711yescharacteristic
187C4⋊C8C2×D711yescharacteristic
188#103—71no—
189#103C2×D711yescharacteristic
190#103C2×D711yescharacteristic
191#102—71no—
192#102—71no—
193#102—71no—
194C22⋊C8—71no—
195C22⋊C8—71no—
196C4×C8—71no—
197C22×C7C22×D411yescharacteristic
198C2×D7—21no—
199C2×D7—21no—
200C2×D7—21no—
201C2×D7—21no—
202C2×D7—21no—
203C2×D7—41no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7#13512yes—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7C2×#4111yescharacteristic
214C4×C72−1+411yescharacteristic
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8C22×D711yescharacteristic
221C2×D4—71no—
222C2×D4—71no—
223C2×D4—141no—
224C2×D4—141no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4—141no—
230SD16—142no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8—142no—
234C2×C8—142no—
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4C22×D711yescharacteristic
251C4⋊C4C22×D711yescharacteristic
252C4⋊C4C22×D711yescharacteristic
253C22⋊C4—71no—
254C22⋊C4—71no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C42—21no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7#54711yescharacteristic
267C2×C7C2×#13511yescharacteristic
268C2×C7#57711yescharacteristic
269D7—22no—
270D7—41no—
271D7—81no—
272C23—71no—
273C23—71no—
274C23—141no—
275Q8—21no—
276Q8—21no—
277D4—142no—
278D4—141no—
279D4—141no—
280D4—282no—
281D4—561no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—282no—
296C2×C4—561no—
297C2×C4—561no—
298C2×C4D4×D711yescharacteristic
299C2×C4D4×D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—21no—
302C8—21no—
303C8—72no—
304C8—141no—
305C7#333711yescharacteristic, complemented, Sylow
306C22—72no—
307C22—141no—
308C22—141no—
309C22—141no—
310C22—141no—
311C22—141no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—141no—
323C4—141no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4#655111yescharacteristic
328C4D7×#4111yescharacteristic
329C2—142no—
330C2—281no—
331C2—561no—
332C2#9032211yescharacteristic
333C2D7×#13511yescharacteristic
334C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radical#90386C2
hypercenter#479D7
hypocenterC7#3337
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
2. #90272C2semidirect
3. C7×#479C2semidirect
4. #89836C2semidirect
5. C4×#1143C2semidirect
6. #89654C2semidirect
7. #89652C2semidirect
8. #89638C2semidirect
9. D7×#106C2semidirect
10. #89619C2semidirect
11. #89606C2semidirect
12. #89587C2semidirect
13. #89581C2semidirect
14. #89579C2semidirect
15. #89579C2semidirect
113. #479D7semidirect
305. C7#3337semidirect

Maximal subgroups

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

subgroupindex
#903862
#902722
C7×#4792
#898362
C4×#11432
#896542
#896522
#896382
D7×#1062
#896192
#896062
#895872
#895812
#895792
#33377