Finite Group Databasebeta

← #11738662

Subgroups — #11738662

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90381C211yesmaximal, characteristic, complemented
2#90276C211yesmaximal, characteristic, complemented
3C7×#479C211yesmaximal, characteristic, complemented
4#89839C211yesmaximal, characteristic, complemented
5#89798C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89644C211yesmaximal, characteristic, complemented
8#89643C211yesmaximal, characteristic, complemented
9#89641C211yesmaximal, characteristic, complemented
10#89615C211yesmaximal, characteristic
11#89600C211yesmaximal, characteristic, complemented
12#89591C211yesmaximal, characteristic, complemented
13#89591C211yesmaximal, characteristic, complemented
14#89572C211yesmaximal, characteristic, complemented
15#89567C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6510C2211yescharacteristic
19#6510C2211yescharacteristic
20C2×#1143C2211yescharacteristic
21C2×D56C2211yescharacteristic
22C2×D56C2211yescharacteristic
23#6460—21no—
24#6459—21no—
25#6459—21no—
26#6457—21no—
27#6457—21no—
28#6456C2211yescharacteristic
29#6456C2211yescharacteristic
30#6456C2211yescharacteristic
31#6456C2211yescharacteristic
32D7×C4⋊C4—21no—
33#6453C2211yescharacteristic
34#6451C2211yescharacteristic
35#6438—21no—
36#6437C2211yescharacteristic
37C4×D28—21no—
38C4×D28C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6389C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3370—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×D28—21no—
62C2×D28—21no—
63C2×D28C2311yescharacteristic
64C2×D28C2311yescharacteristic
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×Dic14—21no—
72C2×Dic14D411yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C42×C7—21no—
82C42×C7C2311yescharacteristic
83#1143—22no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
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⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104D56—22no—
105D56—22no—
106#559—71no—
107#554—71no—
108#490—71no—
109#489—71no—
110#488—71no—
111#479D711yescharacteristic, complemented
112#478—71no—
113#474—71no—
114#463—71no—
115#463—71no—
116#459—71no—
117#446—71no—
118#442—71no—
119#440—71no—
120C4×D8—71no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
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×Dic7C2×D411yescharacteristic
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138D28—21no—
139D28—21no—
140D28—21no—
141D28—21no—
142D28—21no—
143D28—42no—
144D28—81no—
145D28#4112yes—
146C4×D7—42no—
147C4×D7—81no—
148C4×D7—81no—
149C4×D7—81no—
150C4×D7—81no—
151Dic14—41no—
152Dic14—81no—
153C7×C8—21no—
154C7×C8—21no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×SD16—71no—
158C2×D8—71no—
159C2×D8—71no—
160C4⋊Q8—71no—
161#127—71no—
162#125—71no—
163#125C2×D711yescharacteristic
164#124—71no—
165#124—71no—
166#124—141no—
167#123—141no—
168#123—141no—
169C22⋊Q8—141no—
170C22⋊Q8—141no—
171C22⋊Q8—141no—
172C4×Q8—71no—
173C4×Q8C2×D711yescharacteristic
174C4×D4—71no—
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—71no—
179C4×D4—141no—
180C2×C4⋊C4—141no—
181#107—71no—
182#107C2×D711yescharacteristic
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185C4⋊C8—71no—
186C4⋊C8C2×D711yescharacteristic
187#103—71no—
188#103C2×D711yescharacteristic
189#103C2×D711yescharacteristic
190#102—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#97—71no—
198#97—71no—
199C4×C8—71no—
200C22×C7C22×D411yescharacteristic
201C2×D7—22no—
202C2×D7—41no—
203C2×D7—41no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C72−1+411yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223C2×Q8—71no—
224C2×Q8—141no—
225C2×Q8C22×D711yescharacteristic
226C2×D4—71no—
227C2×D4—71no—
228C2×D4—71no—
229C2×D4—141no—
230C2×D4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—141no—
236SD16—142no—
237D8—142no—
238D8—142no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—21no—
266C42—71no—
267C42—71no—
268C42C22×D711yescharacteristic
269C2×C7#54811yescharacteristic
270C2×C7#57511yescharacteristic
271C2×C7#57611yescharacteristic
272D7—42no—
273D7—81no—
274D7—81no—
275C23—141no—
276C23—141no—
277C23—141no—
278Q8—21no—
279Q8—21no—
280Q8—281no—
281Q8—561no—
282D4—72no—
283D4—141no—
284D4—141no—
285D4—141no—
286D4—141no—
287D4—141no—
288D4—282no—
289D4—561no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—71no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—141no—
299C2×C4—282no—
300C2×C4—561no—
301C2×C4—561no—
302C2×C4—561no—
303C2×C4—561no—
304C2×C4D4×D711yescharacteristic
305C2×C4D4×D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—21no—
308C8—21no—
309C8—141no—
310C8—141no—
311C7#337011yescharacteristic, complemented, Sylow
312C22—142no—
313C22—281no—
314C22—281no—
315C22—281no—
316C22—281no—
317C22—281no—
318C22—281no—
319C22—281no—
320C22—281no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—141no—
329C4—141no—
330C4—141no—
331C4—141no—
332C4—281no—
333C4#655111yescharacteristic
334C4D7×#4111yescharacteristic
335C2—282no—
336C2—561no—
337C2—561no—
338C2#9032411yescharacteristic
339C2#9046011yescharacteristic
340C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radical#89615C2
hypercenter#479D7
hypocenterC7#3370
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
1. #90381C2semidirect
2. #90276C2semidirect
3. C7×#479C2semidirect
4. #89839C2semidirect
5. #89798C2semidirect
6. #89662C2semidirect
7. #89644C2semidirect
8. #89643C2semidirect
9. #89641C2semidirect
11. #89600C2semidirect
12. #89591C2semidirect
13. #89591C2semidirect
14. #89572C2semidirect
15. #89567C2semidirect
111. #479D7semidirect
311. C7#3370semidirect

Maximal subgroups

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

subgroupindex
#903812
#902762
C7×#4792
#898392
#897982
#896622
#896442
#896432
#896412
#896152
#896002
#895912
#895722
#895672
#33707