Finite Group Databasebeta

← #11738884

Subgroups — #11738884

Order 896.

348 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#90332C211yesmaximal, characteristic, complemented
2#90205C211yesmaximal, characteristic
3C7×#483C211yesmaximal, characteristic, complemented
4#89821C211yesmaximal, characteristic, complemented
5#89736C211yesmaximal, characteristic, complemented
6#89642C211yesmaximal, characteristic, complemented
7#89638C211yesmaximal, characteristic, complemented
8#89632C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89589C211yesmaximal, characteristic, complemented
11#89586C211yesmaximal, characteristic, complemented
12#89579C211yesmaximal, characteristic, complemented
13#89571C211yesmaximal, characteristic, complemented
14#89509C211yesmaximal, characteristic, complemented
15#89500C211yesmaximal, characteristic, complemented
16C7×C22⋊Q8C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18#6509C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1143C2211yescharacteristic
21#6493—21no—
22#6493C2211yescharacteristic
23C4×C22⋊D7—21no—
24C4×C22⋊D7C2211yescharacteristic
25#6489C2211yescharacteristic
26C2×#1136C2211yescharacteristic
27C2×C8×D7C2211yescharacteristic
28#6461—21no—
29#6461—21no—
30#6458C2211yescharacteristic
31#6457—21no—
32#6456C2211yescharacteristic
33#6455C2211yescharacteristic
34#6455C2211yescharacteristic
35#6453C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450—21no—
38#6448—21no—
39#6442—21no—
40C7×#106C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6391C2211yescharacteristic
51#6389C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3350—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7—21no—
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7D411yescharacteristic
65C2×D28—21no—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×C4×D7C2311yescharacteristic
69C2×Dic14—21no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7D411yescharacteristic
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)⋊C4D411yescharacteristic
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4D411yescharacteristic
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102#1136—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C8×D7—22no—
106#556—71no—
107#553—71no—
108#501—71no—
109#485—71no—
110#483D711yescharacteristic, complemented
111#482—71no—
112#482—71no—
113#479—71no—
114#476—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C23×C7C2×D411yescharacteristic
122C22×D7—21no—
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×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—81no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C22⋊D7—41no—
136C22⋊D7—42no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—21no—
144D28—21no—
145D28—81no—
146C4×D7—21no—
147C4×D7—21no—
148C4×D7—81no—
149Dic14—42no—
150C7×C8—21no—
151C7×C8—21no—
152C7⋊C8—21no—
153C7⋊C8#4112yes—
154C2×SD16—71no—
155C2×M16—71no—
156C22×C8—71no—
157C4⋊Q8—71no—
158#126—141no—
159#126—141no—
160#126—141no—
161#125—71no—
162#124—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8C2×D711yescharacteristic
166#121—71no—
167#121—71no—
168#121—141no—
169C4×Q8—71no—
170C4×Q8—141no—
171C4×D4—71no—
172C4×D4—71no—
173C4×D4—141no—
174#117—71no—
175#117—71no—
176#117—71no—
177C2×C4⋊C4C2×D711yescharacteristic
178#107—71no—
179#107—71no—
180#107—71no—
181#107—71no—
182#106—71no—
183#106—71no—
184#106—71no—
185#106C2×D711yescharacteristic
186#106C2×D711yescharacteristic
187C4⋊C8—71no—
188#103—71no—
189#103C2×D711yescharacteristic
190#103C2×D711yescharacteristic
191#102—71no—
192#102—71no—
193#102—71no—
194C22⋊C8—71no—
195C22⋊C8C2×D711yescharacteristic
196#97—71no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7C22×D411yescharacteristic
201C2×D7—21no—
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7C2×#4111yescharacteristic
213C4×C72−1+411yescharacteristic
214Dic7—21no—
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C2×D4—71no—
224C2×D4—141no—
225C2×D4—141no—
226C22×C4—21no—
227C22×C4—71no—
228C22×C4—141no—
229C22×C4C22×D711yescharacteristic
230SD16—142no—
231M16—142no—
232C2×C8—71no—
233C2×C8—71no—
234C2×C8—142no—
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—71no—
253C22⋊C4—71no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—71no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42—141no—
266C2×C7—41no—
267C2×C7#54711yescharacteristic
268C2×C7#57511yescharacteristic
269C2×C7#57711yescharacteristic
270D7—41no—
271D7—81no—
272C23—71no—
273C23—141no—
274C23D4×D711yescharacteristic
275Q8—21no—
276Q8—21no—
277Q8—282no—
278D4—141no—
279D4—141no—
280D4—281no—
281D4—282no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—71no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—81no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—561no—
299C2×C4D4×D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—21no—
302C8—21no—
303C8—72no—
304C8—141no—
305C7#335011yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—141no—
310C22—141no—
311C22—141no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—141no—
322C4—141no—
323C4—141no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4D7×#4111yescharacteristic
328C4#655411yescharacteristic
329C2—41no—
330C2—281no—
331C2—561no—
332C2#9034611yescharacteristic
333C2#9045311yescharacteristic
334C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#483C2
socleC22×C7C22×D4
prime radical#90205C2
hypercenter#483D7
hypocenterC7#3350
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. #90332C2semidirect
3. C7×#483C2semidirect
4. #89821C2semidirect
5. #89736C2semidirect
6. #89642C2semidirect
7. #89638C2semidirect
8. #89632C2semidirect
9. #89628C2semidirect
10. #89589C2semidirect
11. #89586C2semidirect
12. #89579C2semidirect
13. #89571C2semidirect
14. #89509C2semidirect
15. #89500C2semidirect
110. #483D7semidirect
305. C7#3350semidirect

Maximal subgroups

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

subgroupindex
#903322
#902052
C7×#4832
#898212
#897362
#896422
#896382
#896322
#896282
#895892
#895862
#895792
#895712
#895092
#895002
#33507