Finite Group Databasebeta

← #11737235

Subgroups — #11737235

Order 896.

410 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90271C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic, complemented
3C7×#440C211yesmaximal, characteristic, complemented
4#89948C211yesmaximal, characteristic, complemented
5#89943C211yesmaximal, characteristic, complemented
6#89798C211yesmaximal, characteristic, complemented
7#89786C211yesmaximal, characteristic, complemented
8#89655C211yesmaximal, characteristic, complemented
9#89648C211yesmaximal, characteristic, complemented
10#89579C211yesmaximal, characteristic, complemented
11#89567C211yesmaximal, characteristic, complemented
12#89547C211yesmaximal, characteristic, complemented
13#89536C211yesmaximal, characteristic, complemented
14#89483C211yesmaximal, characteristic, complemented
15D7×#97C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C2×C7×SD16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6510C2211yescharacteristic
21#6509C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6503C2211yescharacteristic
24#6501C2211yescharacteristic
25C2×#1141C2211yescharacteristic
26C4×C22⋊D7—21no—
27C2×D56C2211yescharacteristic
28C2×C8×D7C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6458C2211yescharacteristic
32#6458C2211yescharacteristic
33#6457—21no—
34#6453C2211yescharacteristic
35#6451C2211yescharacteristic
36#6448—21no—
37#6447—21no—
38C4×D28—21no—
39C4×D28C2211yescharacteristic
40C4×D28C2211yescharacteristic
41#6435—21no—
42C42×D7C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6391C2211yescharacteristic
52#6389C2211yescharacteristic
53#6389C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3180—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66#1157—44no—
67C2×D28—21no—
68C2×D28—21no—
69C2×D28C2311yescharacteristic
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7D411yescharacteristic
74C2×C4×D7D411yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C7×SD16—22no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C42×C7—21no—
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87#1143—22no—
88#1141—22no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107D56—22no—
108C8×D7—22no—
109C8×D7—22no—
110#532—71no—
111#529—71no—
112#501—71no—
113#496—71no—
114#490—71no—
115#489—71no—
116#466—71no—
117#466—71no—
118#465—71no—
119#465—71no—
120#442—71no—
121#442—71no—
122#440—71no—
123#440D711yescharacteristic, complemented
124C2×#97—71no—
125C23×C7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7C2×D411yescharacteristic
129C7×Q8—21no—
130C7×Q8—21no—
131C7×D4—21no—
132C7×D4—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—42no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—42no—
141C22⋊D7—42no—
142C22⋊D7—42no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148D28—21no—
149D28—21no—
150D28—21no—
151D28—21no—
152D28—42no—
153D28—42no—
154C4×D7—22no—
155C4×D7—21no—
156C4×D7—21no—
157C4×D7—41no—
158C4×D7—42no—
159C4×D7—42no—
160Dic14—42no—
161C7×C8—21no—
162C7×C8#4112yescomplemented
163C7⋊C8—21no—
164C7⋊C8#4112yescomplemented
165C2×#41—141no—
166C2×SD16—71no—
167C2×SD16C2×D711yescharacteristic
168C2×D8—71no—
169C2×D8—71no—
170C22×C8—71no—
171C22×C8—71no—
172C4⋊Q8—71no—
173#127—71no—
174#126—141no—
175#125—71no—
176#124—71no—
177#124—141no—
178#123—141no—
179C22⋊Q8—71no—
180#121—71no—
181#121—71no—
182#121—71no—
183#121—141no—
184C4×Q8C2×D711yescharacteristic
185C4×D4—71no—
186C4×D4—71no—
187C4×D4—141no—
188C4×D4—141no—
189C4×D4C2×D711yescharacteristic
190#117—141no—
191C2×C42—71no—
192#107—71no—
193#107C2×D711yescharacteristic
194#106—71no—
195#106—71no—
196#103—71no—
197#103C2×D711yescharacteristic
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203#102C2×D711yescharacteristic
204#97—71no—
205#97—71no—
206#97—71no—
207#97C2×D711yescharacteristic
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C22×D411yescharacteristic
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7AGL(1,ℤ/8)12yes—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7C2×#4111yescharacteristic
227C4×C7C2×#4111yescharacteristic
228Dic7—21no—
229Dic7—21no—
230Dic7—21no—
231Dic7—41no—
232Dic7—41no—
233#41—284no—
234C2×Q8—141no—
235C2×Q8C22×D711yescharacteristic
236C2×D4—71no—
237C2×D4—71no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4—141no—
242C2×D4—141no—
243C2×D4C22×D711yescharacteristic
244C22×C4—21no—
245C22×C4—71no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250SD16—22no—
251SD16—142no—
252D8—142no—
253D8—142no—
254C2×C8—71no—
255C2×C8—71no—
256C2×C8—142no—
257C2×C8—142no—
258C2×C8C22×D711yescharacteristic
259C2×C8C22×D711yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—21no—
279C42—71no—
280C42—71no—
281C42—71no—
282C42C22×D711yescharacteristic
283C2×C7—81no—
284C2×C7#53211yescharacteristic
285C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
286C2×C7#57511yescharacteristic
287D7—22no—
288D7—81no—
289D7—81no—
290C23—21no—
291C23—71no—
292C23—141no—
293C23—141no—
294Q8—21no—
295Q8—21no—
296Q8—282no—
297D4—21no—
298D4—21no—
299D4—141no—
300D4—141no—
301D4—141no—
302D4—141no—
303D4—282no—
304D4—282no—
305D4—282no—
306D4—282no—
307D4—282no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—71no—
313C2×C4—71no—
314C2×C4—71no—
315C2×C4—142no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—281no—
321C2×C4—282no—
322C2×C4—282no—
323C2×C4D4×D711yescharacteristic
324C2×C4D4×D711yescharacteristic
325C2×C4C23×D711yescharacteristic
326C8—21no—
327C8—72no—
328C8—141no—
329C8#115712yescomplemented
330C7#318011yescharacteristic, complemented, Sylow
331C22—41no—
332C22—41no—
333C22—41no—
334C22—72no—
335C22—141no—
336C22—141no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22—281no—
342C22—281no—
343C22C2×D4×D711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—141no—
350C4—141no—
351C4—141no—
352C4—281no—
353C4—281no—
354C4C2×#115711yescharacteristic
355C4D7×#4111yescharacteristic
356C2—81no—
357C2—142no—
358C2—561no—
359C2—561no—
360C2#9023911yescharacteristic
361C2D7×AGL(1,ℤ/8)11yescharacteristic
362C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
hypercenter#440D7
hypocenterC7#3180
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90271C2semidirect
2. #90239C2semidirect
3. C7×#440C2semidirect
4. #89948C2semidirect
5. #89943C2semidirect
6. #89798C2semidirect
7. #89786C2semidirect
8. #89655C2semidirect
9. #89648C2semidirect
10. #89579C2semidirect
11. #89567C2semidirect
12. #89547C2semidirect
13. #89536C2semidirect
14. #89483C2semidirect
15. D7×#97C2semidirect
123. #440D7semidirect
162. C7×C8#41semidirect
164. C7⋊C8#41semidirect
329. C8#1157semidirect
330. C7#3180semidirect

Maximal subgroups

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

subgroupindex
#902712
#902392
C7×#4402
#899482
#899432
#897982
#897862
#896552
#896482
#895792
#895672
#895472
#895362
#894832
D7×#972
#31807