Finite Group Databasebeta

← #11742888

Subgroups — #11742888

Order 896.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×Q8C211yesmaximal, characteristic, complemented
2#90508C211yesmaximal, characteristic, complemented
3Q8×C22⋊D7C211yesmaximal, characteristic, complemented
4#90274C213yesmaximal, complemented
5#90273C213yesmaximal, complemented
6#90272C213yesmaximal, complemented
7#90267C211yesmaximal, characteristic, complemented
8#90265C211yesmaximal, characteristic, complemented
9#90263C211yesmaximal, characteristic, complemented
10#90260C211yesmaximal, characteristic, complemented
11#90259C213yesmaximal, complemented
12#90205C213yesmaximal, complemented
13#90202C213yesmaximal, complemented
14#90170C213yesmaximal, complemented
15#90170C213yesmaximal, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C4×C7×Q8C2213yes—
20C4×C7×Q8C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic
22C4×C7×Q8C2213yes—
23C2×C7×C4⋊C4C2213yes—
24C2×C42×C7C2213yes—
25#6510C2213yes—
26#6509C2213yes—
27#6509C2213yes—
28Q8×Dic7C2211yescharacteristic
29Q8×Dic7C2211yescharacteristic
30#6507C2213yes—
31#6493C2213yes—
32#6493C2213yes—
33#6492C2211yescharacteristic
34#6492C2213yes—
35C4×C22⋊D7C2216yes—
36C4×C22⋊D7C2213yes—
37C4×C22⋊D7C2213yes—
38#6487C2213yes—
39#6487C2213yes—
40#6461C2216yes—
41#6460C2213yes—
42#6459C2216yes—
43#6458C2213yes—
44#6457C2216yes—
45#6456C2213yes—
46#6455C2213yes—
47D7×C4⋊C4C2213yes—
48#6453C2213yes—
49#6452C2216yes—
50#6451C2213yes—
51#6450C2213yes—
52#6439C2213yes—
53#6439C2213yes—
54#6439C2216yes—
55#6438C2213yes—
56#6438C2213yes—
57C4×D28C2213yes—
58C4×D28C2213yes—
59#6435C2216yes—
60#6435C2213yes—
61#6435C2213yes—
62#6433C2213yes—
63#6433C2213yes—
64C4×Dic14C2213yes—
65C4×Dic14C2213yes—
66#3516—71nomaximal, Sylow
67C2×C7×Q8—24no—
68C2×C7×Q8C2313yes—
69C2×C7×Q8C2311yescharacteristic
70C22×C4×C7C2313yes—
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2313yes—
73C2×C22⋊D7C2313yes—
74C2×C22⋊D7C2311yescharacteristic
75#1161—24no—
76Q8×D7—24no—
77C2×D28C2313yes—
78C2×D28C2313yes—
79C2×C4×D7C2313yes—
80C2×C4×D7C2313yes—
81C2×C4×D7C2316yes—
82C2×Dic14C2313yes—
83C2×Dic14C2313yes—
84C7×C4⋊C4C2313yes—
85C7×C4⋊C4C2313yes—
86C7×C4⋊C4C2316yes—
87C42×C7C2313yes—
88C42×C7C2313yes—
89C42×C7C2313yes—
90C42×C7C2313yes—
91C22⋊Dic7C2311yescharacteristic
92C22⋊Dic7C2313yes—
93(C2×D7)⋊C4C2311yescharacteristic
94(C2×D7)⋊C4C2313yes—
95(C2×D7)⋊C4C2313yes—
96(C2×D7)⋊C4C2313yes—
97(C2×D7)⋊C4C2313yes—
98(C2×D7)⋊C4C2316yes—
99(C2×D7)⋊C4C2311yescharacteristic
100(C2×D7)⋊C4C2313yes—
101(C2×D7)⋊C4C2316yes—
102(C2×D7)⋊C4C2313yes—
103C4⋊Dic7C2313yes—
104C4⋊Dic7C2313yes—
105C4⋊Dic7C2313yes—
106C4⋊Dic7C2313yes—
107Dic7⋊C4C2311yescharacteristic
108Dic7⋊C4C2313yes—
109Dic7⋊C4C2313yes—
110Dic7⋊C4C2311yescharacteristic
111Dic7⋊C4C2316yes—
112Dic7⋊C4C2313yes—
113Dic7⋊C4C2313yes—
114Dic7⋊C4C2313yes—
115Dic7⋊C4C2313yes—
116Dic7⋊C4C2316yes—
117C4×Dic7C2313yes—
118C4×Dic7C2316yes—
119C4×Dic7C2313yes—
120#557—71no—
121#556—73no—
122#554—73no—
123#553—73no—
124#552—73no—
125D4×Q8—71no—
126#548—71no—
127#548—71no—
128#542—73no—
129#541—73no—
130#529—73no—
131#529—73no—
132#520—71no—
133#519—71no—
134C2×C4×Q8D711yescharacteristic, complemented
135C23×C7C2411yescharacteristic
136C22×D7C2411yescharacteristic
137C22×D7C2413yes—
138C7×Q8—26no—
139C7×Q8#4114yes—
140C2×C4×C7—22no—
141C2×C4×C7—26no—
142C2×C4×C7—43no—
143C2×C4×C7C2413yes—
144C2×C4×C7C2411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C2×C4×C7C2413yes—
147C2×C4×C7C2413yes—
148C2×C4×C7C2413yes—
149C22⋊D7—26no—
150C22⋊D7—22no—
151C2×Dic7C2413yes—
152C2×Dic7C2411yescharacteristic
153C2×Dic7C2411yescharacteristic
154C2×Dic7C2413yes—
155C2×Dic7C2413yes—
156C2×Dic7C2411yescharacteristic
157D28—26no—
158D28—43no—
159C4×D7—26no—
160C4×D7—26no—
161C4×D7—46no—
162Dic14—26no—
163Dic14—43no—
164C2×#41—71no—
165C22×Q8—71no—
166C22×Q8C2×D711yescharacteristic
167C4⋊Q8—73no—
168C4⋊Q8—73no—
169#126—76no—
170#126—76no—
171#126—73no—
172#126—73no—
173#125—76no—
174#125—73no—
175#125—73no—
176#125—73no—
177#124—73no—
178#124—73no—
179#124—73no—
180#123—71no—
181#123—73no—
182#123—76no—
183C22⋊Q8—73no—
184C22⋊Q8—73no—
185C22⋊Q8—73no—
186C22⋊Q8—73no—
187C22⋊Q8—73no—
188C22⋊Q8—76no—
189#121—73no—
190#121—73no—
191#121—73no—
192C4×Q8—71no—
193C4×Q8—71no—
194C4×Q8—73no—
195C4×Q8—73no—
196C4×Q8—73no—
197C4×Q8C2×D711yescharacteristic
198C4×Q8C2×D713yes—
199C4×Q8C2×D711yescharacteristic
200C4×Q8C2×D713yes—
201C4×D4—73no—
202C4×D4—73no—
203C4×D4—76no—
204C4×D4—73no—
205C4×D4—73no—
206C4×D4—73no—
207#117—73no—
208#117—73no—
209#117—73no—
210#117—76no—
211C2×C4⋊C4—73no—
212C2×C4⋊C4C2×D713yes—
213C2×C42C2×D713yes—
214C22×C7—21no—
215C22×C7—21no—
216C22×C72−1+412yes—
217C22×C7C2511yescharacteristic
218C2×D7—21no—
219C2×D7—23no—
220C2×D7—21no—
221C2×D7—23no—
222C2×D7—23no—
223C2×D7—21no—
224C4×C7—23no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—23no—
228C4×C7—23no—
229C4×C7C2×#4116yes—
230Dic7—23no—
231Dic7—21no—
232Dic7—23no—
233Dic7—21no—
234Dic7—21no—
235Dic7—23no—
236#41—144no—
237C2×Q8—24no—
238C2×Q8—73no—
239C2×Q8—73no—
240C2×Q8—144no—
241C2×Q8C22×D713yes—
242C2×Q8C22×D711yescharacteristic
243C2×D4—73no—
244C2×D4—71no—
245C2×D4—73no—
246C2×D4—73no—
247C22×C4—73no—
248C22×C4—73no—
249C22×C4—76no—
250C22×C4C22×D711yescharacteristic
251C22×C4C22×D713yes—
252C22×C4C22×D713yes—
253C4⋊C4—73no—
254C4⋊C4—71no—
255C4⋊C4—73no—
256C4⋊C4—73no—
257C4⋊C4—73no—
258C4⋊C4—76no—
259C4⋊C4—73no—
260C4⋊C4—76no—
261C4⋊C4—73no—
262C4⋊C4—71no—
263C4⋊C4—73no—
264C4⋊C4—73no—
265C4⋊C4—73no—
266C4⋊C4—73no—
267C4⋊C4C22×D713yes—
268C4⋊C4C22×D713yes—
269C4⋊C4C22×D716yes—
270C22⋊C4—71no—
271C22⋊C4—73no—
272C22⋊C4—73no—
273C22⋊C4—71no—
274C22⋊C4—73no—
275C22⋊C4—73no—
276C22⋊C4—73no—
277C22⋊C4—76no—
278C22⋊C4—71no—
279C22⋊C4—76no—
280C22⋊C4—73no—
281C22⋊C4—73no—
282C42—76no—
283C42—73no—
284C42—73no—
285C42C22×D713yes—
286C42C22×D713yes—
287C42C22×D713yes—
288C42C22×D713yes—
289C2×C7—22no—
290C2×C7C22×#4111yescharacteristic
291C2×C7C2×2−1+411yescharacteristic
292C2×C7#58511yescharacteristic
293D7—41no—
294D7—43no—
295C23—71no—
296C23—73no—
297C23C23×D711yescharacteristic
298Q8—26no—
299Q8—146no—
300Q8—283no—
301Q8#115714yes—
302D4—146no—
303D4—142no—
304D4—146no—
305D4—283no—
306C2×C4—22no—
307C2×C4—26no—
308C2×C4—43no—
309C2×C4—71no—
310C2×C4—73no—
311C2×C4—71no—
312C2×C4—73no—
313C2×C4—73no—
314C2×C4—71no—
315C2×C4—146no—
316C2×C4—146no—
317C2×C4—286no—
318C2×C4C23×D713yes—
319C2×C4C23×D711yescharacteristic
320C2×C4C23×D711yescharacteristic
321C2×C4C23×D713yes—
322C2×C4C23×D713yes—
323C2×C4C23×D713yes—
324C7#351611yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—141no—
328C22—143no—
329C22—141no—
330C22—143no—
331C22—143no—
332C22—141no—
333C22#655112yes—
334C22C24×D711yescharacteristic
335C4—23no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—141no—
341C4—143no—
342C4—143no—
343C4—141no—
344C4—143no—
345C4—141no—
346C4C2×#115716yes—
347C2—22no—
348C2—281no—
349C2—283no—
350C2C22×#115711yescharacteristic
351C2C2×#655111yescharacteristic
352C2#9061611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC2×C4×C7×Q8C2
hypercenterC2×C4×Q8D7
hypocenterC7#3516
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×Q8C2semidirect
2. #90508C2semidirect
3. Q8×C22⋊D7C2semidirect
4. #90274C2semidirect
5. #90273C2semidirect
6. #90272C2semidirect
7. #90267C2semidirect
8. #90265C2semidirect
9. #90263C2semidirect
10. #90260C2semidirect
11. #90259C2semidirect
12. #90205C2semidirect
13. #90202C2semidirect
14. #90170C2semidirect
15. #90170C2semidirect
134. C2×C4×Q8D7semidirect
324. C7#3516semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×Q82
#905082
Q8×C22⋊D72
#902742
#902732
#902722
#902672
#902652
#902632
#902602
#902592
#902052
#902022
#901702
#35167