Finite Group Databasebeta

← #11744094

Subgroups — #11744094

Order 896.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#557C211yesmaximal, characteristic, complemented
2#90425C213yesmaximal, complemented
3#90424C213yesmaximal, complemented
4#90389C213yesmaximal, complemented
5#90386C213yesmaximal, complemented
6#90385C213yesmaximal, complemented
7#90382C213yesmaximal, complemented
8#90276C213yesmaximal, complemented
9#90271C213yesmaximal, complemented
10#90268C211yesmaximal, characteristic, complemented
11#90267C211yesmaximal, characteristic, complemented
12#90267C211yesmaximal, characteristic, complemented
13#90265C211yesmaximal, characteristic, complemented
14C4×#1161C211yesmaximal, characteristic, complemented
15#90263C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C7×C4⋊Q8C2213yes—
20C7×#125C2213yes—
21C7×#125C2213yes—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×Q8C2213yes—
26#6510C2216yes—
27#6510C2213yes—
28#6509C2213yes—
29Q8×Dic7C2211yescharacteristic
30#6461C2213yes—
31#6461C2216yes—
32#6461C2211yescharacteristic
33#6460C2213yes—
34#6460C2213yes—
35#6458C2213yes—
36#6458C2216yes—
37#6458C2213yes—
38#6458C2213yes—
39#6458C2213yes—
40#6458C2213yes—
41#6457C2216yes—
42#6457C2216yes—
43#6457C2213yes—
44#6457C2213yes—
45#6456C2216yes—
46#6456C2213yes—
47#6456C2213yes—
48#6455C2216yes—
49#6455C2213yes—
50#6455C2213yes—
51D7×C4⋊C4C2213yes—
52#6453C2213yes—
53#6450C2213yes—
54#6438C2213yes—
55#6438C2213yes—
56#6437C2213yes—
57C4×D28C2213yes—
58C4×D28C2213yes—
59C4×D28C2213yes—
60C4×D28C2213yes—
61C4×D28C2213yes—
62#6435C2213yes—
63#6435C2213yes—
64C42×D7C2213yes—
65C4×Dic14C2213yes—
66#3524—71nomaximal, Sylow
67C2×C7×Q8C2313yes—
68C2×C7×Q8C2311yescharacteristic
69#1161—24no—
70#1161—24no—
71Q8×D7—24no—
72C2×D28C2313yes—
73C2×D28C2313yes—
74C2×D28C2313yes—
75C2×D28C2313yes—
76C2×D28C2313yes—
77C2×D28C2316yes—
78C2×C4×D7C2313yes—
79C2×C4×D7C2313yes—
80C2×C4×D7C2313yes—
81C2×C4×D7C2313yes—
82C2×C4×D7C2313yes—
83C2×C4×D7C2316yes—
84C2×Dic14C2313yes—
85C7×C4⋊C4C2313yes—
86C7×C4⋊C4C2313yes—
87C7×C4⋊C4C2313yes—
88C7×C4⋊C4C2313yes—
89C7×C4⋊C4C2316yes—
90C7×C4⋊C4C2313yes—
91C7×C4⋊C4C2311yescharacteristic
92C42×C7C2313yes—
93C42×C7C2313yes—
94C42×C7C2313yes—
95(C2×D7)⋊C4C2311yescharacteristic
96(C2×D7)⋊C4C2313yes—
97(C2×D7)⋊C4C2313yes—
98(C2×D7)⋊C4C2313yes—
99(C2×D7)⋊C4C2311yescharacteristic
100(C2×D7)⋊C4C2311yescharacteristic
101(C2×D7)⋊C4C2313yes—
102(C2×D7)⋊C4C2313yes—
103(C2×D7)⋊C4C2313yes—
104(C2×D7)⋊C4C2316yes—
105(C2×D7)⋊C4C2313yes—
106(C2×D7)⋊C4C2316yes—
107(C2×D7)⋊C4C2316yes—
108C4⋊Dic7C2313yes—
109C4⋊Dic7C2311yescharacteristic
110C4⋊Dic7C2313yes—
111C4⋊Dic7C2313yes—
112Dic7⋊C4C2313yes—
113Dic7⋊C4C2311yescharacteristic
114Dic7⋊C4C2313yes—
115Dic7⋊C4C2313yes—
116C4×Dic7C2316yes—
117C4×Dic7C2311yescharacteristic
118C4×Dic7C2313yes—
119C4×Dic7C2313yes—
120#559—73no—
121#557D711yescharacteristic, complemented
122#555—73no—
123#553—73no—
124#552—73no—
125#550—71no—
126#548—71no—
127#548—71no—
128#542—73no—
129#540—73no—
130#529—73no—
131#529—73no—
132#520—71no—
133#519—71no—
134C4×#41—71no—
135C22×D7C2411yescharacteristic
136C22×D7C2413yes—
137C22×D7C2413yes—
138C7×Q8—43no—
139C7×Q8#4114yes—
140C2×C4×C7C2413yes—
141C2×C4×C7C2413yes—
142C2×C4×C7C2411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C2×C4×C7C2413yes—
145C2×C4×C7C2411yescharacteristic
146C2×C4×C7C2413yes—
147C2×Dic7C2411yescharacteristic
148C2×Dic7C2413yes—
149C2×Dic7C2411yescharacteristic
150C2×Dic7C2413yes—
151C2×Dic7C2411yescharacteristic
152D28—26no—
153D28—26no—
154D28—43no—
155D28—43no—
156D28—43no—
157D28—46no—
158C4×D7—26no—
159C4×D7—26no—
160C4×D7—26no—
161C4×D7—43no—
162C4×D7—46no—
163C4×D7—43no—
164Dic14—26no—
165C2×#41—71no—
166C2×#41—71no—
167C22×Q8—71no—
168C4⋊Q8C2×D713yes—
169#127—73no—
170#126—71no—
171#126—73no—
172#126—76no—
173#125—73no—
174#125C2×D713yes—
175#125C2×D713yes—
176#124—73no—
177#124—76no—
178#124—73no—
179#124—73no—
180#123—76no—
181#123—73no—
182#123—73no—
183#123—76no—
184C22⋊Q8—73no—
185C22⋊Q8—73no—
186C22⋊Q8—73no—
187#121—73no—
188#121—73no—
189#121—73no—
190#121—73no—
191#121—76no—
192#121—73no—
193C4×Q8—71no—
194C4×Q8—73no—
195C4×Q8—73no—
196C4×Q8C2×D711yescharacteristic
197C4×Q8C2×D711yescharacteristic
198C4×Q8C2×D711yescharacteristic
199C4×Q8C2×D713yes—
200C4×D4—73no—
201C4×D4—73no—
202C4×D4—76no—
203C4×D4—73no—
204C4×D4—73no—
205C4×D4—73no—
206C4×D4—73no—
207C4×D4—73no—
208#117—73no—
209#117—73no—
210#117—76no—
211#117—73no—
212#117—73no—
213C2×C4⋊C4—73no—
214C2×C42—73no—
215C22×C7C2511yescharacteristic
216C2×D7—21no—
217C2×D7—23no—
218C2×D7—23no—
219C2×D7—23no—
220C2×D7—23no—
221C2×D7—23no—
222C2×D7—21no—
223C2×D7—23no—
224C2×D7—21no—
225C4×C7—21no—
226C4×C7—23no—
227C4×C7—23no—
228C4×C7—21no—
229C4×C7—23no—
230C4×C7—21no—
231C4×C7C2×#4116yes—
232Dic7—23no—
233Dic7—21no—
234Dic7—21no—
235Dic7—21no—
236Dic7—23no—
237#41—144no—
238#41—144no—
239C2×Q8—73no—
240C2×Q8—144no—
241C2×Q8C22×D713yes—
242C2×Q8C22×D711yescharacteristic
243C2×D4—73no—
244C2×D4—73no—
245C2×D4—73no—
246C2×D4—73no—
247C2×D4—76no—
248C2×D4—73no—
249C22×C4—73no—
250C22×C4—73no—
251C22×C4—73no—
252C22×C4—73no—
253C22×C4—73no—
254C22×C4—76no—
255C4⋊C4—73no—
256C4⋊C4—71no—
257C4⋊C4—73no—
258C4⋊C4—73no—
259C4⋊C4—73no—
260C4⋊C4—73no—
261C4⋊C4—71no—
262C4⋊C4—73no—
263C4⋊C4C22×D713yes—
264C4⋊C4C22×D713yes—
265C4⋊C4C22×D713yes—
266C4⋊C4C22×D716yes—
267C4⋊C4C22×D713yes—
268C4⋊C4C22×D711yescharacteristic
269C4⋊C4C22×D713yes—
270C22⋊C4—71no—
271C22⋊C4—73no—
272C22⋊C4—73no—
273C22⋊C4—73no—
274C22⋊C4—76no—
275C22⋊C4—73no—
276C22⋊C4—76no—
277C22⋊C4—71no—
278C22⋊C4—73no—
279C22⋊C4—71no—
280C22⋊C4—73no—
281C22⋊C4—76no—
282C22⋊C4—73no—
283C42—71no—
284C42—73no—
285C42—73no—
286C42—76no—
287C42C22×D713yes—
288C42C22×D713yes—
289C42C22×D713yes—
290C2×C7C22×#4111yescharacteristic
291C2×C7#58511yescharacteristic
292C2×C7#58511yescharacteristic
293D7—41no—
294D7—43no—
295D7—43no—
296C23—71no—
297C23—73no—
298C23—73no—
299Q8—43no—
300Q8—146no—
301Q8#116114yes—
302D4—146no—
303D4—146no—
304D4—283no—
305D4—286no—
306D4—283no—
307D4—283no—
308C2×C4—71no—
309C2×C4—71no—
310C2×C4—73no—
311C2×C4—73no—
312C2×C4—71no—
313C2×C4—146no—
314C2×C4—146no—
315C2×C4—146no—
316C2×C4—283no—
317C2×C4—283no—
318C2×C4—286no—
319C2×C4C23×D713yes—
320C2×C4C23×D711yescharacteristic
321C2×C4C23×D713yes—
322C2×C4C23×D711yescharacteristic
323C2×C4C23×D711yescharacteristic
324C2×C4C23×D713yes—
325C2×C4C23×D713yes—
326C7#352411yescharacteristic, complemented, Sylow
327C22—141no—
328C22—143no—
329C22—143no—
330C22—143no—
331C22—141no—
332C22—143no—
333C22—143no—
334C22—143no—
335C22—141no—
336C22C24×D711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—141no—
344C4—141no—
345C4—143no—
346C4—141no—
347C4—143no—
348C4C2×#116116yes—
349C2—281no—
350C2—283no—
351C2—283no—
352C2C22×#116111yescharacteristic
353C2#9061611yescharacteristic
354C2#9062011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#557C2
hypercenter#557D7
hypocenterC7#3524
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C7×#557C2semidirect
2. #90425C2semidirect
3. #90424C2semidirect
4. #90389C2semidirect
5. #90386C2semidirect
6. #90385C2semidirect
7. #90382C2semidirect
8. #90276C2semidirect
9. #90271C2semidirect
10. #90268C2semidirect
11. #90267C2semidirect
12. #90267C2semidirect
13. #90265C2semidirect
14. C4×#1161C2semidirect
15. #90263C2semidirect
121. #557D7semidirect
326. C7#3524semidirect

Maximal subgroups

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

subgroupindex
C7×#5572
#904252
#904242
#903892
#903862
#903852
#903822
#902762
#902712
#902682
#902672
#902652
C4×#11612
#902632
#35247