Finite Group Databasebeta

← #11743792

Subgroups — #11743792

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×#548C211yesmaximal, characteristic, complemented
2#90525C211yesmaximal, characteristic, complemented
3#90518C211yesmaximal, characteristic, complemented
4#90508C211yesmaximal, characteristic, complemented
5#90503C211yesmaximal, characteristic, complemented
6#90372C213yesmaximal, complemented
7#90371C213yesmaximal, complemented
8#90334C213yesmaximal, complemented
9#90333C213yesmaximal, complemented
10#90310C213yesmaximal, complemented
11#90307C213yesmaximal, complemented
12C4×#1161C211yesmaximal, characteristic, complemented
13#90260C211yesmaximal, characteristic, complemented
14#90256C213yesmaximal, complemented
15#90248C213yesmaximal, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C2×#1161C2211yescharacteristic
19C7×#124C2213yes—
20C7×C22⋊Q8C2213yes—
21C7×#121C2213yes—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4C2213yes—
24#6510C2213yes—
25#6509C2213yes—
26Q8×Dic7C2213yes—
27Q8×Dic7C2213yes—
28Q8×Dic7C2211yescharacteristic
29Q8×Dic7C2211yescharacteristic
30Q8×Dic7C2211yescharacteristic
31#6503C2213yes—
32#6502C2216yes—
33#6501C2213yes—
34#6501C2213yes—
35#6499C2213yes—
36D4×Dic7C2213yes—
37D4×Dic7C2213yes—
38#6493C2213yes—
39#6493C2213yes—
40C4×C22⋊D7C2213yes—
41C4×C22⋊D7C2213yes—
42#6489C2213yes—
43#6489C2213yes—
44#6489C2213yes—
45C2×C4⋊Dic7C2213yes—
46#6461C2216yes—
47#6460C2213yes—
48#6456C2213yes—
49#6455C2213yes—
50#6453C2213yes—
51#6453C2213yes—
52#6451C2213yes—
53#6449C2216yes—
54#6448C2216yes—
55#6446C2216yes—
56#6446C2213yes—
57#6446C2213yes—
58(C22⋊D7)⋊C4C2216yes—
59#6442C2213yes—
60#6442C2211yescharacteristic
61#6440C2216yes—
62C4×D28C2213yes—
63C42×D7C2213yes—
64#6433C2213yes—
65C4×Dic14C2213yes—
66#3524—71nomaximal, Sylow
67C7×#41—24no—
68C2×C7×Q8—24no—
69C2×C7×Q8C2313yes—
70C2×C7×Q8C2311yescharacteristic
71C2×C7×D4C2313yes—
72C2×C7×D4C2313yes—
73C22×C4×C7C2313yes—
74C22×C4×C7C2313yes—
75C2×C22⋊D7C2316yes—
76C2×C22⋊D7C2313yes—
77C2×C22⋊D7C2313yes—
78C22×Dic7C2316yes—
79#1161—24no—
80C2×D28C2313yes—
81C2×C4×D7C2313yes—
82C2×C4×D7C2313yes—
83C2×C4×D7C2313yes—
84C2×Dic14C2313yes—
85C7×C4⋊C4C2313yes—
86C7×C4⋊C4C2313yes—
87C7×C22⋊C4C2316yes—
88C7×C22⋊C4C2313yes—
89C7×C22⋊C4C2311yescharacteristic
90C42×C7C2313yes—
91C22⋊Dic7C2313yes—
92C22⋊Dic7C2313yes—
93C22⋊Dic7C2311yescharacteristic
94C22⋊Dic7C2311yescharacteristic
95C22⋊Dic7C2313yes—
96C22⋊Dic7C2313yes—
97(C2×D7)⋊C4C2316yes—
98(C2×D7)⋊C4C2316yes—
99(C2×D7)⋊C4C2313yes—
100(C2×D7)⋊C4C2313yes—
101C4⋊Dic7C2313yes—
102C4⋊Dic7C2313yes—
103C4⋊Dic7C2313yes—
104C4⋊Dic7C2313yes—
105C4⋊Dic7C2313yes—
106C4⋊Dic7C2311yescharacteristic
107C4⋊Dic7C2313yes—
108C4⋊Dic7C2313yes—
109Dic7⋊C4C2311yescharacteristic
110Dic7⋊C4C2313yes—
111Dic7⋊C4C2316yes—
112Dic7⋊C4C2313yes—
113Dic7⋊C4C2311yescharacteristic
114C4×Dic7C2313yes—
115C4×Dic7C2313yes—
116C4×Dic7C2311yescharacteristic
117C4×Dic7C2313yes—
118C4×Dic7C2313yes—
119C4×Dic7C2316yes—
120#559—73no—
121#557—71no—
122#555—73no—
123#553—73no—
124#552—73no—
125#550—71no—
126#548—71no—
127#548D711yescharacteristic, complemented
128#542—73no—
129#540—73no—
130#529—73no—
131#529—73no—
132#520—71no—
133#519—71no—
134C4×#41—71no—
135C23×C7C2411yescharacteristic
136C23×C7C2413yes—
137C22×D7C2413yes—
138C7×Q8—26no—
139C7×Q8#4114yes—
140C7×D4—26no—
141C7×D4—43no—
142C2×C4×C7—26no—
143C2×C4×C7—26no—
144C2×C4×C7C2413yes—
145C2×C4×C7C2411yescharacteristic
146C2×C4×C7C2413yes—
147C2×C4×C7C2411yescharacteristic
148C2×C4×C7C2413yes—
149C22⋊D7—46no—
150C22⋊D7—43no—
151C22⋊D7—43no—
152C2×Dic7—46no—
153C2×Dic7C2411yescharacteristic
154C2×Dic7C2413yes—
155C2×Dic7C2411yescharacteristic
156C2×Dic7C2413yes—
157C2×Dic7C2411yescharacteristic
158C2×Dic7C2411yescharacteristic
159C2×Dic7C2413yes—
160D28—26no—
161C4×D7—26no—
162C4×D7—43no—
163C4×D7—43no—
164Dic14—43no—
165C2×#41—71no—
166C2×#41C2×D711yescharacteristic
167C22×Q8C2×D711yescharacteristic
168C4⋊Q8—73no—
169#127—73no—
170#126—76no—
171#126—71no—
172#126—73no—
173#125—73no—
174#125—73no—
175#125—73no—
176#124—73no—
177#124—73no—
178#124—76no—
179#124C2×D713yes—
180#123—76no—
181#123—73no—
182#123—73no—
183#123—76no—
184C22⋊Q8—73no—
185C22⋊Q8—73no—
186C22⋊Q8C2×D713yes—
187#121—73no—
188#121—73no—
189#121—76no—
190#121—73no—
191#121—73no—
192#121C2×D713yes—
193C4×Q8—71no—
194C4×Q8—73no—
195C4×Q8—71no—
196C4×Q8—73no—
197C4×Q8—71no—
198C4×Q8—73no—
199C4×Q8C2×D711yescharacteristic
200C4×D4—73no—
201C4×D4—73no—
202C4×D4—73no—
203C4×D4—73no—
204C4×D4—73no—
205C4×D4—76no—
206C4×D4—73no—
207C4×D4C2×D713yes—
208#117—73no—
209#117—73no—
210#117—73no—
211#117—73no—
212#117—76no—
213C2×C4⋊C4—73no—
214C2×C42—73no—
215C22×C7—21no—
216C22×C7—23no—
217C22×C7—23no—
218C22×C7—21no—
219C22×C7—23no—
220C22×C7—21no—
221C22×C7C2511yescharacteristic
222C2×D7—23no—
223C2×D7—23no—
224C2×D7—23no—
225C4×C7—21no—
226C4×C7—23no—
227C4×C7—23no—
228C4×C7—21no—
229C4×C7C2×#4116yes—
230Dic7—21no—
231Dic7—21no—
232Dic7—23no—
233Dic7—21no—
234Dic7—23no—
235Dic7—23no—
236Dic7—21no—
237#41—24no—
238#41—144no—
239C2×Q8—24no—
240C2×Q8—73no—
241C2×Q8C22×D713yes—
242C2×Q8C22×D711yescharacteristic
243C2×D4—73no—
244C2×D4—76no—
245C2×D4—73no—
246C2×D4—73no—
247C2×D4C22×D713yes—
248C2×D4C22×D713yes—
249C22×C4—76no—
250C22×C4—73no—
251C22×C4—73no—
252C22×C4—73no—
253C22×C4C22×D713yes—
254C22×C4C22×D713yes—
255C4⋊C4—73no—
256C4⋊C4—76no—
257C4⋊C4—71no—
258C4⋊C4—73no—
259C4⋊C4—71no—
260C4⋊C4—73no—
261C4⋊C4—73no—
262C4⋊C4—73no—
263C4⋊C4—71no—
264C4⋊C4—73no—
265C4⋊C4—73no—
266C4⋊C4—73no—
267C4⋊C4—73no—
268C4⋊C4C22×D713yes—
269C4⋊C4C22×D713yes—
270C22⋊C4—71no—
271C22⋊C4—73no—
272C22⋊C4—73no—
273C22⋊C4—73no—
274C22⋊C4—73no—
275C22⋊C4—71no—
276C22⋊C4—76no—
277C22⋊C4—73no—
278C22⋊C4—76no—
279C22⋊C4—73no—
280C22⋊C4C22×D716yes—
281C22⋊C4C22×D713yes—
282C22⋊C4C22×D711yescharacteristic
283C42—71no—
284C42—73no—
285C42—73no—
286C42—76no—
287C42—73no—
288C42—73no—
289C42C22×D713yes—
290C2×C7—41no—
291C2×C7—43no—
292C2×C7C22×#4111yescharacteristic
293C2×C7#58511yescharacteristic
294C2×C7#58511yescharacteristic
295D7—43no—
296C23—73no—
297C23C23×D711yescharacteristic
298C23C23×D713yes—
299Q8—26no—
300Q8—283no—
301Q8#115914yes—
302D4—26no—
303D4—43no—
304D4—146no—
305D4—286no—
306D4—283no—
307D4—283no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—73no—
313C2×C4—71no—
314C2×C4—73no—
315C2×C4—71no—
316C2×C4—73no—
317C2×C4—146no—
318C2×C4—283no—
319C2×C4—283no—
320C2×C4—286no—
321C2×C4C23×D713yes—
322C2×C4C23×D711yescharacteristic
323C2×C4C23×D713yes—
324C2×C4C23×D711yescharacteristic
325C2×C4C23×D713yes—
326C7#352411yescharacteristic, complemented, Sylow
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22—143no—
334C22—143no—
335C22—143no—
336C22C24×D711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—141no—
342C4—141no—
343C4—143no—
344C4—141no—
345C4—143no—
346C4—143no—
347C4—141no—
348C4C2×#115916yes—
349C2—41no—
350C2—43no—
351C2—283no—
352C2C22×#115911yescharacteristic
353C2#9061611yescharacteristic
354C2#9062011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#548C2
hypercenter#548D7
hypocenterC7#3524
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C7×#548C2semidirect
2. #90525C2semidirect
3. #90518C2semidirect
4. #90508C2semidirect
5. #90503C2semidirect
6. #90372C2semidirect
7. #90371C2semidirect
8. #90334C2semidirect
9. #90333C2semidirect
10. #90310C2semidirect
11. #90307C2semidirect
12. C4×#1161C2semidirect
13. #90260C2semidirect
14. #90256C2semidirect
15. #90248C2semidirect
127. #548D7semidirect
326. C7#3524semidirect

Maximal subgroups

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

subgroupindex
C7×#5482
#905252
#905182
#905082
#905032
#903722
#903712
#903342
#903332
#903102
#903072
C4×#11612
#902602
#902562
#902482
#35247