Finite Group Databasebeta

← #11744090

Subgroups — #11744090

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
1C7×#557C211yesmaximal, characteristic, complemented
2#90424C213yesmaximal, complemented
3#90421C213yesmaximal, complemented
4#90388C213yesmaximal, complemented
5#90386C213yesmaximal, complemented
6#90385C213yesmaximal, complemented
7#90382C213yesmaximal, complemented
8#90275C213yesmaximal, complemented
9#90271C213yesmaximal, complemented
10#90267C211yesmaximal, characteristic, complemented
11#90267C211yesmaximal, characteristic, complemented
12Q8×D28C211yesmaximal, characteristic, complemented
13#90265C211yesmaximal, characteristic, complemented
14#90263C211yesmaximal, characteristic, complemented
15C4×Q8×D7C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C7×C4⋊Q8C2213yes—
20C7×#125C2213yes—
21C7×#125C2213yes—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×Q8C2213yes—
26#6510C2213yes—
27#6509C2213yes—
28#6509C2216yes—
29Q8×Dic7C2211yescharacteristic
30#6461C2216yes—
31#6461C2213yes—
32#6461C2216yes—
33#6461C2213yes—
34#6460C2213yes—
35#6460C2213yes—
36#6460C2213yes—
37#6460C2213yes—
38#6458C2213yes—
39#6458C2213yes—
40#6458C2213yes—
41#6457C2213yes—
42#6457C2216yes—
43#6457C2211yescharacteristic
44#6456C2216yes—
45#6456C2213yes—
46#6455C2216yes—
47#6455C2213yes—
48D7×C4⋊C4C2213yes—
49D7×C4⋊C4C2213yes—
50#6453C2216yes—
51#6453C2213yes—
52#6450C2213yes—
53#6450C2213yes—
54#6438C2213yes—
55#6438C2213yes—
56C4×D28C2213yes—
57C4×D28C2213yes—
58C4×D28C2213yes—
59C4×D28C2213yes—
60#6435C2213yes—
61#6435C2213yes—
62C42×D7C2213yes—
63#6432C2213yes—
64C4×Dic14C2213yes—
65C4×Dic14C2213yes—
66#3516—71nomaximal, Sylow
67C2×C7×Q8C2313yes—
68C2×C7×Q8C2311yescharacteristic
69#1161—24no—
70Q8×D7—24no—
71Q8×D7—24no—
72C2×D28C2311yescharacteristic
73C2×D28C2313yes—
74C2×D28C2313yes—
75C2×D28C2313yes—
76C2×C4×D7C2311yescharacteristic
77C2×C4×D7C2313yes—
78C2×C4×D7C2313yes—
79C2×C4×D7C2313yes—
80C2×C4×D7C2313yes—
81C2×C4×D7C2316yes—
82C2×Dic14C2313yes—
83C2×Dic14C2313yes—
84C7×C4⋊C4C2313yes—
85C7×C4⋊C4C2313yes—
86C7×C4⋊C4C2316yes—
87C7×C4⋊C4C2313yes—
88C7×C4⋊C4C2313yes—
89C7×C4⋊C4C2313yes—
90C7×C4⋊C4C2311yescharacteristic
91C42×C7C2313yes—
92C42×C7C2313yes—
93C42×C7C2313yes—
94(C2×D7)⋊C4C2311yescharacteristic
95(C2×D7)⋊C4C2313yes—
96(C2×D7)⋊C4C2313yes—
97(C2×D7)⋊C4C2313yes—
98(C2×D7)⋊C4C2311yescharacteristic
99(C2×D7)⋊C4C2311yescharacteristic
100(C2×D7)⋊C4C2313yes—
101(C2×D7)⋊C4C2313yes—
102(C2×D7)⋊C4C2313yes—
103(C2×D7)⋊C4C2313yes—
104(C2×D7)⋊C4C2316yes—
105(C2×D7)⋊C4C2316yes—
106C4⋊Dic7C2313yes—
107C4⋊Dic7C2313yes—
108C4⋊Dic7C2316yes—
109C4⋊Dic7C2313yes—
110C4⋊Dic7C2313yes—
111Dic7⋊C4C2313yes—
112Dic7⋊C4C2316yes—
113Dic7⋊C4C2311yescharacteristic
114Dic7⋊C4C2313yes—
115Dic7⋊C4C2313yes—
116C4×Dic7C2313yes—
117C4×Dic7C2316yes—
118C4×Dic7C2313yes—
119C4×Dic7C2313yes—
120#557D711yescharacteristic, complemented
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×Q8—71no—
135C22×D7C2411yescharacteristic
136C22×D7C2413yes—
137C22×D7C2411yescharacteristic
138C7×Q8—43no—
139C7×Q8#4114yes—
140C2×C4×C7C2411yescharacteristic
141C2×C4×C7C2413yes—
142C2×C4×C7C2413yes—
143C2×C4×C7C2411yescharacteristic
144C2×C4×C7C2411yescharacteristic
145C2×C4×C7C2413yes—
146C2×C4×C7C2413yes—
147C2×Dic7C2411yescharacteristic
148C2×Dic7C2413yes—
149C2×Dic7C2413yes—
150C2×Dic7C2413yes—
151C2×Dic7C2411yescharacteristic
152D28—22no—
153D28—26no—
154D28—26no—
155D28—43no—
156C4×D7—22no—
157C4×D7—26no—
158C4×D7—26no—
159C4×D7—26no—
160C4×D7—46no—
161C4×D7—43no—
162Dic14—26no—
163Dic14—26no—
164C2×#41—71no—
165C22×Q8—71no—
166C22×Q8—71no—
167C4⋊Q8—73no—
168C4⋊Q8C2×D713yes—
169#126—76no—
170#126—73no—
171#126—73no—
172#126—76no—
173#125—73no—
174#125—76no—
175#125C2×D713yes—
176#125C2×D713yes—
177#124—73no—
178#124—73no—
179#124—73no—
180#123—71no—
181#123—73no—
182#123—76no—
183C22⋊Q8—73no—
184C22⋊Q8—76no—
185C22⋊Q8—73no—
186C22⋊Q8—73no—
187C22⋊Q8—73no—
188C22⋊Q8—73no—
189#121—73no—
190#121—73no—
191#121—73no—
192C4×Q8—71no—
193C4×Q8—73no—
194C4×Q8—73no—
195C4×Q8—73no—
196C4×Q8—73no—
197C4×Q8C2×D711yescharacteristic
198C4×Q8C2×D711yescharacteristic
199C4×Q8C2×D711yescharacteristic
200C4×Q8C2×D713yes—
201C4×D4—73no—
202C4×D4—76no—
203C4×D4—73no—
204C4×D4—73no—
205C4×D4—73no—
206C4×D4—73no—
207#117—73no—
208#117—76no—
209#117—73no—
210#117—73no—
211C2×C4⋊C4—73no—
212C2×C4⋊C4—73no—
213C2×C42—73no—
214C22×C7C2511yescharacteristic
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—23no—
219C2×D7—21no—
220C2×D7—23no—
221C2×D7—23no—
222C2×D7—21no—
223C2×D72−1+412yes—
224C4×C7—21no—
225C4×C7—23no—
226C4×C7—21no—
227C4×C7—23no—
228C4×C7—23no—
229C4×C7—21no—
230C4×C7C2×#4116yes—
231Dic7—23no—
232Dic7—23no—
233Dic7—21no—
234Dic7—21no—
235Dic7—23no—
236#41—144no—
237C2×Q8—73no—
238C2×Q8—73no—
239C2×Q8—144no—
240C2×Q8—144no—
241C2×Q8C22×D713yes—
242C2×Q8C22×D711yescharacteristic
243C2×D4—73no—
244C2×D4—71no—
245C2×D4—73no—
246C2×D4—73no—
247C22×C4—71no—
248C22×C4—73no—
249C22×C4—73no—
250C22×C4—73no—
251C22×C4—73no—
252C22×C4—76no—
253C4⋊C4—73no—
254C4⋊C4—76no—
255C4⋊C4—71no—
256C4⋊C4—73no—
257C4⋊C4—73no—
258C4⋊C4—73no—
259C4⋊C4—76no—
260C4⋊C4—73no—
261C4⋊C4—73no—
262C4⋊C4—73no—
263C4⋊C4C22×D713yes—
264C4⋊C4C22×D713yes—
265C4⋊C4C22×D713yes—
266C4⋊C4C22×D713yes—
267C4⋊C4C22×D716yes—
268C4⋊C4C22×D711yescharacteristic
269C4⋊C4C22×D713yes—
270C22⋊C4—73no—
271C22⋊C4—71no—
272C22⋊C4—73no—
273C22⋊C4—73no—
274C22⋊C4—73no—
275C22⋊C4—76no—
276C22⋊C4—73no—
277C22⋊C4—71no—
278C22⋊C4—71no—
279C22⋊C4—76no—
280C22⋊C4—73no—
281C22⋊C4—73no—
282C42—73no—
283C42—73no—
284C42—73no—
285C42—76no—
286C42C22×D713yes—
287C42C22×D713yes—
288C42C22×D713yes—
289C2×C7C22×#4111yescharacteristic
290C2×C7C2×2−1+411yescharacteristic
291C2×C7#58511yescharacteristic
292D7—22no—
293D7—41no—
294D7—43no—
295C23—71no—
296C23—71no—
297C23—73no—
298Q8—43no—
299Q8—146no—
300Q8—146no—
301Q8#116114yes—
302D4—146no—
303D4—142no—
304D4—146no—
305D4—283no—
306C2×C4—71no—
307C2×C4—73no—
308C2×C4—73no—
309C2×C4—73no—
310C2×C4—71no—
311C2×C4—142no—
312C2×C4—146no—
313C2×C4—146no—
314C2×C4—146no—
315C2×C4—286no—
316C2×C4—283no—
317C2×C4C23×D711yescharacteristic
318C2×C4C23×D713yes—
319C2×C4C23×D713yes—
320C2×C4C23×D711yescharacteristic
321C2×C4C23×D711yescharacteristic
322C2×C4C23×D713yes—
323C2×C4C23×D713yes—
324C7#351611yescharacteristic, complemented, Sylow
325C22—72no—
326C22—141no—
327C22—141no—
328C22—141no—
329C22—143no—
330C22—143no—
331C22—141no—
332C22—141no—
333C22—143no—
334C22C24×D711yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—141no—
342C4—143no—
343C4—143no—
344C4—143no—
345C4—141no—
346C4C2×#116116yes—
347C2—142no—
348C2—281no—
349C2—283no—
350C2C22×#116111yescharacteristic
351C2#9061611yescharacteristic
352C2D7×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#557C2
hypercenter#557D7
hypocenterC7#3516
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. #90424C2semidirect
3. #90421C2semidirect
4. #90388C2semidirect
5. #90386C2semidirect
6. #90385C2semidirect
7. #90382C2semidirect
8. #90275C2semidirect
9. #90271C2semidirect
10. #90267C2semidirect
11. #90267C2semidirect
12. Q8×D28C2semidirect
13. #90265C2semidirect
14. #90263C2semidirect
15. C4×Q8×D7C2semidirect
120. #557D7semidirect
324. C7#3516semidirect

Maximal subgroups

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

subgroupindex
C7×#5572
#904242
#904212
#903882
#903862
#903852
#903822
#902752
#902712
#902672
Q8×D282
#902652
#902632
C4×Q8×D72
#35167