Finite Group Databasebeta

← #11742976

Subgroups — #11742976

Order 896.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#520C211yesmaximal, characteristic, complemented
2#90525C211yesmaximal, characteristic, complemented
3#90522C211yesmaximal, characteristic, complemented
4#90267C211yesmaximal, characteristic, complemented
5#90265C211yesmaximal, characteristic, complemented
6C4×Q8×D7C211yesmaximal, characteristic, complemented
7Q8×Dic14C211yesmaximal, characteristic, complemented
8#90252C213yesmaximal, complemented
9#90250C213yesmaximal, complemented
10#90242C213yesmaximal, complemented
11#90239C213yesmaximal, complemented
12#90225C213yesmaximal, complemented
13#90219C213yesmaximal, complemented
14#90202C213yesmaximal, complemented
15#90198C213yesmaximal, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C2×#1157C2216yes—
20C4×C7×Q8C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic
22C4×C7×D4C2213yes—
23C4×C7×D4C2213yes—
24C7×#117C2213yes—
25C2×C7×C4⋊C4C2213yes—
26Q8×Dic7C2211yescharacteristic
27Q8×Dic7C2211yescharacteristic
28#6503C2213yes—
29#6501C2213yes—
30#6501C2213yes—
31#6499C2213yes—
32#6493C2213yes—
33#6492C2216yes—
34C4×C22⋊D7C2213yes—
35C4×C22⋊D7C2213yes—
36C4×C22⋊D7C2216yes—
37#6487C2213yes—
38#6487C2216yes—
39#6461C2213yes—
40#6460C2213yes—
41#6459C2216yes—
42#6459C2211yescharacteristic
43#6458C2213yes—
44#6457C2213yes—
45#6456C2213yes—
46D7×C4⋊C4C2213yes—
47D7×C4⋊C4C2213yes—
48#6452C2213yes—
49#6451C2213yes—
50#6451C2213yes—
51#6450C2213yes—
52#6448C2213yes—
53#6448C2216yes—
54#6447C2213yes—
55#6446C2213yes—
56#6446C2216yes—
57#6442C2213yes—
58#6438C2213yes—
59C4×D28C2213yes—
60C4×D28C2213yes—
61#6435C2213yes—
62C42×D7C2213yes—
63#6432C2213yes—
64C4×Dic14C2213yes—
65C4×Dic14C2213yes—
66#3522—71nomaximal, Sylow
67C7×#41—24no—
68C2×C7×Q8C2311yescharacteristic
69C2×C7×D4C2313yes—
70C22×C4×C7C2313yes—
71C22×C4×C7C2316yes—
72C2×C22⋊D7C2313yes—
73C2×C22⋊D7C2313yes—
74C2×C22⋊D7C2316yes—
75#1161—24no—
76Q8×D7—24no—
77#1157—412no—
78C2×D28C2313yes—
79C2×D28C2313yes—
80C2×C4×D7C2311yescharacteristic
81C2×C4×D7C2313yes—
82C2×C4×D7C2313yes—
83C2×C4×D7C2313yes—
84C2×C4×D7C2316yes—
85C2×Dic14C2311yescharacteristic
86C2×Dic14C2313yes—
87C2×Dic14C2316yes—
88C7×C4⋊C4C2313yes—
89C7×C4⋊C4C2313yes—
90C7×C4⋊C4C2313yes—
91C7×C4⋊C4C2311yescharacteristic
92C7×C22⋊C4C2313yes—
93C7×C22⋊C4C2313yes—
94C42×C7C2313yes—
95C42×C7C2313yes—
96C22⋊Dic7C2313yes—
97C22⋊Dic7C2313yes—
98C22⋊Dic7C2316yes—
99(C2×D7)⋊C4C2313yes—
100(C2×D7)⋊C4C2313yes—
101(C2×D7)⋊C4C2311yescharacteristic
102(C2×D7)⋊C4C2313yes—
103(C2×D7)⋊C4C2316yes—
104(C2×D7)⋊C4C2311yescharacteristic
105(C2×D7)⋊C4C2313yes—
106C4⋊Dic7C2313yes—
107C4⋊Dic7C2313yes—
108C4⋊Dic7C2313yes—
109C4⋊Dic7C2313yes—
110Dic7⋊C4C2313yes—
111Dic7⋊C4C2316yes—
112Dic7⋊C4C2313yes—
113Dic7⋊C4C2311yescharacteristic
114Dic7⋊C4C2313yes—
115Dic7⋊C4C2311yescharacteristic
116Dic7⋊C4C2313yes—
117C4×Dic7C2313yes—
118C4×Dic7C2313yes—
119C4×Dic7C2313yes—
120C4×Dic7C2313yes—
121#559—73no—
122Q82—71no—
123#554—73no—
124#550—71no—
125#548—71no—
126#548—71no—
127#547—73no—
128#547—73no—
129#542—73no—
130#542—73no—
131#532—73no—
132#529—73no—
133#520—71no—
134#520D711yescharacteristic, complemented
135C2×C4×Q8—71no—
136C23×C7C2413yes—
137C22×D7C2413yes—
138C22×D7C2411yescharacteristic
139C7×Q8#4114yescomplemented
140C7×D4—26no—
141C2×C4×C7—26no—
142C2×C4×C7—46no—
143C2×C4×C7C2411yescharacteristic
144C2×C4×C7C2413yes—
145C2×C4×C7C2411yescharacteristic
146C2×C4×C7C2413yes—
147C2×C4×C7C2411yescharacteristic
148C2×C4×C7C2413yes—
149C22⋊D7—26no—
150C22⋊D7—26no—
151C22⋊D7—46no—
152C2×Dic7C2411yescharacteristic
153C2×Dic7C2413yes—
154C2×Dic7C2413yes—
155C2×Dic7C2413yes—
156C2×Dic7C2411yescharacteristic
157C2×Dic7C2411yescharacteristic
158D28—26no—
159D28—26no—
160C4×D7—26no—
161C4×D7—26no—
162C4×D7—26no—
163C4×D7—22no—
164C4×D7—46no—
165Dic14—26no—
166Dic14—22no—
167Dic14—46no—
168C2×#41—71no—
169C2×#41—76no—
170C2×#41C2×D711yescharacteristic
171C22×Q8—71no—
172C4⋊Q8—73no—
173C4⋊Q8—73no—
174C4⋊Q8—73no—
175#127—73no—
176#126—73no—
177#126—73no—
178#125—73no—
179#124—73no—
180#124—73no—
181#124—76no—
182#124—73no—
183#123—76no—
184#123—73no—
185#123—73no—
186#123—76no—
187C22⋊Q8—76no—
188C22⋊Q8—73no—
189C22⋊Q8—71no—
190C22⋊Q8—76no—
191C22⋊Q8—73no—
192#121—73no—
193#121—73no—
194#121—73no—
195#121—73no—
196#121—73no—
197C4×Q8—71no—
198C4×Q8—71no—
199C4×Q8—73no—
200C4×Q8—73no—
201C4×Q8—73no—
202C4×Q8C2×D711yescharacteristic
203C4×Q8C2×D711yescharacteristic
204C4×D4—73no—
205C4×D4—73no—
206C4×D4—76no—
207C4×D4—73no—
208C4×D4—73no—
209C4×D4—73no—
210C4×D4C2×D713yes—
211C4×D4C2×D713yes—
212#117—73no—
213#117C2×D713yes—
214C2×C4⋊C4—73no—
215C2×C4⋊C4—73no—
216C2×C4⋊C4C2×D713yes—
217C2×C42—73no—
218C22×C7—23no—
219C22×C7—23no—
220C22×C7—23no—
221C22×C7C2511yescharacteristic
222C2×D7—21no—
223C2×D7—21no—
224C2×D7—23no—
225C2×D7—23no—
226C2×D7—23no—
227C2×D7H(5,𝔽2)12yes—
228C4×C7—21no—
229C4×C7—23no—
230C4×C7—23no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7C2×#4116yes—
234Dic7—23no—
235Dic7—21no—
236Dic7—23no—
237Dic7—23no—
238Dic7—21no—
239Dic72−1+412yes—
240#41—24no—
241#41—144no—
242#41—2812no—
243C2×Q8—71no—
244C2×Q8—73no—
245C2×Q8—76no—
246C2×Q8—144no—
247C2×Q8C22×D711yescharacteristic
248C2×D4—73no—
249C2×D4—73no—
250C2×D4—76no—
251C2×D4—73no—
252C2×D4—73no—
253C2×D4C22×D713yes—
254C22×C4—71no—
255C22×C4—73no—
256C22×C4—73no—
257C22×C4—73no—
258C22×C4—76no—
259C22×C4C22×D713yes—
260C22×C4C22×D716yes—
261C4⋊C4—73no—
262C4⋊C4—76no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—73no—
266C4⋊C4—73no—
267C4⋊C4—73no—
268C4⋊C4—73no—
269C4⋊C4—73no—
270C4⋊C4—73no—
271C4⋊C4—73no—
272C4⋊C4C22×D713yes—
273C4⋊C4C22×D713yes—
274C4⋊C4C22×D713yes—
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—73no—
277C22⋊C4—76no—
278C22⋊C4—73no—
279C22⋊C4—73no—
280C22⋊C4—73no—
281C22⋊C4—71no—
282C22⋊C4—71no—
283C22⋊C4—73no—
284C22⋊C4—76no—
285C22⋊C4—73no—
286C22⋊C4C22×D713yes—
287C22⋊C4C22×D713yes—
288C42—73no—
289C42—73no—
290C42—73no—
291C42—73no—
292C42C22×D713yes—
293C42C22×D713yes—
294C2×C7—43no—
295C2×C7C22×#4111yescharacteristic
296C2×C7C2×H(5,𝔽2)11yescharacteristic
297C2×C7C2×2−1+411yescharacteristic
298D7—22no—
299D7—43no—
300C23—71no—
301C23—73no—
302C23C23×D713yes—
303Q8—142no—
304Q8—146no—
305Q8—286no—
306Q8#115714yescomplemented
307D4—26no—
308D4—146no—
309D4—146no—
310D4—146no—
311D4—146no—
312D4—286no—
313C2×C4—26no—
314C2×C4—46no—
315C2×C4—71no—
316C2×C4—71no—
317C2×C4—73no—
318C2×C4—73no—
319C2×C4—73no—
320C2×C4—71no—
321C2×C4—142no—
322C2×C4—146no—
323C2×C4—146no—
324C2×C4—146no—
325C2×C4—286no—
326C2×C4C23×D711yescharacteristic
327C2×C4C23×D713yes—
328C2×C4C23×D711yescharacteristic
329C2×C4C23×D713yes—
330C2×C4C23×D711yescharacteristic
331C2×C4C23×D713yes—
332C7#352211yescharacteristic, complemented, Sylow
333C22—23no—
334C22—23no—
335C22—23no—
336C22—72no—
337C22—141no—
338C22—141no—
339C22—143no—
340C22—143no—
341C22—143no—
342C22C24×D711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—72no—
349C4—141no—
350C4—143no—
351C4—143no—
352C4—143no—
353C4—141no—
354C4C2×#115716yes—
355C2—43no—
356C2—142no—
357C2—283no—
358C2C22×#115711yescharacteristic
359C2D7×H(5,𝔽2)11yescharacteristic
360C2D7×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#520C2
hypercenter#520D7
hypocenterC7#3522
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C7×#520C2semidirect
2. #90525C2semidirect
3. #90522C2semidirect
4. #90267C2semidirect
5. #90265C2semidirect
6. C4×Q8×D7C2semidirect
7. Q8×Dic14C2semidirect
8. #90252C2semidirect
9. #90250C2semidirect
10. #90242C2semidirect
11. #90239C2semidirect
12. #90225C2semidirect
13. #90219C2semidirect
14. #90202C2semidirect
15. #90198C2semidirect
134. #520D7semidirect
139. C7×Q8#41semidirect
306. Q8#1157semidirect
332. C7#3522semidirect

Maximal subgroups

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

subgroupindex
C7×#5202
#905252
#905222
#902672
#902652
C4×Q8×D72
Q8×Dic142
#902522
#902502
#902422
#902392
#902252
#902192
#902022
#901982
#35227