Finite Group Databasebeta

← #1768×D71 #31995326

Subgroups — #1768×D71 #31995326

Order 18176.

1246 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
1C2×C4⋊C8×D71C212yesmaximal
2C2×#97×D71C211yesmaximal, characteristic
3C2×C4×C8×D71C211yesmaximal, characteristic
4#376×D71C211yesmaximal, characteristic
5#22588200C211yesmaximal, characteristic, complemented
6#22588199C212yesmaximal, complemented
7#22588198C211yesmaximal, characteristic, complemented
8#22588197C211yesmaximal, characteristic, complemented
9#22588196C212yesmaximal, complemented
10#336×D71C212yesmaximal
11C71×#1768C211yesmaximal, characteristic, complemented
12C22×C8×D71C2212yes—
13C22×C8×D71C2212yes—
14C2×C42×D71C2211yescharacteristic
15C2×C42×D71C2211yescharacteristic
16C2×C42×D71C2211yescharacteristic
17C2×C4⋊(C71⋊C8)C2212yes—
18C2×Dic71⋊C8C2212yes—
19C2×Dic71⋊C8C2212yes—
20C4⋊C8×D71C4116yescomplemented
21C2×#14556039C2211yescharacteristic
22C2×#14556038C2212yes—
23#97×D71—24no—
24C2×C4×C71⋊C8C2211yescharacteristic
25C2×C8×Dic71C2212yes—
26C4×C8×D71—24no—
27#17985845C2211yescharacteristic
28#17985845C2211yescharacteristic
29#17985845C2211yescharacteristic
30#17985844C414yes—
31#95×D71C414yes—
32#17985754C2212yes—
33#17985753C2212yes—
34#17985753C2212yes—
35C2×C4⋊C8×C71C2212yes—
36C2×#97×C71C2211yescharacteristic
37C2×C4×C8×C71C2211yescharacteristic
38#376×C71C2211yescharacteristic
39#336×C71C2212yes—
40C22×C4×D71C2×C412yes—
41C22×C4×D71C2×C412yes—
42C22×C4×D71C2×C412yes—
43C22×C4×D71C2311yescharacteristic
44C22×C71⋊C8C2312yes—
45C22×C71⋊C8C2312yes—
46C2×C8×D71—24no—
47C2×C8×D71—24no—
48C2×C8×D71—24no—
49C2×C8×D71—24no—
50C2×C8×D71C2×C418yes—
51C2×C8×D71C2×C418yes—
52C2×C4×Dic71C2×C412yes—
53C2×C4×Dic71C2×C412yes—
54C2×C4×Dic71C2×C412yes—
55C2×C4×Dic71C2311yescharacteristic
56C2×C4×Dic71C2311yescharacteristic
57C2×C4×Dic71C2311yescharacteristic
58C42×D71—24no—
59C42×D71C2×C418yes—
60C42×D71D414yes—
61C42×D71Q814yes—
62C4⋊(C71⋊C8)C2×C418yescomplemented
63Dic71⋊C8C2×C418yescomplemented
64Dic71⋊C8C2×C418yescomplemented
65#14556039—22no—
66#14556038—24no—
67C4×C71⋊C8—22no—
68C8×Dic71—24no—
69C71⋊#95C2×C414yes—
70C71⋊#95C2×C414yes—
71C71⋊#95C2×C414yes—
72C22×C8×C71C2312yes—
73C22×C8×C71C2312yes—
74C2×C42×C71C2311yescharacteristic
75C2×C42×C71C2311yescharacteristic
76C2×C42×C71C2311yescharacteristic
77C4⋊C8×C71C2×C418yescomplemented
78#97×C71—22no—
79C4×C8×C71—22no—
80#95×C71C2×C414yes—
81C23×D71C22×C411yescharacteristic
82C22×Dic71C22×C412yes—
83C22×Dic71C22×C412yes—
84C22×Dic71C22×C412yes—
85C22×Dic71C22×C411yescharacteristic
86C2×C4×D71—28no—
87C2×C4×D71—28no—
88C2×C4×D71—24no—
89C2×C4×D71—28no—
90C2×C4×D71C4218yes—
91C2×C4×D71C4⋊C418yes—
92C2×C4×D71C22×C414yes—
93C2×C4×D71C2×D412yes—
94C2×C4×D71C2×Q812yes—
95C2×C4×D71#4114yes—
96C2×C71⋊C8—22no—
97C2×C71⋊C8—22no—
98C2×C71⋊C8—22no—
99C2×C71⋊C8—22no—
100C2×C71⋊C8C22×C414yes—
101C2×C71⋊C8C22×C414yes—
102C8×D71—216no—
103C8×D71—216no—
104C4×Dic71—22no—
105C4×Dic71—24no—
106C4×Dic71C4218yes—
107C4×Dic71C4⋊C418yes—
108C4×Dic71C22×C414yes—
109C4×Dic71C2×D412yes—
110C4×Dic71C2×Q812yes—
111C22×C4×C71C22×C412yes—
112C22×C4×C71C22×C412yes—
113C22×C4×C71C22×C412yes—
114C22×C4×C71C2411yescharacteristic
115C2×C8×C71—22no—
116C2×C8×C71—22no—
117C2×C8×C71—22no—
118C2×C8×C71—22no—
119C2×C8×C71C22×C414yes—
120C2×C8×C71C22×C414yes—
121C42×C71—22no—
122C42×C71C22×C414yes—
123C42×C71C2×D412yes—
124C42×C71C2×Q812yes—
125C23×C71C23×C411yescharacteristic
126C22×D71C2×C4212yes—
127C22×D71C2×C4⋊C412yes—
128C22×D71#11712yes—
129C22×D71#13118yes—
130C2×C4×C71—24no—
131C2×C4×C71—24no—
132C2×C4×C71—22no—
133C2×C4×C71—24no—
134C2×C4×C71C2×C4214yes—
135C2×C4×C71C2×C4⋊C414yes—
136C2×C4×C71C23×C412yes—
137C2×C4×C71C22×D411yescharacteristic
138C2×C4×C71C22×Q811yescharacteristic
139C2×C4×C71C2×#4112yes—
140C2×Dic71—24no—
141C2×Dic71—24no—
142C2×Dic71—22no—
143C2×Dic71—24no—
144C2×Dic71C2×C4214yes—
145C2×Dic71C2×C4212yes—
146C2×Dic71C2×C4⋊C414yes—
147C2×Dic71C2×C4⋊C412yes—
148C2×Dic71#11712yes—
149C4×D71—216no—
150C4×D71—216no—
151C4×D71—48no—
152C4×D71C4×D418yes—
153C4×D71C4×Q818yes—
154C8×C71—24no—
155C8×C71—24no—
156C71⋊C8—24no—
157C71⋊C8—24no—
158C22×C71C22×C4211yescharacteristic
159C22×C71C22×C4⋊C411yescharacteristic
160C22×C71C2×#11711yescharacteristic
161C22×C71C2×#13114yes—
162C2×D71C4×C4⋊C414yes—
163C2×D71#40518yes—
164C2×D71#42418yes—
165C2×D71#43318yes—
166C4×C71—24no—
167C4×C71—24no—
168C4×C71—42no—
169C4×C71C2×C4×D412yes—
170C4×C71C2×C4×Q812yes—
171Dic71—24no—
172Dic71—24no—
173Dic71—42no—
174Dic71C4×C4⋊C414yes—
175C2×#1768—711nomaximal, Sylow
176C2×C71C2×C4×C4⋊C411yescharacteristic
177C2×C71C2×#40512yes—
178C2×C71C2×#42412yes—
179C2×C71C2×#43312yes—
180D71#176818yesdirect product factor
181C22×C4⋊C8—712no—
182C22×#97—711no—
183C22×C4×C8—711no—
184C2×#376—711no—
185#1768—712no—
186#1768—711no—
187#1768—711no—
188#1768—711no—
189#1768—712no—
190#1768D7111yescharacteristic, direct product factor
191C2×#336—712no—
192C71C2×#176811yescharacteristic, complemented, Sylow
193C23×C8—712no—
194C23×C8—712no—
195C22×C42—711no—
196C22×C42—711no—
197C22×C42—711no—
198C2×C4⋊C8—712no—
199C2×C4⋊C8—712no—
200C2×C4⋊C8—712no—
201C2×C4⋊C8—7116no—
202C2×C4⋊C8C2×D7112yes—
203C2×#97—712no—
204C2×#97—711no—
205C2×#97—1424no—
206C2×#97C2×D7111yescharacteristic
207C2×C4×C8—712no—
208C2×C4×C8—711no—
209C2×C4×C8—1424no—
210C2×C4×C8C2×D7111yescharacteristic
211#376—711no—
212#376—711no—
213#376—711no—
214#376—714no—
215#376C2×D7111yescharacteristic
216C2×#95—714no—
217#336—712no—
218#336—712no—
219#336—712no—
220#336C2×D7112yes—
221C23×C4—711no—
222C23×C4—712no—
223C23×C4—712no—
224C23×C4—712no—
225C22×C8—712no—
226C22×C8—712no—
227C22×C8—718no—
228C22×C8—718no—
229C22×C8—1424no—
230C22×C8—1424no—
231C22×C8—1424no—
232C22×C8—1424no—
233C22×C8C22×D7112yes—
234C22×C8C22×D7112yes—
235C2×C42—712no—
236C2×C42—712no—
237C2×C42—711no—
238C2×C42—711no—
239C2×C42—718no—
240C2×C42—714no—
241C2×C42—714no—
242C2×C42—712no—
243C2×C42—711no—
244C2×C42—1424no—
245C2×C42C22×D7111yescharacteristic
246C2×C42C22×D7111yescharacteristic
247C2×C42C22×D7111yescharacteristic
248C4⋊C8—718no—
249C4⋊C8—718no—
250C4⋊C8—718no—
251C4⋊C8C4×D7118yescomplemented
252#97—22no—
253#97—1424no—
254#97—1422no—
255C4×C8—22no—
256C4×C8—1424no—
257C4×C8—1422no—
258#95—714no—
259#95—714no—
260#95—714no—
261#95C4×D7114yes—
262C24—711no—
263C22×C4—711no—
264C22×C4—714no—
265C22×C4—712no—
266C22×C4—712no—
267C22×C4—714no—
268C22×C4—712no—
269C22×C4—712no—
270C22×C4—718no—
271C22×C4—718no—
272C22×C4—712no—
273C22×C4—1424no—
274C22×C4—1428no—
275C22×C4—1428no—
276C22×C4—1428no—
277C22×C4C2×C4×D7112yes—
278C22×C4C2×C4×D7112yes—
279C22×C4C2×C4×D7112yes—
280C22×C4C23×D7111yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—714no—
286C2×C8—714no—
287C2×C8—14216no—
288C2×C8—14216no—
289C2×C8—1422no—
290C2×C8—1422no—
291C2×C8—1422no—
292C2×C8—1422no—
293C2×C8C2×C4×D7114yes—
294C2×C8C2×C4×D7114yes—
295C42—22no—
296C42—714no—
297C42—712no—
298C42—712no—
299C42—718no—
300C42—718no—
301C42—1424no—
302C42—1422no—
303C42C2×C4×D7114yes—
304C42D4×D7112yes—
305C42Q8×D7112yes—
306C23—712no—
307C23—712no—
308C23—712no—
309C23—718no—
310C23C22×C4×D7111yescharacteristic
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—2848no—
316C2×C4—712no—
317C2×C4—712no—
318C2×C4—712no—
319C2×C4—718no—
320C2×C4—718no—
321C2×C4—714no—
322C2×C4—714no—
323C2×C4—14216no—
324C2×C4—14216no—
325C2×C4—1422no—
326C2×C4—1424no—
327C2×C4—1424no—
328C2×C4—1424no—
329C2×C4C42×D7114yes—
330C2×C4C4⋊C4×D7114yes—
331C2×C4C22×C4×D7112yes—
332C2×C4C2×D4×D7111yescharacteristic
333C2×C4C2×Q8×D7111yescharacteristic
334C2×C4#41×D7112yes—
335C8—24no—
336C8—24no—
337C8—1424no—
338C8—1424no—
339C22—718no—
340C22—718no—
341C22—718no—
342C22—714no—
343C22C2×C42×D7111yescharacteristic
344C22C2×C4⋊C4×D7111yescharacteristic
345C22#117×D7111yescharacteristic
346C22#131×D7114yes—
347C4—24no—
348C4—24no—
349C4—42no—
350C4—2842no—
351C4—714no—
352C4—1424no—
353C4—1424no—
354C4C4×D4×D7112yes—
355C4C4×Q8×D7112yes—
356C2—718no—
357C2C4×C4⋊C4×D7111yescharacteristic
358C2#405×D7112yes—
359C2#424×D7112yes—
360C2#433×D7112yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutatorC22×C71C22×C42
FittingC71×#1768C2
socleC23×C71C23×C4
prime radicalC23×D71C22×C4
cosocleC22×C4×C71C24
hypercenter#1768D71
hypocenterC71C2×#1768
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1768

Complements

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

NH ≅ G/Ntype
5. #22588200C2semidirect
6. #22588199C2semidirect
7. #22588198C2semidirect
8. #22588197C2semidirect
9. #22588196C2semidirect
11. C71×#1768C2semidirect
20. C4⋊C8×D71C4semidirect
62. C4⋊(C71⋊C8)C2×C4semidirect
63. Dic71⋊C8C2×C4semidirect
64. Dic71⋊C8C2×C4semidirect
77. C4⋊C8×C71C2×C4semidirect
180. D71#1768direct (N × H)
190. #1768D71direct (N × H)
192. C71C2×#1768semidirect
251. C4⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C8×D712
C2×#97×D712
C2×C4×C8×D712
#376×D712
#225882002
#225881992
#225881982
#225881972
#225881962
#336×D712
C71×#17682
C2×#176871