Finite Group Databasebeta

← #1992×D71 #32002015

Subgroups — #1992×D71 #32002015

Order 18176.

1294 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D71C211yesmaximal, characteristic
2C2×#97×D71C211yesmaximal, characteristic
3#384×D71C211yesmaximal, characteristic
4#22589310C211yesmaximal, characteristic, complemented
5#22589309C211yesmaximal, characteristic, complemented
6#22589308C211yesmaximal, characteristic, complemented
7#22589307C214yesmaximal, complemented
8#340×D71C214yesmaximal
9C71×#1992C211yesmaximal, characteristic, complemented
10C22×C8×D71C2212yes—
11C2×#14556120C2211yescharacteristic
12C2×#14556119C2212yes—
13C2×#14556118—22no—
14#125×D71—24no—
15#125×D71C418yescomplemented
16C2×C4⋊C4×D71—22no—
17C2×C4⋊C4×D71—22no—
18C2×C4⋊C4×D71C2212yes—
19C2×C4⋊C4×D71C2212yes—
20C2×C42×D71C2211yescharacteristic
21C2×#14556039C2211yescharacteristic
22C2×#14556038C2212yes—
23#97×D71—24no—
24#17985869C2211yescharacteristic
25#17985868—22no—
26#17985868C2212yes—
27#17985762C2214yes—
28#17985761C2214yes—
29#17985761C2214yes—
30C2×#125×C71C2211yescharacteristic
31C2×#97×C71C2211yescharacteristic
32#384×C71C2211yescharacteristic
33#340×C71C2214yes—
34C22×C4×D71—22no—
35C22×C4×D71—22no—
36C22×C4×D71D412yes—
37C22×C4×D71C2311yescharacteristic
38C22×C71⋊C8C2312yes—
39C2×C8×D71—24no—
40C2×C8×D71—24no—
41C2×C8×D71Q818yes—
42#14556120—22no—
43#14556120C2×C414yescomplemented
44#14556119—24no—
45#14556119C2×C418yescomplemented
46#14556118—216no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71C2312yes—
50C2×C4⋊Dic71C2312yes—
51C2×Dic71⋊C4—22no—
52C2×Dic71⋊C4—22no—
53C2×Dic71⋊C4—22no—
54C2×Dic71⋊C4—24no—
55C2×Dic71⋊C4—22no—
56C4⋊C4×D71—216no—
57C4⋊C4×D71—24no—
58C4⋊C4×D71—28no—
59C4⋊C4×D71—48no—
60C4⋊C4×D71C2×C418yes—
61C2×C4×Dic71D412yes—
62C2×C4×Dic71C2311yescharacteristic
63C42×D71C2×C414yes—
64C42×D71D414yes—
65#14556039—22no—
66#14556038—24no—
67C22×C8×C71C2312yes—
68#125×C71—22no—
69#125×C71C2×C414yescomplemented
70C2×C4⋊C4×C71—22no—
71C2×C4⋊C4×C71—22no—
72C2×C4⋊C4×C71C2312yes—
73C2×C4⋊C4×C71C2312yes—
74C2×C42×C71C2311yescharacteristic
75#97×C71—22no—
76C23×D71C2×D411yescharacteristic
77C22×Dic71—22no—
78C22×Dic71—22no—
79C22×Dic71C2×D412yes—
80C22×Dic71C2×D411yescharacteristic
81C2×C4×D71—24no—
82C2×C4×D71—24no—
83C2×C4×D71—28no—
84C2×C4×D71—44no—
85C2×C4×D71—44no—
86C2×C4×D71—44no—
87C2×C4×D71—48no—
88C2×C4×D71C22⋊C418yes—
89C2×C4×D71C22×C412yes—
90C2×C4×D71C2×D412yes—
91C2×C4×D71C2×Q814yes—
92C2×C4×D71#4114yes—
93C2×C71⋊C8—22no—
94C2×C71⋊C8—22no—
95C2×C71⋊C8C2×Q814yes—
96C8×D71—216no—
97C4⋊Dic71—22no—
98C4⋊Dic71—28no—
99C4⋊Dic71—24no—
100C4⋊Dic71—44no—
101C4⋊Dic71C22×C414yes—
102Dic71⋊C4—28no—
103Dic71⋊C4—28no—
104Dic71⋊C4—28no—
105Dic71⋊C4—48no—
106Dic71⋊C4—44no—
107C4×Dic71C22⋊C418yes—
108C4×Dic71C22×C412yes—
109C4×Dic71C2×D412yes—
110C22×C4×C71—22no—
111C22×C4×C71—22no—
112C22×C4×C71C2×D412yes—
113C22×C4×C71C2411yescharacteristic
114C2×C8×C71—22no—
115C2×C8×C71—22no—
116C2×C8×C71C2×Q814yes—
117C4⋊C4×C71—28no—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—24no—
120C4⋊C4×C71—44no—
121C4⋊C4×C71C22×C414yes—
122C42×C71C22×C412yes—
123C42×C71C2×D412yes—
124C23×C71C22×D411yescharacteristic
125C22×D71C2×C22⋊C412yes—
126C22×D71#12412yes—
127C22×D71C4⋊Q812yes—
128C22×D71AGL(1,ℤ/8)14yes—
129C22×D71#13714yes—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—24no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
136C2×C4×C71—44no—
137C2×C4×C71C2×C22⋊C414yes—
138C2×C4×C71C23×C411yescharacteristic
139C2×C4×C71C22×D411yescharacteristic
140C2×C4×C71C22×Q812yes—
141C2×C4×C71C2×#4112yes—
142C2×Dic71—22no—
143C2×Dic71—22no—
144C2×Dic71—24no—
145C2×Dic71—42no—
146C2×Dic71—42no—
147C2×Dic71—42no—
148C2×Dic71—44no—
149C2×Dic71C2×C22⋊C414yes—
150C2×Dic71C2×C22⋊C412yes—
151C2×Dic71#12412yes—
152C2×Dic71C4⋊Q812yes—
153C4×D71—216no—
154C4×D71—416no—
155C4×D71—88no—
156C4×D71C4×Q818yes—
157C4×D71C22⋊Q818yes—
158C8×C71—24no—
159C71⋊C8—24no—
160C22×C71C22×C22⋊C411yescharacteristic
161C22×C71C2×#12411yescharacteristic
162C22×C71C2×C4⋊Q811yescharacteristic
163C22×C71C2×AGL(1,ℤ/8)12yes—
164C22×C71C2×#13712yes—
165C2×D71#39114yes—
166C2×D71#41718yes—
167C2×D71#49014yes—
168C2×D71#49114yes—
169C2×D71#50118yes—
170C4×C71—24no—
171C4×C71—44no—
172C4×C71—82no—
173C4×C71C2×C4×Q812yes—
174C4×C71C2×C22⋊Q812yes—
175Dic71—24no—
176Dic71—44no—
177Dic71—82no—
178Dic71#39114yes—
179C2×#1992—711nomaximal, Sylow
180C2×C71C2×#39111yescharacteristic
181C2×C71C2×#41712yes—
182C2×C71C2×#49011yescharacteristic
183C2×C71C2×#49111yescharacteristic
184C2×C71C2×#50112yes—
185D71#199218yesdirect product factor
186C22×#125—711no—
187C22×#97—711no—
188C2×#384—711no—
189#1992—711no—
190#1992—711no—
191#1992—714no—
192#1992—711no—
193#1992D7111yescharacteristic, direct product factor
194C2×#340—714no—
195C71C2×#199211yescharacteristic, complemented, Sylow
196C23×C8—712no—
197C2×#125—718no—
198C2×#125—712no—
199C2×#125—711no—
200C2×#125—1422no—
201C2×#125—1424no—
202C2×#125C2×D7111yescharacteristic
203C22×C4⋊C4—712no—
204C22×C4⋊C4—712no—
205C22×C4⋊C4—1422no—
206C22×C4⋊C4—1422no—
207C22×C42—711no—
208C2×#97—712no—
209C2×#97—711no—
210C2×#97—1424no—
211C2×#97C2×D7111yescharacteristic
212#384—712no—
213#384—711no—
214#384—1422no—
215#384C2×D7111yescharacteristic
216#340—714no—
217#340—714no—
218#340—714no—
219#340C2×D7114yes—
220C23×C4—711no—
221C23×C4—712no—
222C23×C4—1422no—
223C23×C4—1422no—
224C22×C8—712no—
225C22×C8—718no—
226C22×C8—1424no—
227C22×C8—1424no—
228C22×C8C22×D7112yes—
229#125—22no—
230#125—718no—
231#125—714no—
232#125—14216no—
233#125—1424no—
234#125—1422no—
235#125C4×D7114yescomplemented
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—2848no—
239C2×C4⋊C4—712no—
240C2×C4⋊C4—718no—
241C2×C4⋊C4—712no—
242C2×C4⋊C4—1428no—
243C2×C4⋊C4—1424no—
244C2×C4⋊C4—14216no—
245C2×C4⋊C4—1422no—
246C2×C4⋊C4—1424no—
247C2×C4⋊C4—1422no—
248C2×C4⋊C4—1422no—
249C2×C4⋊C4—1422no—
250C2×C4⋊C4—1422no—
251C2×C4⋊C4—1422no—
252C2×C4⋊C4C22×D7112yes—
253C2×C4⋊C4C22×D7112yes—
254C2×C42—712no—
255C2×C42—711no—
256C2×C42—714no—
257C2×C42—714no—
258C2×C42C22×D7111yescharacteristic
259#97—22no—
260#97—1424no—
261#97—1422no—
262C24—711no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—2844no—
266C22×C4—2844no—
267C22×C4—2848no—
268C22×C4—2844no—
269C22×C4—711no—
270C22×C4—712no—
271C22×C4—712no—
272C22×C4—714no—
273C22×C4—714no—
274C22×C4—712no—
275C22×C4—718no—
276C22×C4—1428no—
277C22×C4—1424no—
278C22×C4—1424no—
279C22×C4—1422no—
280C22×C4—1422no—
281C22×C4D4×D7112yes—
282C22×C4C23×D7111yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—714no—
286C2×C8—14216no—
287C2×C8—1422no—
288C2×C8—1422no—
289C2×C8Q8×D7114yes—
290C4⋊C4—22no—
291C4⋊C4—28no—
292C4⋊C4—24no—
293C4⋊C4—44no—
294C4⋊C4—2844no—
295C4⋊C4—2844no—
296C4⋊C4—2848no—
297C4⋊C4—714no—
298C4⋊C4—1428no—
299C4⋊C4—1428no—
300C4⋊C4—1428no—
301C4⋊C4—1424no—
302C4⋊C4—1422no—
303C4⋊C4—1428no—
304C4⋊C4C2×C4×D7114yes—
305C42—712no—
306C42—712no—
307C42—718no—
308C42C2×C4×D7112yes—
309C42D4×D7112yes—
310C23—712no—
311C23—712no—
312C23—714no—
313C23—714no—
314C23—712no—
315C23C2×D4×D7111yescharacteristic
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—28416no—
324C2×C4—2842no—
325C2×C4—2842no—
326C2×C4—2842no—
327C2×C4—2844no—
328C2×C4—5688no—
329C2×C4—712no—
330C2×C4—712no—
331C2×C4—712no—
332C2×C4—718no—
333C2×C4—718no—
334C2×C4—714no—
335C2×C4—14216no—
336C2×C4—1424no—
337C2×C4—1422no—
338C2×C4—1422no—
339C2×C4C22⋊C4×D7114yes—
340C2×C4C22×C4×D7111yescharacteristic
341C2×C4C2×D4×D7111yescharacteristic
342C2×C4C2×Q8×D7112yes—
343C2×C4#41×D7112yes—
344C8—24no—
345C8—1424no—
346C22—714no—
347C22—714no—
348C22—714no—
349C22—718no—
350C22—718no—
351C22C2×C22⋊C4×D7111yescharacteristic
352C22#124×D7111yescharacteristic
353C22C4⋊Q8×D7111yescharacteristic
354C22AGL(1,ℤ/8)×D7112yes—
355C22#137×D7112yes—
356C4—24no—
357C4—44no—
358C4—82no—
359C4—2844no—
360C4—5682no—
361C4—714no—
362C4—1424no—
363C4C4×Q8×D7112yes—
364C4C22⋊Q8×D7112yes—
365C2—718no—
366C2#391×D7111yescharacteristic
367C2#417×D7112yes—
368C2#490×D7111yescharacteristic
369C2#491×D7111yescharacteristic
370C2#501×D7112yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1992C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1992D71
hypocenterC71C2×#1992
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
4. #22589310C2semidirect
5. #22589309C2semidirect
6. #22589308C2semidirect
7. #22589307C2semidirect
9. C71×#1992C2semidirect
15. #125×D71C4semidirect
43. #14556120C2×C4semidirect
45. #14556119C2×C4semidirect
69. #125×C71C2×C4semidirect
185. D71#1992direct (N × H)
193. #1992D71direct (N × H)
195. C71C2×#1992semidirect
235. #125C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D712
C2×#97×D712
#384×D712
#225893102
#225893092
#225893082
#225893072
#340×D712
C71×#19922
C2×#199271