Finite Group Databasebeta

← #2608×D71 #32168879

Subgroups — #2608×D71 #32168879

Order 18176.

2270 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22594010C211yesmaximal, characteristic, complemented
2#22594009C214yesmaximal, complemented
3#22594008C214yesmaximal, complemented
4#22594007C212yesmaximal, complemented
5#22594006C214yesmaximal, complemented
6#396×D71C214yesmaximal
7#389×D71C214yesmaximal, complemented
8#387×D71C214yesmaximal, complemented
9#381×D71C211yesmaximal, characteristic
10C4×C22⋊C4×D71C212yesmaximal
11C71×#2608C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C2×C4⋊C4×D71C2214yes—
14C2×C4⋊C4×D71C2214yes—
15C2×C4⋊C4×D71C2214yes—
16C2×C22⋊C4×D71C2214yes—
17C2×C22⋊C4×D71C2212yes—
18C2×C42×D71C2212yes—
19C2×C42×D71C2212yes—
20#17985923C2214yes—
21#17985922C2214yes—
22#17985922C2214yes—
23#17985922C2214yes—
24#17985921C2214yes—
25#17985921C2214yes—
26#17985921C2214yes—
27#17985898C2214yescomplemented
28#17985897C2214yescomplemented
29#17985896C2214yescomplemented
30#17985896C2214yescomplemented
31#17985895C2214yescomplemented
32#17985894C2214yescomplemented
33#17985894C2214yescomplemented
34#17985886C2214yescomplemented
35#17985885C2214yescomplemented
36#17985884C2214yescomplemented
37#17985883C2214yescomplemented
38#17985882C2214yescomplemented
39#17985881C2214yescomplemented
40#17985880C2214yescomplemented
41#17985862C2211yescharacteristic
42#17985861C2214yes—
43#17985860C2212yes—
44C22⋊C4×Dic71C2214yes—
45C4×(C2×D71)⋊C4C2212yes—
46C4×(C2×D71)⋊C4C2212yes—
47C4×C22⋊Dic71C2212yes—
48#17985846C2214yes—
49#95×D71C2214yes—
50#95×D71C2214yes—
51#95×D71C2214yes—
52#396×C71C2214yes—
53#389×C71C2214yescomplemented
54#387×C71C2214yescomplemented
55#381×C71C2211yescharacteristic
56C4×C22⋊C4×C71C2212yes—
57C24×D71C2311yescharacteristic
58C23×Dic71C2311yescharacteristic
59C22×C4×D71—44no—
60C22×C4×D71Q818yes—
61C22×C4×D71C2314yes—
62C22×C4×D71C2314yes—
63C22×C4×D71C2312yes—
64C22×C4×D71C2314yes—
65C22×C4×D71C2312yes—
66C2×C4⋊Dic71C2314yes—
67C2×C4⋊Dic71C2314yes—
68C2×C4⋊Dic71C2314yes—
69C2×Dic71⋊C4C2314yes—
70C2×Dic71⋊C4C2314yes—
71C2×Dic71⋊C4C2314yes—
72C2×Dic71⋊C4C2314yes—
73C2×Dic71⋊C4C2314yes—
74C2×Dic71⋊C4C2314yes—
75C4⋊C4×D71—216no—
76C4⋊C4×D71—216no—
77C4⋊C4×D71—48no—
78C2×C22⋊Dic71C2314yes—
79C2×C22⋊Dic71C2312yes—
80C2×(C2×D71)⋊C4C2314yes—
81C2×(C2×D71)⋊C4C2314yes—
82C2×(C2×D71)⋊C4C2314yes—
83C22⋊C4×D71—216no—
84C22⋊C4×D71—44no—
85C2×C4×Dic71C2314yes—
86C2×C4×Dic71C2314yes—
87C2×C4×Dic71C2312yes—
88C2×C4×Dic71C2312yes—
89C42×D71—44no—
90C42×D71—44no—
91C71⋊#95C2314yes—
92C71⋊#95C2314yes—
93C71⋊#95C2314yes—
94C71⋊#95C2314yes—
95C71⋊#95C2314yes—
96C71⋊#95C2314yes—
97C71⋊#95C2314yes—
98C71⋊#95C2314yes—
99C71⋊#95C2314yes—
100C23×C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2314yes—
102C2×C4⋊C4×C71C2314yes—
103C2×C4⋊C4×C71C2314yes—
104C2×C22⋊C4×C71C2314yes—
105C2×C22⋊C4×C71C2312yes—
106C2×C42×C71C2312yes—
107C2×C42×C71C2312yes—
108#95×C71C2314yes—
109#95×C71C2314yes—
110#95×C71C2314yes—
111C23×D71—28no—
112C23×D71—24no—
113C23×D71C2×Q814yes—
114C23×D71C2411yescharacteristic
115C23×D71C2411yescharacteristic
116C22×Dic71—42no—
117C22×Dic71C2×Q814yes—
118C22×Dic71C2414yes—
119C22×Dic71C2414yes—
120C22×Dic71C2412yes—
121C22×Dic71C2414yes—
122C2×C4×D71—28no—
123C2×C4×D71—24no—
124C2×C4×D71—24no—
125C2×C4×D71—28no—
126C2×C4×D71—28no—
127C2×C4×D71—28no—
128C2×C4×D71—28no—
129C2×C4×D71—28no—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71—416no—
133C2×C4×D71—42no—
134C2×C4×D71—42no—
135C2×C4×D71C2×Q818yes—
136C2×C4×D71C2×Q818yes—
137C2×C4×D71#41116yes—
138C4⋊Dic71—28no—
139C4⋊Dic71—28no—
140C4⋊Dic71—44no—
141Dic71⋊C4—28no—
142Dic71⋊C4—28no—
143Dic71⋊C4—28no—
144Dic71⋊C4—28no—
145Dic71⋊C4—44no—
146Dic71⋊C4—44no—
147C22⋊Dic71—28no—
148C22⋊Dic71—42no—
149(C2×D71)⋊C4—28no—
150(C2×D71)⋊C4—28no—
151(C2×D71)⋊C4—44no—
152C4×Dic71—42no—
153C4×Dic71—44no—
154C4×Dic71—44no—
155C4×Dic71—42no—
156C24×C71C2411yescharacteristic
157C22×C4×C71—42no—
158C22×C4×C71C2×Q814yes—
159C22×C4×C71C2414yes—
160C22×C4×C71C2414yes—
161C22×C4×C71C2412yes—
162C22×C4×C71C2414yes—
163C4⋊C4×C71—28no—
164C4⋊C4×C71—28no—
165C4⋊C4×C71—44no—
166C22⋊C4×C71—28no—
167C22⋊C4×C71—42no—
168C42×C71—42no—
169C42×C71—42no—
170C23×C71—24no—
171C23×C71—22no—
172C23×C71C22×Q812yes—
173C23×C71C2511yescharacteristic
174C22×D71—24no—
175C22×D71—22no—
176C22×D71—216no—
177C22×D71—28no—
178C22×D71—416no—
179C22×D71C22×Q812yes—
180C22×D71C22×Q812yes—
181C22×D71C2×#4118yes—
182C22×D71H(5,𝔽2)14yes—
183C2×C4×C71—24no—
184C2×C4×C71—24no—
185C2×C4×C71—22no—
186C2×C4×C71—22no—
187C2×C4×C71—24no—
188C2×C4×C71—24no—
189C2×C4×C71—24no—
190C2×C4×C71—24no—
191C2×C4×C71—24no—
192C2×C4×C71—24no—
193C2×C4×C71—44no—
194C2×C4×C71C22×Q814yes—
195C2×C4×C71C2×#4118yes—
196C2×Dic71—24no—
197C2×Dic71—24no—
198C2×Dic71—22no—
199C2×Dic71—22no—
200C2×Dic71—24no—
201C2×Dic71—24no—
202C2×Dic71—24no—
203C2×Dic71—24no—
204C2×Dic71—24no—
205C2×Dic71—24no—
206C2×Dic71—44no—
207C2×Dic71C22×Q814yes—
208C2×Dic71C2×#4118yes—
209C4×D71—216no—
210C4×D71—232no—
211C4×D71—48no—
212C4×D71—416no—
213C4×D71—416no—
214C22×C71—24no—
215C22×C71—22no—
216C22×C71—44no—
217C22×C71C23×Q811yescharacteristic
218C22×C71C22×#4114yes—
219C22×C71C2×H(5,𝔽2)12yes—
220C2×D71—24no—
221C2×D71—22no—
222C2×D71—44no—
223C2×D71—416no—
224C2×D71#53318yes—
225C2×D71#54314yes—
226C2×D71#553116yes—
227C4×C71—24no—
228C4×C71—28no—
229C4×C71—44no—
230C4×C71—44no—
231Dic71—24no—
232Dic71—28no—
233Dic71—44no—
234Dic71—44no—
235C2×#2608—711nomaximal, Sylow
236C2×C71—42no—
237C2×C71C2×#53312yes—
238C2×C71C2×#54311yescharacteristic
239C2×C71C2×#55314yes—
240D71—42no—
241D71#260818yesdirect product factor
242#2608—714no—
243#2608—714no—
244#2608—714no—
245#2608—711no—
246#2608—712no—
247#2608D7111yescharacteristic, direct product factor
248C2×#396—714no—
249C2×#389—714no—
250C2×#387—714no—
251C2×#381—711no—
252C2×C4×C22⋊C4—712no—
253C71C2×#260811yescharacteristic, complemented, Sylow
254C24×C4—711no—
255C22×C4⋊C4—714no—
256C22×C4⋊C4—714no—
257C22×C4⋊C4—714no—
258C22×C22⋊C4—712no—
259C22×C22⋊C4—714no—
260C22×C42—712no—
261C22×C42—712no—
262#396—714no—
263#396—714no—
264#396—714no—
265#396—714no—
266#396—714no—
267#396—714no—
268#396—714no—
269#396C2×D7114yes—
270#389—714no—
271#389—714no—
272#389—714no—
273#389—714no—
274#389—714no—
275#389—714no—
276#389—714no—
277#389C2×D7114yescomplemented
278#387—714no—
279#387—714no—
280#387—714no—
281#387—714no—
282#387—714no—
283#387—714no—
284#387—714no—
285#387C2×D7114yescomplemented
286#381—711no—
287#381—712no—
288#381—714no—
289#381C2×D7111yescharacteristic
290C4×C22⋊C4—712no—
291C4×C22⋊C4—714no—
292C4×C22⋊C4—712no—
293C4×C22⋊C4—712no—
294C4×C22⋊C4—714no—
295C4×C22⋊C4C2×D7112yes—
296C2×#95—714no—
297C2×#95—714no—
298C2×#95—714no—
299C25—711no—
300C23×C4—2844no—
301C23×C4—711no—
302C23×C4—718no—
303C23×C4—712no—
304C23×C4—714no—
305C23×C4—712no—
306C23×C4—714no—
307C23×C4—714no—
308C23×C4C22×D7111yescharacteristic
309C2×C4⋊C4—2848no—
310C2×C4⋊C4—714no—
311C2×C4⋊C4—714no—
312C2×C4⋊C4—714no—
313C2×C4⋊C4—714no—
314C2×C4⋊C4—714no—
315C2×C4⋊C4—714no—
316C2×C4⋊C4—714no—
317C2×C4⋊C4—714no—
318C2×C4⋊C4—714no—
319C2×C4⋊C4—14216no—
320C2×C4⋊C4—14216no—
321C2×C4⋊C4C22×D7114yes—
322C2×C4⋊C4C22×D7114yes—
323C2×C4⋊C4C22×D7114yes—
324C2×C22⋊C4—2844no—
325C2×C22⋊C4—712no—
326C2×C22⋊C4—714no—
327C2×C22⋊C4—714no—
328C2×C22⋊C4—714no—
329C2×C22⋊C4—714no—
330C2×C22⋊C4—14216no—
331C2×C22⋊C4C22×D7114yes—
332C2×C22⋊C4C22×D7112yes—
333C2×C42—2844no—
334C2×C42—2844no—
335C2×C42—714no—
336C2×C42—712no—
337C2×C42—714no—
338C2×C42—712no—
339C2×C42C22×D7112yes—
340C2×C42C22×D7112yes—
341#95—714no—
342#95—714no—
343#95—714no—
344#95—714no—
345#95—714no—
346#95—714no—
347#95—714no—
348#95—714no—
349#95—714no—
350#95C22×D7114yes—
351#95C22×D7114yes—
352#95C22×D7114yes—
353C24—711no—
354C24—714no—
355C24—711no—
356C24—1424no—
357C24—1428no—
358C24C23×D7111yescharacteristic
359C22×C4—42no—
360C22×C4—2842no—
361C22×C4—2842no—
362C22×C4—28416no—
363C22×C4—2842no—
364C22×C4—714no—
365C22×C4—718no—
366C22×C4—712no—
367C22×C4—714no—
368C22×C4—718no—
369C22×C4—7116no—
370C22×C4—714no—
371C22×C4—714no—
372C22×C4—1428no—
373C22×C4—1428no—
374C22×C4—1428no—
375C22×C4—1428no—
376C22×C4—1428no—
377C22×C4—1428no—
378C22×C4—1428no—
379C22×C4—1424no—
380C22×C4—1424no—
381C22×C4—1428no—
382C22×C4Q8×D7114yes—
383C22×C4C23×D7114yes—
384C22×C4C23×D7114yes—
385C22×C4C23×D7112yes—
386C22×C4C23×D7114yes—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—44no—
390C4⋊C4—2844no—
391C4⋊C4—2844no—
392C4⋊C4—2844no—
393C4⋊C4—1428no—
394C4⋊C4—1428no—
395C4⋊C4—1428no—
396C4⋊C4—1428no—
397C4⋊C4—1428no—
398C4⋊C4—1428no—
399C22⋊C4—28no—
400C22⋊C4—42no—
401C22⋊C4—2844no—
402C22⋊C4—2842no—
403C22⋊C4—1428no—
404C22⋊C4—1428no—
405C22⋊C4—1428no—
406C42—42no—
407C42—42no—
408C42—2844no—
409C42—2842no—
410C42—2842no—
411C42—2844no—
412C23—24no—
413C23—22no—
414C23—28416no—
415C23—712no—
416C23—718no—
417C23—714no—
418C23—712no—
419C23—1428no—
420C23—14216no—
421C23—1424no—
422C23—1422no—
423C23C2×Q8×D7112yes—
424C23C24×D7111yescharacteristic
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—44no—
436C2×C4—2848no—
437C2×C4—28416no—
438C2×C4—28416no—
439C2×C4—2844no—
440C2×C4—714no—
441C2×C4—718no—
442C2×C4—14216no—
443C2×C4—14232no—
444C2×C4—1424no—
445C2×C4—1424no—
446C2×C4—1424no—
447C2×C4—1424no—
448C2×C4—1424no—
449C2×C4—1424no—
450C2×C4—1424no—
451C2×C4—1424no—
452C2×C4—1422no—
453C2×C4—1422no—
454C2×C4C2×Q8×D7114yes—
455C2×C4#41×D7118yes—
456C22—22no—
457C22—24no—
458C22—44no—
459C22—28416no—
460C22—2844no—
461C22—714no—
462C22—7116no—
463C22—718no—
464C22—1422no—
465C22—1424no—
466C22C22×Q8×D7111yescharacteristic
467C22C2×#41×D7114yes—
468C22H(5,𝔽2)×D7112yes—
469C4—24no—
470C4—28no—
471C4—44no—
472C4—44no—
473C4—2844no—
474C4—2844no—
475C4—1424no—
476C4—1428no—
477C2—42no—
478C2—2842no—
479C2—718no—
480C2#533×D7112yes—
481C2#543×D7111yescharacteristic
482C2#553×D7114yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2608C2
prime radicalC24×D71C23
hypercenter#2608D71
hypocenterC71C2×#2608
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2608

Complements

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

NH ≅ G/Ntype
1. #22594010C2semidirect
2. #22594009C2semidirect
3. #22594008C2semidirect
4. #22594007C2semidirect
5. #22594006C2semidirect
7. #389×D71C2semidirect
8. #387×D71C2semidirect
11. C71×#2608C2semidirect
27. #17985898C22semidirect
28. #17985897C22semidirect
29. #17985896C22semidirect
30. #17985896C22semidirect
31. #17985895C22semidirect
32. #17985894C22semidirect
33. #17985894C22semidirect
34. #17985886C22semidirect
35. #17985885C22semidirect
36. #17985884C22semidirect
37. #17985883C22semidirect
38. #17985882C22semidirect
39. #17985881C22semidirect
40. #17985880C22semidirect
53. #389×C71C22semidirect
54. #387×C71C22semidirect
241. D71#2608direct (N × H)
247. #2608D71direct (N × H)
253. C71C2×#2608semidirect
277. #389C2×D71semidirect
285. #387C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225940102
#225940092
#225940082
#225940072
#225940062
#396×D712
#389×D712
#387×D712
#381×D712
C4×C22⋊C4×D712
C71×#26082
C2×#260871