Finite Group Databasebeta

← C71×#14326 #31835198

Subgroups — C71×#14326 #31835198

Order 18176.

1194 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C71C211yesmaximal, characteristic
2C2×#389×C71C211yesmaximal, characteristic
3C2×#387×C71C211yesmaximal, characteristic
4C71×#1445C211yesmaximal, characteristic
5C71×#1443C212yesmaximal
6C71×#1442C212yesmaximal
7C71×#1442C211yesmaximal, characteristic
8C71×#1441C211yesmaximal, characteristic
9C71×#1436C212yesmaximal
10C71×#1436C212yesmaximal
11C4×#95×C71C211yesmaximal, characteristic
12C22×C4⋊C4×C71C2212yes—
13C22×C4⋊C4×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16C22×C4⋊C4×C71C2211yescharacteristic
17C22×C42×C71C2212yes—
18C22×C42×C71C2211yescharacteristic
19#400×C71C418yescomplemented
20#389×C71—24no—
21#387×C71—24no—
22C2×#95×C71C2211yescharacteristic
23C2×#95×C71C2211yescharacteristic
24C2×#95×C71C2211yescharacteristic
25C2×#95×C71C2211yescharacteristic
26C2×#95×C71C2212yes—
27C2×#95×C71C2211yescharacteristic
28C2×#95×C71C2212yes—
29C2×#95×C71C2212yes—
30C2×#95×C71C2212yes—
31C2×#95×C71C2212yes—
32C2×#95×C71C2212yes—
33C2×#95×C71C2212yes—
34C2×#95×C71C2212yes—
35C2×#95×C71C2211yescharacteristic
36C2×#95×C71C2212yes—
37C2×#95×C71C2212yes—
38C23×C4×C71C2312yes—
39C23×C4×C71C2311yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C23×C4×C71C2312yes—
42C23×C4×C71C2312yes—
43C23×C4×C71C2311yescharacteristic
44C23×C4×C71C2311yescharacteristic
45C23×C4×C71C2311yescharacteristic
46C23×C4×C71C2312yes—
47C23×C4×C71C2311yescharacteristic
48C23×C4×C71C2311yescharacteristic
49C2×C4⋊C4×C71—22no—
50C2×C4⋊C4×C71—22no—
51C2×C4⋊C4×C71—22no—
52C2×C4⋊C4×C71—22no—
53C2×C4⋊C4×C71—22no—
54C2×C4⋊C4×C71—22no—
55C2×C4⋊C4×C71—24no—
56C2×C4⋊C4×C71—24no—
57C2×C4⋊C4×C71—24no—
58C2×C4⋊C4×C71—42no—
59C2×C4⋊C4×C71—42no—
60C2×C4⋊C4×C71C2×C414yes—
61C2×C4⋊C4×C71C2×C414yes—
62C2×C42×C71—24no—
63C2×C42×C71—42no—
64C2×C42×C71—42no—
65C2×C42×C71—42no—
66C2×C42×C71D414yes—
67#95×C71—22no—
68#95×C71—22no—
69#95×C71—22no—
70#95×C71—22no—
71#95×C71—24no—
72#95×C71—24no—
73#95×C71—22no—
74#95×C71—24no—
75#95×C71—24no—
76#95×C71—42no—
77#95×C71—42no—
78#95×C71—42no—
79#95×C71—42no—
80#95×C71C2×C414yes—
81#95×C71C2×C418yes—
82#95×C71C2×C418yes—
83C24×C71C2411yescharacteristic
84C22×C4×C71—22no—
85C22×C4×C71—21no—
86C22×C4×C71—22no—
87C22×C4×C71—22no—
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71—22no—
91C22×C4×C71—21no—
92C22×C4×C71—22no—
93C22×C4×C71—22no—
94C22×C4×C71—22no—
95C22×C4×C71—22no—
96C22×C4×C71—22no—
97C22×C4×C71—22no—
98C22×C4×C71—22no—
99C22×C4×C71—22no—
100C22×C4×C71—21no—
101C22×C4×C71—21no—
102C22×C4×C71—22no—
103C22×C4×C71—22no—
104C22×C4×C71—21no—
105C22×C4×C71—22no—
106C22×C4×C71—22no—
107C22×C4×C71—22no—
108C22×C4×C71—22no—
109C22×C4×C71—22no—
110C22×C4×C71—22no—
111C22×C4×C71—21no—
112C22×C4×C71—21no—
113C22×C4×C71—22no—
114C22×C4×C71—21no—
115C22×C4×C71—22no—
116C22×C4×C71—22no—
117C22×C4×C71—21no—
118C22×C4×C71—21no—
119C22×C4×C71—22no—
120C22×C4×C71—22no—
121C22×C4×C71—21no—
122C22×C4×C71—21no—
123C22×C4×C71—22no—
124C22×C4×C71—22no—
125C22×C4×C71—22no—
126C22×C4×C71—22no—
127C22×C4×C71—22no—
128C22×C4×C71—22no—
129C22×C4×C71—21no—
130C22×C4×C71—22no—
131C22×C4×C71—22no—
132C22×C4×C71—22no—
133C22×C4×C71—21no—
134C22×C4×C71—22no—
135C22×C4×C71—21no—
136C22×C4×C71—22no—
137C22×C4×C71—21no—
138C22×C4×C71C22×C414yes—
139C22×C4×C71C22×C414yes—
140C22×C4×C71C22×C412yes—
141C22×C4×C71C22×C412yes—
142C22×C4×C71C22×C412yes—
143C22×C4×C71C2×D412yes—
144C22×C4×C71C2×D412yes—
145C22×C4×C71C2×D412yes—
146C22×C4×C71#4114yes—
147C22×C4×C71#4112yes—
148C22×C4×C71#4114yes—
149C4⋊C4×C71—28no—
150C4⋊C4×C71—28no—
151C4⋊C4×C71—44no—
152C4⋊C4×C71—44no—
153C4⋊C4×C71—48no—
154C42×C71—44no—
155C42×C71—48no—
156C23×C71C23×C411yescharacteristic
157C23×C71C22×D411yescharacteristic
158C23×C71C2×#4111yescharacteristic
159C23×C71C2×#4112yes—
160C23×C71C2×#4112yes—
161C23×C71H(5,𝔽2)12yes—
162C23×C71H(5,𝔽2)11yescharacteristic
163C23×C712−1+411yescharacteristic
164C23×C712−1+411yescharacteristic
165C23×C712−1+412yes—
166C23×C712−1+411yescharacteristic
167C2×C4×C71—24no—
168C2×C4×C71—22no—
169C2×C4×C71—22no—
170C2×C4×C71—22no—
171C2×C4×C71—22no—
172C2×C4×C71—24no—
173C2×C4×C71—24no—
174C2×C4×C71—24no—
175C2×C4×C71—22no—
176C2×C4×C71—22no—
177C2×C4×C71—24no—
178C2×C4×C71—24no—
179C2×C4×C71—24no—
180C2×C4×C71—24no—
181C2×C4×C71—22no—
182C2×C4×C71—24no—
183C2×C4×C71—22no—
184C2×C4×C71—24no—
185C2×C4×C71—24no—
186C2×C4×C71—24no—
187C2×C4×C71—22no—
188C2×C4×C71—22no—
189C2×C4×C71—22no—
190C2×C4×C71—24no—
191C2×C4×C71—24no—
192C2×C4×C71—42no—
193C2×C4×C71—42no—
194C2×C4×C71—42no—
195C2×C4×C71—42no—
196C2×C4×C71—41no—
197C2×C4×C71—41no—
198C2×C4×C71—41no—
199C2×C4×C71—41no—
200C2×C4×C71—42no—
201C2×C4×C71—42no—
202C2×C4×C71—42no—
203C2×C4×C71—42no—
204C2×C4×C71—42no—
205C2×C4×C71—42no—
206C2×C4×C71—41no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—41no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—42no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—42no—
221C2×C4×C71—41no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71C4×D414yes—
231C2×C4×C71#12414yes—
232C22×C71C2×C4×D411yescharacteristic
233C22×C71C4×#4112yes—
234C22×C71#51911yescharacteristic
235C22×C71#52012yes—
236C22×C71#52011yescharacteristic
237C22×C71C2×#12411yescharacteristic
238C22×C71#52912yes—
239C22×C71#53211yescharacteristic
240C22×C71#53411yescharacteristic
241C22×C71#53611yescharacteristic
242C22×C71#53611yescharacteristic
243C22×C71#53711yescharacteristic
244C22×C71#53812yes—
245C22×C71#53912yes—
246C22×C71#53912yes—
247C22×C71#53912yes—
248C22×C71#53912yes—
249C22×C71#54111yescharacteristic
250C22×C71#54112yes—
251C22×C71#54112yes—
252C22×C71#54112yes—
253C22×C71#54211yescharacteristic
254C22×C71#54212yes—
255C4×C71—24no—
256C4×C71—24no—
257C4×C71—42no—
258C4×C71—42no—
259C4×C71—44no—
260C4×C71—44no—
261C4×C71—42no—
262C4×C71—44no—
263C4×C71—82no—
264C4×C71—81no—
265C4×C71—81no—
266#14326C7111yesmaximal, characteristic, direct product factor, Sylow
267C2×C71C4×#12411yescharacteristic
268C2×C71#232211yescharacteristic
269C2×C71#232311yescharacteristic
270C2×C71#233412yes—
271C2×C71#233612yes—
272C2×C71#263911yescharacteristic
273C2×C71#264211yescharacteristic
274C2×C71#265212yes—
275C2×C71#265512yes—
276C2×C71#266311yescharacteristic
277C2×C71#266411yescharacteristic
278C2×#400C2×C7111yescharacteristic
279C2×#389C2×C7111yescharacteristic
280C2×#387C2×C7111yescharacteristic
281#1445C2×C7111yescharacteristic
282#1443C2×C7112yes—
283#1442C2×C7111yescharacteristic
284#1442C2×C7112yes—
285#1441C2×C7111yescharacteristic
286#1436C2×C7112yes—
287#1436C2×C7112yes—
288C4×#95C2×C7111yescharacteristic
289C71#1432611yescharacteristic, direct product factor, Sylow
290C22×C4⋊C4C22×C7112yes—
291C22×C4⋊C4C22×C7111yescharacteristic
292C22×C4⋊C4C22×C7111yescharacteristic
293C22×C4⋊C4C22×C7111yescharacteristic
294C22×C4⋊C4C22×C7111yescharacteristic
295C22×C42C22×C7112yes—
296C22×C42C22×C7111yescharacteristic
297#400C4×C7118yescomplemented
298#389—24no—
299#387—24no—
300C2×#95C22×C7111yescharacteristic
301C2×#95C22×C7112yes—
302C2×#95C22×C7111yescharacteristic
303C2×#95C22×C7111yescharacteristic
304C2×#95C22×C7112yes—
305C2×#95C22×C7112yes—
306C2×#95C22×C7112yes—
307C2×#95C22×C7111yescharacteristic
308C2×#95C22×C7111yescharacteristic
309C2×#95C22×C7112yes—
310C2×#95C22×C7112yes—
311C2×#95C22×C7112yes—
312C2×#95C22×C7112yes—
313C2×#95C22×C7111yescharacteristic
314C2×#95C22×C7112yes—
315C2×#95C22×C7112yes—
316C23×C4C23×C7112yes—
317C23×C4C23×C7111yescharacteristic
318C23×C4C23×C7111yescharacteristic
319C23×C4C23×C7112yes—
320C23×C4C23×C7111yescharacteristic
321C23×C4C23×C7111yescharacteristic
322C23×C4C23×C7112yes—
323C23×C4C23×C7111yescharacteristic
324C23×C4C23×C7112yes—
325C23×C4C23×C7111yescharacteristic
326C23×C4C23×C7111yescharacteristic
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4C2×C4×C7114yes—
339C2×C4⋊C4C2×C4×C7114yes—
340C2×C42—24no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42D4×C7114yes—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95C2×C4×C7114yes—
359#95C2×C4×C7118yes—
360#95C2×C4×C7118yes—
361C24C24×C7111yescharacteristic
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4C22×C4×C7114yes—
417C22×C4C22×C4×C7114yes—
418C22×C4C22×C4×C7112yes—
419C22×C4C22×C4×C7112yes—
420C22×C4C22×C4×C7112yes—
421C22×C4C2×D4×C7112yes—
422C22×C4C2×D4×C7112yes—
423C22×C4C2×D4×C7112yes—
424C22×C4#41×C7114yes—
425C22×C4#41×C7112yes—
426C22×C4#41×C7114yes—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C42—44no—
433C42—48no—
434C23C23×C4×C7111yescharacteristic
435C23C22×D4×C7111yescharacteristic
436C23C2×#41×C7111yescharacteristic
437C23C2×#41×C7112yes—
438C23C2×#41×C7112yes—
439C23H(5,𝔽2)×C7112yes—
440C23H(5,𝔽2)×C7111yescharacteristic
441C232−1+4×C7111yescharacteristic
442C232−1+4×C7111yescharacteristic
443C232−1+4×C7112yes—
444C232−1+4×C7111yescharacteristic
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4C4×D4×C7114yes—
509C2×C4#124×C7114yes—
510C22C2×C4×D4×C7111yescharacteristic
511C22C4×#41×C7112yes—
512C22#519×C7111yescharacteristic
513C22#520×C7112yes—
514C22#520×C7111yescharacteristic
515C22C2×#124×C7111yescharacteristic
516C22#529×C7112yes—
517C22#532×C7111yescharacteristic
518C22#534×C7111yescharacteristic
519C22#536×C7111yescharacteristic
520C22#536×C7111yescharacteristic
521C22#537×C7111yescharacteristic
522C22#538×C7112yes—
523C22#539×C7112yes—
524C22#539×C7112yes—
525C22#539×C7112yes—
526C22#539×C7112yes—
527C22#541×C7111yescharacteristic
528C22#541×C7112yes—
529C22#541×C7112yes—
530C22#541×C7112yes—
531C22#542×C7111yescharacteristic
532C22#542×C7112yes—
533C4—24no—
534C4—24no—
535C4—44no—
536C4—42no—
537C4—42no—
538C4—44no—
539C4—44no—
540C4—42no—
541C4—82no—
542C4—81no—
543C4—81no—
544C2C4×#124×C7111yescharacteristic
545C2C71×#232211yescharacteristic
546C2C71×#232311yescharacteristic
547C2C71×#233412yes—
548C2C71×#233612yes—
549C2C71×#263911yescharacteristic
550C2C71×#264211yescharacteristic
551C2C71×#265212yes—
552C2C71×#265512yes—
553C2C71×#266311yescharacteristic
554C2C71×#266411yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C71C24
commutatorC23C23×C4×C71
Frattini, cosocleC24C24×C71
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C71 ◁ G

Complements

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

NH ≅ G/Ntype
19. #400×C71C4semidirect
266. #14326C71direct (N × H)
289. C71#14326direct (N × H)
297. #400C4×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#400×C712
C2×#389×C712
C2×#387×C712
C71×#14452
C71×#14432
C71×#14422
C71×#14412
C71×#14362
C4×#95×C712
#1432671