Finite Group Databasebeta

← #32055501

Subgroups — #32055501

Order 18176.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22591215C211yesmaximal, characteristic
2#22589902C211yesmaximal, characteristic, complemented
3#22589841C211yesmaximal, characteristic, complemented
4#22589831C211yesmaximal, characteristic, complemented
5#22589675C211yesmaximal, characteristic, complemented
6#22589298C211yesmaximal, characteristic, complemented
7#22589096C211yesmaximal, characteristic, complemented
8#22588922C211yesmaximal, characteristic, complemented
9#22588915C211yesmaximal, characteristic, complemented
10#22588539C211yesmaximal, characteristic
11#22588525C211yesmaximal, characteristic
12#22588424C211yesmaximal, characteristic
13#22588371C211yesmaximal, characteristic
14#22588061C211yesmaximal, characteristic
15C71×#2441C211yesmaximal, characteristic
16C22×#11765849C2211yescharacteristic
17C22×C4⋊Dic71C2211yescharacteristic
18C2×#14556061C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25#17985897C2211yescharacteristic
26#17985897C2211yescharacteristic
27#17985882—21no—
28#17985882—21no—
29#17985859C2211yescharacteristic
30#17985857—21no—
31C4×C4⋊Dic71C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C22⋊C4×Dic71—21no—
34#17985761C2211yescharacteristic
35#17985761C2211yescharacteristic
36#17985761C2211yescharacteristic
37#17985761C2211yescharacteristic
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C2×#125×C71C2211yescharacteristic
49C2×#117×C71C2211yescharacteristic
50#399×C71—21no—
51#396×C71—21no—
52#389×C71—21no—
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#387×C71—21no—
56#384×C71C2211yescharacteristic
57#380×C71C2211yescharacteristic
58C4×C22⋊C4×C71C2211yescharacteristic
59C23×Dic71—21no—
60C2×#11765849—22no—
61C2×#11765849—22no—
62C2×#11765849—41no—
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C2×C4⋊Dic71—21no—
68C2×C4⋊Dic71—22no—
69C2×C4⋊Dic71Q814yes—
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×C22⋊Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—21no—
79#14556061—41no—
80#14556061—41no—
81#14556059—22no—
82#14556059—22no—
83C4⋊(C71⋊C8)—22no—
84C4⋊(C71⋊C8)—41no—
85C22⋊(C71⋊C8)—22no—
86C71⋊#95—21no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C23×C4×C71C2311yescharacteristic
92#125×C71—22no—
93#125×C71—41no—
94#117×C71—22no—
95#117×C71—41no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C22⋊C4×C71—21no—
107C2×C22⋊C4×C71—21no—
108C2×C22⋊C4×C71D411yescharacteristic
109C2×C22⋊C4×C71D411yescharacteristic
110C2×C42×C71—21no—
111C2×C42×C71C2311yescharacteristic
112C2×C42×C71C2311yescharacteristic
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71D411yescharacteristic
117#95×C71D411yescharacteristic
118C22×Dic71—21no—
119C22×Dic71—21no—
120C22×Dic71—24no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—81no—
124C22×Dic71—81no—
125#11765849—44no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8#4112yes—
138C4⋊Dic71—24no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71C2×Q812yes—
146C4⋊Dic71C2×Q812yes—
147Dic71⋊C4—81no—
148Dic71⋊C4—81no—
149C22⋊Dic71—42no—
150C4×Dic71—81no—
151C4×Dic71—81no—
152C24×C71C2×D411yescharacteristic
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×Q812yes—
167C22×C4×C71#4112yes—
168C22×C4×C71C2411yescharacteristic
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—42no—
177C4⋊C4×C71—42no—
178C4⋊C4×C71—42no—
179C4⋊C4×C71—42no—
180C4⋊C4×C71—81no—
181C4⋊C4×C71—81no—
182C4⋊C4×C71#4112yes—
183C22⋊C4×C71—22no—
184C22⋊C4×C71—41no—
185C22⋊C4×C71—42no—
186C22⋊C4×C71—81no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—21no—
190C42×C71—21no—
191C42×C71—42no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71C22⋊Q812yes—
199C23×C71C22×D411yescharacteristic
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—22no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71C22⋊Q812yes—
230C2×C4×C71#13712yes—
231C2×C4×C71C22×Q811yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71H(5,𝔽2)11yescharacteristic
236C2×C4×C712−1+411yescharacteristic
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—84no—
250C71⋊C8—22no—
251C71⋊C8—41no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71—41no—
261C22×C71C2×C22⋊Q811yescharacteristic
262C22×C71#54611yescharacteristic
263C22×C71#54711yescharacteristic
264C22×C71C2×#13711yescharacteristic
265C22×C71#57511yescharacteristic
266C22×C71#57611yescharacteristic
267C22×C71#57711yescharacteristic
268C4×C71—22no—
269C4×C71—22no—
270C4×C71—41no—
271C4×C71—42no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71—81no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71#53311yescharacteristic
278C4×C71#54411yescharacteristic
279C4×C71#55111yescharacteristic
280C4×C71#55211yescharacteristic
281Dic71—42no—
282Dic71—42no—
283Dic71—81no—
284Dic71—81no—
285#26018—711nomaximal, Sylow
286C2×C71—42no—
287C2×C71#243811yescharacteristic
288C2×C71#307811yescharacteristic
289C2×C71#308111yescharacteristic
290C2×C71#330911yescharacteristic
291C2×C71#333111yescharacteristic
292C2×C71#334611yescharacteristic
293C2×C71#335111yescharacteristic
294#2441D7111yescharacteristic
295#2345—711no—
296#2100—711no—
297#2088—711no—
298#2086—711no—
299#2059—711no—
300#1989—711no—
301#1947—711no—
302#1920—711no—
303#1919—711no—
304#1854—711no—
305#1851—711no—
306#1823—711no—
307#1809—711no—
308#1736—711no—
309C71#2601811yescharacteristic, complemented, Sylow
310C22×M16—711no—
311C2×#125C2×D7111yescharacteristic
312C2×#117C2×D7111yescharacteristic
313C22×C4⋊C4—711no—
314C2×#107—711no—
315C2×#107—711no—
316C2×#106—711no—
317C2×#106—711no—
318C2×C4⋊C8—711no—
319C2×C4⋊C8—711no—
320C2×C22⋊C8—711no—
321#399—21no—
322#396—21no—
323#389—21no—
324#389—711no—
325#389—711no—
326#389C2×D7111yescharacteristic
327#389C2×D7111yescharacteristic
328#387—21no—
329#387—1421no—
330#387—1421no—
331#384C2×D7111yescharacteristic
332#380—711no—
333#380—1421no—
334#380C2×D7111yescharacteristic
335C4×C4⋊C4—711no—
336C4×C4⋊C4—711no—
337C4×C22⋊C4—1421no—
338C4×C22⋊C4C2×D7111yescharacteristic
339#340—711no—
340#340—711no—
341#340—711no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#336—711no—
352#336—711no—
353C23×C4—1421no—
354C23×C4C22×D7111yescharacteristic
355C2×M16—2841no—
356C2×M16—1422no—
357C2×M16—1422no—
358C22×C8—711no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362#125—22no—
363#125—41no—
364#117—22no—
365#117—41no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—714no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4—1422no—
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C4⋊C4C22×D7111yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1421no—
388C2×C22⋊C4C22⋊D7111yescharacteristic
389C2×C22⋊C4C22⋊D7111yescharacteristic
390C2×C42—21no—
391C2×C42—1421no—
392C2×C42—1421no—
393C2×C42C22×D7111yescharacteristic
394C2×C42C22×D7111yescharacteristic
395#107—2841no—
396#107—2841no—
397#106—1422no—
398#106—1422no—
399C4⋊C8—2841no—
400C4⋊C8—1422no—
401C22⋊C8—1422no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1421no—
406#95—1421no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95C22⋊D7111yescharacteristic
411#95C22⋊D7111yescharacteristic
412C24C2×C22⋊D7111yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—5681no—
422C22×C4—5681no—
423C22×C4—1421no—
424C22×C4—1421no—
425C22×C4—1421no—
426C22×C4—1421no—
427C22×C4—1424no—
428C22×C4C2×C22⋊D7111yescharacteristic
429C22×C4C2×C22⋊D7111yescharacteristic
430C22×C4C2×C22⋊D7111yescharacteristic
431C22×C4C2×C22⋊D7111yescharacteristic
432C22×C4C2×C22⋊D7111yescharacteristic
433C22×C4Q8×D7112yes—
434C22×C4#1176586612yes—
435C22×C4C23×D7111yescharacteristic
436M16—2844no—
437C2×C8—712no—
438C2×C8—1421no—
439C2×C8—1421no—
440C2×C8—1421no—
441C2×C8—1421no—
442C2×C8—1421no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—5681no—
463C4⋊C4—5681no—
464C4⋊C4—712no—
465C4⋊C4—712no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1424no—
473C4⋊C4#1176586412yes—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—2842no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—5681no—
485C42—5681no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1455610412yes—
493C23C22×C22⋊D7111yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—5684no—
534C2×C4—1422no—
535C2×C4—1422no—
536C2×C4#1455610412yes—
537C2×C4#1455615812yes—
538C2×C4C2×Q8×D7111yescharacteristic
539C2×C4#41×D7111yescharacteristic
540C2×C4C2×#1176586411yescharacteristic
541C2×C4C2×#1176586611yescharacteristic
542C2×C4#1455617311yescharacteristic
543C2×C4#1455617511yescharacteristic
544C8—2841no—
545C8—2841no—
546C8—2841no—
547C8—1422no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1455610411yescharacteristic
556C22#1798664311yescharacteristic
557C22#1798664811yescharacteristic
558C22C2×#1455615811yescharacteristic
559C22#1798679711yescharacteristic
560C22#1798680211yescharacteristic
561C22#1798680711yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—2842no—
572C4—2842no—
573C4—5681no—
574C4—5681no—
575C4#1798655111yescharacteristic
576C4#1798662811yescharacteristic
577C4#1798667611yescharacteristic
578C4#1798669111yescharacteristic
579C2—42no—
580C2#2259201211yescharacteristic
581C2#2259814111yescharacteristic
582C2#2259816511yescharacteristic
583C2#2260034611yescharacteristic
584C2#2260065611yescharacteristic
585C2#2260082511yescharacteristic
586C2#2260090011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2441C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2441D71
hypocenterC71#26018
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. #22589902C2semidirect
3. #22589841C2semidirect
4. #22589831C2semidirect
5. #22589675C2semidirect
6. #22589298C2semidirect
7. #22589096C2semidirect
8. #22588922C2semidirect
9. #22588915C2semidirect
309. C71#26018semidirect

Maximal subgroups

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

subgroupindex
#225912152
#225899022
#225898412
#225898312
#225896752
#225892982
#225890962
#225889222
#225889152
#225885392
#225885252
#225884242
#225883712
#225880612
C71×#24412
#2601871