Finite Group Databasebeta

← #31997349

Subgroups — #31997349

Order 18176.

1426 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D71C211yesmaximal, characteristic
2C2×#17986054C211yesmaximal, characteristic, complemented
3C2×#17986050C211yesmaximal, characteristic, complemented
4#22588407C212yesmaximal, complemented
5#22588323C212yesmaximal, complemented
6#22588322C212yesmaximal, complemented
7#22588249C212yesmaximal
8#22588109C211yesmaximal, characteristic, complemented
9#22588107C211yesmaximal, characteristic, complemented
10#22588106C211yesmaximal, characteristic, complemented
11C71×#1750C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C2×#14556087C2211yescharacteristic
14C2×#14556086C411yescharacteristic
15C2×#14556086C411yescharacteristic
16C2×#14556085C2211yescharacteristic
17C2×#14556085C2211yescharacteristic
18C2×#14556084C411yescharacteristic
19C2×#14556084C411yescharacteristic
20#117×D71—24no—
21#117×D71—24no—
22C2×C4⋊C4×D71C411yescharacteristic
23C2×C4⋊C4×D71C411yescharacteristic
24C2×C22⋊C4×D71C411yescharacteristic
25C2×C22⋊C4×D71C411yescharacteristic
26C2×C42×D71C2211yescharacteristic
27C2×C42×D71C2211yescharacteristic
28#17986054—24no—
29#17986050—24no—
30C2×Dic71⋊C8C2212yescomplemented
31C2×Dic71⋊C8C2212yescomplemented
32C2×C22⋊(C71⋊C8)C2212yes—
33C2×(C2×D71)⋊C8C2212yes—
34C2×(C2×D71)⋊C8C2212yes—
35C2×(C2×D71)⋊C8C2212yes—
36C2×(C2×D71)⋊C8C2212yes—
37C2×#14556039C2211yescharacteristic, complemented
38C2×C4×C71⋊C8C2211yescharacteristic, complemented
39#17985754C2212yescomplemented
40#17985753C2212yescomplemented
41#17985753C2212yescomplemented
42C2×#117×C71C2211yescharacteristic
43C2×C22⋊C8×C71C2212yes—
44C2×#97×C71C2211yescharacteristic, complemented
45C2×C4×C8×C71C2211yescharacteristic, complemented
46#336×C71C2212yescomplemented
47C24×D71C2×C411yescharacteristic
48C23×Dic71C2×C411yescharacteristic
49C22×C4×D71—22no—
50C22×C4×D71—22no—
51C22×C4×D71—24no—
52C22×C4×D71—24no—
53C22×C4×D71C2×C411yescharacteristic
54C22×C4×D71C2×C411yescharacteristic
55C22×C4×D71C2×C411yescharacteristic
56C22×C4×D71C2×C411yescharacteristic
57C22×C4×D71C2×C411yescharacteristic
58C22×C4×D71C2×C411yescharacteristic
59C22×C4×D71C2311yescharacteristic
60C22×C4×D71C2311yescharacteristic
61C22×C71⋊C8C2312yes—
62C22×C71⋊C8C2312yes—
63#14556087—22no—
64#14556087—22no—
65#14556086—24no—
66#14556086—24no—
67#14556085—22no—
68#14556085—22no—
69#14556085—22no—
70#14556085—22no—
71#14556084—24no—
72#14556084—24no—
73C2×C4⋊Dic71C2×C411yescharacteristic
74C2×C4⋊Dic71C2×C411yescharacteristic
75C2×Dic71⋊C4C2×C411yescharacteristic
76C2×Dic71⋊C4C2×C411yescharacteristic
77C2×Dic71⋊C4C2×C411yescharacteristic
78C2×Dic71⋊C4C2×C411yescharacteristic
79C4⋊C4×D71—24no—
80C4⋊C4×D71—24no—
81C2×C22⋊Dic71C2×C411yescharacteristic
82C2×C22⋊Dic71C2×C411yescharacteristic
83C2×(C2×D71)⋊C4C2×C411yescharacteristic
84C2×(C2×D71)⋊C4C2×C411yescharacteristic
85C2×(C2×D71)⋊C4C2×C411yescharacteristic
86C2×(C2×D71)⋊C4C2×C411yescharacteristic
87C22⋊C4×D71—24no—
88C22⋊C4×D71—24no—
89C2×C4×Dic71C2×C411yescharacteristic
90C2×C4×Dic71C2×C411yescharacteristic
91C2×C4×Dic71C2×C411yescharacteristic
92C2×C4×Dic71C2×C411yescharacteristic
93C2×C4×Dic71C2311yescharacteristic
94C2×C4×Dic71C2311yescharacteristic
95C42×D71—22no—
96C42×D71—22no—
97C42×D71D414yes—
98C42×D71D414yes—
99Dic71⋊C8—24no—
100Dic71⋊C8—24no—
101C22⋊(C71⋊C8)—24no—
102(C2×D71)⋊C8—24no—
103(C2×D71)⋊C8—24no—
104(C2×D71)⋊C8—24no—
105(C2×D71)⋊C8—24no—
106#14556039—22no—
107C4×C71⋊C8—22no—
108C23×C4×C71C2311yescharacteristic
109C22×C8×C71C2312yes—
110C22×C8×C71C2312yes—
111#117×C71—22no—
112#117×C71—22no—
113C2×C4⋊C4×C71C2×C411yescharacteristic
114C2×C4⋊C4×C71C2×C411yescharacteristic
115C2×C22⋊C4×C71C2×C411yescharacteristic
116C2×C22⋊C4×C71C2×C411yescharacteristic
117C2×C42×C71C2311yescharacteristic
118C2×C42×C71C2311yescharacteristic
119C22⋊C8×C71—24no—
120#97×C71—22no—
121C4×C8×C71—22no—
122C23×D71—22no—
123C23×D71—22no—
124C23×D71—22no—
125C23×D71—24no—
126C23×D71—42no—
127C23×D71C22×C411yescharacteristic
128C23×D71C22×C411yescharacteristic
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71C22×C411yescharacteristic
133C22×Dic71C22×C411yescharacteristic
134C22×Dic71C22×C411yescharacteristic
135C22×Dic71C22×C411yescharacteristic
136C22×Dic71C22×C411yescharacteristic
137C22×Dic71C22×C411yescharacteristic
138C2×C4×D71—216no—
139C2×C4×D71—216no—
140C2×C4×D71—22no—
141C2×C4×D71—22no—
142C2×C4×D71—24no—
143C2×C4×D71—22no—
144C2×C4×D71—22no—
145C2×C4×D71—21no—
146C2×C4×D71—21no—
147C2×C4×D71—22no—
148C2×C4×D71—22no—
149C2×C4×D71—22no—
150C2×C4×D71—22no—
151C2×C4×D71—22no—
152C2×C4×D71—22no—
153C2×C4×D71—22no—
154C2×C4×D71—24no—
155C2×C4×D71—42no—
156C2×C4×D71—42no—
157C2×C4×D71C22⋊C414yes—
158C2×C4×D71C22⋊C414yes—
159C2×C4×D71C2×D412yes—
160C2×C4×D71C2×D412yes—
161C2×C4×D71#4114yes—
162C2×C71⋊C8—22no—
163C2×C71⋊C8—22no—
164C2×C71⋊C8—22no—
165C2×C71⋊C8—22no—
166C2×C71⋊C8—22no—
167C2×C71⋊C8—22no—
168C4⋊Dic71—22no—
169C4⋊Dic71—22no—
170Dic71⋊C4—22no—
171Dic71⋊C4—22no—
172Dic71⋊C4—22no—
173Dic71⋊C4—22no—
174C22⋊Dic71—22no—
175C22⋊Dic71—22no—
176(C2×D71)⋊C4—22no—
177(C2×D71)⋊C4—22no—
178(C2×D71)⋊C4—22no—
179(C2×D71)⋊C4—22no—
180C4×Dic71—21no—
181C4×Dic71—21no—
182C4×Dic71—22no—
183C4×Dic71—22no—
184C4×Dic71C22⋊C414yes—
185C4×Dic71C22⋊C414yes—
186C4×Dic71C2×D412yes—
187C4×Dic71C2×D412yes—
188C24×C71C22×C411yescharacteristic
189C22×C4×C71—21no—
190C22×C4×C71—21no—
191C22×C4×C71—22no—
192C22×C4×C71C22×C411yescharacteristic
193C22×C4×C71C22×C411yescharacteristic
194C22×C4×C71C22×C411yescharacteristic
195C22×C4×C71C22×C411yescharacteristic
196C22×C4×C71C22×C411yescharacteristic
197C22×C4×C71#4114yes—
198C22×C4×C71C2411yescharacteristic
199C2×C8×C71—22no—
200C2×C8×C71—22no—
201C2×C8×C71—22no—
202C2×C8×C71—22no—
203C2×C8×C71—22no—
204C2×C8×C71—22no—
205C4⋊C4×C71—22no—
206C4⋊C4×C71—22no—
207C22⋊C4×C71—22no—
208C22⋊C4×C71—22no—
209C42×C71—21no—
210C42×C71—21no—
211C42×C71C2×D412yes—
212C42×C71C2×D412yes—
213C23×C71—21no—
214C23×C71—21no—
215C23×C71—22no—
216C23×C71—22no—
217C23×C71#11712yes—
218C23×C71C23×C411yescharacteristic
219C22×D71—21no—
220C22×D71—21no—
221C22×D71—22no—
222C22×D71—28no—
223C22×D71—21no—
224C22×D71—21no—
225C22×D71—22no—
226C22×D71—22no—
227C22×D71—48no—
228C22×D71—48no—
229C22×D71—44no—
230C22×D71—44no—
231C22×D71C2×C22⋊C412yes—
232C22×D71#11712yes—
233C22×D71#13114yes—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—22no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—21no—
241C2×C4×C71—21no—
242C2×C4×C71—22no—
243C2×C4×C71—22no—
244C2×C4×C71C2×C22⋊C412yes—
245C2×C4×C71C2×C22⋊C412yes—
246C2×C4×C71#11712yes—
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×D411yescharacteristic
249C2×C4×C71C2×#4112yes—
250C2×C4×C71C2×#4112yes—
251C2×Dic71—28no—
252C2×Dic71—21no—
253C2×Dic71—21no—
254C2×Dic71—21no—
255C2×Dic71—21no—
256C2×Dic71—22no—
257C2×Dic71—22no—
258C2×Dic71—21no—
259C2×Dic71—21no—
260C2×Dic71—22no—
261C2×Dic71—22no—
262C2×Dic71C2×C22⋊C412yes—
263C2×Dic71C2×C22⋊C412yes—
264C2×Dic71C2×C22⋊C412yes—
265C2×Dic71#11712yes—
266C4×D71—28no—
267C4×D71—28no—
268C4×D71—28no—
269C4×D71—24no—
270C4×D71—24no—
271C4×D71—24no—
272C4×D71—24no—
273C4×D71—44no—
274C4×D71—44no—
275C4×D71—44no—
276C8×C71—42no—
277C8×C71—42no—
278C71⋊C8—42no—
279C71⋊C8—42no—
280C22×C71—24no—
281C22×C71—22no—
282C22×C71—41no—
283C22×C71—41no—
284C22×C71—42no—
285C22×C71C22×C22⋊C411yescharacteristic
286C22×C71C2×#11711yescharacteristic
287C22×C71C2×#11711yescharacteristic
288C22×C71C2×#13112yes—
289C22×C71#56812yes—
290C2×D71—22no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71—22no—
294C2×D71—24no—
295C2×D71—22no—
296C2×D71—416no—
297C2×D71—41no—
298C2×D71—41no—
299C2×D71—42no—
300C2×D71#40814yes—
301C4×C71—22no—
302C4×C71—22no—
303C4×C71—22no—
304C4×C71—41no—
305C4×C71—41no—
306C4×C71#52512yes—
307C4×C71#52512yes—
308Dic71—22no—
309Dic71—22no—
310Dic71—22no—
311Dic71—22no—
312Dic71—41no—
313Dic71—41no—
314#20717—711nomaximal, Sylow
315C2×C71—42no—
316C2×C71#228011yescharacteristic
317C2×C71C2×#40811yescharacteristic
318C2×C71#287911yescharacteristic
319C2×C71#292412yes—
320C2×C71#292412yes—
321D71—24no—
322D71—42no—
323C22×#117—711no—
324C2×#432—711no—
325C2×#431—711no—
326#1817—712no—
327#1798—712no—
328#1798—712no—
329#1779—712no—
330#1750—711no—
331#1750—711no—
332#1750—711no—
333#1750D7111yescharacteristic, complemented
334C71#2071711yescharacteristic, complemented, Sylow
335C24×C4—711no—
336C2×#117—711no—
337C2×#117—711no—
338C2×#117—711no—
339C2×#117—711no—
340C2×#117—711no—
341C2×#117—711no—
342C2×#117—711no—
343C2×#117—1424no—
344C2×#117—1424no—
345C2×#117C2×D7111yescharacteristic
346C22×C4⋊C4—711no—
347C22×C4⋊C4—711no—
348C22×C22⋊C4—711no—
349C22×C22⋊C4—711no—
350C22×C42—711no—
351C22×C42—711no—
352#432—1424no—
353#431—1424no—
354C2×C4⋊C8—712no—
355C2×C4⋊C8—712no—
356C2×C22⋊C8—712no—
357C2×C22⋊C8—712no—
358C2×C22⋊C8—712no—
359C2×C22⋊C8—712no—
360C2×C22⋊C8—712no—
361C2×C22⋊C8C2×D7112yes—
362C2×#97—711no—
363C2×#97C2×D7111yescharacteristic, complemented
364C2×C4×C8—711no—
365C2×C4×C8C2×D7111yescharacteristic, complemented
366#336—712no—
367#336—712no—
368#336—712no—
369#336C2×D7112yescomplemented
370C25—711no—
371C23×C4—711no—
372C23×C4—711no—
373C23×C4—711no—
374C23×C4—711no—
375C23×C4—711no—
376C23×C4—711no—
377C23×C4—711no—
378C23×C4—711no—
379C23×C4—711no—
380C23×C4—1422no—
381C23×C4—1422no—
382C23×C4—1424no—
383C23×C4—1424no—
384C23×C4C22×D7111yescharacteristic
385C22×C8—712no—
386C22×C8—712no—
387C22×C8C22×D7112yes—
388C22×C8C22×D7112yes—
389#117—22no—
390#117—22no—
391#117—1422no—
392#117—1422no—
393#117—1424no—
394#117—1424no—
395#117—1424no—
396#117—1424no—
397#117—1422no—
398#117—1422no—
399#117—1422no—
400#117—1422no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—711no—
403C2×C4⋊C4—711no—
404C2×C4⋊C4—711no—
405C2×C4⋊C4—711no—
406C2×C4⋊C4—711no—
407C2×C4⋊C4—1424no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4C4×D7111yescharacteristic
410C2×C4⋊C4C4×D7111yescharacteristic
411C2×C22⋊C4—711no—
412C2×C22⋊C4—711no—
413C2×C22⋊C4—711no—
414C2×C22⋊C4—711no—
415C2×C22⋊C4—711no—
416C2×C22⋊C4—711no—
417C2×C22⋊C4—1424no—
418C2×C22⋊C4—1424no—
419C2×C22⋊C4C4×D7111yescharacteristic
420C2×C22⋊C4C4×D7111yescharacteristic
421C2×C42—711no—
422C2×C42—711no—
423C2×C42—711no—
424C2×C42—714no—
425C2×C42—714no—
426C2×C42—711no—
427C2×C42—711no—
428C2×C42—711no—
429C2×C42—1422no—
430C2×C42—1422no—
431C2×C42C22×D7111yescharacteristic
432C2×C42C22×D7111yescharacteristic
433C4⋊C8—1424no—
434C4⋊C8—1424no—
435C22⋊C8—24no—
436C22⋊C8—1424no—
437C22⋊C8—1424no—
438C22⋊C8—1424no—
439C22⋊C8—1424no—
440C22⋊C8—1424no—
441#97—22no—
442#97—1422no—
443C4×C8—22no—
444C4×C8—1422no—
445C24—2842no—
446C24—711no—
447C24—711no—
448C24—1424no—
449C24—1422no—
450C24—1422no—
451C24—1422no—
452C24C2×C4×D7111yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—2842no—
457C22×C4—2842no—
458C22×C4—714no—
459C22×C4—711no—
460C22×C4—711no—
461C22×C4—711no—
462C22×C4—711no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—714no—
466C22×C4—714no—
467C22×C4—711no—
468C22×C4—711no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1424no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1424no—
476C22×C4—1422no—
477C22×C4—14216no—
478C22×C4—14216no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1421no—
483C22×C4—1421no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4C2×C4×D7111yescharacteristic
490C22×C4C2×C4×D7111yescharacteristic
491C22×C4C2×C4×D7111yescharacteristic
492C22×C4C2×C4×D7111yescharacteristic
493C22×C4C2×C4×D7111yescharacteristic
494C22×C4#1176586514yes—
495C22×C4C23×D7111yescharacteristic
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—1422no—
503C2×C8—1422no—
504C2×C8—1422no—
505C2×C8—1422no—
506C2×C8—1422no—
507C2×C8—1422no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—1422no—
511C4⋊C4—1422no—
512C4⋊C4—1422no—
513C4⋊C4—1422no—
514C4⋊C4—1422no—
515C4⋊C4—1422no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—1422no—
519C22⋊C4—1422no—
520C22⋊C4—1422no—
521C22⋊C4—1422no—
522C22⋊C4—1422no—
523C22⋊C4—1422no—
524C42—21no—
525C42—21no—
526C42—712no—
527C42—712no—
528C42—714no—
529C42—714no—
530C42—1422no—
531C42—1422no—
532C42—1421no—
533C42—1421no—
534C42D4×D7112yes—
535C42D4×D7112yes—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—2848no—
541C23—2844no—
542C23—2844no—
543C23—2848no—
544C23—712no—
545C23—714no—
546C23—712no—
547C23—1428no—
548C23—1421no—
549C23—1421no—
550C23—1422no—
551C23—1422no—
552C23—1421no—
553C23—1421no—
554C23—1422no—
555C23#1455608512yes—
556C23C22×C4×D7111yescharacteristic
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—2844no—
568C2×C4—2844no—
569C2×C4—2844no—
570C2×C4—712no—
571C2×C4—712no—
572C2×C4—712no—
573C2×C4—712no—
574C2×C4—1428no—
575C2×C4—1428no—
576C2×C4—1428no—
577C2×C4—1421no—
578C2×C4—1421no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—1428no—
582C2×C4—1421no—
583C2×C4—1421no—
584C2×C4—1421no—
585C2×C4—1421no—
586C2×C4—1422no—
587C2×C4—1422no—
588C2×C4—1424no—
589C2×C4—1424no—
590C2×C4—1424no—
591C2×C4—1424no—
592C2×C4C22⋊C4×D7112yes—
593C2×C4C22⋊C4×D7112yes—
594C2×C4#1455608512yes—
595C2×C4C2×D4×D7111yescharacteristic
596C2×C4C2×D4×D7111yescharacteristic
597C2×C4#41×D7112yes—
598C2×C4C2×#1176586512yes—
599C8—42no—
600C8—42no—
601C8—2842no—
602C8—2842no—
603C22—22no—
604C22—24no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—28416no—
609C22—2842no—
610C22—2841no—
611C22—2841no—
612C22—714no—
613C22—1422no—
614C22—1424no—
615C22—1422no—
616C22—1424no—
617C22—1424no—
618C22—1422no—
619C22C2×C22⋊C4×D7111yescharacteristic
620C22#117×D7111yescharacteristic
621C22C2×#1455608511yescharacteristic
622C22#131×D7112yes—
623C22#1798676412yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—2841no—
630C4—2841no—
631C4—1422no—
632C4—1422no—
633C4—1422no—
634C4—1422no—
635C4#1798650812yes—
636C4#1798650812yes—
637C2—42no—
638C2—2842no—
639C2—1424no—
640C2#2259072111yescharacteristic
641C2#408×D7111yescharacteristic
642C2#2259678211yescharacteristic
643C2#2259699512yes—
644C2#2259699612yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC71×#1750C2
prime radicalC24×D71C2×C4
cosocleC22×C4×C71C24
hypercenter#1750D71
hypocenterC71#20717
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#17986054C2semidirect
3. C2×#17986050C2semidirect
4. #22588407C2semidirect
5. #22588323C2semidirect
6. #22588322C2semidirect
8. #22588109C2semidirect
9. #22588107C2semidirect
10. #22588106C2semidirect
11. C71×#1750C2semidirect
30. C2×Dic71⋊C8C22semidirect
31. C2×Dic71⋊C8C22semidirect
37. C2×#14556039C22semidirect
38. C2×C4×C71⋊C8C22semidirect
39. #17985754C22semidirect
40. #17985753C22semidirect
41. #17985753C22semidirect
44. C2×#97×C71C22semidirect
45. C2×C4×C8×C71C22semidirect
46. #336×C71C22semidirect
333. #1750D71semidirect
334. C71#20717semidirect
363. C2×#97C2×D71semidirect
365. C2×C4×C8C2×D71semidirect
369. #336C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D712
C2×#179860542
C2×#179860502
#225884072
#225883232
#225883222
#225882492
#225881092
#225881072
#225881062
C71×#17502
#2071771