Finite Group Databasebeta

← C2×#20257467 #27124485

Subgroups — C2×#20257467 #27124485

Order 12032.

7758 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257467C2132yesmaximal, direct product factor
2C2×#15689856C213yesmaximal, complemented
3C2×Q8×Dic94C211yesmaximal, characteristic, complemented
4C2×#15689828C213yesmaximal, complemented
5C2×#15689792C211yesmaximal, characteristic, complemented
6C2×#15689785C211yesmaximal, characteristic, complemented
7C2×#15689780C211yesmaximal, characteristic, complemented
8C2×#15689778C213yesmaximal, complemented
9C2×#15689774C213yesmaximal, complemented
10C2×#15689744C213yesmaximal, complemented
11C2×#15689738C213yesmaximal, complemented
12C2×#15689670C213yesmaximal, complemented
13C2×#15689651C213yesmaximal, complemented
14C2×#15689605C211yesmaximal, characteristic, complemented
15C2×C4×Q8×D47C211yesmaximal, characteristic, complemented
16C2×C47×#520C211yesmaximal, characteristic, complemented
17C22×#10625354C2211yescharacteristic
18C22×#10625350C2216yes—
19C22×Q8×D47C2211yescharacteristic
20#15689856C22148yescomplemented
21Q8×Dic94C22116yescomplemented
22#15689828C22148yescomplemented
23#15689792C22116yescomplemented
24#15689785C22116yescomplemented
25#15689780C22116yescomplemented
26#15689778C22148yescomplemented
27#15689774C22148yescomplemented
28#15689744C22148yescomplemented
29#15689738C22148yescomplemented
30#15689670C22148yescomplemented
31C2×#13094024C2213yes—
32C2×#13094023C2213yes—
33C2×#13094023C2213yes—
34C2×#13094020C2213yes—
35#15689651C22148yescomplemented
36C2×#13094019C2213yes—
37C2×#13094017C2213yes—
38C2×#13094014C2213yes—
39C2×#13094012C2213yes—
40C2×#13094011C2213yes—
41C2×#13094010C2213yes—
42C2×#13094010C2216yes—
43C2×#13094009C2213yes—
44C2×#13094007C2216yes—
45C2×#13094006C2213yes—
46C2×#13094006C2216yes—
47C2×#13094004C2213yes—
48C2×#13094003C2216yes—
49C2×#13094003C2213yes—
50C2×#13094002C2216yes—
51C2×#13094002C2211yescharacteristic
52C2×#13093999C2213yes—
53C2×#13093998C2213yes—
54C2×#13093997C2213yes—
55C2×#13093995C2213yes—
56C2×#13093995C2213yes—
57#15689605C22116yescomplemented
58C2×Q8×Dic47C2211yescharacteristic
59C2×Q8×Dic47C2211yescharacteristic
60C2×C4×Dic94C2213yes—
61C2×C4×Dic94C2213yes—
62C2×#13093989C2213yes—
63C4×Q8×D47C22116yescomplemented
64C2×C4×D188C2213yes—
65C2×C4×D188C2213yes—
66C2×#13093986C2213yes—
67C2×C4×C22⋊D47C2213yes—
68C2×C4×C22⋊D47C2213yes—
69C2×C4×C22⋊D47C2216yes—
70C2×#13093981C2213yes—
71C2×C4⋊C4×D47C2213yes—
72C2×C4⋊C4×D47C2213yes—
73C2×C42×D47C2213yes—
74C22×#41×C47C2211yescharacteristic
75C47×#520C22116yescomplemented
76C2×C4×Q8×C47C2211yescharacteristic
77C2×C4×Q8×C47C2211yescharacteristic
78C2×C4×D4×C47C2213yes—
79C2×C4×D4×C47C2213yes—
80C2×#117×C47C2213yes—
81C22×C4⋊C4×C47C2213yes—
82C2×#10625354—28no—
83C2×#10625354C2318yes—
84C2×#10625350—424no—
85C2×#10625350C23148yes—
86C22×Dic94C2313yes—
87C22×Dic94C2311yescharacteristic
88C22×Dic94C2316yes—
89C2×Q8×D47—28no—
90C2×Q8×D47C2318yes—
91C22×D188C2313yes—
92C22×D188C2313yes—
93C22×C22⋊D47C2313yes—
94C22×C22⋊D47C2313yes—
95C22×C22⋊D47C2316yes—
96C22×C4×D47C2311yescharacteristic
97C22×C4×D47C2313yes—
98C22×C4×D47C2313yes—
99C22×C4×D47C2313yes—
100C22×C4×D47C2316yes—
101#13094024C23124yes—
102#13094023C23124yes—
103#13094023C23124yes—
104#13094020C23124yes—
105#13094019C23124yes—
106#13094017C23124yes—
107#13094014C23124yes—
108#13094012C23124yes—
109#13094011C23124yes—
110#13094010C23148yes—
111#13094010C23124yes—
112#13094009C23124yes—
113#13094007C23148yes—
114#13094006C23148yes—
115#13094006C23124yes—
116#13094004C23124yes—
117#13094003C23148yes—
118#13094003C23124yes—
119#13094002C23148yes—
120#13094002C2318yes—
121#13093999C23124yes—
122#13093998C23124yes—
123#13093997C23124yes—
124#13093995C23124yes—
125#13093995C23124yes—
126Q8×Dic47C2318yes—
127Q8×Dic47C2318yes—
128C4×Dic94C23124yes—
129C4×Dic94C23124yes—
130#13093989C23124yes—
131C4×D188C23124yes—
132C4×D188C23124yes—
133#13093986C23124yes—
134C4×C22⋊D47C23124yes—
135C4×C22⋊D47C23124yes—
136C4×C22⋊D47C23148yes—
137#13093981C23124yes—
138C2×C4⋊Dic47C2313yes—
139C2×C4⋊Dic47C2313yes—
140C2×C4⋊Dic47C2313yes—
141C2×C4⋊Dic47C2313yes—
142C2×Dic47⋊C4C2313yes—
143C2×Dic47⋊C4C2316yes—
144C2×Dic47⋊C4C2311yescharacteristic
145C2×Dic47⋊C4C2311yescharacteristic
146C2×Dic47⋊C4C2313yes—
147C2×Dic47⋊C4C2313yes—
148C2×Dic47⋊C4C2313yes—
149C4⋊C4×D47C23124yes—
150C4⋊C4×D47C23124yes—
151C2×C22⋊Dic47C2313yes—
152C2×C22⋊Dic47C2316yes—
153C2×C22⋊Dic47C2313yes—
154C2×(C2×D47)⋊C4C2313yes—
155C2×(C2×D47)⋊C4C2313yes—
156C2×(C2×D47)⋊C4C2311yescharacteristic
157C2×(C2×D47)⋊C4C2313yes—
158C2×(C2×D47)⋊C4C2316yes—
159C2×(C2×D47)⋊C4C2311yescharacteristic
160C2×(C2×D47)⋊C4C2313yes—
161C2×C4×Dic47C2313yes—
162C2×C4×Dic47C2313yes—
163C2×C4×Dic47C2313yes—
164C2×C4×Dic47C2313yes—
165C42×D47C23124yes—
166C2×#41×C47—28no—
167C2×#41×C47C2318yes—
168C22×Q8×C47C2311yescharacteristic
169C22×D4×C47C2313yes—
170C23×C4×C47C2313yes—
171C23×C4×C47C2316yes—
172C4×Q8×C47C2318yes—
173C4×Q8×C47C2318yes—
174C4×D4×C47C23124yes—
175C4×D4×C47C23124yes—
176#117×C47C23124yes—
177C2×C4⋊C4×C47C23124yes—
178C2×C4⋊C4×C47C2313yes—
179C2×C4⋊C4×C47C2313yes—
180C2×C4⋊C4×C47C2313yes—
181C2×C4⋊C4×C47C2311yescharacteristic
182C2×C22⋊C4×C47C2313yes—
183C2×C22⋊C4×C47C2313yes—
184C2×C42×C47C2313yes—
185C2×C42×C47C2313yes—
186C24×C47C2413yes—
187C23×D47C2413yes—
188C23×D47C2411yescharacteristic
189#41×C47—232no—
190C2×Q8×C47#4118yescomplemented
191C2×Q8×C47C2414yes—
192C2×D4×C47—212no—
193C2×D4×C47C24112yes—
194C22×C4×C47—212no—
195C22×C4×C47—412no—
196C22×C4×C47C2411yescharacteristic
197C22×C4×C47C2411yescharacteristic
198C22×C4×C47C2413yes—
199C22×C4×C47C2413yes—
200C22×C4×C47C2411yescharacteristic
201C22×C4×C47C24112yes—
202C22×C4×C47C24124yes—
203C22×C4×C47C2413yes—
204C2×C22⋊D47—212no—
205C2×C22⋊D47—212no—
206C2×C22⋊D47—412no—
207C2×C22⋊D47C24112yes—
208C2×C22⋊D47C24112yes—
209C2×C22⋊D47C24124yes—
210C22×Dic47C2411yescharacteristic
211C22×Dic47C2413yes—
212C22×Dic47C2413yes—
213C22×Dic47C2413yes—
214C22×Dic47C2411yescharacteristic
215C22×Dic47C2411yescharacteristic
216#10625354—232no—
217Q8×D47—232no—
218#10625350—496no—
219C2×D188—212no—
220C2×D188—212no—
221C2×D188C24112yes—
222C2×D188C24112yes—
223C2×C4×D47—212no—
224C2×C4×D47—212no—
225C2×C4×D47—212no—
226C2×C4×D47—24no—
227C2×C4×D47—412no—
228C2×C4×D47C2414yes—
229C2×C4×D47C24112yes—
230C2×C4×D47C24112yes—
231C2×C4×D47C24112yes—
232C2×C4×D47C24124yes—
233C2×Dic94—212no—
234C2×Dic94—24no—
235C2×Dic94—412no—
236C2×Dic94C24112yes—
237C2×Dic94C2414yes—
238C2×Dic94C24124yes—
239C4⋊C4×C47C24112yes—
240C4⋊C4×C47C24112yes—
241C4⋊C4×C47C24112yes—
242C4⋊C4×C47C2414yes—
243C22⋊C4×C47C24112yes—
244C22⋊C4×C47C24112yes—
245C42×C47C24112yes—
246C42×C47C24112yes—
247C22⋊Dic47C24112yes—
248C22⋊Dic47C24124yes—
249C22⋊Dic47C24112yes—
250(C2×D47)⋊C4C24112yes—
251(C2×D47)⋊C4C24112yes—
252(C2×D47)⋊C4C2414yes—
253(C2×D47)⋊C4C24112yes—
254(C2×D47)⋊C4C24124yes—
255(C2×D47)⋊C4C2414yes—
256(C2×D47)⋊C4C24112yes—
257C4⋊Dic47C24112yes—
258C4⋊Dic47C24112yes—
259C4⋊Dic47C24112yes—
260C4⋊Dic47C24112yes—
261Dic47⋊C4C24112yes—
262Dic47⋊C4C24124yes—
263Dic47⋊C4C2414yes—
264Dic47⋊C4C2414yes—
265Dic47⋊C4C24112yes—
266Dic47⋊C4C24112yes—
267Dic47⋊C4C24112yes—
268C4×Dic47C24112yes—
269C4×Dic47C24112yes—
270C4×Dic47C24112yes—
271C4×Dic47C24112yes—
272C23×C47—26no—
273C23×C47—26no—
274C23×C47—26no—
275C23×C47C2516yes—
276C23×C47C2511yescharacteristic
277C22×D47—22no—
278C22×D47—22no—
279C22×D47—26no—
280C22×D47—26no—
281C22×D47—26no—
282C22×D47H(5,𝔽2)14yes—
283C22×D47C2516yes—
284C22×D47C2512yes—
285Q8×C47C2×#41116yescomplemented
286D4×C47—224no—
287C2×C4×C47—22no—
288C2×C4×C47—26no—
289C2×C4×C47—26no—
290C2×C4×C47—22no—
291C2×C4×C47—22no—
292C2×C4×C47—224no—
293C2×C4×C47—424no—
294C2×C4×C47C2×#41112yes—
295C2×C4×C47C2512yes—
296C2×C4×C47C2516yes—
297C2×C4×C47C2512yes—
298C2×C4×C47C2516yes—
299C2×C4×C47C2512yes—
300C2×C4×C47C2516yes—
301C22⋊D47—224no—
302C22⋊D47—224no—
303C22⋊D47—424no—
304C2×Dic47—26no—
305C2×Dic47—22no—
306C2×Dic47—26no—
307C2×Dic47—26no—
308C2×Dic47—22no—
309C2×Dic472−1+414yes—
310C2×Dic47C2512yes—
311C2×Dic47C2516yes—
312C2×Dic47C2516yes—
313C2×Dic47C2516yes—
314C2×Dic47C2512yes—
315C2×Dic47C2512yes—
316D188—224no—
317D188—224no—
318C4×D47—224no—
319C4×D47—224no—
320C4×D47—224no—
321C4×D47—28no—
322C4×D47—424no—
323Dic94—224no—
324Dic94—28no—
325Dic94—424no—
326C2×#3522—471nomaximal, Sylow
327C22×C47—26no—
328C22×C47—26no—
329C22×C47—26no—
330C22×C47—412no—
331C22×C47C22×#4112yes—
332C22×C47C2×H(5,𝔽2)12yes—
333C22×C47C2×2−1+412yes—
334C22×C47C2611yescharacteristic
335C2×D47—22no—
336C2×D47—22no—
337C2×D47—28no—
338C2×D47—26no—
339C2×D47—26no—
340C2×D47—26no—
341C2×D47—412no—
342C2×D47C2×H(5,𝔽2)14yes—
343C4×C47—22no—
344C4×C47—26no—
345C4×C47—26no—
346C4×C47—22no—
347C4×C47—22no—
348C4×C47C22×#41112yes—
349Dic47—26no—
350Dic47—22no—
351Dic47—26no—
352Dic47—26no—
353Dic47—22no—
354Dic47C2×2−1+414yes—
355#3522—4732no—
356C2×#559—473no—
357C2×Q82—471no—
358C2×#554—473no—
359C2×#550—471no—
360C2×#548—471no—
361C2×#548—471no—
362C2×#547—473no—
363C2×#547—473no—
364C2×#542—473no—
365C2×#542—473no—
366C2×#532—473no—
367C2×#529—473no—
368C2×#520—471no—
369C2×#520D4711yescharacteristic, complemented
370C22×C4×Q8—471no—
371C2×C47—46no—
372C2×C47#352214yescomplemented
373C2×C47C23×#4111yescharacteristic
374C2×C47C22×H(5,𝔽2)11yescharacteristic
375C2×C47C22×2−1+411yescharacteristic
376D47—24no—
377D47—46no—
378C22×#41—471no—
379C22×#41—476no—
380C22×#41C2×D4711yescharacteristic
381C23×Q8—471no—
382#559—4748no—
383Q82—4716no—
384#554—4748no—
385#550—4716no—
386#548—4716no—
387#548—4716no—
388#547—4748no—
389#547—4748no—
390#542—4748no—
391#542—4748no—
392#532—4748no—
393C2×C4⋊Q8—473no—
394C2×C4⋊Q8—473no—
395C2×C4⋊Q8—473no—
396C2×#127—473no—
397#529—4748no—
398C2×#126—473no—
399C2×#126—473no—
400C2×#125—473no—
401C2×#124—473no—
402C2×#124—473no—
403C2×#124—476no—
404C2×#124—473no—
405C2×#123—476no—
406C2×#123—473no—
407C2×#123—473no—
408C2×#123—476no—
409C2×C22⋊Q8—476no—
410C2×C22⋊Q8—473no—
411C2×C22⋊Q8—471no—
412C2×C22⋊Q8—476no—
413C2×C22⋊Q8—473no—
414C2×#121—473no—
415C2×#121—473no—
416C2×#121—473no—
417C2×#121—473no—
418C2×#121—473no—
419#520—4716no—
420#520C2×D47116yescomplemented
421C2×C4×Q8—471no—
422C2×C4×Q8—471no—
423C2×C4×Q8—4716no—
424C2×C4×Q8—473no—
425C2×C4×Q8—473no—
426C2×C4×Q8—473no—
427C2×C4×Q8C2×D4711yescharacteristic
428C2×C4×Q8C2×D4711yescharacteristic
429C2×C4×D4—473no—
430C2×C4×D4—473no—
431C2×C4×D4—476no—
432C2×C4×D4—473no—
433C2×C4×D4—473no—
434C2×C4×D4—473no—
435C2×C4×D4C2×D4713yes—
436C2×C4×D4C2×D4713yes—
437C2×#117—473no—
438C2×#117C2×D4713yes—
439C22×C4⋊C4—473no—
440C22×C4⋊C4—473no—
441C22×C4⋊C4C2×D4713yes—
442C22×C42—473no—
443C47C2×#352211yescharacteristic, complemented, Sylow
444C2×#41—28no—
445C2×#41—478no—
446C2×#41—4748no—
447C2×#41—948no—
448C2×#41—18824no—
449C2×#41C22×D4718yes—
450C22×Q8—478no—
451C22×Q8—471no—
452C22×Q8—473no—
453C22×Q8—476no—
454C22×Q8—948no—
455C22×Q8C22×D4711yescharacteristic
456C22×D4—473no—
457C22×D4—473no—
458C22×D4—476no—
459C22×D4—473no—
460C22×D4—473no—
461C22×D4C22×D4713yes—
462C23×C4—473no—
463C23×C4—473no—
464C23×C4—473no—
465C23×C4—471no—
466C23×C4—476no—
467C23×C4C22×D4713yes—
468C23×C4C22×D4716yes—
469C4⋊Q8—4724no—
470C4⋊Q8—4724no—
471C4⋊Q8—4724no—
472#127—4724no—
473#126—4724no—
474#126—4724no—
475#125—4724no—
476#124—4724no—
477#124—4724no—
478#124—4748no—
479#124—4724no—
480#123—4748no—
481#123—4724no—
482#123—4724no—
483#123—4748no—
484C22⋊Q8—4748no—
485C22⋊Q8—4724no—
486C22⋊Q8—478no—
487C22⋊Q8—4748no—
488C22⋊Q8—4724no—
489#121—4724no—
490#121—4724no—
491#121—4724no—
492#121—4724no—
493#121—4724no—
494C4×Q8—478no—
495C4×Q8—478no—
496C4×Q8—4724no—
497C4×Q8—4724no—
498C4×Q8—4724no—
499C4×Q8C22×D4718yes—
500C4×Q8C22×D4718yes—
501C4×D4—4724no—
502C4×D4—4724no—
503C4×D4—4748no—
504C4×D4—4724no—
505C4×D4—4724no—
506C4×D4—4724no—
507C4×D4C22×D47124yes—
508C4×D4C22×D47124yes—
509#117—4724no—
510#117C22×D47124yes—
511C2×C4⋊C4—473no—
512C2×C4⋊C4—473no—
513C2×C4⋊C4—473no—
514C2×C4⋊C4—476no—
515C2×C4⋊C4—471no—
516C2×C4⋊C4—471no—
517C2×C4⋊C4—473no—
518C2×C4⋊C4—473no—
519C2×C4⋊C4—473no—
520C2×C4⋊C4—473no—
521C2×C4⋊C4—473no—
522C2×C4⋊C4—4724no—
523C2×C4⋊C4—4724no—
524C2×C4⋊C4C22×D4713yes—
525C2×C4⋊C4C22×D4713yes—
526C2×C4⋊C4C22×D4713yes—
527C2×C4⋊C4C22×D47124yes—
528C2×C4⋊C4C22×D4711yescharacteristic
529C2×C22⋊C4—473no—
530C2×C22⋊C4—476no—
531C2×C22⋊C4—473no—
532C2×C22⋊C4—473no—
533C2×C22⋊C4—473no—
534C2×C22⋊C4—471no—
535C2×C22⋊C4—471no—
536C2×C22⋊C4—473no—
537C2×C22⋊C4—476no—
538C2×C22⋊C4—473no—
539C2×C22⋊C4C22×D4713yes—
540C2×C22⋊C4C22×D4713yes—
541C2×C42—473no—
542C2×C42—473no—
543C2×C42—473no—
544C2×C42—473no—
545C2×C42—4724no—
546C2×C42C22×D4713yes—
547C2×C42C22×D4713yes—
548C24—471no—
549C24—473no—
550C24C23×D4713yes—
551#41—232no—
552#41—9432no—
553#41—18896no—
554C2×Q8—474no—
555C2×Q8—4712no—
556C2×Q8—4724no—
557C2×Q8—944no—
558C2×Q8—9432no—
559C2×Q8—9412no—
560C2×Q8—18812no—
561C2×Q8#1062535018yescomplemented
562C2×Q8C23×D4714yes—
563C2×D4—212no—
564C2×D4—4712no—
565C2×D4—4712no—
566C2×D4—4724no—
567C2×D4—4712no—
568C2×D4—4712no—
569C2×D4—9412no—
570C2×D4—9412no—
571C2×D4—9412no—
572C2×D4—9412no—
573C2×D4—18812no—
574C2×D4C23×D47112yes—
575C22×C4—212no—
576C22×C4—412no—
577C22×C4—471no—
578C22×C4—471no—
579C22×C4—473no—
580C22×C4—473no—
581C22×C4—473no—
582C22×C4—471no—
583C22×C4—474no—
584C22×C4—4712no—
585C22×C4—4712no—
586C22×C4—4712no—
587C22×C4—4724no—
588C22×C4—944no—
589C22×C4—9412no—
590C22×C4—9412no—
591C22×C4—9412no—
592C22×C4—18812no—
593C22×C4C23×D47112yes—
594C22×C4C23×D4711yescharacteristic
595C22×C4C23×D4713yes—
596C22×C4C23×D4711yescharacteristic
597C22×C4C23×D4713yes—
598C22×C4C23×D4711yescharacteristic
599C22×C4C23×D47124yes—
600C22×C4C23×D4713yes—
601C4⋊C4—4712no—
602C4⋊C4—4724no—
603C4⋊C4—474no—
604C4⋊C4—474no—
605C4⋊C4—4712no—
606C4⋊C4—4712no—
607C4⋊C4—4712no—
608C4⋊C4—4712no—
609C4⋊C4—4712no—
610C4⋊C4—4712no—
611C4⋊C4—4712no—
612C4⋊C4C23×D47112yes—
613C4⋊C4C23×D47112yes—
614C4⋊C4C23×D47112yes—
615C4⋊C4C23×D4714yes—
616C22⋊C4—4712no—
617C22⋊C4—4724no—
618C22⋊C4—4712no—
619C22⋊C4—4712no—
620C22⋊C4—4712no—
621C22⋊C4—474no—
622C22⋊C4—474no—
623C22⋊C4—4712no—
624C22⋊C4—4724no—
625C22⋊C4—4712no—
626C22⋊C4C23×D47112yes—
627C22⋊C4C23×D47112yes—
628C42—4712no—
629C42—4712no—
630C42—4712no—
631C42—4712no—
632C42C23×D47112yes—
633C42C23×D47112yes—
634C23—26no—
635C23—26no—
636C23—26no—
637C23—474no—
638C23—472no—
639C23—476no—
640C23—942no—
641C23—942no—
642C23—946no—
643C23—946no—
644C23—946no—
645C23C24×D4716yes—
646C23C24×D4711yescharacteristic
647Q8—948no—
648Q8—9424no—
649Q8—18824no—
650Q8C2×#10625350116yescomplemented
651D4—224no—
652D4—9424no—
653D4—9424no—
654D4—9424no—
655D4—9424no—
656D4—18824no—
657C2×C4—22no—
658C2×C4—26no—
659C2×C4—26no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—224no—
663C2×C4—424no—
664C2×C4—474no—
665C2×C4—472no—
666C2×C4—472no—
667C2×C4—476no—
668C2×C4—476no—
669C2×C4—476no—
670C2×C4—472no—
671C2×C4—948no—
672C2×C4—9424no—
673C2×C4—942no—
674C2×C4—946no—
675C2×C4—946no—
676C2×C4—946no—
677C2×C4—942no—
678C2×C4—9424no—
679C2×C4—9424no—
680C2×C4—18824no—
681C2×C4C2×#10625350112yes—
682C2×C4C24×D4712yes—
683C2×C4C24×D4716yes—
684C2×C4C24×D4712yes—
685C2×C4C24×D4716yes—
686C2×C4C24×D4712yes—
687C2×C4C24×D4716yes—
688C22—26no—
689C22—26no—
690C22—26no—
691C22—412no—
692C22—474no—
693C22—942no—
694C22—942no—
695C22—948no—
696C22—946no—
697C22—946no—
698C22—946no—
699C22—18812no—
700C22C22×#1062535012yes—
701C22H(5,𝔽2)×D4712yes—
702C222−1+4×D4712yes—
703C22C25×D4711yescharacteristic
704C4—22no—
705C4—26no—
706C4—26no—
707C4—22no—
708C4—22no—
709C4—474no—
710C4—942no—
711C4—946no—
712C4—946no—
713C4—946no—
714C4—942no—
715C4C22×#10625350112yes—
716C2—46no—
717C2—944no—
718C2—1886no—
719C2#2025746714yesdirect product factor
720C2C23×#1062535011yescharacteristic
721C2C2×H(5,𝔽2)×D4711yescharacteristic
722C2C2×2−1+4×D4711yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#520C2
socleC23×C47C25
hypercenterC2×#520D47
hypocenterC47C2×#3522
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. #20257467C2direct (N × H)
2. C2×#15689856C2semidirect
3. C2×Q8×Dic94C2semidirect
4. C2×#15689828C2semidirect
5. C2×#15689792C2semidirect
6. C2×#15689785C2semidirect
7. C2×#15689780C2semidirect
8. C2×#15689778C2semidirect
9. C2×#15689774C2semidirect
10. C2×#15689744C2semidirect
11. C2×#15689738C2semidirect
12. C2×#15689670C2semidirect
13. C2×#15689651C2semidirect
14. C2×#15689605C2semidirect
15. C2×C4×Q8×D47C2semidirect
16. C2×C47×#520C2semidirect
20. #15689856C22semidirect
21. Q8×Dic94C22semidirect
22. #15689828C22semidirect
23. #15689792C22semidirect
24. #15689785C22semidirect
25. #15689780C22semidirect
26. #15689778C22semidirect
27. #15689774C22semidirect
28. #15689744C22semidirect
29. #15689738C22semidirect
30. #15689670C22semidirect
35. #15689651C22semidirect
57. #15689605C22semidirect
63. C4×Q8×D47C22semidirect
75. C47×#520C22semidirect
190. C2×Q8×C47#41semidirect
285. Q8×C47C2×#41semidirect
369. C2×#520D47semidirect
372. C2×C47#3522semidirect
420. #520C2×D47semidirect
443. C47C2×#3522semidirect
561. C2×Q8#10625350semidirect
650. Q8C2×#10625350semidirect
719. C2#20257467direct (N × H)

Maximal subgroups

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

subgroupindex
#202574672
C2×#156898562
C2×Q8×Dic942
C2×#156898282
C2×#156897922
C2×#156897852
C2×#156897802
C2×#156897782
C2×#156897742
C2×#156897442
C2×#156897382
C2×#156896702
C2×#156896512
C2×#156896052
C2×C4×Q8×D472
C2×C47×#5202
C2×#352247