Finite Group Databasebeta

← C2×#22602419 #32905659

Subgroups — C2×#22602419 #32905659

Order 18176.

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#22602419C2132yesmaximal, direct product factor
2C2×#17986732C213yesmaximal, complemented
3C2×Q8×Dic142C211yesmaximal, characteristic, complemented
4C2×#17986704C213yesmaximal, complemented
5C2×#17986668C211yesmaximal, characteristic, complemented
6C2×#17986661C211yesmaximal, characteristic, complemented
7C2×#17986656C211yesmaximal, characteristic, complemented
8C2×#17986654C213yesmaximal, complemented
9C2×#17986650C213yesmaximal, complemented
10C2×#17986620C213yesmaximal, complemented
11C2×#17986614C213yesmaximal, complemented
12C2×#17986546C213yesmaximal, complemented
13C2×#17986527C213yesmaximal, complemented
14C2×#17986481C211yesmaximal, characteristic, complemented
15C2×C4×Q8×D71C211yesmaximal, characteristic, complemented
16C2×#520×C71C211yesmaximal, characteristic, complemented
17C22×#11765866C2211yescharacteristic
18C22×#11765865C2216yes—
19C22×Q8×D71C2211yescharacteristic
20#17986732C22148yescomplemented
21Q8×Dic142C22116yescomplemented
22#17986704C22148yescomplemented
23#17986668C22116yescomplemented
24#17986661C22116yescomplemented
25#17986656C22116yescomplemented
26#17986654C22148yescomplemented
27#17986650C22148yescomplemented
28#17986620C22148yescomplemented
29#17986614C22148yescomplemented
30#17986546C22148yescomplemented
31C2×#14556128C2213yes—
32C2×#14556127C2213yes—
33C2×#14556127C2213yes—
34C2×#14556124C2213yes—
35#17986527C22148yescomplemented
36C2×#14556123C2213yes—
37C2×#14556121C2213yes—
38C2×#14556118C2213yes—
39C2×#14556116C2213yes—
40C2×#14556115C2213yes—
41C2×#14556114C2213yes—
42C2×#14556114C2216yes—
43C2×#14556113C2213yes—
44C2×#14556111C2216yes—
45C2×#14556110C2213yes—
46C2×#14556110C2216yes—
47C2×#14556108C2213yes—
48C2×#14556107C2216yes—
49C2×#14556107C2213yes—
50C2×#14556106C2216yes—
51C2×#14556106C2211yescharacteristic
52C2×#14556103C2213yes—
53C2×#14556102C2213yes—
54C2×#14556101C2213yes—
55C2×#14556099C2213yes—
56C2×#14556099C2213yes—
57#17986481C22116yescomplemented
58C2×Q8×Dic71C2211yescharacteristic
59C2×Q8×Dic71C2211yescharacteristic
60C2×C4×Dic142C2213yes—
61C2×C4×Dic142C2213yes—
62C2×#14556093C2213yes—
63C4×Q8×D71C22116yescomplemented
64C2×C4×D284C2213yes—
65C2×C4×D284C2213yes—
66C2×#14556090C2213yes—
67C2×C4×C22⋊D71C2213yes—
68C2×C4×C22⋊D71C2213yes—
69C2×C4×C22⋊D71C2216yes—
70C2×#14556085C2213yes—
71C2×C4⋊C4×D71C2213yes—
72C2×C4⋊C4×D71C2213yes—
73C2×C42×D71C2213yes—
74C22×#41×C71C2211yescharacteristic
75#520×C71C22116yescomplemented
76C2×C4×Q8×C71C2211yescharacteristic
77C2×C4×Q8×C71C2211yescharacteristic
78C2×C4×D4×C71C2213yes—
79C2×C4×D4×C71C2213yes—
80C2×#117×C71C2213yes—
81C22×C4⋊C4×C71C2213yes—
82C2×#11765866—28no—
83C2×#11765866C2318yes—
84C2×#11765865—424no—
85C2×#11765865C23148yes—
86C22×Dic142C2313yes—
87C22×Dic142C2311yescharacteristic
88C22×Dic142C2316yes—
89C2×Q8×D71—28no—
90C2×Q8×D71C2318yes—
91C22×D284C2313yes—
92C22×D284C2313yes—
93C22×C22⋊D71C2313yes—
94C22×C22⋊D71C2313yes—
95C22×C22⋊D71C2316yes—
96C22×C4×D71C2313yes—
97C22×C4×D71C2313yes—
98C22×C4×D71C2313yes—
99C22×C4×D71C2311yescharacteristic
100C22×C4×D71C2316yes—
101#14556128C23124yes—
102#14556127C23124yes—
103#14556127C23124yes—
104#14556124C23124yes—
105#14556123C23124yes—
106#14556121C23124yes—
107#14556118C23124yes—
108#14556116C23124yes—
109#14556115C23124yes—
110#14556114C23148yes—
111#14556114C23124yes—
112#14556113C23124yes—
113#14556111C23148yes—
114#14556110C23148yes—
115#14556110C23124yes—
116#14556108C23124yes—
117#14556107C23148yes—
118#14556107C23124yes—
119#14556106C23148yes—
120#14556106C2318yes—
121#14556103C23124yes—
122#14556102C23124yes—
123#14556101C23124yes—
124#14556099C23124yes—
125#14556099C23124yes—
126Q8×Dic71C2318yes—
127Q8×Dic71C2318yes—
128C4×Dic142C23124yes—
129C4×Dic142C23124yes—
130#14556093C23124yes—
131C4×D284C23124yes—
132C4×D284C23124yes—
133#14556090C23124yes—
134C4×C22⋊D71C23124yes—
135C4×C22⋊D71C23124yes—
136C4×C22⋊D71C23148yes—
137#14556085C23124yes—
138C2×C4⋊Dic71C2313yes—
139C2×C4⋊Dic71C2313yes—
140C2×C4⋊Dic71C2313yes—
141C2×C4⋊Dic71C2313yes—
142C2×Dic71⋊C4C2313yes—
143C2×Dic71⋊C4C2316yes—
144C2×Dic71⋊C4C2311yescharacteristic
145C2×Dic71⋊C4C2311yescharacteristic
146C2×Dic71⋊C4C2313yes—
147C2×Dic71⋊C4C2313yes—
148C2×Dic71⋊C4C2313yes—
149C4⋊C4×D71C23124yes—
150C4⋊C4×D71C23124yes—
151C2×C22⋊Dic71C2313yes—
152C2×C22⋊Dic71C2316yes—
153C2×C22⋊Dic71C2313yes—
154C2×(C2×D71)⋊C4C2313yes—
155C2×(C2×D71)⋊C4C2313yes—
156C2×(C2×D71)⋊C4C2311yescharacteristic
157C2×(C2×D71)⋊C4C2313yes—
158C2×(C2×D71)⋊C4C2316yes—
159C2×(C2×D71)⋊C4C2311yescharacteristic
160C2×(C2×D71)⋊C4C2313yes—
161C2×C4×Dic71C2313yes—
162C2×C4×Dic71C2313yes—
163C2×C4×Dic71C2313yes—
164C2×C4×Dic71C2313yes—
165C42×D71C23124yes—
166C2×#41×C71—28no—
167C2×#41×C71C2318yes—
168C22×Q8×C71C2311yescharacteristic
169C22×D4×C71C2313yes—
170C23×C4×C71C2313yes—
171C23×C4×C71C2316yes—
172C4×Q8×C71C2318yes—
173C4×Q8×C71C2318yes—
174C4×D4×C71C23124yes—
175C4×D4×C71C23124yes—
176#117×C71C23124yes—
177C2×C4⋊C4×C71C23124yes—
178C2×C4⋊C4×C71C2313yes—
179C2×C4⋊C4×C71C2313yes—
180C2×C4⋊C4×C71C2313yes—
181C2×C4⋊C4×C71C2311yescharacteristic
182C2×C22⋊C4×C71C2313yes—
183C2×C22⋊C4×C71C2313yes—
184C2×C42×C71C2313yes—
185C2×C42×C71C2313yes—
186C23×D71C2413yes—
187C23×D71C2411yescharacteristic
188#11765866—232no—
189#11765865—496no—
190C2×Dic142—212no—
191C2×Dic142—24no—
192C2×Dic142—412no—
193C2×Dic142C24112yes—
194C2×Dic142C2414yes—
195C2×Dic142C24124yes—
196Q8×D71—232no—
197C2×D284—212no—
198C2×D284—212no—
199C2×D284C24112yes—
200C2×D284C24112yes—
201C2×C22⋊D71—212no—
202C2×C22⋊D71—212no—
203C2×C22⋊D71—412no—
204C2×C22⋊D71C24112yes—
205C2×C22⋊D71C24112yes—
206C2×C22⋊D71C24124yes—
207C22×Dic71C2411yescharacteristic
208C22×Dic71C2413yes—
209C22×Dic71C2413yes—
210C22×Dic71C2413yes—
211C22×Dic71C2411yescharacteristic
212C22×Dic71C2411yescharacteristic
213C2×C4×D71—212no—
214C2×C4×D71—212no—
215C2×C4×D71—212no—
216C2×C4×D71—24no—
217C2×C4×D71—412no—
218C2×C4×D71C24112yes—
219C2×C4×D71C24112yes—
220C2×C4×D71C24112yes—
221C2×C4×D71C2414yes—
222C2×C4×D71C24124yes—
223C4⋊Dic71C24112yes—
224C4⋊Dic71C24112yes—
225C4⋊Dic71C24112yes—
226C4⋊Dic71C24112yes—
227Dic71⋊C4C24112yes—
228Dic71⋊C4C24124yes—
229Dic71⋊C4C2414yes—
230Dic71⋊C4C2414yes—
231Dic71⋊C4C24112yes—
232Dic71⋊C4C24112yes—
233Dic71⋊C4C24112yes—
234C22⋊Dic71C24112yes—
235C22⋊Dic71C24124yes—
236C22⋊Dic71C24112yes—
237(C2×D71)⋊C4C24112yes—
238(C2×D71)⋊C4C24112yes—
239(C2×D71)⋊C4C2414yes—
240(C2×D71)⋊C4C24112yes—
241(C2×D71)⋊C4C24124yes—
242(C2×D71)⋊C4C2414yes—
243(C2×D71)⋊C4C24112yes—
244C4×Dic71C24112yes—
245C4×Dic71C24112yes—
246C4×Dic71C24112yes—
247C4×Dic71C24112yes—
248C24×C71C2413yes—
249#41×C71—232no—
250C2×Q8×C71#4118yescomplemented
251C2×Q8×C71C2414yes—
252C2×D4×C71—212no—
253C2×D4×C71C24112yes—
254C22×C4×C71—212no—
255C22×C4×C71—412no—
256C22×C4×C71C2413yes—
257C22×C4×C71C2411yescharacteristic
258C22×C4×C71C2413yes—
259C22×C4×C71C2411yescharacteristic
260C22×C4×C71C2411yescharacteristic
261C22×C4×C71C24112yes—
262C22×C4×C71C24124yes—
263C22×C4×C71C2413yes—
264C4⋊C4×C71C24112yes—
265C4⋊C4×C71C24112yes—
266C4⋊C4×C71C24112yes—
267C4⋊C4×C71C2414yes—
268C22⋊C4×C71C24112yes—
269C22⋊C4×C71C24112yes—
270C42×C71C24112yes—
271C42×C71C24112yes—
272C23×C71—26no—
273C23×C71—26no—
274C23×C71—26no—
275C23×C71C2516yes—
276C23×C71C2511yescharacteristic
277C22×D71—22no—
278C22×D71—22no—
279C22×D71—26no—
280C22×D71—26no—
281C22×D71—26no—
282C22×D71H(5,𝔽2)14yes—
283C22×D71C2516yes—
284C22×D71C2512yes—
285Q8×C71C2×#41116yescomplemented
286D4×C71—224no—
287C2×C4×C71—22no—
288C2×C4×C71—26no—
289C2×C4×C71—26no—
290C2×C4×C71—22no—
291C2×C4×C71—22no—
292C2×C4×C71—224no—
293C2×C4×C71—424no—
294C2×C4×C71C2×#41112yes—
295C2×C4×C71C2512yes—
296C2×C4×C71C2512yes—
297C2×C4×C71C2516yes—
298C2×C4×C71C2516yes—
299C2×C4×C71C2512yes—
300C2×C4×C71C2516yes—
301C22⋊D71—224no—
302C22⋊D71—224no—
303C22⋊D71—424no—
304C2×Dic71—22no—
305C2×Dic71—26no—
306C2×Dic71—26no—
307C2×Dic71—26no—
308C2×Dic71—22no—
309C2×Dic712−1+414yes—
310C2×Dic71C2512yes—
311C2×Dic71C2516yes—
312C2×Dic71C2516yes—
313C2×Dic71C2516yes—
314C2×Dic71C2512yes—
315C2×Dic71C2512yes—
316D284—224no—
317D284—224no—
318C4×D71—224no—
319C4×D71—224no—
320C4×D71—224no—
321C4×D71—28no—
322C4×D71—424no—
323Dic142—224no—
324Dic142—28no—
325Dic142—424no—
326C22×C71—26no—
327C22×C71—26no—
328C22×C71—26no—
329C22×C71—412no—
330C22×C71C22×#4112yes—
331C22×C71C2×H(5,𝔽2)12yes—
332C22×C71C2×2−1+412yes—
333C22×C71C2611yescharacteristic
334C2×D71—22no—
335C2×D71—22no—
336C2×D71—28no—
337C2×D71—26no—
338C2×D71—26no—
339C2×D71—26no—
340C2×D71—412no—
341C2×D71C2×H(5,𝔽2)14yes—
342C4×C71—22no—
343C4×C71—26no—
344C4×C71—26no—
345C4×C71—22no—
346C4×C71—22no—
347C4×C71C22×#41112yes—
348Dic71—26no—
349Dic71—22no—
350Dic71—26no—
351Dic71—26no—
352Dic71—22no—
353Dic71C2×2−1+414yes—
354C2×#3522—711nomaximal, Sylow
355C2×C71—46no—
356C2×C71#352214yescomplemented
357C2×C71C23×#4111yescharacteristic
358C2×C71C22×H(5,𝔽2)11yescharacteristic
359C2×C71C22×2−1+411yescharacteristic
360D71—24no—
361D71—46no—
362#3522—7132no—
363C2×#559—713no—
364C2×Q82—711no—
365C2×#554—713no—
366C2×#550—711no—
367C2×#548—711no—
368C2×#548—711no—
369C2×#547—713no—
370C2×#547—713no—
371C2×#542—713no—
372C2×#542—713no—
373C2×#532—713no—
374C2×#529—713no—
375C2×#520—711no—
376C2×#520D7111yescharacteristic, complemented
377C22×C4×Q8—711no—
378C71C2×#352211yescharacteristic, complemented, Sylow
379C22×#41—711no—
380C22×#41—716no—
381C22×#41C2×D7111yescharacteristic
382C23×Q8—711no—
383#559—7148no—
384Q82—7116no—
385#554—7148no—
386#550—7116no—
387#548—7116no—
388#548—7116no—
389#547—7148no—
390#547—7148no—
391#542—7148no—
392#542—7148no—
393#532—7148no—
394C2×C4⋊Q8—713no—
395C2×C4⋊Q8—713no—
396C2×C4⋊Q8—713no—
397C2×#127—713no—
398#529—7148no—
399C2×#126—713no—
400C2×#126—713no—
401C2×#125—713no—
402C2×#124—713no—
403C2×#124—713no—
404C2×#124—716no—
405C2×#124—713no—
406C2×#123—716no—
407C2×#123—713no—
408C2×#123—713no—
409C2×#123—716no—
410C2×C22⋊Q8—716no—
411C2×C22⋊Q8—713no—
412C2×C22⋊Q8—711no—
413C2×C22⋊Q8—716no—
414C2×C22⋊Q8—713no—
415C2×#121—713no—
416C2×#121—713no—
417C2×#121—713no—
418C2×#121—713no—
419C2×#121—713no—
420#520—7116no—
421#520C2×D71116yescomplemented
422C2×C4×Q8—711no—
423C2×C4×Q8—711no—
424C2×C4×Q8—7116no—
425C2×C4×Q8—713no—
426C2×C4×Q8—713no—
427C2×C4×Q8—713no—
428C2×C4×Q8C2×D7111yescharacteristic
429C2×C4×Q8C2×D7111yescharacteristic
430C2×C4×D4—713no—
431C2×C4×D4—713no—
432C2×C4×D4—716no—
433C2×C4×D4—713no—
434C2×C4×D4—713no—
435C2×C4×D4—713no—
436C2×C4×D4C2×D7113yes—
437C2×C4×D4C2×D7113yes—
438C2×#117—713no—
439C2×#117C2×D7113yes—
440C22×C4⋊C4—713no—
441C22×C4⋊C4—713no—
442C22×C4⋊C4C2×D7113yes—
443C22×C42—713no—
444C2×#41—28no—
445C2×#41—28424no—
446C2×#41—718no—
447C2×#41—7148no—
448C2×#41—1428no—
449C2×#41C22×D7118yes—
450C22×Q8—718no—
451C22×Q8—716no—
452C22×Q8—711no—
453C22×Q8—713no—
454C22×Q8—1428no—
455C22×Q8C22×D7111yescharacteristic
456C22×D4—716no—
457C22×D4—713no—
458C22×D4—713no—
459C22×D4—713no—
460C22×D4—713no—
461C22×D4C22×D7113yes—
462C23×C4—716no—
463C23×C4—713no—
464C23×C4—713no—
465C23×C4—713no—
466C23×C4—711no—
467C23×C4C22×D7113yes—
468C23×C4C22×D7116yes—
469C4⋊Q8—7124no—
470C4⋊Q8—7124no—
471C4⋊Q8—7124no—
472#127—7124no—
473#126—7124no—
474#126—7124no—
475#125—7124no—
476#124—7124no—
477#124—7124no—
478#124—7148no—
479#124—7124no—
480#123—7148no—
481#123—7124no—
482#123—7124no—
483#123—7148no—
484C22⋊Q8—7148no—
485C22⋊Q8—7124no—
486C22⋊Q8—718no—
487C22⋊Q8—7148no—
488C22⋊Q8—7124no—
489#121—7124no—
490#121—7124no—
491#121—7124no—
492#121—7124no—
493#121—7124no—
494C4×Q8—718no—
495C4×Q8—718no—
496C4×Q8—7124no—
497C4×Q8—7124no—
498C4×Q8—7124no—
499C4×Q8C22×D7118yes—
500C4×Q8C22×D7118yes—
501C4×D4—7124no—
502C4×D4—7124no—
503C4×D4—7148no—
504C4×D4—7124no—
505C4×D4—7124no—
506C4×D4—7124no—
507C4×D4C22×D71124yes—
508C4×D4C22×D71124yes—
509#117—7124no—
510#117C22×D71124yes—
511C2×C4⋊C4—713no—
512C2×C4⋊C4—713no—
513C2×C4⋊C4—713no—
514C2×C4⋊C4—716no—
515C2×C4⋊C4—711no—
516C2×C4⋊C4—711no—
517C2×C4⋊C4—713no—
518C2×C4⋊C4—713no—
519C2×C4⋊C4—713no—
520C2×C4⋊C4—713no—
521C2×C4⋊C4—713no—
522C2×C4⋊C4—7124no—
523C2×C4⋊C4—7124no—
524C2×C4⋊C4C22×D7113yes—
525C2×C4⋊C4C22×D7113yes—
526C2×C4⋊C4C22×D7113yes—
527C2×C4⋊C4C22×D71124yes—
528C2×C4⋊C4C22×D7111yescharacteristic
529C2×C22⋊C4—713no—
530C2×C22⋊C4—716no—
531C2×C22⋊C4—713no—
532C2×C22⋊C4—713no—
533C2×C22⋊C4—713no—
534C2×C22⋊C4—711no—
535C2×C22⋊C4—711no—
536C2×C22⋊C4—713no—
537C2×C22⋊C4—716no—
538C2×C22⋊C4—713no—
539C2×C22⋊C4C22×D7113yes—
540C2×C22⋊C4C22×D7113yes—
541C2×C42—713no—
542C2×C42—713no—
543C2×C42—713no—
544C2×C42—713no—
545C2×C42—7124no—
546C2×C42C22×D7113yes—
547C2×C42C22×D7113yes—
548C24—711no—
549C24—713no—
550C24C23×D7113yes—
551#41—232no—
552#41—28496no—
553#41—14232no—
554C2×Q8—28412no—
555C2×Q8—7124no—
556C2×Q8—714no—
557C2×Q8—7112no—
558C2×Q8—1424no—
559C2×Q8—14232no—
560C2×Q8—14212no—
561C2×Q8#1176586518yescomplemented
562C2×Q8C23×D7114yes—
563C2×D4—212no—
564C2×D4—28412no—
565C2×D4—7124no—
566C2×D4—7112no—
567C2×D4—7112no—
568C2×D4—7112no—
569C2×D4—7112no—
570C2×D4—14212no—
571C2×D4—14212no—
572C2×D4—14212no—
573C2×D4—14212no—
574C2×D4C23×D71112yes—
575C22×C4—212no—
576C22×C4—412no—
577C22×C4—28412no—
578C22×C4—7124no—
579C22×C4—711no—
580C22×C4—711no—
581C22×C4—713no—
582C22×C4—713no—
583C22×C4—713no—
584C22×C4—711no—
585C22×C4—714no—
586C22×C4—7112no—
587C22×C4—7112no—
588C22×C4—7112no—
589C22×C4—1424no—
590C22×C4—14212no—
591C22×C4—14212no—
592C22×C4—14212no—
593C22×C4C23×D71112yes—
594C22×C4C23×D7111yescharacteristic
595C22×C4C23×D7113yes—
596C22×C4C23×D7111yescharacteristic
597C22×C4C23×D7113yes—
598C22×C4C23×D7111yescharacteristic
599C22×C4C23×D71124yes—
600C22×C4C23×D7113yes—
601C4⋊C4—7112no—
602C4⋊C4—7124no—
603C4⋊C4—714no—
604C4⋊C4—714no—
605C4⋊C4—7112no—
606C4⋊C4—7112no—
607C4⋊C4—7112no—
608C4⋊C4—7112no—
609C4⋊C4—7112no—
610C4⋊C4—7112no—
611C4⋊C4—7112no—
612C4⋊C4C23×D71112yes—
613C4⋊C4C23×D71112yes—
614C4⋊C4C23×D71112yes—
615C4⋊C4C23×D7114yes—
616C22⋊C4—7112no—
617C22⋊C4—7124no—
618C22⋊C4—7112no—
619C22⋊C4—7112no—
620C22⋊C4—7112no—
621C22⋊C4—714no—
622C22⋊C4—714no—
623C22⋊C4—7112no—
624C22⋊C4—7124no—
625C22⋊C4—7112no—
626C22⋊C4C23×D71112yes—
627C22⋊C4C23×D71112yes—
628C42—7112no—
629C42—7112no—
630C42—7112no—
631C42—7112no—
632C42C23×D71112yes—
633C42C23×D71112yes—
634C23—26no—
635C23—26no—
636C23—26no—
637C23—714no—
638C23—712no—
639C23—716no—
640C23—1426no—
641C23—1426no—
642C23—1426no—
643C23—1422no—
644C23—1422no—
645C23C24×D7116yes—
646C23C24×D7111yescharacteristic
647Q8—28424no—
648Q8—1428no—
649Q8—14224no—
650Q8C2×#11765865116yescomplemented
651D4—224no—
652D4—28424no—
653D4—14224no—
654D4—14224no—
655D4—14224no—
656D4—14224no—
657C2×C4—22no—
658C2×C4—26no—
659C2×C4—26no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—224no—
663C2×C4—424no—
664C2×C4—28424no—
665C2×C4—714no—
666C2×C4—712no—
667C2×C4—712no—
668C2×C4—716no—
669C2×C4—716no—
670C2×C4—716no—
671C2×C4—712no—
672C2×C4—1428no—
673C2×C4—14224no—
674C2×C4—1422no—
675C2×C4—1426no—
676C2×C4—1426no—
677C2×C4—1426no—
678C2×C4—1422no—
679C2×C4—14224no—
680C2×C4—14224no—
681C2×C4C2×#11765865112yes—
682C2×C4C24×D7112yes—
683C2×C4C24×D7116yes—
684C2×C4C24×D7112yes—
685C2×C4C24×D7116yes—
686C2×C4C24×D7112yes—
687C2×C4C24×D7116yes—
688C22—26no—
689C22—26no—
690C22—26no—
691C22—412no—
692C22—28412no—
693C22—714no—
694C22—1426no—
695C22—1426no—
696C22—1426no—
697C22—1422no—
698C22—1422no—
699C22—1428no—
700C22C22×#1176586512yes—
701C22H(5,𝔽2)×D7112yes—
702C222−1+4×D7112yes—
703C22C25×D7111yescharacteristic
704C4—22no—
705C4—26no—
706C4—26no—
707C4—22no—
708C4—22no—
709C4—714no—
710C4—1422no—
711C4—1426no—
712C4—1426no—
713C4—1426no—
714C4—1422no—
715C4C22×#11765865112yes—
716C2—46no—
717C2—2846no—
718C2—1424no—
719C2#2260241914yesdirect product factor
720C2C23×#1176586511yescharacteristic
721C2C2×H(5,𝔽2)×D7111yescharacteristic
722C2C2×2−1+4×D7111yescharacteristic
subgroupG/H
centerC23C24×D71
commutator, cosocleC22×C71C26
FrattiniC22C25×D71
FittingC2×#520×C71C2
socleC23×C71C25
hypercenterC2×#520D71
hypocenterC71C2×#3522
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. #22602419C2direct (N × H)
2. C2×#17986732C2semidirect
3. C2×Q8×Dic142C2semidirect
4. C2×#17986704C2semidirect
5. C2×#17986668C2semidirect
6. C2×#17986661C2semidirect
7. C2×#17986656C2semidirect
8. C2×#17986654C2semidirect
9. C2×#17986650C2semidirect
10. C2×#17986620C2semidirect
11. C2×#17986614C2semidirect
12. C2×#17986546C2semidirect
13. C2×#17986527C2semidirect
14. C2×#17986481C2semidirect
15. C2×C4×Q8×D71C2semidirect
16. C2×#520×C71C2semidirect
20. #17986732C22semidirect
21. Q8×Dic142C22semidirect
22. #17986704C22semidirect
23. #17986668C22semidirect
24. #17986661C22semidirect
25. #17986656C22semidirect
26. #17986654C22semidirect
27. #17986650C22semidirect
28. #17986620C22semidirect
29. #17986614C22semidirect
30. #17986546C22semidirect
35. #17986527C22semidirect
57. #17986481C22semidirect
63. C4×Q8×D71C22semidirect
75. #520×C71C22semidirect
250. C2×Q8×C71#41semidirect
285. Q8×C71C2×#41semidirect
356. C2×C71#3522semidirect
376. C2×#520D71semidirect
378. C71C2×#3522semidirect
421. #520C2×D71semidirect
561. C2×Q8#11765865semidirect
650. Q8C2×#11765865semidirect
719. C2#22602419direct (N × H)

Maximal subgroups

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

subgroupindex
#226024192
C2×#179867322
C2×Q8×Dic1422
C2×#179867042
C2×#179866682
C2×#179866612
C2×#179866562
C2×#179866542
C2×#179866502
C2×#179866202
C2×#179866142
C2×#179865462
C2×#179865272
C2×#179864812
C2×C4×Q8×D712
C2×#520×C712
C2×#352271