Finite Group Databasebeta

← #13513616

Subgroups — #13513616

Order 1792.

3232 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90267C211yesmaximal, characteristic
2C2×C4×Q8×D7C211yesmaximal, characteristic, complemented
3C2×C4×#1161C211yesmaximal, characteristic, complemented
4C7×#2351C211yesmaximal, characteristic, complemented
5Q8×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
6#11739902C211yesmaximal, characteristic, complemented
7C4×Q8×Dic7C211yesmaximal, characteristic, complemented
8#11733651C213yesmaximal, complemented
9#11733429C213yesmaximal, complemented
10#11733410C213yesmaximal, complemented
11#11733409C213yesmaximal, complemented
12#11731674C213yesmaximal, complemented
13#11731594C213yesmaximal, complemented
14#11731100C213yesmaximal, complemented
15C4×#6456C213yesmaximal, complemented
16C22×#1161C2211yescharacteristic
17C22×Q8×D7C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×Q8C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23#90267C4116yescomplemented
24C4×#1161—28no—
25C4×Q8×D7—28no—
26C2×#6460C2213yes—
27C2×#6458C2213yes—
28C2×#6456C2213yes—
29C2×#6456C2213yes—
30C2×#6456C2213yes—
31C2×#6455C2213yes—
32C2×D7×C4⋊C4C2213yes—
33C2×#6450C2213yes—
34C2×#6438C2213yes—
35C2×C4×D28C2213yes—
36C2×C4×D28C2213yes—
37C2×C42×D7C2213yes—
38C2×C42×D7C2213yes—
39C2×C4×Dic14C2213yes—
40C7×#389C2213yes—
41C7×#389C2213yes—
42C7×#382C2213yes—
43C4×C7×C4⋊C4C2213yes—
44#89762C2216yes—
45#89761C2216yes—
46#89760C2213yes—
47#89759C2213yes—
48#89759C2213yes—
49C4⋊C4×Dic7C2213yes—
50C4⋊C4×Dic7C2213yes—
51C4⋊C4×Dic7C2213yes—
52C4⋊C4×Dic7C2213yes—
53#89746C2213yes—
54#89711C2213yes—
55C4×(C2×D7)⋊C4C2213yes—
56C4×(C2×D7)⋊C4C2213yes—
57C4×C4⋊Dic7C2213yes—
58C42×Dic7C2213yes—
59#89701C2213yes—
60#89441C2216yes—
61#89441C2216yes—
62#89438C2211yescharacteristic
63#89438C2213yes—
64#89438C2213yes—
65#89438C2213yes—
66#89438C2216yes—
67#89437C2216yes—
68#89426C2216yes—
69C4×#548—71nomaximal, Sylow
70C22×C7×Q8C2311yescharacteristic
71C2×#1161—24no—
72C2×#1161—24no—
73C2×#1161C2×C418yescomplemented
74C2×Q8×D7—24no—
75C2×Q8×D7—24no—
76C2×Q8×D7C2×C418yescomplemented
77C22×D28C2313yes—
78C22×D28C2313yes—
79C22×C4×D7C2313yes—
80C22×C4×D7C2313yes—
81C22×C4×D7C2313yes—
82C22×C4×D7C2313yes—
83C22×C4×D7C2313yes—
84C22×C4×D7C2311yescharacteristic
85C22×Dic14C2313yes—
86C4×C7×Q8—24no—
87C4×C7×Q8—24no—
88C4×C7×Q8C2×C418yescomplemented
89C2×C7×C4⋊C4C2313yes—
90C2×C7×C4⋊C4C2313yes—
91C2×C7×C4⋊C4C2313yes—
92C2×C7×C4⋊C4C2313yes—
93C2×C7×C4⋊C4C2313yes—
94C2×C42×C7C2313yes—
95C2×C42×C7C2313yes—
96C2×C42×C7C2313yes—
97Q8×Dic7—24no—
98Q8×Dic7D418yescomplemented
99C2×(C2×D7)⋊C4C2316yes—
100C2×(C2×D7)⋊C4C2313yes—
101C2×(C2×D7)⋊C4C2313yes—
102C2×(C2×D7)⋊C4C2313yes—
103C2×(C2×D7)⋊C4C2316yes—
104C2×(C2×D7)⋊C4C2311yescharacteristic
105C2×(C2×D7)⋊C4C2313yes—
106C2×(C2×D7)⋊C4C2316yes—
107C2×(C2×D7)⋊C4C2311yescharacteristic
108C2×C4⋊Dic7C2313yes—
109C2×C4⋊Dic7C2313yes—
110C2×C4⋊Dic7C2313yes—
111C2×Dic7⋊C4C2316yes—
112C2×C4×Dic7C2313yes—
113C2×C4×Dic7C2311yescharacteristic
114C2×C4×Dic7C2313yes—
115C2×C4×Dic7C2313yes—
116C2×C4×Dic7C2313yes—
117C2×C4×Dic7C2313yes—
118C2×C4×Dic7C2311yescharacteristic
119C2×C4×Dic7C2313yes—
120C2×C4×Dic7C2313yes—
121C2×C4×Dic7C2313yes—
122#6460C2×C4124yescomplemented
123#6458C2×C4124yescomplemented
124#6456—212no—
125#6456—212no—
126#6456—212no—
127#6455—212no—
128D7×C4⋊C4—212no—
129#6450—212no—
130#6438C2×C4124yescomplemented
131C4×D28—212no—
132C4×D28C2×C4124yescomplemented
133C42×D7—212no—
134C42×D7—212no—
135C4×Dic14—212no—
136C7×#95C2316yes—
137C7×#95C2311yescharacteristic
138C7×#95C2313yes—
139C7⋊#95C2313yes—
140C7⋊#95C2316yes—
141C7⋊#95C2316yes—
142C7⋊#95C2313yes—
143C7⋊#95C2313yes—
144C7⋊#95C2311yescharacteristic
145C2×#548—71no—
146C2×C4×#41—71no—
147C22×C4×Q8—71no—
148#2367—73no—
149#2362—73no—
150#2361—73no—
151#2352—73no—
152#2351D711yescharacteristic, complemented
153#2341—71no—
154Q8×C22⋊C4—71no—
155C4×#124—73no—
156C4×C22⋊Q8—73no—
157C4×#121—73no—
158C42×Q8—71no—
159C42×D4—73no—
160C23×D7C2413yes—
161C23×D7C2411yescharacteristic
162C2×C7×Q8C22×C414yes—
163C2×C7×Q8C2×D414yes—
164C2×C7×Q8#4114yes—
165C22×C4×C7C2413yes—
166C22×C4×C7C2413yes—
167C22×C4×C7C2413yes—
168C22×C4×C7C2411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C22×C4×C7C2411yescharacteristic
171C22×C4×C7C2413yes—
172C22×Dic7C2413yes—
173C22×Dic7C2411yescharacteristic
174C22×Dic7C2413yes—
175C22×Dic7C2413yes—
176C22×Dic7C2411yescharacteristic
177C22×Dic7C2411yescharacteristic
178#1161—232no—
179Q8×D7—232no—
180C2×D28—26no—
181C2×D28—26no—
182C2×D28—26no—
183C2×D28—26no—
184C2×D28C22×C4112yes—
185C2×D28C22×C4112yes—
186C2×C4×D7—26no—
187C2×C4×D7—26no—
188C2×C4×D7—26no—
189C2×C4×D7—26no—
190C2×C4×D7—22no—
191C2×C4×D7—22no—
192C2×C4×D7—26no—
193C2×C4×D7—26no—
194C2×C4×D7—26no—
195C2×C4×D7—26no—
196C2×C4×D7—26no—
197C2×C4×D7—26no—
198C2×C4×D7—26no—
199C2×C4×D7—43no—
200C2×C4×D7—43no—
201C2×C4×D7C22×C4112yes—
202C2×C4×D7C22×C4112yes—
203C2×C4×D7#4114yescomplemented
204C2×Dic14—26no—
205C2×Dic14—26no—
206C2×Dic14C22×C4112yes—
207C7×C4⋊C4—26no—
208C7×C4⋊C4—26no—
209C7×C4⋊C4—26no—
210C7×C4⋊C4—26no—
211C7×C4⋊C4C22×C4112yes—
212C42×C7—26no—
213C42×C7—26no—
214C42×C7C22×C4112yes—
215(C2×D7)⋊C4—26no—
216(C2×D7)⋊C4—26no—
217(C2×D7)⋊C4—212no—
218(C2×D7)⋊C4—26no—
219(C2×D7)⋊C4—46no—
220(C2×D7)⋊C4C22×C414yes—
221(C2×D7)⋊C4C22×C4112yes—
222(C2×D7)⋊C4C22×C4124yes—
223(C2×D7)⋊C4#4114yescomplemented
224C4⋊Dic7—26no—
225C4⋊Dic7C22×C4112yes—
226C4⋊Dic7C2×D4112yes—
227Dic7⋊C4—212no—
228C4×Dic7—26no—
229C4×Dic7—26no—
230C4×Dic7—22no—
231C4×Dic7—26no—
232C4×Dic7—26no—
233C4×Dic7—26no—
234C4×Dic7—26no—
235C4×Dic7—22no—
236C4×Dic7—43no—
237C4×Dic7C2×D4112yes—
238C22×#41—71no—
239C23×Q8—71no—
240#548—716no—
241C2×#124—73no—
242C2×C22⋊Q8—73no—
243C2×#121—73no—
244C4×#41—148no—
245C2×C4×Q8—71no—
246C2×C4×Q8—71no—
247C2×C4×Q8—73no—
248C2×C4×Q8—73no—
249C2×C4×Q8—148no—
250C2×C4×Q8C2×D711yescharacteristic
251C2×C4×Q8C2×D711yescharacteristic
252C2×C4×Q8C2×D711yescharacteristic
253C2×C4×D4—73no—
254C2×C4×D4—73no—
255C2×C4×D4—73no—
256C2×C4×D4—73no—
257C2×C4×D4—73no—
258C2×#117—73no—
259C22×C4⋊C4—73no—
260C22×C42—73no—
261C22×C42—73no—
262#391—73no—
263#391—73no—
264#390—73no—
265#390—73no—
266#389C2×D713yes—
267#389C2×D713yes—
268#388—76no—
269#388—76no—
270#388—76no—
271#387—76no—
272#386—76no—
273#385—76no—
274#382C2×D713yes—
275#380—73no—
276#380—73no—
277C4×C4⋊C4—73no—
278C4×C4⋊C4—73no—
279C4×C4⋊C4—73no—
280C4×C4⋊C4—73no—
281C4×C4⋊C4—73no—
282C4×C4⋊C4C2×D713yes—
283C4×C22⋊C4—73no—
284C4×C22⋊C4—73no—
285C4×C22⋊C4—73no—
286C4×C22⋊C4—73no—
287C4×C22⋊C4—71no—
288C4×C22⋊C4—73no—
289C4×C22⋊C4—76no—
290C43—73no—
291C23×C7C2511yescharacteristic
292C22×D7—21no—
293C22×D7—21no—
294C22×D7—21no—
295C22×D7—21no—
296C22×D7—23no—
297C22×D7—23no—
298C22×D7—23no—
299C22×D7—23no—
300C22×D7—23no—
301C22×D7—23no—
302C22×D7C23×C416yes—
303C22×D7C23×C412yes—
304C22×D7C2×#4112yes—
305C22×D7C2×#4112yes—
306C7×Q8C4×D4116yescomplemented
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—23no—
310C2×C4×C7—23no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—23no—
316C2×C4×C7—23no—
317C2×C4×C7—23no—
318C2×C4×C7—23no—
319C2×C4×C7—23no—
320C2×C4×C7—23no—
321C2×C4×C7C23×C416yes—
322C2×C4×C7C23×C412yes—
323C2×C4×C7C23×C416yes—
324C2×C4×C7C22×D416yes—
325C2×C4×C7C2×#4116yes—
326C2×C4×C7C2×#4112yes—
327C2×C4×C7C2×#4112yes—
328C2×Dic7—21no—
329C2×Dic7—23no—
330C2×Dic7—21no—
331C2×Dic7—23no—
332C2×Dic7—23no—
333C2×Dic7—21no—
334C2×Dic7—21no—
335C2×Dic7—23no—
336C2×Dic7—23no—
337C2×Dic7—23no—
338C2×Dic7—23no—
339C2×Dic7C23×C412yes—
340C2×Dic7C23×C416yes—
341C2×Dic7C22×D416yes—
342C2×Dic7C22×D412yes—
343C2×Dic7C2×#4112yes—
344C2×Dic7C2×#4112yes—
345C2×Dic72−1+412yes—
346D28—224no—
347D28—224no—
348C4×D7—224no—
349C4×D7—224no—
350C4×D7—28no—
351C4×D7—412no—
352C4×D7—412no—
353C4×D7—412no—
354Dic14—224no—
355C2×#41—78no—
356C2×#41—144no—
357C2×#41—144no—
358C22×Q8—78no—
359C22×Q8—73no—
360C22×Q8—144no—
361C22×Q8—144no—
362C22×Q8C22×D711yescharacteristic
363C22×D4—73no—
364C22×D4—73no—
365C23×C4—73no—
366C23×C4—73no—
367C23×C4—71no—
368C23×C4—73no—
369C23×C4—73no—
370C23×C4—73no—
371#124—724no—
372C22⋊Q8—724no—
373#121—724no—
374C4×Q8—24no—
375C4×Q8—24no—
376C4×Q8—78no—
377C4×Q8—1412no—
378C4×Q8—1412no—
379C4×Q8—144no—
380C4×Q8C4×D718yescomplemented
381C4×D4—724no—
382C4×D4—1412no—
383C4×D4—1412no—
384C4×D4—1412no—
385C4×D4—1412no—
386#117—1412no—
387C2×C4⋊C4—73no—
388C2×C4⋊C4—73no—
389C2×C4⋊C4—76no—
390C2×C4⋊C4—73no—
391C2×C4⋊C4—1412no—
392C2×C4⋊C4C22×D713yes—
393C2×C4⋊C4C22×D713yes—
394C2×C4⋊C4C22×D713yes—
395C2×C4⋊C4C22×D713yes—
396C2×C4⋊C4C22×D713yes—
397C2×C22⋊C4—71no—
398C2×C22⋊C4—71no—
399C2×C22⋊C4—73no—
400C2×C22⋊C4—76no—
401C2×C22⋊C4—76no—
402C2×C22⋊C4—73no—
403C2×C22⋊C4—73no—
404C2×C22⋊C4—73no—
405C2×C22⋊C4—76no—
406C2×C42—73no—
407C2×C42—73no—
408C2×C42—73no—
409C2×C42—73no—
410C2×C42—73no—
411C2×C42—73no—
412C2×C42—73no—
413C2×C42—71no—
414C2×C42—71no—
415C2×C42—73no—
416C2×C42—1412no—
417C2×C42—1412no—
418C2×C42C22×D713yes—
419C2×C42C22×D713yes—
420C2×C42C22×D713yes—
421#95—71no—
422#95—73no—
423#95—73no—
424#95—73no—
425#95—76no—
426#95—76no—
427#95C22×D711yescharacteristic
428#95C22×D713yes—
429#95C22×D716yes—
430C22×C7C24×C411yescharacteristic
431C22×C7C23×D411yescharacteristic
432C22×C7C22×#4111yescharacteristic
433C22×C7C22×#4111yescharacteristic
434C22×C7C22×#4111yescharacteristic
435C22×C7C2×2−1+411yescharacteristic
436C22×C7#58511yescharacteristic
437C2×D7—22no—
438C2×D7—22no—
439C2×D7—22no—
440C2×D7—22no—
441C2×D7—22no—
442C2×D7—22no—
443C2×D7—26no—
444C2×D7—26no—
445C2×D7—26no—
446C2×D7—43no—
447C2×D7—43no—
448C2×D7—43no—
449C2×D7—43no—
450C2×D7C4×#4114yescomplemented
451C4×C7—26no—
452C4×C7—22no—
453C4×C7—22no—
454C4×C7—22no—
455C4×C7—43no—
456C4×C7—43no—
457C4×C7C2×C4×D4112yes—
458Dic7—26no—
459Dic7—22no—
460Dic7—26no—
461Dic7—22no—
462Dic7—43no—
463Dic7#54814yescomplemented
464C24—71no—
465C24—73no—
466#41—1432no—
467C2×Q8—712no—
468C2×Q8—1432no—
469C2×Q8—146no—
470C2×Q8—146no—
471C2×Q8C2×C4×D714yes—
472C2×Q8D4×D714yes—
473C2×Q8#116114yes—
474C2×D4—712no—
475C2×D4—712no—
476C2×D4—146no—
477C2×D4—146no—
478C2×D4—146no—
479C2×D4—146no—
480C22×C4—71no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—73no—
484C22×C4—73no—
485C22×C4—712no—
486C22×C4—712no—
487C22×C4—74no—
488C22×C4—73no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—146no—
492C22×C4—146no—
493C22×C4—146no—
494C22×C4—146no—
495C22×C4—146no—
496C22×C4—146no—
497C22×C4—146no—
498C22×C4—146no—
499C22×C4—146no—
500C22×C4—146no—
501C22×C4—146no—
502C22×C4—283no—
503C22×C4—283no—
504C22×C4C23×D713yes—
505C22×C4C23×D713yes—
506C22×C4C23×D713yes—
507C22×C4C23×D711yescharacteristic
508C22×C4C23×D711yescharacteristic
509C22×C4C23×D711yescharacteristic
510C22×C4C23×D713yes—
511C4⋊C4—26no—
512C4⋊C4—26no—
513C4⋊C4—26no—
514C4⋊C4—26no—
515C4⋊C4—712no—
516C4⋊C4—712no—
517C4⋊C4—1412no—
518C4⋊C4—146no—
519C4⋊C4C2×C4×D7112yes—
520C22⋊C4—74no—
521C22⋊C4—712no—
522C22⋊C4—724no—
523C22⋊C4—74no—
524C22⋊C4—1412no—
525C22⋊C4—146no—
526C22⋊C4—146no—
527C22⋊C4—146no—
528C22⋊C4—286no—
529C42—26no—
530C42—26no—
531C42—712no—
532C42—142no—
533C42—142no—
534C42—146no—
535C42—146no—
536C42—146no—
537C42—146no—
538C42—146no—
539C42—146no—
540C42—283no—
541C42C2×C4×D7112yes—
542C2×C7C22×C4×D411yescharacteristic
543C2×C7C2×C4×#4111yescharacteristic
544C2×C7C4×2−1+411yescharacteristic
545C2×C7C2×#54811yescharacteristic
546C2×C7D4×#4111yescharacteristic
547C2×C7#348011yescharacteristic
548C2×C7#348211yescharacteristic
549D7—24no—
550D7—46no—
551C23—72no—
552C23—72no—
553C23—72no—
554C23—76no—
555C23—141no—
556C23—141no—
557C23—141no—
558C23—141no—
559C23—143no—
560C23—143no—
561C23—143no—
562C23—143no—
563C23—143no—
564C23—143no—
565C23C24×D711yescharacteristic
566Q8—1424no—
567Q8#6456116yescomplemented
568D4—1424no—
569D4—1424no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—23no—
579C2×C4—23no—
580C2×C4—23no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—23no—
584C2×C4—72no—
585C2×C4—72no—
586C2×C4—72no—
587C2×C4—76no—
588C2×C4—72no—
589C2×C4—76no—
590C2×C4—72no—
591C2×C4—148no—
592C2×C4—141no—
593C2×C4—143no—
594C2×C4—141no—
595C2×C4—141no—
596C2×C4—143no—
597C2×C4—143no—
598C2×C4—141no—
599C2×C4—143no—
600C2×C4—1424no—
601C2×C4—1424no—
602C2×C4—143no—
603C2×C4—143no—
604C2×C4—143no—
605C2×C4—2812no—
606C2×C4—2812no—
607C2×C4—2812no—
608C2×C4C22×C4×D712yes—
609C2×C4C22×C4×D716yes—
610C2×C4C22×C4×D716yes—
611C2×C4C2×D4×D716yes—
612C2×C4C2×#116116yes—
613C2×C4D7×#4112yes—
614C2×C4D7×#4112yes—
615C7C4×#54811yescharacteristic, complemented, Sylow
616C22—74no—
617C22—142no—
618C22—142no—
619C22—142no—
620C22—142no—
621C22—142no—
622C22—142no—
623C22—146no—
624C22—146no—
625C22—146no—
626C22—283no—
627C22—283no—
628C22—283no—
629C22—283no—
630C22C23×C4×D711yescharacteristic
631C22C22×D4×D711yescharacteristic
632C22C22×#116111yescharacteristic
633C22C2×D7×#4111yescharacteristic
634C22C2×D7×#4111yescharacteristic
635C22D7×2−1+411yescharacteristic
636C22#9062011yescharacteristic
637C4—22no—
638C4—22no—
639C4—26no—
640C4—22no—
641C4—43no—
642C4—43no—
643C4—74no—
644C4—146no—
645C4—142no—
646C4—146no—
647C4—142no—
648C4—283no—
649C4C2×#6456112yes—
650C2—144no—
651C2—286no—
652C2C22×#645611yescharacteristic
653C2C4×D7×#4111yescharacteristic
654C2#1174294111yescharacteristic
655C2#1174377811yescharacteristic
656C2D7×#54811yescharacteristic
657C2#1174408411yescharacteristic
658C2#1174408511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2351C2
socle, cosocleC23×C7C25
prime radicalC2×#90267C2
hypercenter#2351D7
hypocenterC7C4×#548
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×C4×Q8×D7C2semidirect
3. C2×C4×#1161C2semidirect
4. C7×#2351C2semidirect
5. Q8×(C2×D7)⋊C4C2semidirect
6. #11739902C2semidirect
7. C4×Q8×Dic7C2semidirect
8. #11733651C2semidirect
9. #11733429C2semidirect
10. #11733410C2semidirect
11. #11733409C2semidirect
12. #11731674C2semidirect
13. #11731594C2semidirect
14. #11731100C2semidirect
15. C4×#6456C2semidirect
23. #90267C4semidirect
73. C2×#1161C2×C4semidirect
76. C2×Q8×D7C2×C4semidirect
88. C4×C7×Q8C2×C4semidirect
98. Q8×Dic7D4semidirect
122. #6460C2×C4semidirect
123. #6458C2×C4semidirect
130. #6438C2×C4semidirect
132. C4×D28C2×C4semidirect
152. #2351D7semidirect
203. C2×C4×D7#41semidirect
223. (C2×D7)⋊C4#41semidirect
306. C7×Q8C4×D4semidirect
380. C4×Q8C4×D7semidirect
450. C2×D7C4×#41semidirect
463. Dic7#548semidirect
567. Q8#6456semidirect
615. C7C4×#548semidirect

Maximal subgroups

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

subgroupindex
C2×#902672
C2×C4×Q8×D72
C2×C4×#11612
C7×#23512
Q8×(C2×D7)⋊C42
#117399022
C4×Q8×Dic72
#117336512
#117334292
#117334102
#117334092
#117316742
#117315942
#117311002
C4×#64562
C4×#5487