Finite Group Databasebeta

← #13514744

Subgroups — #13514744

Order 1792.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90265C211yesmaximal, characteristic
2C2×#90265C211yesmaximal, characteristic
3C2×C4×#1161C211yesmaximal, characteristic
4C7×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#11739906C211yesmaximal, characteristic, complemented
6#11739905C211yesmaximal, characteristic, complemented
7Q8×Dic7⋊C4C211yesmaximal, characteristic, complemented
8#11733648C213yesmaximal, complemented
9#11733647C213yesmaximal, complemented
10#11733640C213yesmaximal, complemented
11D7×#389C213yesmaximal, complemented
12#11731805C213yesmaximal, complemented
13#11731804C213yesmaximal, complemented
14#11731215C213yesmaximal, complemented
15#11731200C213yesmaximal, complemented
16C22×#1161C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×Q8C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23#90265—28no—
24#90265C4116yescomplemented
25C4×#1161—28no—
26C2×#6456C2213yes—
27C2×#6456C2213yes—
28C2×#6456C2213yes—
29C2×#6455C2213yes—
30C2×D7×C4⋊C4C2213yes—
31C2×D7×C4⋊C4C2216yes—
32C2×D7×C4⋊C4C2213yes—
33C2×C4×D28C2213yes—
34C2×C4×D28C2213yes—
35C2×C4×D28C2213yes—
36C2×#6435C2213yes—
37C2×#6435C2213yes—
38C2×C42×D7C2213yes—
39C7×#389C2213yes—
40C7×#389C2213yes—
41C7×#384C2213yes—
42C4×C7×C4⋊C4C2213yes—
43#89761C2213yes—
44#89761C2213yes—
45#89761C2216yes—
46#89761C2216yes—
47#89759C2216yes—
48#89757C2213yes—
49#89757C2213yes—
50#89749C2213yes—
51#89748C2213yes—
52#89745C2213yes—
53#89745C2213yes—
54#89744C2211yescharacteristic
55#89744C2213yes—
56#89744C2216yes—
57#89708C2213yes—
58C4×C4⋊Dic7C2213yes—
59C4×Dic7⋊C4C2213yes—
60C4×Dic7⋊C4C2213yes—
61#89700C2213yes—
62#89700C2213yes—
63#89440C2213yes—
64#89440C2213yes—
65#89439C2216yes—
66D7×#95C2216yes—
67#89419C2213yes—
68#89419C2213yes—
69#34250—71nomaximal, Sylow
70C22×C7×Q8C2311yescharacteristic
71C2×#1161C2×C418yes—
72C2×#1161D418yes—
73C2×#1161Q818yes—
74C22×D28C2313yes—
75C22×C4×D7C2313yes—
76C22×C4×D7C2313yes—
77C22×C4×D7C2316yes—
78C22×C4×D7C2316yes—
79C22×C4×D7C2313yes—
80C4×C7×Q8—24no—
81C4×C7×Q8—24no—
82C4×C7×Q8C2×C418yescomplemented
83C2×C7×C4⋊C4C2313yes—
84C2×C7×C4⋊C4C2313yes—
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2316yes—
87C2×C7×C4⋊C4C2313yes—
88C2×C7×C4⋊C4C2313yes—
89C2×C42×C7C2313yes—
90C2×C42×C7C2313yes—
91C2×C42×C7C2313yes—
92Q8×Dic7—24no—
93Q8×Dic7—24no—
94Q8×Dic7C2×C418yescomplemented
95C2×(C2×D7)⋊C4C2313yes—
96C2×(C2×D7)⋊C4C2313yes—
97C2×(C2×D7)⋊C4C2316yes—
98C2×(C2×D7)⋊C4C2313yes—
99C2×(C2×D7)⋊C4C2313yes—
100C2×C4⋊Dic7C2313yes—
101C2×C4⋊Dic7C2313yes—
102C2×C4⋊Dic7C2313yes—
103C2×C4⋊Dic7C2313yes—
104C2×C4⋊Dic7C2313yes—
105C2×Dic7⋊C4C2311yescharacteristic
106C2×Dic7⋊C4C2313yes—
107C2×Dic7⋊C4C2313yes—
108C2×Dic7⋊C4C2311yescharacteristic
109C2×Dic7⋊C4C2313yes—
110C2×Dic7⋊C4C2316yes—
111C2×Dic7⋊C4C2313yes—
112C2×Dic7⋊C4C2311yescharacteristic
113C2×Dic7⋊C4C2313yes—
114C2×C4×Dic7C2311yescharacteristic
115C2×C4×Dic7C2313yes—
116C2×C4×Dic7C2313yes—
117C2×C4×Dic7C2313yes—
118C2×C4×Dic7C2313yes—
119#6456—212no—
120#6456—212no—
121#6456C2×C4124yescomplemented
122#6455—212no—
123D7×C4⋊C4—212no—
124D7×C4⋊C4—412no—
125D7×C4⋊C4C2×C4124yescomplemented
126C4×D28—212no—
127C4×D28—212no—
128C4×D28C2×C4124yescomplemented
129#6435—212no—
130#6435C2×C4124yescomplemented
131C42×D7—212no—
132C7×#95C2313yes—
133C7×#95C2313yes—
134C7⋊#95C2313yes—
135C7⋊#95C2313yes—
136C7⋊#95C2316yes—
137C7⋊#95C2316yes—
138C7⋊#95C2311yescharacteristic
139C7⋊#95C2313yes—
140C7⋊#95C2311yescharacteristic
141C7⋊#95C2313yes—
142C2×#520—71no—
143C2×#520—71no—
144C2×C4×#41—71no—
145#2351—71no—
146#2351—71no—
147Q8×C4⋊C4—71no—
148Q8×C4⋊C4D711yescharacteristic, complemented
149#2349—73no—
150#2349—73no—
151D4×C4⋊C4—73no—
152D4×C4⋊C4—73no—
153#2317—73no—
154#2316—73no—
155#2296—73no—
156C2×#389—73no—
157C23×D7C2413yes—
158C2×C7×Q8C22×C414yes—
159C2×C7×Q8C2×D414yes—
160C2×C7×Q8C2×Q814yes—
161C22×C4×C7C2413yes—
162C22×C4×C7C2411yescharacteristic
163C22×C4×C7C2413yes—
164C22×C4×C7C2411yescharacteristic
165C22×C4×C7C2413yes—
166C22×C4×C7C2411yescharacteristic
167C22×C4×C7C2413yes—
168C22×Dic7C2413yes—
169C22×Dic7C2411yescharacteristic
170C22×Dic7C2413yes—
171C22×Dic7C2411yescharacteristic
172C22×Dic7C2413yes—
173C22×Dic7C2411yescharacteristic
174C22×Dic7C2411yescharacteristic
175#1161C4⋊C4164yescomplemented
176C2×D28C22×C4112yes—
177C2×D28C2×D4112yes—
178C2×D28C2×Q8112yes—
179C2×C4×D7—26no—
180C2×C4×D7—26no—
181C2×C4×D7—26no—
182C2×C4×D7—26no—
183C2×C4×D7—212no—
184C2×C4×D7—46no—
185C2×C4×D7—46no—
186C2×C4×D7—43no—
187C2×C4×D7—43no—
188C2×C4×D7—46no—
189C2×C4×D7—46no—
190C2×C4×D7C22×C4124yes—
191C2×C4×D7C22×C4112yes—
192C2×C4×D7C2×D4112yes—
193C2×C4×D7C2×Q8112yes—
194C7×C4⋊C4—26no—
195C7×C4⋊C4—26no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—46no—
198C7×C4⋊C4—43no—
199C7×C4⋊C4C22×C4112yes—
200C42×C7—26no—
201C42×C7—26no—
202C42×C7C22×C4112yes—
203(C2×D7)⋊C4—26no—
204(C2×D7)⋊C4—26no—
205(C2×D7)⋊C4—212no—
206(C2×D7)⋊C4C22×C4112yes—
207(C2×D7)⋊C4C22×C4112yes—
208C4⋊Dic7—26no—
209C4⋊Dic7—26no—
210C4⋊Dic7—26no—
211C4⋊Dic7—43no—
212C4⋊Dic7C22×C4112yes—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—26no—
216Dic7⋊C4—26no—
217Dic7⋊C4—46no—
218Dic7⋊C4—43no—
219Dic7⋊C4—43no—
220Dic7⋊C4C22×C414yes—
221Dic7⋊C4C22×C4112yes—
222C4×Dic7—26no—
223C4×Dic7—26no—
224C4×Dic7—22no—
225C4×Dic7—26no—
226C4×Dic7C22×C4112yes—
227C22×#41—71no—
228#520—716no—
229#520—148no—
230C4×#41—148no—
231C2×C4×Q8—71no—
232C2×C4×Q8—71no—
233C2×C4×Q8—71no—
234C2×C4×Q8C2×D711yescharacteristic
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×Q8C2×D711yescharacteristic
237C2×C4×D4—73no—
238C2×C4×D4—73no—
239C2×C4×D4—73no—
240C2×C4×D4—73no—
241C2×C4×D4—73no—
242C2×C4×D4—73no—
243C2×#117—73no—
244C2×#117—73no—
245C2×#117—73no—
246C22×C4⋊C4—73no—
247C22×C4⋊C4—73no—
248C22×C4⋊C4—76no—
249C22×C42—73no—
250#389—76no—
251#389—73no—
252#389—73no—
253#389—73no—
254#389—73no—
255#389—73no—
256#389—71no—
257#389—73no—
258#389—73no—
259#389—73no—
260#389—73no—
261#389C2×D713yes—
262#389C2×D713yes—
263#385—76no—
264#385—76no—
265#385—73no—
266#385—73no—
267#385—73no—
268#385—73no—
269#384—73no—
270#384C2×D713yes—
271#382—73no—
272#382—73no—
273#380—76no—
274#380—76no—
275C4×C4⋊C4—73no—
276C4×C4⋊C4—73no—
277C4×C4⋊C4—73no—
278C4×C4⋊C4C2×D713yes—
279C2×#95—76no—
280C23×C7C2511yescharacteristic
281C22×D7—23no—
282C22×D7—23no—
283C22×D7—23no—
284C22×D7—23no—
285C22×D7C23×C416yes—
286C22×D7C22×D416yes—
287C22×D7C22×Q816yes—
288C7×Q8C2×C4⋊C4116yescomplemented
289C2×C4×C7—21no—
290C2×C4×C7—23no—
291C2×C4×C7—21no—
292C2×C4×C7—23no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—23no—
296C2×C4×C7—21no—
297C2×C4×C7—23no—
298C2×C4×C7—23no—
299C2×C4×C7—23no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—23no—
303C2×C4×C7—23no—
304C2×C4×C7C23×C412yes—
305C2×C4×C7C23×C416yes—
306C2×C4×C7C23×C416yes—
307C2×C4×C7C22×D416yes—
308C2×C4×C7C22×Q816yes—
309C2×C4×C72−1+412yes—
310C2×Dic7—21no—
311C2×Dic7—23no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—23no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—23no—
318C2×Dic7—23no—
319C2×Dic7—23no—
320C2×Dic7—21no—
321C2×Dic7—23no—
322C2×Dic7—23no—
323C2×Dic7—23no—
324C2×Dic7—21no—
325C2×Dic7C23×C412yes—
326C2×Dic7C23×C416yes—
327C2×Dic7C23×C412yes—
328C2×Dic7C22×D412yes—
329C2×Dic7C22×Q812yes—
330C2×Dic7H(5,𝔽2)12yes—
331D28C2×C4⋊C4148yescomplemented
332C4×D7—424no—
333C4×D7—412no—
334C4×D7—412no—
335C4×D7—812no—
336C4×D7C2×C4⋊C4148yescomplemented
337C2×#41—78no—
338C2×#41—78no—
339C2×#41—78no—
340C22×Q8C22×D711yescharacteristic
341C22×D4—73no—
342C23×C4—73no—
343C23×C4—76no—
344C23×C4—73no—
345C23×C4—73no—
346C23×C4—76no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×Q8—78no—
350C4×Q8—144no—
351C4×Q8—144no—
352C4×Q8C4×D718yescomplemented
353C4×D4—724no—
354C4×D4—724no—
355C4×D4—1412no—
356C4×D4—1412no—
357C4×D4—1412no—
358C4×D4—1412no—
359#117—724no—
360#117—1412no—
361#117—1412no—
362C2×C4⋊C4—73no—
363C2×C4⋊C4—73no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—724no—
366C2×C4⋊C4—73no—
367C2×C4⋊C4—73no—
368C2×C4⋊C4—73no—
369C2×C4⋊C4—73no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—73no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—76no—
374C2×C4⋊C4—73no—
375C2×C4⋊C4—73no—
376C2×C4⋊C4—73no—
377C2×C4⋊C4—1412no—
378C2×C4⋊C4—2812no—
379C2×C4⋊C4C22×D713yes—
380C2×C4⋊C4C22×D713yes—
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C4⋊C4C22×D716yes—
383C2×C4⋊C4C22×D713yes—
384C2×C4⋊C4C22×D713yes—
385C2×C22⋊C4—73no—
386C2×C22⋊C4—73no—
387C2×C22⋊C4—76no—
388C2×C22⋊C4—73no—
389C2×C22⋊C4—73no—
390C2×C42—73no—
391C2×C42—73no—
392C2×C42—73no—
393C2×C42—73no—
394C2×C42—71no—
395C2×C42—1412no—
396C2×C42C22×D713yes—
397C2×C42C22×D713yes—
398C2×C42C22×D713yes—
399#95—73no—
400#95—76no—
401#95—76no—
402#95—71no—
403#95—71no—
404#95—73no—
405#95—73no—
406#95—73no—
407#95C22×D713yes—
408#95C22×D713yes—
409C22×C7C24×C411yescharacteristic
410C22×C7C23×D411yescharacteristic
411C22×C7C23×Q811yescharacteristic
412C22×C7C2×H(5,𝔽2)11yescharacteristic
413C22×C7C2×2−1+411yescharacteristic
414C22×C7#58511yescharacteristic
415C22×C7#58511yescharacteristic
416C2×D7—26no—
417C2×D7—26no—
418C2×D7—26no—
419C2×D7—26no—
420C2×D7—26no—
421C2×D7—26no—
422C2×D7C22×C4⋊C4112yes—
423C4×C7—26no—
424C4×C7—22no—
425C4×C7—22no—
426C4×C7—43no—
427C4×C7—43no—
428C4×C7—41no—
429C4×C7C22×C4⋊C4112yes—
430Dic7—26no—
431Dic7—22no—
432Dic7—22no—
433Dic7—43no—
434Dic7—43no—
435Dic7—41no—
436Dic7C22×C4⋊C414yes—
437C24—73no—
438#41—764no—
439C2×Q8C2×C4×D714yes—
440C2×Q8D4×D714yes—
441C2×Q8Q8×D714yes—
442C2×D4—712no—
443C2×D4—712no—
444C2×D4—712no—
445C22×C4—71no—
446C22×C4—712no—
447C22×C4—712no—
448C22×C4—712no—
449C22×C4—71no—
450C22×C4—73no—
451C22×C4—71no—
452C22×C4—724no—
453C22×C4—73no—
454C22×C4—71no—
455C22×C4—73no—
456C22×C4—146no—
457C22×C4—146no—
458C22×C4—146no—
459C22×C4—1412no—
460C22×C4—146no—
461C22×C4—286no—
462C22×C4—286no—
463C22×C4—283no—
464C22×C4—283no—
465C22×C4—286no—
466C22×C4—286no—
467C22×C4C23×D713yes—
468C22×C4C23×D711yescharacteristic
469C22×C4C23×D713yes—
470C22×C4C23×D711yescharacteristic
471C22×C4C23×D713yes—
472C22×C4C23×D711yescharacteristic
473C22×C4C23×D713yes—
474C4⋊C4—26no—
475C4⋊C4—26no—
476C4⋊C4—22no—
477C4⋊C4—46no—
478C4⋊C4—43no—
479C4⋊C4—712no—
480C4⋊C4—74no—
481C4⋊C4—712no—
482C4⋊C4—146no—
483C4⋊C4—146no—
484C4⋊C4—142no—
485C4⋊C4—146no—
486C4⋊C4—146no—
487C4⋊C4—146no—
488C4⋊C4—142no—
489C4⋊C4—283no—
490C4⋊C4—283no—
491C4⋊C4—286no—
492C4⋊C4—283no—
493C4⋊C4C2×C4×D7112yes—
494C22⋊C4—712no—
495C22⋊C4—712no—
496C22⋊C4—146no—
497C22⋊C4—1412no—
498C22⋊C4—146no—
499C42—26no—
500C42—26no—
501C42—712no—
502C42—142no—
503C42—146no—
504C42—146no—
505C42—146no—
506C42C2×C4×D7112yes—
507C2×C7C23×C4⋊C411yescharacteristic
508C2×C7C4×H(5,𝔽2)11yescharacteristic
509C2×C7C4×2−1+411yescharacteristic
510C2×C7#348811yescharacteristic
511C2×C7#348911yescharacteristic
512C2×C7#350311yescharacteristic
513C2×C7#350411yescharacteristic
514D7—212no—
515C23—76no—
516C23—76no—
517C23—76no—
518C23—143no—
519C23—143no—
520C23—143no—
521C23—143no—
522C23C24×D711yescharacteristic
523Q8D7×C4⋊C4116yescomplemented
524D4—748no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—23no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—23no—
533C2×C4—23no—
534C2×C4—21no—
535C2×C4—23no—
536C2×C4—23no—
537C2×C4—23no—
538C2×C4—23no—
539C2×C4—21no—
540C2×C4—72no—
541C2×C4—72no—
542C2×C4—72no—
543C2×C4—748no—
544C2×C4—72no—
545C2×C4—76no—
546C2×C4—72no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—143no—
552C2×C4—143no—
553C2×C4—141no—
554C2×C4—143no—
555C2×C4—143no—
556C2×C4—141no—
557C2×C4—143no—
558C2×C4—143no—
559C2×C4—143no—
560C2×C4—141no—
561C2×C4—143no—
562C2×C4—2824no—
563C2×C4—2812no—
564C2×C4—2812no—
565C2×C4—5612no—
566C2×C4C22×C4×D712yes—
567C2×C4C22×C4×D716yes—
568C2×C4C22×C4×D716yes—
569C2×C4C2×D4×D716yes—
570C2×C4C2×Q8×D716yes—
571C2×C4#655112yes—
572C7#3425011yescharacteristic, complemented, Sylow
573C22—712no—
574C22—146no—
575C22—146no—
576C22—146no—
577C22—146no—
578C22—146no—
579C22—146no—
580C22C23×C4×D711yescharacteristic
581C22C22×D4×D711yescharacteristic
582C22C22×Q8×D711yescharacteristic
583C22C2×#655111yescharacteristic
584C22#9061611yescharacteristic
585C22D7×H(5,𝔽2)11yescharacteristic
586C22#9062011yescharacteristic
587C4—22no—
588C4—22no—
589C4—26no—
590C4—41no—
591C4—43no—
592C4—43no—
593C4—74no—
594C4—142no—
595C4—142no—
596C4—146no—
597C4—283no—
598C4—281no—
599C4—283no—
600C4C2×D7×C4⋊C4112yes—
601C2—1412no—
602C2C22×D7×C4⋊C411yescharacteristic
603C2C4×#655111yescharacteristic
604C2#1174295411yescharacteristic
605C2#1174381311yescharacteristic
606C2#1174383711yescharacteristic
607C2#1174408711yescharacteristic
608C2#1174410911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×Q8×C4⋊C4C2
socle, cosocleC23×C7C25
prime radicalC22×#1161C22
hypercenterQ8×C4⋊C4D7
hypocenterC7#34250
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C7×Q8×C4⋊C4C2semidirect
5. #11739906C2semidirect
6. #11739905C2semidirect
7. Q8×Dic7⋊C4C2semidirect
8. #11733648C2semidirect
9. #11733647C2semidirect
10. #11733640C2semidirect
11. D7×#389C2semidirect
12. #11731805C2semidirect
13. #11731804C2semidirect
14. #11731215C2semidirect
15. #11731200C2semidirect
24. #90265C4semidirect
82. C4×C7×Q8C2×C4semidirect
94. Q8×Dic7C2×C4semidirect
121. #6456C2×C4semidirect
125. D7×C4⋊C4C2×C4semidirect
128. C4×D28C2×C4semidirect
130. #6435C2×C4semidirect
148. Q8×C4⋊C4D7semidirect
175. #1161C4⋊C4semidirect
288. C7×Q8C2×C4⋊C4semidirect
331. D28C2×C4⋊C4semidirect
336. C4×D7C2×C4⋊C4semidirect
352. C4×Q8C4×D7semidirect
523. Q8D7×C4⋊C4semidirect
572. C7#34250semidirect

Maximal subgroups

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

subgroupindex
C2×#902652
C2×C4×#11612
C7×Q8×C4⋊C42
#117399062
#117399052
Q8×Dic7⋊C42
#117336482
#117336472
#117336402
D7×#3892
#117318052
#117318042
#117312152
#117312002
#342507