Finite Group Databasebeta

← #14217981

Subgroups — #14217981

Order 1792.

2882 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#518C211yesmaximal, characteristic, complemented
2C22×#6495C211yesmaximal, characteristic
3#11742799C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5#11739988C214yesmaximal, complemented
6#11739983C214yesmaximal, complemented
7#11739982C218yesmaximal, complemented
8C4×#6495C218yesmaximal, complemented
9#11739658C211yesmaximal, characteristic, complemented
10#11739652C211yesmaximal, characteristic, complemented
11#11739647C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C7×#518—22no—
15C7×#518—22no—
16C7×#518C2214yes—
17C7×#518C2214yescomplemented
18C2×C4×C7×D4—22no—
19C2×C4×C7×D4C2214yescomplemented
20C2×C7×#117C2211yescharacteristic
21C2×C7×#117C2211yescharacteristic
22C22×C7×C22⋊C4—22no—
23C22×#1142C2212yes—
24C2×#6495C4116yescomplemented
25C2×#6495C2214yes—
26C2×#6495C2214yes—
27C22×#1141C2212yes—
28C2×#6489C2211yescharacteristic
29C22×#1136C2211yescharacteristic
30C22×#1136C2211yescharacteristic
31#90214C2214yescomplemented
32#90213—22no—
33#90212C2212yescomplemented
34D7×#117—22no—
35#90207C2212yescomplemented
36#90195—22no—
37#90177—22no—
38C4×#1157C2214yescomplemented
39#89987C2214yes—
40#89987C2214yescomplemented
41C2×#6406C2214yescomplemented
42#89790C22116yescomplemented
43C4×#1142C22116yescomplemented
44#89786C22116yescomplemented
45C4×#1141C22116yescomplemented
46#89776C2212yescomplemented
47#89774C2214yescomplemented
48#89773C2212yescomplemented
49#89770C2214yescomplemented
50#89769C2214yescomplemented
51#89769C2214yescomplemented
52#89767C2214yescomplemented
53#89738C2214yescomplemented
54#89736C2214yescomplemented
55C4×#1136C2214yescomplemented
56#60168—71nomaximal, Sylow
57C25×C7—21no—
58C22×C7×D4—21no—
59C22×C7×D4—21no—
60C22×C7×D4—21no—
61C22×C7×D4—24no—
62C22×C7×D4C2×C418yescomplemented
63C22×C7×D4C2312yes—
64C22×C7×D4C2312yes—
65C22×C7×D4C2312yes—
66C23×C4×C7—24no—
67C23×C4×C7C2311yescharacteristic
68C22×C22⋊D7—21no—
69C2×#1157—21no—
70C2×#1157—21no—
71C2×#1157—24no—
72C2×#1157C2×C418yescomplemented
73C2×#1157C2312yes—
74C2×#1157C2312yes—
75C22×D28C2311yescharacteristic
76C22×C4×D7—21no—
77C22×Dic14C2311yescharacteristic
78C4×C7×D4—28no—
79C4×C7×D4—28no—
80C4×C7×D4—28no—
81C4×C7×D4C23116yes—
82C7×#117—21no—
83C7×#117—21no—
84C7×#117D412yes—
85C7×#117D412yes—
86C7×#117C2312yes—
87C7×#117C2312yes—
88C7×#117C2312yes—
89C7×#117C2312yes—
90C2×C7×C4⋊C4D412yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—24no—
94C2×C7×C22⋊C4—28no—
95C2×C7×C22⋊C4—28no—
96C2×C7×C22⋊C4D412yes—
97C2×C42×C7—21no—
98C2×C42×C7C2312yes—
99C2×C22⋊Dic7—21no—
100C2×#1142C2×C4116yescomplemented
101C2×#1142C2318yes—
102#6495C2×C4132yescomplemented
103#6495C2×C4132yescomplemented
104C2×#1141C2×C4116yescomplemented
105C2×#1141C2318yes—
106C4×C22⋊D7—28no—
107#6489—21no—
108#6489—21no—
109#6489C2312yes—
110#6489C2312yes—
111C2×C4⋊Dic7C2312yes—
112C2×C4×Dic7—21no—
113C2×#1136—24no—
114C2×#1136C2×C418yescomplemented
115C2×#1136C2312yes—
116C2×#1136C2312yes—
117C2×#1136C2312yes—
118C2×#1136C2312yes—
119C22×C7⋊C8C2312yes—
120C22×C7⋊C8C2312yes—
121#6456—24no—
122#6455—24no—
123D7×C4⋊C4—24no—
124#6450—24no—
125(C22⋊D7)⋊C4—28no—
126D7×C22⋊C4—24no—
127#6440—24no—
128C4×D28C2318yescomplemented
129#6435—24no—
130C42×D7—24no—
131C4×Dic14C2318yescomplemented
132#6406C23116yes—
133#6384C2318yescomplemented
134#6383C2318yescomplemented
135#6382C2318yescomplemented
136#6381C2318yescomplemented
137#6377C2318yescomplemented
138C4×C7⋊C8C2318yescomplemented
139C22×AGL(1,ℤ/8)—71no—
140#3428—71no—
141C2×#518D711yescharacteristic, complemented
142#2949—74no—
143#2948—74no—
144#2946—78no—
145C4×AGL(1,ℤ/8)—78no—
146#2911—71no—
147#2896—71no—
148C2×#418—71no—
149#2873—71no—
150C24×C7—22no—
151C24×C7—28no—
152C24×C7—22no—
153C24×C7—22no—
154C24×C7—84no—
155C24×C7C2×D411yescharacteristic
156C23×D7—21no—
157C2×C7×D4—28no—
158C2×C7×D4—28no—
159C2×C7×D4—28no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—22no—
163C2×C7×D4—24no—
164C2×C7×D4—24no—
165C2×C7×D4—24no—
166C2×C7×D4C22×C418yes—
167C2×C7×D4C22×C418yes—
168C2×C7×D4C22×C418yes—
169C2×C7×D4C2414yes—
170C22×C4×C7—22no—
171C22×C4×C7—216no—
172C22×C4×C7—88no—
173C22×C4×C7C22×C414yes—
174C22×C4×C7C2×D414yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7#4114yes—
179C22×C4×C7C2411yescharacteristic
180C22×C4×C7C2411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C2×C22⋊D7—28no—
183C2×C22⋊D7—22no—
184C2×C22⋊D7—22no—
185C22×Dic7—21no—
186C22×Dic7—22no—
187#1157—28no—
188#1157—28no—
189#1157—24no—
190#1157—24no—
191#1157C22×C418yes—
192#1157C22×C418yescomplemented
193C2×D28—21no—
194C2×D28—22no—
195C2×D28C22×C414yescomplemented
196C2×D28C2412yes—
197C2×C4×D7—28no—
198C2×C4×D7—22no—
199C2×C4×D7—22no—
200C2×C4×D7—28no—
201C2×Dic14—21no—
202C2×Dic14—22no—
203C2×Dic14C22×C414yescomplemented
204C2×Dic14C2412yes—
205C7×C4⋊C4—22no—
206C7×C4⋊C4C2×D414yes—
207C7×C4⋊C4C2×D414yes—
208C7×C4⋊C4C2414yes—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—216no—
212C7×C22⋊C4C2×D414yes—
213C7×C22⋊C4C2×D414yes—
214C42×C7—22no—
215C42×C7—22no—
216C42×C7C2×D414yes—
217C42×C7C2414yes—
218C22⋊Dic7—22no—
219C22⋊Dic7—22no—
220#1142C22×C4132yescomplemented
221#1141C22×C4132yescomplemented
222(C2×D7)⋊C4—28no—
223C4⋊Dic7—22no—
224C4⋊Dic7C2414yes—
225Dic7⋊C4—28no—
226C4×Dic7—28no—
227C4×Dic7—22no—
228C4×Dic7—22no—
229#1136—24no—
230#1136—24no—
231#1136C22×C418yes—
232#1136C22×C418yescomplemented
233C2×C7⋊C8—24no—
234C2×C7⋊C8C22×C418yescomplemented
235C2×C7⋊C8C2414yes—
236C2×C7⋊C8C2414yes—
237C22×#41—71no—
238C23×D4C2×D711yescharacteristic
239C2×AGL(1,ℤ/8)—716no—
240C2×AGL(1,ℤ/8)—74no—
241C2×AGL(1,ℤ/8)—74no—
242C22×SD16—72no—
243C22×D8—72no—
244C22×M16—71no—
245C22×M16—71no—
246#520—74no—
247#520—142no—
248#519—72no—
249#518—22no—
250#518—22no—
251#518—72no—
252#518—142no—
253#518C2×D714yes—
254#518C2×D714yescomplemented
255C4×#41—74no—
256C4×#41—142no—
257C2×C4×D4—22no—
258C2×C4×D4C2×D714yescomplemented
259C2×#117—71no—
260C2×#117—142no—
261C2×#117C2×D711yescharacteristic
262C2×#117C2×D711yescharacteristic
263C22×C22⋊C4—22no—
264#442—716no—
265#440—716no—
266C4×SD16—716no—
267C4×D8—716no—
268#428—74no—
269#428—74no—
270#427—74no—
271#419—72no—
272#418—74no—
273#418—74no—
274#418—72no—
275#417—74no—
276#416—74no—
277C2×#102—74no—
278#405—74no—
279#405—74no—
280C4×M16—74no—
281C23×C7—28no—
282C23×C7—24no—
283C23×C7—28no—
284C23×C7—28no—
285C23×C7—42no—
286C23×C7—48no—
287C23×C7—84no—
288C23×C7—84no—
289C23×C7—816no—
290C23×C7C4×D414yes—
291C23×C7C22×D411yescharacteristic
292C23×C7C22×D411yescharacteristic
293C23×C7C22×D411yescharacteristic
294C22×D7—22no—
295C22×D7—24no—
296C22×D7—82no—
297C7×D4—28no—
298C7×D4—28no—
299C7×D4—28no—
300C7×D4—24no—
301C7×D4C23×C418yes—
302C2×C4×C7—24no—
303C2×C4×C7—44no—
304C2×C4×C7—44no—
305C2×C4×C7—816no—
306C2×C4×C7C4×D414yes—
307C2×C4×C7C4×D414yes—
308C2×C4×C7C4×D414yes—
309C2×C4×C7C23×C412yes—
310C2×C4×C7C23×C412yes—
311C2×C4×C7C23×C412yes—
312C2×C4×C7C22×D418yes—
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C2×#4112yes—
317C2×C4×C7C2×#4112yes—
318C2×C4×C7C2×#4112yes—
319C2×C4×C7C2511yescharacteristic
320C22⋊D7—28no—
321C22⋊D7—28no—
322C2×Dic7—24no—
323C2×Dic7—22no—
324C2×Dic7—24no—
325C2×Dic7—82no—
326D28—24no—
327D28—22no—
328D28C23×C414yes—
329C4×D7—28no—
330C4×D7—28no—
331C4×D7—88no—
332Dic14—24no—
333Dic14—22no—
334Dic14C23×C414yes—
335C7⋊C8—24no—
336C7⋊C8C23×C418yes—
337C25—21no—
338C2×#41—78no—
339C2×#41—72no—
340C2×#41—72no—
341C2×#41—144no—
342C2×#41—141no—
343C2×#41—141no—
344C22×Q8—71no—
345C22×D4—24no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4—71no—
350C22×D4—141no—
351C22×D4C4×D718yescomplemented
352C22×D4C22×D712yes—
353C22×D4C22×D712yes—
354C22×D4C22×D712yes—
355C23×C4—24no—
356C23×C4—141no—
357C23×C4C22×D711yescharacteristic
358AGL(1,ℤ/8)—732no—
359AGL(1,ℤ/8)—732no—
360C2×SD16—716no—
361C2×SD16—78no—
362C2×D8—716no—
363C2×D8—78no—
364C2×M16—78no—
365C2×M16—72no—
366C2×M16—72no—
367C2×M16—72no—
368C2×M16—72no—
369C2×M16—144no—
370C22×C8—72no—
371C22×C8—72no—
372C4×Q8—78no—
373C4×Q8—144no—
374C4×D4—28no—
375C4×D4—28no—
376C4×D4—28no—
377C4×D4—78no—
378C4×D4—148no—
379C4×D4—144no—
380C4×D4—148no—
381C4×D4C22×D7116yes—
382#117—21no—
383#117—21no—
384#117—72no—
385#117—72no—
386#117—141no—
387#117—141no—
388#117—144no—
389#117—144no—
390#117—144no—
391#117C22⋊D712yes—
392#117C22⋊D712yes—
393#117C22×D712yes—
394#117C22×D712yes—
395#117C22×D712yes—
396#117C22×D712yes—
397C2×C4⋊C4—72no—
398C2×C4⋊C4—144no—
399C2×C4⋊C4C22⋊D712yes—
400C2×C4⋊C4C22×D712yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—28no—
404C2×C22⋊C4—28no—
405C2×C22⋊C4—141no—
406C2×C22⋊C4—144no—
407C2×C22⋊C4C22⋊D712yes—
408C2×C42—21no—
409C2×C42—141no—
410C2×C42—144no—
411C2×C42C22×D712yes—
412#107—78no—
413#106—78no—
414#103—78no—
415#102—716no—
416#102—78no—
417#97—78no—
418C4×C8—78no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—28no—
423C22×C7—44no—
424C22×C7—416no—
425C22×C7—48no—
426C22×C7—88no—
427C22×C7—88no—
428C22×C7C2×C4×D412yes—
429C22×C7C2×C4×D412yes—
430C22×C7C2×C4×D412yes—
431C22×C7C23×D411yescharacteristic
432C2×D7—24no—
433C2×D7—44no—
434C2×D7—84no—
435C4×C7—28no—
436C4×C7—44no—
437C4×C7C2×C4×D412yes—
438C4×C7C2×C4×D412yes—
439C4×C7C2×C4×D412yes—
440C4×C7C24×C411yescharacteristic
441C4×C7C22×#4111yescharacteristic
442Dic7—24no—
443Dic7—44no—
444C24—22no—
445C24—28no—
446C24—22no—
447C24—22no—
448C24—84no—
449C24—141no—
450C24C2×C22⋊D711yescharacteristic
451#41—78no—
452#41—78no—
453#41—148no—
454#41—148no—
455#41—144no—
456#41—144no—
457C2×Q8—74no—
458C2×Q8—72no—
459C2×Q8—142no—
460C2×Q8—141no—
461C2×D4—28no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—74no—
471C2×D4—72no—
472C2×D4—148no—
473C2×D4—142no—
474C2×D4—142no—
475C2×D4—142no—
476C2×D4—141no—
477C2×D4C2×C4×D718yes—
478C2×D4C2×C4×D718yes—
479C2×D4C2×C4×D718yes—
480C2×D4C23×D714yes—
481C22×C4—22no—
482C22×C4—216no—
483C22×C4—88no—
484C22×C4—141no—
485C22×C4—148no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—148no—
490C22×C4C2×C4×D714yes—
491C22×C4#115714yes—
492C22×C4C2×C22⋊D714yes—
493C22×C4C2×C22⋊D711yescharacteristic
494C22×C4C2×C22⋊D711yescharacteristic
495C22×C4C2×C22⋊D711yescharacteristic
496C22×C4C23×D711yescharacteristic
497C22×C4C23×D711yescharacteristic
498C22×C4C23×D711yescharacteristic
499SD16—732no—
500D8—732no—
501M16—78no—
502M16—78no—
503M16—144no—
504M16—144no—
505C2×C8—78no—
506C2×C8—74no—
507C2×C8—74no—
508C2×C8—144no—
509C4⋊C4—22no—
510C4⋊C4—74no—
511C4⋊C4—148no—
512C4⋊C4—142no—
513C4⋊C4C2×C22⋊D714yes—
514C4⋊C4C2×C22⋊D714yes—
515C4⋊C4C23×D714yes—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—216no—
519C22⋊C4—142no—
520C22⋊C4—142no—
521C22⋊C4—148no—
522C22⋊C4C2×C22⋊D714yes—
523C22⋊C4C2×C22⋊D714yes—
524C42—22no—
525C42—22no—
526C42—142no—
527C42—142no—
528C42—148no—
529C42C2×C22⋊D714yes—
530C42C23×D714yes—
531C2×C7—22no—
532C2×C7—22no—
533C2×C7—22no—
534C2×C7—48no—
535C2×C7C22×C4×D411yescharacteristic
536C2×C7#358512yes—
537D7—44no—
538C23—28no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—42no—
543C23—48no—
544C23—84no—
545C23—84no—
546C23—816no—
547C23—142no—
548C23—144no—
549C23—562no—
550C23C4×C22⋊D714yes—
551C23C22×C22⋊D711yescharacteristic
552C23C22×C22⋊D711yescharacteristic
553C23C22×C22⋊D711yescharacteristic
554Q8—74no—
555Q8—144no—
556Q8—142no—
557D4—28no—
558D4—28no—
559D4—28no—
560D4—24no—
561D4—74no—
562D4—148no—
563D4—148no—
564D4—144no—
565D4—142no—
566D4C22×C4×D718yes—
567C2×C4—24no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—816no—
571C2×C4—144no—
572C2×C4—142no—
573C2×C4—148no—
574C2×C4—148no—
575C2×C4—144no—
576C2×C4—568no—
577C2×C4—562no—
578C2×C4C4×C22⋊D714yes—
579C2×C4C4×C22⋊D714yes—
580C2×C4C4×C22⋊D714yes—
581C2×C4C22×C4×D712yes—
582C2×C4C22×C4×D712yes—
583C2×C4C22×C4×D712yes—
584C2×C4C2×#115712yes—
585C2×C4C2×#115712yes—
586C2×C4C2×#115712yes—
587C2×C4C22×C22⋊D718yes—
588C2×C4C22×C22⋊D711yescharacteristic
589C2×C4C22×C22⋊D711yescharacteristic
590C2×C4C22×C22⋊D711yescharacteristic
591C2×C4C24×D711yescharacteristic
592C8—78no—
593C8—144no—
594C7#6016811yescharacteristic, complemented, Sylow
595C22—22no—
596C22—22no—
597C22—22no—
598C22—28no—
599C22—44no—
600C22—416no—
601C22—48no—
602C22—88no—
603C22—88no—
604C22—144no—
605C22—284no—
606C22—564no—
607C22C2×C4×C22⋊D712yes—
608C22C2×C4×C22⋊D712yes—
609C22C2×C4×C22⋊D712yes—
610C22C23×C22⋊D711yescharacteristic
611C4—28no—
612C4—44no—
613C4—144no—
614C4—284no—
615C4C2×C4×C22⋊D712yes—
616C4C2×C4×C22⋊D712yes—
617C4C2×C4×C22⋊D712yes—
618C4C23×C4×D711yescharacteristic
619C4C22×#115711yescharacteristic
620C2—22no—
621C2—22no—
622C2—22no—
623C2—48no—
624C2—284no—
625C2C22×C4×C22⋊D711yescharacteristic
626C2#1174434512yes—
subgroupG/H
centerC22C23×C22⋊D7
commutatorC4×C7C24×C4
FrattiniC2×C4C24×D7
FittingC2×C7×#518C2
socleC22×C7C23×D4
prime radicalC22×#6495C2
cosocleC2×C4×C7C25
hypercenterC2×#518D7
hypocenterC7#60168
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
1. C2×C7×#518C2semidirect
3. #11742799C2semidirect
4. C2×#89987C2semidirect
5. #11739988C2semidirect
6. #11739983C2semidirect
7. #11739982C2semidirect
8. C4×#6495C2semidirect
9. #11739658C2semidirect
10. #11739652C2semidirect
11. #11739647C2semidirect
17. C7×#518C22semidirect
19. C2×C4×C7×D4C22semidirect
24. C2×#6495C4semidirect
31. #90214C22semidirect
33. #90212C22semidirect
35. #90207C22semidirect
38. C4×#1157C22semidirect
40. #89987C22semidirect
41. C2×#6406C22semidirect
42. #89790C22semidirect
43. C4×#1142C22semidirect
44. #89786C22semidirect
45. C4×#1141C22semidirect
46. #89776C22semidirect
47. #89774C22semidirect
48. #89773C22semidirect
49. #89770C22semidirect
50. #89769C22semidirect
51. #89769C22semidirect
52. #89767C22semidirect
53. #89738C22semidirect
54. #89736C22semidirect
55. C4×#1136C22semidirect
62. C22×C7×D4C2×C4semidirect
72. C2×#1157C2×C4semidirect
100. C2×#1142C2×C4semidirect
102. #6495C2×C4semidirect
103. #6495C2×C4semidirect
104. C2×#1141C2×C4semidirect
114. C2×#1136C2×C4semidirect
128. C4×D28C23semidirect
131. C4×Dic14C23semidirect
133. #6384C23semidirect
134. #6383C23semidirect
135. #6382C23semidirect
136. #6381C23semidirect
137. #6377C23semidirect
138. C4×C7⋊C8C23semidirect
141. C2×#518D7semidirect
192. #1157C22×C4semidirect
195. C2×D28C22×C4semidirect
203. C2×Dic14C22×C4semidirect
220. #1142C22×C4semidirect
221. #1141C22×C4semidirect
232. #1136C22×C4semidirect
234. C2×C7⋊C8C22×C4semidirect
254. #518C2×D7semidirect
258. C2×C4×D4C2×D7semidirect
351. C22×D4C4×D7semidirect
594. C7#60168semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5182
C22×#64952
#117427992
C2×#899872
#117399882
#117399832
#117399822
C4×#64952
#117396582
#117396522
#117396472
#601687