Finite Group Databasebeta

← #13357467

Subgroups — #13357467

Order 1792.

1638 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90523C211yesmaximal, characteristic
2C2×C7×#417C211yesmaximal, characteristic, complemented
3C2×#90004C211yesmaximal, characteristic, complemented
4C2×#89905C211yesmaximal, characteristic, complemented
5C2×M16×Dic7C211yesmaximal, characteristic, complemented
6C2×#89749C211yesmaximal, characteristic, complemented
7C2×#89738C211yesmaximal, characteristic, complemented
8#11727982C212yesmaximal, complemented
9#11727980C212yesmaximal, complemented
10#11727914C212yesmaximal, complemented
11#11727912C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16#90523—24no—
17#90523C418yescomplemented
18C2×#6510C2211yescharacteristic
19C2×#6507C2211yescharacteristic
20C2×#6503C2211yescharacteristic
21C2×#6502—22no—
22C2×#6499C2211yescharacteristic
23C2×C4×C22⋊D7—22no—
24C22×C4⋊Dic7C2211yescharacteristic
25C22×C4×Dic7C2211yescharacteristic
26C22×#1136C2211yescharacteristic
27C7×#417—24no—
28C2×C7×#103C2212yes—
29C2×C7×#102C2212yes—
30#90004—24no—
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33C2×C7⋊#95—22no—
34#89905—24no—
35M16×Dic7—24no—
36C2×#6395C2212yes—
37C2×#6390C2212yes—
38C2×#6389C2212yes—
39C2×C8×Dic7C2212yes—
40#89749—22no—
41#89749C2212yes—
42#89749C2212yes—
43C2×#6384C2212yes—
44#89738—24no—
45C2×#6383C2212yes—
46#21201—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41C2×C414yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C22×C22⋊D7—21no—
60C22×C22⋊D7—22no—
61C23×Dic7—21no—
62C23×Dic7D412yes—
63C2×#1157—22no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157—22no—
67C2×#1157—22no—
68C2×#1157C2×C414yes—
69C22×D28C2311yescharacteristic
70C22×C4×D7—21no—
71C22×Dic14C2311yescharacteristic
72C2×C7×M16—22no—
73C2×C7×M16—22no—
74C2×C7×M16D414yescomplemented
75C22×C7×C8C2312yes—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2312yes—
80C2×C22⋊Dic7—22no—
81#6510—22no—
82#6510C2×C414yescomplemented
83#6507—22no—
84#6507C2×C414yescomplemented
85#6503—22no—
86#6503C2×C414yescomplemented
87#6502—216no—
88#6499—22no—
89#6499C2×C414yescomplemented
90C4×C22⋊D7—216no—
91C2×(C2×D7)⋊C4—22no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2312yes—
96C2×Dic7⋊C4—22no—
97C2×C4×Dic7C2×C414yes—
98C2×C4×Dic7D412yes—
99C2×C4×Dic7D414yes—
100C2×C4×Dic7C2311yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136D414yescomplemented
105C22×C7⋊C8C2312yes—
106C7×#103—24no—
107C7×#102—24no—
108#6409—24no—
109#6406—24no—
110C7⋊#95—24no—
111C7⋊#95—24no—
112#6395—24no—
113#6390—24no—
114#6389—24no—
115C8×Dic7—24no—
116#6384—24no—
117#6383—24no—
118C2×#532—71no—
119C2×#417—71no—
120C2×#417—71no—
121C2×#417—71no—
122C2×#417D711yescharacteristic, complemented
123C2×C4×M16—71no—
124C2×#382—71no—
125#1961—72no—
126#1960—72no—
127#1958—72no—
128#1958—72no—
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C23×D7—21no—
132C7×#41—28no—
133C7×#41—28no—
134C7×#41—44no—
135C7×#41—44no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8C22×C412yes—
142C2×C7×D4—24no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—42no—
149C2×C7×D4—42no—
150C2×C7×D4C22×C412yes—
151C22×C4×C7—21no—
152C22×C4×C7—24no—
153C22×C4×C7—22no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—44no—
157C22×C4×C7—42no—
158C22×C4×C7C22×C412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7#4112yes—
164C22×C4×C7#4112yes—
165C22×C4×C7C2411yescharacteristic
166C2×C22⋊D7—24no—
167C2×C22⋊D7—28no—
168C2×C22⋊D7—44no—
169C2×C22⋊D7—44no—
170C2×C22⋊D7—42no—
171C2×C22⋊D7—42no—
172C22×Dic7—21no—
173C22×Dic7—22no—
174C22×Dic7—42no—
175C22×Dic7—42no—
176C22×Dic7—44no—
177C22×Dic7—42no—
178C22×Dic7C22⋊C418yes—
179C22×Dic7C2×D412yes—
180C22×Dic7C2×D412yes—
181#1157—28no—
182#1157—28no—
183#1157—44no—
184#1157—44no—
185C2×D28—21no—
186C2×D28—21no—
187C2×D28—21no—
188C2×D28—21no—
189C2×D28—21no—
190C2×D28C22×C412yes—
191C2×C4×D7—24no—
192C2×C4×D7—42no—
193C2×C4×D7—42no—
194C2×Dic14—21no—
195C2×Dic14—21no—
196C2×Dic14—21no—
197C2×Dic14—21no—
198C2×Dic14—21no—
199C2×Dic14C22×C412yes—
200C7×M16—28no—
201C2×C7×C8—22no—
202C2×C7×C8—22no—
203C2×C7×C8C2×D414yes—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—42no—
207C22⋊Dic7—28no—
208(C2×D7)⋊C4—28no—
209C4⋊Dic7—22no—
210C4⋊Dic7—22no—
211C4⋊Dic7—42no—
212Dic7⋊C4—28no—
213C4×Dic7C22⋊C418yes—
214C4×Dic7C22×C412yes—
215C4×Dic7C22×C412yes—
216C4×Dic7C2×D412yes—
217C4×Dic7C2×D412yes—
218#1136—28no—
219C2×C7⋊C8—22no—
220C2×C7⋊C8—22no—
221C2×C7⋊C8C2×D414yes—
222C22×#41—71no—
223C22×#41C2×D711yescharacteristic
224C22×M16—71no—
225C22×M16C2×D711yescharacteristic
226#532—78no—
227#532—144no—
228C2×C4⋊Q8—71no—
229C2×#127—71no—
230C2×#124—71no—
231C2×#124—71no—
232C2×#121—142no—
233C2×C4×D4—142no—
234C22×C4⋊C4—71no—
235C22×C4⋊C4C2×D711yescharacteristic
236C22×C42—71no—
237#417—24no—
238#417—144no—
239#417—144no—
240#417—144no—
241C2×#103—72no—
242C2×#103—72no—
243C2×#103—72no—
244C2×#103C2×D712yes—
245C2×#102—72no—
246C2×#102—72no—
247C2×#102—72no—
248C2×#102C2×D712yes—
249C4×M16—144no—
250C2×#97—72no—
251C2×C4×C8—72no—
252#382—72no—
253#382—72no—
254#382—142no—
255C2×#95—142no—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7C2×C22⋊C412yes—
264C23×C7#12412yes—
265C23×C7#12712yes—
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—42no—
271C22×D7—42no—
272C7×Q8—22no—
273C7×Q8—22no—
274C7×Q8—41no—
275C7×Q8—41no—
276C7×D4—22no—
277C7×D4—22no—
278C7×D4—48no—
279C7×D4—41no—
280C7×D4—41no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—48no—
285C2×C4×C7—42no—
286C2×C4×C7—44no—
287C2×C4×C7—84no—
288C2×C4×C7C2×C22⋊C412yes—
289C2×C4×C7C4×D414yes—
290C2×C4×C7C4×D414yes—
291C2×C4×C7#12114yes—
292C2×C4×C7#12114yes—
293C2×C4×C7#12412yes—
294C2×C4×C7#12712yes—
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C22⋊D7—416no—
302C22⋊D7—48no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—48no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—42no—
312C2×Dic7—44no—
313C2×Dic7—84no—
314C2×Dic7C2×C22⋊C414yes—
315C2×Dic7C2×C22⋊C414yes—
316D28—22no—
317D28—22no—
318D28—41no—
319D28—41no—
320C4×D7—48no—
321Dic14—22no—
322Dic14—22no—
323Dic14—41no—
324Dic14—41no—
325C7×C8—24no—
326C7⋊C8—24no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—74no—
333C2×#41—142no—
334C2×#41—142no—
335C2×#41—142no—
336C2×#41—142no—
337C2×#41—142no—
338C2×#41C4×D714yes—
339C22×Q8—71no—
340C22×Q8C22×D711yescharacteristic
341C22×D4—21no—
342C22×D4—71no—
343C22×D4—142no—
344C22×D4—141no—
345C22×D4C22×D711yescharacteristic
346C23×C4—21no—
347C23×C4—21no—
348C23×C4—72no—
349C23×C4—141no—
350C23×C4—141no—
351C23×C4C22×D711yescharacteristic
352C2×M16—22no—
353C2×M16—22no—
354C2×M16—74no—
355C2×M16—142no—
356C2×M16—142no—
357C2×M16C22⋊D714yescomplemented
358C22×C8—72no—
359C22×C8C22×D712yes—
360C4⋊Q8—74no—
361C4⋊Q8—142no—
362#127—74no—
363#127—142no—
364#124—74no—
365#124—74no—
366#124—142no—
367#124—142no—
368#121—1416no—
369C4×D4—1416no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—72no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—142no—
376C2×C4⋊C4—142no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4C22×D712yes—
379C2×C22⋊C4—142no—
380C2×C22⋊C4—142no—
381C2×C42—74no—
382C2×C42—72no—
383C2×C42—74no—
384C2×C42—71no—
385C2×C42—71no—
386#103—24no—
387#103—144no—
388#103—144no—
389#103—144no—
390#102—24no—
391#102—144no—
392#102—144no—
393#102—144no—
394#97—144no—
395C4×C8—144no—
396#95—144no—
397#95—144no—
398C22×C7—24no—
399C22×C7—24no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—44no—
403C22×C7—42no—
404C22×C7—82no—
405C22×C7—81no—
406C22×C7—81no—
407C22×C7#39014yes—
408C22×C7C22×C22⋊C411yescharacteristic
409C22×C7C2×#12411yescharacteristic
410C22×C7C2×#12711yescharacteristic
411C22×C7#57512yes—
412C22×C7#57512yes—
413C2×D7—44no—
414C2×D7—42no—
415C2×D7—82no—
416C2×D7—81no—
417C2×D7—81no—
418C4×C7—42no—
419C4×C7—82no—
420C4×C7#39014yes—
421C4×C7C2×C4×D411yescharacteristic
422C4×C7C2×C4×D411yescharacteristic
423C4×C7C2×#12111yescharacteristic
424C4×C7C2×#12111yescharacteristic
425Dic7—24no—
426Dic7—24no—
427Dic7—42no—
428Dic7—82no—
429C24—21no—
430C24—141no—
431C24D4×D711yescharacteristic
432#41—28no—
433#41—28no—
434#41—44no—
435#41—44no—
436#41—148no—
437#41—148no—
438#41—284no—
439#41—284no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—72no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—141no—
450C2×Q8—141no—
451C2×Q8C2×C4×D712yes—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—72no—
461C2×D4—148no—
462C2×D4—144no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—141no—
466C2×D4—141no—
467C2×D4—141no—
468C2×D4—284no—
469C2×D4—284no—
470C2×D4—282no—
471C2×D4—282no—
472C2×D4C2×C4×D712yes—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—72no—
481C22×C4—72no—
482C22×C4—78no—
483C22×C4—141no—
484C22×C4—144no—
485C22×C4—142no—
486C22×C4—282no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4—282no—
490C22×C4—284no—
491C22×C4—282no—
492C22×C4C2×C4×D712yes—
493C22×C4#115712yes—
494C22×C4D4×D711yescharacteristic
495C22×C4D4×D712yes—
496C22×C4D4×D712yes—
497C22×C4#115912yes—
498C22×C4C2×C22⋊D712yes—
499C22×C4C23×D711yescharacteristic
500M16—28no—
501M16—148no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—74no—
505C2×C8—142no—
506C2×C8—142no—
507C2×C8C2×C22⋊D714yes—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—148no—
512C4⋊C4—142no—
513C4⋊C4—142no—
514C4⋊C4—282no—
515C22⋊C4—148no—
516C22⋊C4—148no—
517C42—78no—
518C42—72no—
519C42—72no—
520C42—72no—
521C42—72no—
522C2×C7—24no—
523C2×C7—82no—
524C2×C7C2×#39011yescharacteristic
525C2×C7#289612yes—
526C2×C7#314212yes—
527C2×C7#315311yescharacteristic
528C2×C7#315311yescharacteristic
529D7—82no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—142no—
538C23—281no—
539C23—282no—
540C23—282no—
541C23—281no—
542C23D7×C22⋊C412yes—
543C23#644812yes—
544C23#650312yes—
545C23C2×D4×D711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—142no—
551Q8—142no—
552Q8—281no—
553Q8—281no—
554D4—22no—
555D4—22no—
556D4—48no—
557D4—41no—
558D4—41no—
559D4—142no—
560D4—142no—
561D4—2816no—
562D4—288no—
563D4—281no—
564D4—281no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—74no—
572C2×C4—74no—
573C2×C4—84no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—288no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—288no—
583C2×C4—284no—
584C2×C4—282no—
585C2×C4—564no—
586C2×C4D7×C22⋊C412yes—
587C2×C4(C22⋊D7)⋊C414yes—
588C2×C4#644714yes—
589C2×C4#644812yes—
590C2×C4C4×C22⋊D714yes—
591C2×C4#650214yes—
592C2×C4#650312yes—
593C2×C4C22×C4×D711yescharacteristic
594C2×C4C2×#115711yescharacteristic
595C2×C4C2×D4×D711yescharacteristic
596C2×C4C2×D4×D711yescharacteristic
597C2×C4C2×#115911yescharacteristic
598C2×C4C22×C22⋊D711yescharacteristic
599C8—24no—
600C8—144no—
601C7#2120111yescharacteristic, complemented, Sylow
602C22—24no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—42no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—284no—
612C22—282no—
613C22—562no—
614C22—561no—
615C22—561no—
616C22#8972914yes—
617C22C2×D7×C22⋊C411yescharacteristic
618C22C2×#644811yescharacteristic
619C22#9046412yes—
620C22#9046812yes—
621C22C2×#650311yescharacteristic
622C4—42no—
623C4—82no—
624C4—144no—
625C4—144no—
626C4—282no—
627C4—562no—
628C4#8972914yes—
629C4C2×(C22⋊D7)⋊C411yescharacteristic
630C4C2×#644711yescharacteristic
631C4C2×C4×C22⋊D711yescharacteristic
632C4C2×#650211yescharacteristic
633C2—24no—
634C2—82no—
635C2—562no—
636C2#1173632412yes—
637C2#1173632612yes—
638C2C2×#8972911yescharacteristic
639C2#1174138311yescharacteristic
640C2#1174141811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC2×#90523C2
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#21201
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×C7×#417C2semidirect
3. C2×#90004C2semidirect
4. C2×#89905C2semidirect
5. C2×M16×Dic7C2semidirect
6. C2×#89749C2semidirect
7. C2×#89738C2semidirect
8. #11727982C2semidirect
9. #11727980C2semidirect
10. #11727914C2semidirect
11. #11727912C2semidirect
17. #90523C4semidirect
74. C2×C7×M16D4semidirect
82. #6510C2×C4semidirect
84. #6507C2×C4semidirect
86. #6503C2×C4semidirect
89. #6499C2×C4semidirect
104. C2×#1136D4semidirect
122. C2×#417D7semidirect
357. C2×M16C22⋊D7semidirect
601. C7#21201semidirect

Maximal subgroups

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

subgroupindex
C2×#905232
C2×C7×#4172
C2×#900042
C2×#899052
C2×M16×Dic72
C2×#897492
C2×#897382
#117279822
#117279802
#117279142
#117279122
#212017