Finite Group Databasebeta

← #14238550

Subgroups — #14238550

Order 1792.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2#11742856C211yesmaximal, characteristic, complemented
3#11741525C214yesmaximal, complemented
4#11740975C214yesmaximal, complemented
5#11740969C214yesmaximal, complemented
6#11740956C214yesmaximal, complemented
7#11740955C214yesmaximal, complemented
8#11740667C211yesmaximal, characteristic, complemented
9#11740658C211yesmaximal, characteristic, complemented
10#11740182C214yesmaximal, complemented
11#11739759C211yesmaximal, characteristic, complemented
12#11739753C211yesmaximal, characteristic, complemented
13#11739746C211yesmaximal, characteristic, complemented
14C22×C7×#41C2212yes—
15C7×#547—22no—
16C7×#547—22no—
17C7×#547C2214yes—
18C7×#547C2214yes—
19C2×C7×C4⋊Q8C2211yescharacteristic
20C2×C7×#123—22no—
21C2×C7×C22⋊Q8C2212yes—
22C2×C7×#121—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C22×C7×C4⋊C4—21no—
26C2×#6513C2218yescomplemented
27#90496—21no—
28#90482C2214yes—
29#90254C2212yescomplemented
30#90250—22no—
31#90249—22no—
32#90244C2212yescomplemented
33#90242C2212yescomplemented
34#90233C2212yescomplemented
35#90195—21no—
36#90185—22no—
37#90168C2211yescharacteristic
38#90161C2211yescharacteristic
39#90011C2218yescomplemented
40#90010C2218yescomplemented
41#90005C2218yescomplemented
42#90000C2214yes—
43#89999C2214yes—
44#89990C2212yes—
45#89989C2212yes—
46C2×C22⋊(C7⋊C8)C2214yes—
47#89864C2212yescomplemented
48#89862C2214yescomplemented
49#89857C2212yescomplemented
50#89855C2214yescomplemented
51#89853C2212yescomplemented
52#89853C2212yescomplemented
53#89823C2218yescomplemented
54#89820C2218yescomplemented
55#89819C2218yescomplemented
56#89817C2218yescomplemented
57#89791C2212yescomplemented
58C4×#1142C2214yescomplemented
59#89787C2212yescomplemented
60C4×#1141C2214yescomplemented
61D4×C7⋊C8C2214yescomplemented
62D4×C7⋊C8C2214yescomplemented
63#89780C2212yescomplemented
64#89779C2212yescomplemented
65#89743C2212yes—
66#89739C2212yes—
67#89697C2211yescharacteristic
68#89697C2211yescharacteristic
69#61565—71nomaximal, Sylow
70C2×C7×#41—24no—
71C2×C7×#41—48no—
72C2×C7×#41C2318yes—
73C22×C7×Q8C2312yes—
74C22×C7×D4—22no—
75C22×C7×D4D411yescharacteristic
76C22×C7×D4C2311yescharacteristic
77C23×C4×C7—22no—
78C23×C4×C7—21no—
79C23×C4×C7C2312yes—
80C2×#1157—21no—
81C2×#1157C2314yes—
82C7×C4⋊Q8—21no—
83C7×C4⋊Q8—21no—
84C7×C4⋊Q8C2312yes—
85C7×C4⋊Q8C2312yes—
86C7×#123—28no—
87C7×#123—28no—
88C7×C22⋊Q8—24no—
89C7×C22⋊Q8C2318yes—
90C7×#121—24no—
91C7×#121—24no—
92C4×C7×D4—21no—
93C4×C7×D4—21no—
94C4×C7×D4D412yes—
95C4×C7×D4D412yes—
96C4×C7×D4C2312yes—
97C4×C7×D4C2312yes—
98C4×C7×D4C2312yes—
99C4×C7×D4C2312yes—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4—22no—
102C2×C7×C4⋊C4—21no—
103C2×C7×C4⋊C4—24no—
104C2×C7×C4⋊C4—24no—
105C2×C7×C4⋊C4D411yescharacteristic
106C2×C7×C4⋊C4C2311yescharacteristic
107C2×C7×C22⋊C4—22no—
108C2×C7×C22⋊C4—21no—
109C2×C7×C22⋊C4D412yes—
110C2×C42×C7C2311yescharacteristic
111#6516—24no—
112#6513—416no—
113C2×#1144C2318yes—
114C2×#1143C2318yes—
115#6503—22no—
116#6501—22no—
117#6501—22no—
118#6499—22no—
119C2×#1142C2314yes—
120C2×#1141C2314yes—
121#6493—21no—
122#6493C2312yes—
123#6493C2314yes—
124#6492—24no—
125C4×C22⋊D7—24no—
126C4×C22⋊D7—24no—
127#6489—21no—
128#6489C2314yes—
129#6487—21no—
130#6487C2312yes—
131#6487C2314yes—
132C22×C7⋊C8C2318yes—
133#6456—22no—
134#6455—22no—
135D7×C4⋊C4—22no—
136#6450—22no—
137#6448—24no—
138#6447—24no—
139#6446—24no—
140#6442—24no—
141#6438—21no—
142#6437C2311yescharacteristic
143C4×D28—21no—
144C4×D28C2312yes—
145#6433—21no—
146#6432C2311yescharacteristic
147#6432C2311yescharacteristic
148#6432C2311yescharacteristic
149C4×Dic14—21no—
150C4×Dic14C2312yes—
151#6409C2318yes—
152#6409C2318yes—
153#6406C2314yes—
154#6406C2314yes—
155C22⋊(C7⋊C8)C2312yes—
156C22⋊(C7⋊C8)C2312yes—
157C22⋊(C7⋊C8)C2318yes—
158#6384C2314yes—
159#6383C2314yes—
160#6382C2314yes—
161#6381C2314yes—
162C4⋊(C7⋊C8)C2312yes—
163C4⋊(C7⋊C8)C2312yes—
164C4×C7⋊C8C2312yes—
165C4×C7⋊C8C2312yes—
166#3506—71no—
167C2×#547D711yescharacteristic, complemented
168#3303—74no—
169#3302—74no—
170#3284—74no—
171#3283—74no—
172#3209—71no—
173#3206—71no—
174#3190—74no—
175#3101—71no—
176#3100—71no—
177#3002—74no—
178#2972—71no—
179C24×C7—21no—
180C24×C7C2×D412yes—
181C7×#41—216no—
182C7×#41—48no—
183C7×#41—416no—
184C2×C7×Q8—22no—
185C2×C7×Q8—44no—
186C2×C7×Q8C2414yes—
187C2×C7×D4—24no—
188C2×C7×D4—28no—
189C2×C7×D4—24no—
190C2×C7×D4—21no—
191C2×C7×D4—48no—
192C2×C7×D4C2×D412yes—
193C2×C7×D4C2×D412yes—
194C2×C7×D4C2412yes—
195C22×C4×C7—21no—
196C22×C4×C7—22no—
197C22×C4×C7—22no—
198C22×C4×C7—22no—
199C22×C4×C7—28no—
200C22×C4×C7—44no—
201C22×C4×C7—48no—
202C22×C4×C7—44no—
203C22×C4×C7C2×D411yescharacteristic
204C22×C4×C7C2×D412yes—
205C22×C4×C7C2×D414yes—
206C22×C4×C7C2×D411yescharacteristic
207C22×C4×C7C2411yescharacteristic
208C22×C4×C7C2414yes—
209C2×C22⋊D7—22no—
210C2×C22⋊D7—24no—
211C2×C22⋊D7—24no—
212#1157—48no—
213#1157—44no—
214C2×D28—21no—
215C2×D28—21no—
216C2×D28C2412yes—
217C2×C4×D7—22no—
218C2×C4×D7—24no—
219C2×Dic14—21no—
220C2×Dic14—21no—
221C2×Dic14C2412yes—
222C7×C4⋊C4—22no—
223C7×C4⋊C4—22no—
224C7×C4⋊C4—28no—
225C7×C4⋊C4—22no—
226C7×C4⋊C4—22no—
227C7×C4⋊C4—21no—
228C7×C4⋊C4C2×D412yes—
229C7×C4⋊C4C2×D412yes—
230C7×C4⋊C4C2412yes—
231C7×C22⋊C4—28no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4C2×D414yes—
235C7×C22⋊C4C2×D414yes—
236C42×C7C2×D411yescharacteristic
237C42×C7C2×D411yescharacteristic
238C42×C7C2411yescharacteristic
239C42×C7C2411yescharacteristic
240C22⋊Dic7—24no—
241C22⋊Dic7—24no—
242C22⋊Dic7—22no—
243#1144—48no—
244#1143—48no—
245#1142—28no—
246#1141—28no—
247(C2×D7)⋊C4—22no—
248(C2×D7)⋊C4—22no—
249(C2×D7)⋊C4—24no—
250C4⋊Dic7—21no—
251C4⋊Dic7—21no—
252C4⋊Dic7—21no—
253C4⋊Dic7—21no—
254C4⋊Dic7C2412yes—
255C4⋊Dic7C2412yes—
256Dic7⋊C4—22no—
257Dic7⋊C4—22no—
258Dic7⋊C4—24no—
259C4×Dic7—22no—
260C4×Dic7—24no—
261C2×C7⋊C8—48no—
262C2×C7⋊C8C2414yes—
263C2×C7⋊C8C2414yes—
264C22×#41C2×D712yes—
265C2×#135—78no—
266#559—142no—
267#556—72no—
268#555—72no—
269#553—142no—
270#550—72no—
271#547—22no—
272#547—22no—
273#547—72no—
274#547C2×D714yes—
275#547C2×D714yes—
276#538—142no—
277#535—141no—
278#533—71no—
279#532—71no—
280C2×C4⋊Q8C2×D711yescharacteristic
281#525—74no—
282C2×#123—22no—
283C2×C22⋊Q8C2×D712yes—
284C2×#121—21no—
285#520—141no—
286C2×C4×D4C2×D711yescharacteristic
287C2×C4×D4C2×D711yescharacteristic
288C22×C4⋊C4—21no—
289#494—72no—
290#492—72no—
291#487—72no—
292#487—72no—
293#485—78no—
294#483—72no—
295#480—72no—
296#476—72no—
297#474—72no—
298#467—78no—
299#465—78no—
300#462—74no—
301#461—74no—
302#461—72no—
303#459—72no—
304#452—78no—
305#452—78no—
306#451—74no—
307#450—72no—
308#449—78no—
309#448—74no—
310#447—72no—
311C4×SD16—74no—
312C4×D8—74no—
313C8×D4—74no—
314C8×D4—74no—
315#431—71no—
316#431—71no—
317#416—78no—
318C2×C22⋊C8—74no—
319C23×C7—24no—
320C23×C7—22no—
321C23×C7—24no—
322C23×C7—42no—
323C23×C7—42no—
324C23×C7—42no—
325C23×C7—42no—
326C23×C7C22×D414yes—
327C23×C7C22×D411yescharacteristic
328C22×D7—22no—
329C7×Q8—28no—
330C7×Q8—44no—
331C7×D4—24no—
332C7×D4—24no—
333C7×D4—24no—
334C7×D4—22no—
335C7×D4—416no—
336C7×D4H(5,𝔽2)14yes—
337C2×C4×C7—28no—
338C2×C4×C7—22no—
339C2×C4×C7—22no—
340C2×C4×C7—24no—
341C2×C4×C7—42no—
342C2×C4×C7—44no—
343C2×C4×C7—41no—
344C2×C4×C7—416no—
345C2×C4×C7—44no—
346C2×C4×C7—42no—
347C2×C4×C7—44no—
348C2×C4×C7—44no—
349C2×C4×C7#13514yescomplemented
350C2×C4×C7C22×D414yes—
351C2×C4×C7C22×D411yescharacteristic
352C2×C4×C7C22×D411yescharacteristic
353C2×C4×C7C22×D411yescharacteristic
354C2×C4×C7C22×D411yescharacteristic
355C2×C4×C7C22×D411yescharacteristic
356C2×C4×C7C2511yescharacteristic
357C22⋊D7—44no—
358C22⋊D7—84no—
359C22⋊D7—84no—
360C2×Dic7—22no—
361C2×Dic7—22no—
362C2×Dic7—22no—
363D28—24no—
364D28—42no—
365D28—81no—
366C4×D7—44no—
367C4×D7—84no—
368Dic14—24no—
369Dic14—42no—
370Dic14—81no—
371C7⋊C8—24no—
372C7⋊C8—24no—
373C2×#41—24no—
374C2×#41—48no—
375C2×#41—74no—
376C2×#41—141no—
377C2×#41C22×D718yes—
378C22×Q8C22×D712yes—
379C22×D4—22no—
380C22×D4C22⋊D711yescharacteristic
381C22×D4C22×D711yescharacteristic
382C23×C4—21no—
383C23×C4—22no—
384C23×C4C22×D712yes—
385#135—2816no—
386C2×Q16—78no—
387C2×SD16—74no—
388C2×SD16—78no—
389C2×D8—74no—
390C22×C8—78no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393C4⋊Q8—71no—
394C4⋊Q8—71no—
395C4⋊Q8—71no—
396C4⋊Q8C22×D712yes—
397C4⋊Q8C22×D712yes—
398#127—71no—
399#127—142no—
400#126—144no—
401#125—141no—
402#124—142no—
403#124—144no—
404#124—141no—
405#123—28no—
406#123—28no—
407#123—144no—
408#123—144no—
409C22⋊Q8—24no—
410C22⋊Q8—72no—
411C22⋊Q8—74no—
412C22⋊Q8—141no—
413C22⋊Q8C22×D718yes—
414#121—24no—
415#121—24no—
416#121—72no—
417#121—74no—
418#121—142no—
419#121—142no—
420#121—141no—
421#121—144no—
422#120—144no—
423C4×Q8—72no—
424C4×Q8—142no—
425C4×Q8—141no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—72no—
429C4×D4—144no—
430C4×D4—144no—
431C4×D4—142no—
432C4×D4—141no—
433C4×D4C22⋊D712yes—
434C4×D4C22⋊D712yes—
435C4×D4C22×D712yes—
436C4×D4C22×D712yes—
437C4×D4C22×D712yes—
438C4×D4C22×D712yes—
439#117—74no—
440#117—141no—
441#117—142no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—24no—
446C2×C4⋊C4—24no—
447C2×C4⋊C4—142no—
448C2×C4⋊C4C22⋊D711yescharacteristic
449C2×C4⋊C4C22×D711yescharacteristic
450C2×C22⋊C4—22no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4C22⋊D712yes—
453C2×C42C22×D711yescharacteristic
454#107—74no—
455#106—74no—
456C4⋊C8—72no—
457C4⋊C8—72no—
458#103—78no—
459#103—78no—
460#103—74no—
461#102—74no—
462#102—74no—
463#102—74no—
464C22⋊C8—72no—
465C22⋊C8—72no—
466C22⋊C8—78no—
467C4×C8—72no—
468C4×C8—72no—
469C22×C7—21no—
470C22×C7—21no—
471C22×C7—24no—
472C22×C7—24no—
473C22×C7—24no—
474C22×C7—24no—
475C22×C7—48no—
476C22×C7—42no—
477C22×C7—42no—
478C22×C7—84no—
479C22×C7C2×#13512yes—
480C22×C7C23×D411yescharacteristic
481C2×D7—42no—
482C2×D7—42no—
483C2×D7—42no—
484C4×C7—21no—
485C4×C7—21no—
486C4×C7—24no—
487C4×C7—21no—
488C4×C7—44no—
489C4×C7—42no—
490C4×C7C2×#13512yes—
491C4×C7C2×#13512yes—
492C4×C7C2×H(5,𝔽2)12yes—
493Dic7—42no—
494Dic7—42no—
495Dic7—42no—
496C24—21no—
497C24C2×C22⋊D712yes—
498#41—216no—
499#41—48no—
500#41—416no—
501#41—288no—
502#41—284no—
503C2×Q8—22no—
504C2×Q8—44no—
505C2×Q8—72no—
506C2×Q8—141no—
507C2×Q8—141no—
508C2×Q8C23×D714yes—
509C2×D4—24no—
510C2×D4—28no—
511C2×D4—24no—
512C2×D4—21no—
513C2×D4—48no—
514C2×D4—72no—
515C2×D4—142no—
516C2×D4—144no—
517C2×D4—144no—
518C2×D4—141no—
519C2×D4—141no—
520C2×D4C2×C22⋊D712yes—
521C2×D4C2×C22⋊D712yes—
522C2×D4C23×D712yes—
523C22×C4—21no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—28no—
528C22×C4—44no—
529C22×C4—48no—
530C22×C4—44no—
531C22×C4—142no—
532C22×C4—144no—
533C22×C4C2×C22⋊D711yescharacteristic
534C22×C4C2×C22⋊D712yes—
535C22×C4C2×C22⋊D714yes—
536C22×C4C2×C22⋊D711yescharacteristic
537C22×C4C23×D711yescharacteristic
538C22×C4C23×D714yes—
539Q16—288no—
540SD16—148no—
541SD16—288no—
542D8—148no—
543C2×C8—74no—
544C2×C8—74no—
545C2×C8—288no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—28no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—72no—
553C4⋊C4—72no—
554C4⋊C4—144no—
555C4⋊C4—142no—
556C4⋊C4—142no—
557C4⋊C4—141no—
558C4⋊C4—141no—
559C4⋊C4—141no—
560C4⋊C4—141no—
561C4⋊C4C2×C22⋊D712yes—
562C4⋊C4C2×C22⋊D712yes—
563C4⋊C4C23×D712yes—
564C22⋊C4—28no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—144no—
568C22⋊C4—142no—
569C22⋊C4—144no—
570C22⋊C4—144no—
571C22⋊C4—142no—
572C22⋊C4—142no—
573C22⋊C4C2×C22⋊D714yes—
574C22⋊C4C2×C22⋊D714yes—
575C42—144no—
576C42—142no—
577C42C2×C22⋊D711yescharacteristic
578C42C2×C22⋊D711yescharacteristic
579C42C23×D711yescharacteristic
580C42C23×D711yescharacteristic
581C2×C7—22no—
582C2×C7—44no—
583C2×C7—44no—
584C2×C7#348411yescharacteristic
585C2×C7C22×#13511yescharacteristic
586C2×C7#358611yescharacteristic
587D7—82no—
588C23—24no—
589C23—22no—
590C23—24no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—142no—
596C23C22×C22⋊D714yes—
597C23C22×C22⋊D711yescharacteristic
598Q8—28no—
599Q8—44no—
600Q8—144no—
601Q8—282no—
602Q8—561no—
603D4—24no—
604D4—24no—
605D4—24no—
606D4—22no—
607D4—416no—
608D4—144no—
609D4—284no—
610D4—282no—
611D4—561no—
612D4—564no—
613D4—564no—
614D4#654814yes—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—416no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—142no—
630C2×C4—284no—
631C2×C4—564no—
632C2×C4#651314yescomplemented
633C2×C4C22×C22⋊D714yes—
634C2×C4C22×C22⋊D711yescharacteristic
635C2×C4C22×C22⋊D711yescharacteristic
636C2×C4C22×C22⋊D711yescharacteristic
637C2×C4C22×C22⋊D711yescharacteristic
638C2×C4C22×C22⋊D711yescharacteristic
639C2×C4C24×D711yescharacteristic
640C8—144no—
641C8—144no—
642C7#6156511yescharacteristic, complemented, Sylow
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—48no—
650C22—42no—
651C22—42no—
652C22—84no—
653C22—282no—
654C22—282no—
655C22—282no—
656C22C2×#651312yes—
657C22C23×C22⋊D711yescharacteristic
658C4—24no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—44no—
663C4—42no—
664C4—282no—
665C4—282no—
666C4—282no—
667C4C2×#651312yes—
668C4C2×#651312yes—
669C4C2×#654812yes—
670C2—22no—
671C2—44no—
672C2—44no—
673C2—562no—
674C2#1174430711yescharacteristic
675C2C22×#651311yescharacteristic
676C2#1174435511yescharacteristic
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#547C2
socleC22×C7C23×D4
hypercenterC2×#547D7
hypocenterC7#61565
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. #11742856C2semidirect
3. #11741525C2semidirect
4. #11740975C2semidirect
5. #11740969C2semidirect
6. #11740956C2semidirect
7. #11740955C2semidirect
8. #11740667C2semidirect
9. #11740658C2semidirect
10. #11740182C2semidirect
11. #11739759C2semidirect
12. #11739753C2semidirect
13. #11739746C2semidirect
26. C2×#6513C22semidirect
29. #90254C22semidirect
32. #90244C22semidirect
33. #90242C22semidirect
34. #90233C22semidirect
39. #90011C22semidirect
40. #90010C22semidirect
41. #90005C22semidirect
47. #89864C22semidirect
48. #89862C22semidirect
49. #89857C22semidirect
50. #89855C22semidirect
51. #89853C22semidirect
52. #89853C22semidirect
53. #89823C22semidirect
54. #89820C22semidirect
55. #89819C22semidirect
56. #89817C22semidirect
57. #89791C22semidirect
58. C4×#1142C22semidirect
59. #89787C22semidirect
60. C4×#1141C22semidirect
61. D4×C7⋊C8C22semidirect
62. D4×C7⋊C8C22semidirect
63. #89780C22semidirect
64. #89779C22semidirect
167. C2×#547D7semidirect
349. C2×C4×C7#135semidirect
632. C2×C4#6513semidirect
642. C7#61565semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
#117428562
#117415252
#117409752
#117409692
#117409562
#117409552
#117406672
#117406582
#117401822
#117397592
#117397532
#117397462
#615657