Finite Group Databasebeta

← #13357654

Subgroups — #13357654

Order 1792.

1566 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6513C211yesmaximal, characteristic
2C7×#2311C211yesmaximal, characteristic, complemented
3C2×#90004C211yesmaximal, characteristic, complemented
4#11739715C211yesmaximal, characteristic, complemented
5C2×#89769C211yesmaximal, characteristic, complemented
6C2×#89738C211yesmaximal, characteristic, complemented
7C22×#6382C211yesmaximal, characteristic, complemented
8#11727432C212yesmaximal, complemented
9#11727430C212yesmaximal, complemented
10#11727402C212yesmaximal, complemented
11#11727400C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6513—24no—
17C2×#6513—24no—
18C2×#6513C418yescomplemented
19C22×#1144C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C22×#1142C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26C7×#391C2212yes—
27C7×#390C2212yes—
28C7×#386—22no—
29C7×#380—22no—
30#90004—24no—
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33#89769—24no—
34#89759—22no—
35C2×#6384C2212yes—
36#89738—24no—
37C2×#6383C2212yes—
38C2×#6382—24no—
39C2×#6382C2212yes—
40C2×#6382C2212yes—
41#89730—22no—
42#89711C2212yes—
43#89701C2212yes—
44C2×#6377C2212yes—
45C2×C4×C7⋊C8C2212yes—
46#21229—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—
60C23×Dic7—21no—
61C2×#1157—22no—
62C2×#1157—22no—
63C2×#1157—22no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157C2×C414yes—
67C22×D28C2311yescharacteristic
68C22×C4×D7—21no—
69C22×Dic14C2311yescharacteristic
70C7×#117—22no—
71C7×#117—22no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4D412yes—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C22⋊C4—24no—
78C2×C7×C22⋊C4D412yes—
79C2×C42×C7C2312yes—
80#6513—232no—
81C2×#1144—22no—
82C2×#1144—22no—
83C2×#1144C2×C414yescomplemented
84C2×#1143—22no—
85C2×#1143—22no—
86C2×#1143C2×C414yescomplemented
87C2×#1142—22no—
88C2×#1142—22no—
89C2×#1142C2×C414yescomplemented
90C2×#1141—22no—
91C2×#1141—22no—
92C2×#1141C2×C414yescomplemented
93C2×(C2×D7)⋊C4—24no—
94C2×C4⋊Dic7—22no—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7C2312yes—
98C2×#1136—22no—
99C2×#1136—22no—
100C2×#1136D414yes—
101C22×C7⋊C8C2×C414yes—
102C22×C7⋊C8D414yes—
103C22×C7⋊C8D414yes—
104C22×C7⋊C8C2312yes—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#95—24no—
108#6409—24no—
109#6406—24no—
110C7⋊#95—24no—
111#6384—24no—
112#6383—24no—
113#6382—24no—
114#6382—24no—
115#6377—24no—
116C4×C7⋊C8—24no—
117C22×#135—71no—
118C22×#106—71no—
119C2×#417—71no—
120C2×#417—71no—
121C2×#405—71no—
122#2311—71no—
123#2311D711yescharacteristic, complemented
124#1972—72no—
125#1971—72no—
126#1968—72no—
127#1968—72no—
128C24×C7—21no—
129C24×C7C2×D411yescharacteristic
130C23×D7—21no—
131C7×#41—28no—
132C7×#41—28no—
133C7×#41—44no—
134C7×#41—44no—
135C2×C7×Q8—21no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8C22×C412yes—
141C2×C7×D4—24no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—42no—
148C2×C7×D4—42no—
149C2×C7×D4C22×C412yes—
150C22×C4×C7—21no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—82no—
157C22×C4×C7C22×C412yes—
158C22×C4×C7C2×D414yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7#4112yes—
164C22×C4×C7#4112yes—
165C22×C4×C7C2411yescharacteristic
166C2×C22⋊D7—24no—
167C2×C22⋊D7—42no—
168C2×C22⋊D7—42no—
169C22×Dic7—21no—
170C22×Dic7—22no—
171C22×Dic7—42no—
172C22×Dic7—82no—
173#1157—28no—
174#1157—28no—
175#1157—44no—
176#1157—44no—
177C2×D28—21no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28—21no—
182C2×D28C22×C412yes—
183C2×C4×D7—24no—
184C2×C4×D7—42no—
185C2×C4×D7—42no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—21no—
191C2×Dic14C22×C412yes—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—42no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—84no—
198C42×C7—22no—
199C42×C7—22no—
200#1144—28no—
201#1143—28no—
202#1142—28no—
203#1141—28no—
204(C2×D7)⋊C4—84no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7—42no—
208#1136—28no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8C22⋊C4116yescomplemented
212C2×C7⋊C8C22×C412yes—
213C2×C7⋊C8C22×C412yes—
214C2×C7⋊C8C2×D414yes—
215C2×C7⋊C8C2×D412yes—
216C2×C7⋊C8C2×D412yes—
217C2×C7⋊C8C2×D412yes—
218C2×C7⋊C8C2×D412yes—
219C22×#41—71no—
220C22×#41C2×D711yescharacteristic
221C2×#135—78no—
222C2×#135—144no—
223C2×#135—144no—
224C22×Q16—71no—
225C22×SD16—71no—
226C22×SD16—71no—
227C22×D8—71no—
228C22×M16—71no—
229C23×C8—71no—
230C2×#117C2×D711yescharacteristic
231C22×C4⋊C4—71no—
232C22×C4⋊C4C2×D711yescharacteristic
233C2×#106—72no—
234C2×#106—72no—
235C2×#106—144no—
236#417—144no—
237#417—144no—
238C2×#103—72no—
239C2×#103—72no—
240C2×#102—72no—
241C2×#102—72no—
242#405—144no—
243C2×#97—72no—
244C2×C4×C8—72no—
245#391—72no—
246#391C2×D712yes—
247#390—72no—
248#390C2×D712yes—
249#386—22no—
250#386—142no—
251#380—22no—
252#380—142no—
253C23×C7—22no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7C4×D412yes—
261C23×C7#12112yes—
262C23×C7#12712yes—
263C23×C7C22×D411yescharacteristic
264C22×D7—22no—
265C22×D7—41no—
266C22×D7—41no—
267C22×D7—42no—
268C22×D7—42no—
269C7×Q8—22no—
270C7×Q8—22no—
271C7×Q8—41no—
272C7×Q8—41no—
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—48no—
276C7×D4—41no—
277C7×D4—41no—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
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⋊C414yes—
289C2×C4×C7C4×D414yes—
290C2×C4×C7C4×D412yes—
291C2×C4×C7#12114yes—
292C2×C4×C7#12112yes—
293C2×C4×C7#12414yes—
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—48no—
302C2×Dic7—22no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—42no—
306C2×Dic7—44no—
307C2×Dic7—84no—
308D28—22no—
309D28—22no—
310D28—41no—
311D28—41no—
312C4×D7—48no—
313Dic14—22no—
314Dic14—22no—
315Dic14—41no—
316Dic14—41no—
317C7⋊C8—24no—
318C7⋊C8C2×C22⋊C414yes—
319C7⋊C8C2×C22⋊C414yes—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41—74no—
326C2×#41—142no—
327C2×#41—142no—
328C2×#41—142no—
329C2×#41—142no—
330C2×#41—142no—
331C2×#41C4×D714yes—
332C22×Q8—71no—
333C22×Q8C22×D711yescharacteristic
334C22×D4—21no—
335C22×D4—71no—
336C22×D4—141no—
337C22×D4C22×D711yescharacteristic
338C23×C4—21no—
339C23×C4—21no—
340C23×C4—141no—
341C23×C4—141no—
342C23×C4C22×D711yescharacteristic
343#135—1432no—
344C2×Q16—74no—
345C2×Q16—142no—
346C2×Q16—142no—
347C2×SD16—74no—
348C2×SD16—74no—
349C2×SD16—142no—
350C2×SD16—142no—
351C2×SD16—142no—
352C2×SD16—142no—
353C2×D8—74no—
354C2×D8—142no—
355C2×D8—142no—
356C2×M16—74no—
357C2×M16—142no—
358C2×M16—142no—
359C22×C8—74no—
360C22×C8—74no—
361C22×C8—74no—
362C22×C8—71no—
363C22×C8—71no—
364C22×C8—72no—
365#117—22no—
366#117—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—72no—
371C2×C4⋊C4—142no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4C22⋊D712yes—
375C2×C4⋊C4C22×D712yes—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—144no—
378C2×C22⋊C4C22⋊D712yes—
379C2×C42C22×D712yes—
380#106—144no—
381#106—144no—
382#103—144no—
383#103—144no—
384#102—144no—
385#102—144no—
386#97—144no—
387C4×C8—144no—
388#95—24no—
389#95—144no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—24no—
393C22×C7—24no—
394C22×C7—44no—
395C22×C7—42no—
396C22×C7—81no—
397C22×C7—81no—
398C22×C7—82no—
399C22×C7#39014yes—
400C22×C7C2×C4×D411yescharacteristic
401C22×C7C2×#12111yescharacteristic
402C22×C7C2×#12711yescharacteristic
403C22×C7#57712yes—
404C22×C7#57712yes—
405C2×D7—44no—
406C2×D7—42no—
407C2×D7—81no—
408C2×D7—81no—
409C2×D7—82no—
410C4×C7—42no—
411C4×C7—44no—
412C4×C7—82no—
413C4×C7#39014yes—
414C4×C7C22×C22⋊C411yescharacteristic
415C4×C7C2×C4×D411yescharacteristic
416C4×C7C2×#12111yescharacteristic
417C4×C7C2×#12411yescharacteristic
418Dic7—42no—
419Dic7—82no—
420C24—21no—
421C24—141no—
422C24C2×C22⋊D711yescharacteristic
423#41—28no—
424#41—28no—
425#41—44no—
426#41—44no—
427#41—148no—
428#41—148no—
429#41—284no—
430#41—284no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—72no—
437C2×Q8—141no—
438C2×Q8—141no—
439C2×Q8—141no—
440C2×Q8—141no—
441C2×Q8—141no—
442C2×Q8C2×C4×D712yes—
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—72no—
452C2×D4—144no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—141no—
458C2×D4—282no—
459C2×D4—282no—
460C2×D4C2×C4×D712yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4—141no—
469C22×C4—144no—
470C22×C4—142no—
471C22×C4—282no—
472C22×C4—282no—
473C22×C4—282no—
474C22×C4—562no—
475C22×C4C2×C4×D712yes—
476C22×C4#115712yes—
477C22×C4D4×D712yes—
478C22×C4D4×D712yes—
479C22×C4D4×D712yes—
480C22×C4#115912yes—
481C22×C4C2×C22⋊D714yes—
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C23×D711yescharacteristic
484Q16—148no—
485SD16—148no—
486SD16—148no—
487D8—148no—
488M16—148no—
489C2×C8—716no—
490C2×C8—72no—
491C2×C8—72no—
492C2×C8—72no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—72no—
496C2×C8—74no—
497C2×C8—142no—
498C2×C8—142no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—282no—
506C22⋊C4—24no—
507C22⋊C4—84no—
508C22⋊C4—564no—
509C42—22no—
510C42—22no—
511C2×C7—24no—
512C2×C7—82no—
513C2×C7C2×#39011yescharacteristic
514C2×C7#295212yes—
515C2×C7#306012yes—
516C2×C7#315411yescharacteristic
517C2×C7#315411yescharacteristic
518D7—82no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—142no—
527C23—281no—
528C23—281no—
529C23—282no—
530C23—282no—
531C23C4×C22⋊D712yes—
532C23#650212yes—
533C23#650312yes—
534C23C22×C22⋊D711yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—142no—
540Q8—142no—
541Q8—281no—
542Q8—281no—
543D4—22no—
544D4—22no—
545D4—48no—
546D4—41no—
547D4—41no—
548D4—142no—
549D4—142no—
550D4—288no—
551D4—281no—
552D4—281no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—84no—
563C2×C4—142no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—288no—
567C2×C4—282no—
568C2×C4—284no—
569C2×C4—564no—
570C2×C4D7×C22⋊C414yes—
571C2×C4(C22⋊D7)⋊C414yes—
572C2×C4#644714yes—
573C2×C4#644814yes—
574C2×C4C4×C22⋊D712yes—
575C2×C4#650212yes—
576C2×C4#650312yes—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C2×#115711yescharacteristic
579C2×C4C2×D4×D711yescharacteristic
580C2×C4C2×D4×D711yescharacteristic
581C2×C4C2×D4×D711yescharacteristic
582C2×C4C2×#115911yescharacteristic
583C8—74no—
584C8—74no—
585C8—144no—
586C7#2122911yescharacteristic, complemented, Sylow
587C22—24no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—44no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22—284no—
597C22—282no—
598C22—561no—
599C22—561no—
600C22—562no—
601C22#8972914yes—
602C22C2×C4×C22⋊D711yescharacteristic
603C22C2×#650211yescharacteristic
604C22C2×#650311yescharacteristic
605C22#9052712yes—
606C22#9052812yes—
607C4—42no—
608C4—44no—
609C4—82no—
610C4—282no—
611C4—562no—
612C4#8972914yes—
613C4C2×D7×C22⋊C411yescharacteristic
614C4C2×(C22⋊D7)⋊C411yescharacteristic
615C4C2×#644711yescharacteristic
616C4C2×#644811yescharacteristic
617C2—24no—
618C2—82no—
619C2—562no—
620C2C2×#8972911yescharacteristic
621C2#1174000312yes—
622C2#1174082611yescharacteristic
623C2#1174084711yescharacteristic
624C2#1174086512yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2311C2
socleC23×C7C22×D4
prime radicalC22×#6513C2
cosocleC22×C4×C7C24
hypercenter#2311D7
hypocenterC7#21229
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C7×#2311C2semidirect
3. C2×#90004C2semidirect
4. #11739715C2semidirect
5. C2×#89769C2semidirect
6. C2×#89738C2semidirect
7. C22×#6382C2semidirect
8. #11727432C2semidirect
9. #11727430C2semidirect
10. #11727402C2semidirect
11. #11727400C2semidirect
18. C2×#6513C4semidirect
83. C2×#1144C2×C4semidirect
86. C2×#1143C2×C4semidirect
89. C2×#1142C2×C4semidirect
92. C2×#1141C2×C4semidirect
123. #2311D7semidirect
211. C2×C7⋊C8C22⋊C4semidirect
586. C7#21229semidirect

Maximal subgroups

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

subgroupindex
C22×#65132
C7×#23112
C2×#900042
#117397152
C2×#897692
C2×#897382
C22×#63822
#117274322
#117274302
#117274022
#117274002
#212297