Finite Group Databasebeta

← #13357659

Subgroups — #13357659

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
2C2×C7×#417C211yesmaximal, characteristic, complemented
3#11741532C211yesmaximal, characteristic, complemented
4C2×#89905C211yesmaximal, characteristic, complemented
5C2×#89894C211yesmaximal, characteristic, complemented
6#11739631C211yesmaximal, characteristic, complemented
7C22×#6381C211yesmaximal, characteristic, complemented
8#11728014C212yesmaximal, complemented
9#11728012C212yesmaximal, complemented
10#11727946C212yesmaximal, complemented
11#11727944C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6513—24no—
17C2×#6513—24no—
18C2×#6513C418yescomplemented
19C22×#1144C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C22×#1142C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C2×#6489C2211yescharacteristic
24C22×C4⋊Dic7C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26C7×#417—24no—
27C2×C7×#103C2212yes—
28C2×C7×#102C2212yes—
29#90002C2212yes—
30#89993C2212yes—
31#89992—22no—
32#89985—22no—
33#89984—22no—
34#89905—24no—
35#89894—24no—
36C2×#6395C2212yes—
37C2×#6390C2212yes—
38C2×#6389C2212yes—
39C2×C8×Dic7C2212yes—
40#89760C2212yes—
41#89759—22no—
42#89746C2212yes—
43C2×#6381—24no—
44C2×#6381C2212yes—
45C2×#6381C2212yes—
46#21230—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
70C2×C7×M16—22no—
71C2×C7×M16—22no—
72C2×C7×M16D414yes—
73C22×C7×C8C2312yes—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4C2312yes—
78C2×C22⋊Dic7—24no—
79C2×C22⋊Dic7D412yes—
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—
94#6489—22no—
95#6489—22no—
96C2×C4⋊Dic7—22no—
97C2×C4⋊Dic7—22no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7D412yes—
100C2×C4⋊Dic7C2312yes—
101C2×C4×Dic7C2312yes—
102C22×C7⋊C8C2×C414yes—
103C22×C7⋊C8D414yes—
104C22×C7⋊C8D414yes—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#103—24no—
108C7×#102—24no—
109C7⋊#95—24no—
110C7⋊#95—24no—
111#6395—24no—
112#6390—24no—
113#6389—24no—
114C8×Dic7—24no—
115#6381—24no—
116#6381—24no—
117C22×#135—71no—
118C22×#107—71no—
119C2×#417—71no—
120C2×#417D711yescharacteristic, complemented
121C2×#405—71no—
122#2311—71no—
123#2311—71no—
124#1973—72no—
125#1973—72no—
126#1970—72no—
127#1969—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×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—24no—
166C2×C22⋊D7—42no—
167C2×C22⋊D7—42no—
168C22×Dic7—21no—
169C22×Dic7—22no—
170C22×Dic7—42no—
171C22×Dic7—82no—
172C22×Dic7C2×D414yes—
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×M16—28no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8C2×D414yes—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—42no—
199C22⋊Dic7—24no—
200C22⋊Dic7—84no—
201#1144—28no—
202#1143—28no—
203#1142—28no—
204#1141—28no—
205(C2×D7)⋊C4—84no—
206C4⋊Dic7—24no—
207C4⋊Dic7—22no—
208C4⋊Dic7—22no—
209C4⋊Dic7—42no—
210C4×Dic7—22no—
211C4×Dic7—22no—
212C2×C7⋊C8C22⋊C4116yescomplemented
213C2×C7⋊C8C22×C412yes—
214C2×C7⋊C8C22×C412yes—
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×M16C2×D711yescharacteristic
229C23×C8—71no—
230C2×#117—71no—
231C22×C4⋊C4—71no—
232C22×C4⋊C4C2×D711yescharacteristic
233C2×#107—72no—
234C2×#107—72no—
235C2×#107—144no—
236#417—24no—
237#417—144no—
238C2×#103—72no—
239C2×#103C2×D712yes—
240C2×#102—72no—
241C2×#102C2×D712yes—
242#405—144no—
243C2×#97—72no—
244C2×C4×C8—72no—
245#391—72no—
246#391—72no—
247#390—72no—
248#390—72no—
249#386—142no—
250#386—142no—
251#380—142no—
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—41no—
280C2×C4×C7—41no—
281C2×C4×C7—48no—
282C2×C4×C7—42no—
283C2×C4×C7—44no—
284C2×C4×C7—84no—
285C2×C4×C7C2×C22⋊C414yes—
286C2×C4×C7C4×D412yes—
287C2×C4×C7C4×D414yes—
288C2×C4×C7#12112yes—
289C2×C4×C7#12114yes—
290C2×C4×C7#12414yes—
291C2×C4×C7#12712yes—
292C2×C4×C7C23×C411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C22⋊D7—48no—
299C2×Dic7—22no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—24no—
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—22no—
357C2×M16—22no—
358C2×M16C22⋊D714yes—
359C22×C8—74no—
360C22×C8—74no—
361C22×C8—74no—
362C22×C8—71no—
363C22×C8—71no—
364C22×C8C22×D712yes—
365#117—142no—
366#117—142no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—72no—
371C2×C4⋊C4—72no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4C22×D712yes—
376C2×C22⋊C4—72no—
377C2×C22⋊C4—144no—
378C2×C22⋊C4—144no—
379C2×C42—72no—
380#107—144no—
381#107—144no—
382#103—24no—
383#103—144no—
384#102—24no—
385#102—144no—
386#97—144no—
387C4×C8—144no—
388#95—144no—
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#57612yes—
404C22×C7#57812yes—
405C2×D7—44no—
406C2×D7—42no—
407C2×D7—81no—
408C2×D7—81no—
409C2×D7—82no—
410C4×C7—42no—
411C4×C7—82no—
412C4×C7#39014yes—
413C4×C7C22×C22⋊C411yescharacteristic
414C4×C7C2×C4×D411yescharacteristic
415C4×C7C2×#12111yescharacteristic
416C4×C7C2×#12411yescharacteristic
417Dic7—42no—
418Dic7—44no—
419Dic7—82no—
420C24—21no—
421C24—141no—
422C24D4×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—74no—
468C22×C4—82no—
469C22×C4—141no—
470C22×C4—144no—
471C22×C4—142no—
472C22×C4—282no—
473C22×C4—282no—
474C22×C4—282no—
475C22×C4—562no—
476C22×C4C2×C4×D712yes—
477C22×C4#115712yes—
478C22×C4D4×D712yes—
479C22×C4D4×D712yes—
480C22×C4D4×D711yescharacteristic
481C22×C4#115912yes—
482C22×C4C2×C22⋊D712yes—
483C22×C4C23×D711yescharacteristic
484Q16—148no—
485SD16—148no—
486SD16—148no—
487D8—148no—
488M16—28no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—716no—
492C2×C8—72no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—72no—
496C2×C8—72no—
497C2×C8—72no—
498C2×C8C2×C22⋊D714yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—144no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—282no—
506C22⋊C4—144no—
507C22⋊C4—564no—
508C22⋊C4—564no—
509C42—142no—
510C42—142no—
511C2×C7—24no—
512C2×C7—82no—
513C2×C7C2×#39011yescharacteristic
514C2×C7#295112yes—
515C2×C7#306112yes—
516C2×C7#315511yescharacteristic
517C2×C7#315611yescharacteristic
518D7—82no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—142no—
527C23—281no—
528C23—282no—
529C23—282no—
530C23—281no—
531C23(C22⋊D7)⋊C412yes—
532C23#644712yes—
533C23#650312yes—
534C23C2×D4×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—41no—
555C2×C4—41no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—84no—
560C2×C4—142no—
561C2×C4—144no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—288no—
567C2×C4—284no—
568C2×C4—282no—
569C2×C4—564no—
570C2×C4D7×C22⋊C414yes—
571C2×C4(C22⋊D7)⋊C412yes—
572C2×C4#644712yes—
573C2×C4#644814yes—
574C2×C4C4×C22⋊D714yes—
575C2×C4#650214yes—
576C2×C4#650312yes—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C2×#115711yescharacteristic
579C2×C4C2×D4×D711yescharacteristic
580C2×C4C2×D4×D711yescharacteristic
581C2×C4C2×#115911yescharacteristic
582C2×C4C22×C22⋊D711yescharacteristic
583C8—24no—
584C8—74no—
585C8—74no—
586C7#2123011yescharacteristic, complemented, Sylow
587C22—24no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—42no—
592C22—44no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—282no—
597C22—284no—
598C22—562no—
599C22—561no—
600C22—561no—
601C22#8972914yes—
602C22C2×(C22⋊D7)⋊C411yescharacteristic
603C22C2×#644711yescharacteristic
604C22#9046512yes—
605C22#9047012yes—
606C22C2×#650311yescharacteristic
607C4—42no—
608C4—82no—
609C4—284no—
610C4—282no—
611C4—562no—
612C4#8972914yes—
613C4C2×D7×C22⋊C411yescharacteristic
614C4C2×#644811yescharacteristic
615C4C2×C4×C22⋊D711yescharacteristic
616C4C2×#650211yescharacteristic
617C2—24no—
618C2—82no—
619C2—562no—
620C2#1173631312yes—
621C2#1173633612yes—
622C2C2×#8972911yescharacteristic
623C2#1174138511yescharacteristic
624C2#1174142011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×#6513C2
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#21230
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. #11741532C2semidirect
4. C2×#89905C2semidirect
5. C2×#89894C2semidirect
6. #11739631C2semidirect
7. C22×#6381C2semidirect
8. #11728014C2semidirect
9. #11728012C2semidirect
10. #11727946C2semidirect
11. #11727944C2semidirect
18. C2×#6513C4semidirect
83. C2×#1144C2×C4semidirect
86. C2×#1143C2×C4semidirect
89. C2×#1142C2×C4semidirect
92. C2×#1141C2×C4semidirect
120. C2×#417D7semidirect
212. C2×C7⋊C8C22⋊C4semidirect
586. C7#21230semidirect

Maximal subgroups

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

subgroupindex
C22×#65132
C2×C7×#4172
#117415322
C2×#899052
C2×#898942
#117396312
C22×#63812
#117280142
#117280122
#117279462
#117279442
#212307