Finite Group Databasebeta

← #13355385

Subgroups — #13355385

Order 1792.

1694 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90516C211yesmaximal, characteristic
2C2×#90207C211yesmaximal, characteristic, complemented
3C7×#2896C211yesmaximal, characteristic
4#11741517C211yesmaximal, characteristic
5C22×#6398C211yesmaximal, characteristic, complemented
6#11729736C211yesmaximal, characteristic, complemented
7#11729328C211yesmaximal, characteristic, complemented
8#11728099C214yesmaximal, complemented
9#11728079C214yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14#90516—24no—
15#90516C418yescomplemented
16C2×#6507C2212yes—
17C2×#6499C2212yes—
18C2×#6498—22no—
19C2×#6489C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C2×#6487—22no—
22C22×#1136C2211yescharacteristic
23#90207—22no—
24#90207—22no—
25#90207—22no—
26#90207C2214yes—
27C2×#6450—22no—
28C2×#6440—22no—
29C2×C4×Dic14—22no—
30C7×#419C2212yes—
31C7×#418C2212yes—
32C7×#417—22no—
33C7×#416—22no—
34#90005—22no—
35#90004—22no—
36#89997C2212yes—
37#89987C2212yes—
38C2×#6398—24no—
39C2×#6398C2214yes—
40#89350C2214yes—
41#89327C2214yes—
42#20954—71nomaximal, Sylow
43C2×C7×#41—22no—
44C2×C7×#41—21no—
45C2×C7×#41—21no—
46C2×C7×#41—21no—
47C2×C7×#41—21no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41C2×C414yes—
51C22×C7×Q8C2311yescharacteristic
52C22×C7×D4—21no—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7C2311yescharacteristic
56C23×Dic7D412yes—
57C22×Dic14—21no—
58C22×Dic14—21no—
59C22×Dic14—21no—
60C22×Dic14—21no—
61C22×Dic14—22no—
62C22×Dic14—22no—
63C22×Dic14C2×C414yes—
64C22×Dic14D412yes—
65C22×Dic14D414yes—
66C22×Dic14C2312yes—
67C2×C7×M16—21no—
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C2×C7×M16C2312yes—
71C22×C7×C8—21no—
72C7×#117—21no—
73C7×#117—21no—
74C7×#117D412yes—
75C7×#117C2312yes—
76C2×C7×C4⋊C4D412yes—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—24no—
82#6507—24no—
83#6507C2×C418yescomplemented
84#6499—24no—
85#6499C2×C418yescomplemented
86#6498—216no—
87#6489—21no—
88#6489—21no—
89#6489—22no—
90#6489C2×C414yes—
91#6489D412yes—
92#6489C2312yes—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7D412yes—
95#6487—216no—
96C2×Dic7⋊C4—24no—
97C2×Dic7⋊C4—24no—
98C2×C4×Dic7—24no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7C2312yes—
101C2×#1136—21no—
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136C2312yes—
105C22×C7⋊C8—21no—
106#6450—28no—
107#6450—28no—
108#6440—28no—
109#6440—28no—
110C4×Dic14—28no—
111C4×Dic14—28no—
112C7×#103—24no—
113C7×#102—24no—
114#6409—24no—
115#6406—24no—
116#6398—28no—
117C2×#536—71no—
118C2×#519—71no—
119#2896—71no—
120#2896D711yescharacteristic
121C22×#101—71no—
122#1888—74no—
123#1886—74no—
124#1733—71no—
125#1733—71no—
126C24×C7—21no—
127C24×C7C2×D411yescharacteristic
128C7×#41—24no—
129C7×#41—24no—
130C7×#41—24no—
131C7×#41—24no—
132C7×#41—42no—
133C7×#41—42no—
134C7×#41—42no—
135C7×#41—42no—
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—41no—
149C2×C7×D4—41no—
150C2×C7×D4—42no—
151C2×C7×D4C22×C412yes—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—24no—
155C22×C4×C7—41no—
156C22×C4×C7—41no—
157C22×C4×C7—42no—
158C22×C4×C7C22×C412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7#4112yes—
166C22×C4×C7#4112yes—
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—24no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7C22⋊C418yes—
174C22×Dic7C2×D414yes—
175C2×Dic14—24no—
176C2×Dic14—24no—
177C2×Dic14—24no—
178C2×Dic14—24no—
179C2×Dic14—22no—
180C2×Dic14—22no—
181C2×Dic14—24no—
182C2×Dic14—24no—
183C2×Dic14—24no—
184C2×Dic14—24no—
185C2×Dic14—24no—
186C2×Dic14—22no—
187C2×Dic14—22no—
188C2×Dic14—22no—
189C2×Dic14—22no—
190C2×Dic14C22⋊C418yes—
191C2×Dic14C22×C414yes—
192C2×Dic14C2×D414yes—
193C7×M16—22no—
194C7×M16—22no—
195C7×M16—42no—
196C2×C7×C8—22no—
197C2×C7×C8—42no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4C2×D414yes—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C42×C7—22no—
203C42×C7—22no—
204C22⋊Dic7—24no—
205C22⋊Dic7—22no—
206C22⋊Dic7—22no—
207C22⋊Dic7—216no—
208C4⋊Dic7—24no—
209C4⋊Dic7—22no—
210C4⋊Dic7C2×D414yes—
211Dic7⋊C4—216no—
212Dic7⋊C4—216no—
213C4×Dic7—216no—
214C4×Dic7—22no—
215C4×Dic7—22no—
216C4×Dic7—22no—
217C4×Dic7C22×C414yes—
218#1136—22no—
219#1136—22no—
220#1136—42no—
221C2×C7⋊C8—22no—
222C2×C7⋊C8—42no—
223C22×#41C2×D711yescharacteristic
224C23×Q8—71no—
225C22×M16—71no—
226C22×M16C2×D711yescharacteristic
227#536—78no—
228#536—144no—
229C2×C4⋊Q8—72no—
230C2×#124—72no—
231C2×#123—142no—
232C2×C22⋊Q8—142no—
233#519—74no—
234#519—142no—
235#519—142no—
236#519—142no—
237C2×C4×Q8—142no—
238C2×C4×Q8—142no—
239C2×#117—71no—
240C2×#117—71no—
241C2×#117—142no—
242C2×#117C2×D711yescharacteristic
243#419—72no—
244#419C2×D712yes—
245#418—72no—
246#418C2×D712yes—
247#417—22no—
248#417—142no—
249#416—22no—
250#416—142no—
251C2×#101—74no—
252C2×#101—144no—
253#339—74no—
254#339—74no—
255C23×C7—22no—
256C23×C7—42no—
257C23×C7—44no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7C2×C22⋊C412yes—
261C23×C7#12011yescharacteristic
262C23×C7#12011yescharacteristic
263C23×C7#12312yes—
264C23×C7C22×D411yescharacteristic
265C7×Q8—22no—
266C7×Q8—22no—
267C7×Q8—41no—
268C7×Q8—41no—
269C7×D4—22no—
270C7×D4—22no—
271C7×D4—44no—
272C7×D4—44no—
273C7×D4—41no—
274C7×D4—41no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—44no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7C2×C22⋊C412yes—
283C2×C4×C7C4×D412yes—
284C2×C4×C7C4×D412yes—
285C2×C4×C7C4×D412yes—
286C2×C4×C7C4×D412yes—
287C2×C4×C7#12011yescharacteristic
288C2×C4×C7#12011yescharacteristic
289C2×C4×C7#12112yes—
290C2×C4×C7#12112yes—
291C2×C4×C7#12112yes—
292C2×C4×C7#12112yes—
293C2×C4×C7#12312yes—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—28no—
303C2×Dic7—28no—
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—44no—
307C2×Dic7—44no—
308C2×Dic7C2×C22⋊C418yes—
309Dic14—24no—
310Dic14—24no—
311Dic14—24no—
312Dic14—24no—
313Dic14—28no—
314Dic14—28no—
315C7×C8—42no—
316C7⋊C8—42no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41—22no—
322C2×#41—21no—
323C2×#41—21no—
324C2×#41C4×D714yes—
325C22×Q8—74no—
326C22×Q8—72no—
327C22×Q8—74no—
328C22×Q8—72no—
329C22×Q8—142no—
330C22×Q8—141no—
331C22×Q8—141no—
332C22×Q8—142no—
333C22×Q8—141no—
334C22×Q8—141no—
335C22×Q8C22×D711yescharacteristic
336C22×D4—21no—
337C22×D4C22×D711yescharacteristic
338C23×C4—21no—
339C23×C4—72no—
340C23×C4C22×D711yescharacteristic
341C2×M16—21no—
342C2×M16—22no—
343C2×M16—22no—
344C2×M16—72no—
345C2×M16—141no—
346C2×M16—142no—
347C2×M16—142no—
348C2×M16C22×D712yes—
349C22×C8—21no—
350C22×C8—141no—
351C4⋊Q8—78no—
352C4⋊Q8—144no—
353#124—78no—
354#124—144no—
355#123—1416no—
356C22⋊Q8—1416no—
357C4×Q8—148no—
358C4×Q8—148no—
359C4×Q8—148no—
360C4×Q8—148no—
361#117—21no—
362#117—21no—
363#117—74no—
364#117—72no—
365#117—72no—
366#117—148no—
367#117—148no—
368#117—141no—
369#117—142no—
370#117—141no—
371#117C22⋊D712yes—
372#117C22×D712yes—
373C2×C4⋊C4—72no—
374C2×C4⋊C4—144no—
375C2×C4⋊C4—144no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22⋊D712yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—144no—
380C2×C22⋊C4—141no—
381C2×C22⋊C4—141no—
382C2×C42—21no—
383C2×C42—72no—
384C2×C42—144no—
385C2×C42—141no—
386#103—24no—
387#103—144no—
388#102—24no—
389#102—144no—
390#101—148no—
391C22×C7—22no—
392C22×C7—22no—
393C22×C7—22no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—44no—
397C22×C7—42no—
398C22×C7—82no—
399C22×C7—82no—
400C22×C7#38612yes—
401C22×C7#38612yes—
402C22×C7C22×C22⋊C411yescharacteristic
403C22×C7C2×#12011yescharacteristic
404C22×C7C2×#12311yescharacteristic
405C4×C7—44no—
406C4×C7—42no—
407C4×C7#38612yes—
408C4×C7#38612yes—
409C4×C7C2×C4×D411yescharacteristic
410C4×C7C2×C4×D411yescharacteristic
411C4×C7C2×#12111yescharacteristic
412C4×C7C2×#12111yescharacteristic
413Dic7—28no—
414Dic7—44no—
415Dic7—44no—
416C24—21no—
417C24D4×D711yescharacteristic
418#41—24no—
419#41—24no—
420#41—24no—
421#41—24no—
422#41—42no—
423#41—42no—
424#41—42no—
425#41—42no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—78no—
432C2×Q8—74no—
433C2×Q8—74no—
434C2×Q8—144no—
435C2×Q8—144no—
436C2×Q8—144no—
437C2×Q8—144no—
438C2×Q8—144no—
439C2×Q8—144no—
440C2×Q8—144no—
441C2×Q8—142no—
442C2×Q8—142no—
443C2×Q8—144no—
444C2×Q8—144no—
445C2×Q8—142no—
446C2×Q8—142no—
447C2×Q8—142no—
448C2×Q8—142no—
449C2×Q8C2×C4×D712yes—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4C2×C4×D712yes—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—78no—
467C22×C4—74no—
468C22×C4—144no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4C2×C4×D712yes—
474C22×C4#115712yes—
475C22×C4D4×D711yescharacteristic
476C22×C4D4×D712yes—
477C22×C4D4×D711yescharacteristic
478C22×C4D4×D711yescharacteristic
479C22×C4#115912yes—
480C22×C4C2×C22⋊D711yescharacteristic
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C23×D711yescharacteristic
483M16—22no—
484M16—22no—
485M16—42no—
486M16—142no—
487M16—142no—
488M16—282no—
489C2×C8—22no—
490C2×C8—42no—
491C2×C8—142no—
492C2×C8—282no—
493C4⋊C4—22no—
494C4⋊C4—74no—
495C4⋊C4—1416no—
496C4⋊C4—1416no—
497C4⋊C4—144no—
498C4⋊C4—142no—
499C4⋊C4C2×C22⋊D714yes—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—1416no—
503C22⋊C4—144no—
504C22⋊C4—142no—
505C22⋊C4—142no—
506C42—22no—
507C42—22no—
508C42—74no—
509C42—1416no—
510C42—142no—
511C42—142no—
512C42—142no—
513C2×C7—21no—
514C2×C7—21no—
515C2×C7—22no—
516C2×C7—22no—
517C2×C7—82no—
518C2×C7C2×#38611yescharacteristic
519C2×C7#302012yes—
520C23—22no—
521C23—42no—
522C23—44no—
523C23—41no—
524C23—41no—
525C23D7×C22⋊C412yes—
526C23#644612yes—
527C23#650011yescharacteristic
528C23#650011yescharacteristic
529C23C2×D4×D711yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—144no—
535Q8—144no—
536Q8—144no—
537Q8—144no—
538Q8—148no—
539Q8—148no—
540D4—22no—
541D4—22no—
542D4—44no—
543D4—44no—
544D4—41no—
545D4—41no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—78no—
554C2×C4—144no—
555C2×C4—144no—
556C2×C4—148no—
557C2×C4—148no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—284no—
561C2×C4—284no—
562C2×C4D7×C22⋊C412yes—
563C2×C4(C22⋊D7)⋊C412yes—
564C2×C4(C22⋊D7)⋊C412yes—
565C2×C4#644612yes—
566C2×C4#644712yes—
567C2×C4#644712yes—
568C2×C4C4×C22⋊D712yes—
569C2×C4C4×C22⋊D712yes—
570C2×C4#650011yescharacteristic
571C2×C4#650011yescharacteristic
572C2×C4#650112yes—
573C2×C4#650112yes—
574C2×C4C22×C4×D711yescharacteristic
575C2×C4C2×#115711yescharacteristic
576C2×C4C2×D4×D711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4C2×#115911yescharacteristic
579C2×C4C22×C22⋊D711yescharacteristic
580C8—42no—
581C8—282no—
582C7#2095411yescharacteristic, complemented, Sylow
583C22—22no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22#8972812yes—
593C22#8972812yes—
594C22C2×D7×C22⋊C411yescharacteristic
595C22C2×#644611yescharacteristic
596C22C2×#650011yescharacteristic
597C4—44no—
598C4—42no—
599C4—148no—
600C4—284no—
601C4—284no—
602C4#8972812yes—
603C4#8972812yes—
604C4C2×(C22⋊D7)⋊C411yescharacteristic
605C4C2×#644711yescharacteristic
606C4C2×C4×C22⋊D711yescharacteristic
607C4C2×#650111yescharacteristic
608C2—21no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—82no—
613C2C2×#8972811yescharacteristic
614C2#1174145112yes—
subgroupG/H
centerC22C2×#6500
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2896C2
socleC22×C7C2×#120
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenter#2896D7
hypocenterC7#20954
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#90207C2semidirect
5. C22×#6398C2semidirect
6. #11729736C2semidirect
7. #11729328C2semidirect
8. #11728099C2semidirect
9. #11728079C2semidirect
15. #90516C4semidirect
83. #6507C2×C4semidirect
85. #6499C2×C4semidirect
582. C7#20954semidirect

Maximal subgroups

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

subgroupindex
C2×#905162
C2×#902072
C7×#28962
#117415172
C22×#63982
#117297362
#117293282
#117280992
#117280792
#209547