Finite Group Databasebeta

← #13371141

Subgroups — #13371141

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2880C211yesmaximal, characteristic
2C2×#90005C211yesmaximal, characteristic
3#11736029C211yesmaximal, characteristic, complemented
4#11729892C211yesmaximal, characteristic
5#11729771C212yesmaximal, complemented
6#11729763C212yesmaximal, complemented
7#11727752C212yesmaximal, complemented
8#11727750C212yesmaximal, complemented
9#11727736C211yesmaximal, characteristic, complemented
10#11727729C211yesmaximal, characteristic, complemented
11#11727702C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C2×C7×#131C2212yes—
14C2×#6489C2211yescharacteristic
15C23×C7⋊C8C2211yescharacteristic
16C7×#408—22no—
17C7×#408C2212yes—
18C7×#407—21no—
19C7×#407C2211yescharacteristic
20C2×C7×C22⋊C8—21no—
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×#6411C2212yes—
23C2×#6411C2212yes—
24#90005C411yescharacteristic, complemented
25#90005C411yescharacteristic, complemented
26#90005C411yescharacteristic, complemented
27#90005C411yescharacteristic, complemented
28#90005C2211yescharacteristic
29#90005C2211yescharacteristic
30#90005C2211yescharacteristic
31#90005C2211yescharacteristic
32C2×#6409C411yescharacteristic, complemented
33C2×#6409C411yescharacteristic, complemented
34C2×#6406C411yescharacteristic, complemented
35C2×#6406C411yescharacteristic, complemented
36C4×C22⋊Dic7C2211yescharacteristic
37#89979C2211yescharacteristic
38C2×#6396C2212yescomplemented
39#89874C2212yescomplemented
40#89873C2212yescomplemented
41C2×#6389C2212yescomplemented
42#89492—22no—
43#89491—21no—
44#89490—21no—
45#89353C2212yescomplemented
46#89350C2212yescomplemented
47#89350C2212yescomplemented
48#89345C2212yescomplemented
49#22586—71nomaximal, Sylow
50C2×C7×#41—21no—
51C2×C7×#41—21no—
52C2×C7×#41—24no—
53C2×C7×#41—42no—
54C2×C7×#41C2×C411yescharacteristic
55C2×C7×#41C2×C411yescharacteristic
56C2×C7×#41C2311yescharacteristic
57C2×C7×#41C2311yescharacteristic
58C22×C7×Q8C2×C411yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4C2×C411yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C7×#131—24no—
64C7×#131—42no—
65C2×C7×M16—22no—
66C2×C7×M16C2312yes—
67C22×C7×C8—22no—
68C22×C7×C8C2312yes—
69C2×C22⋊Dic7—21no—
70C2×C22⋊Dic7—21no—
71#6489—21no—
72#6489—21no—
73#6489C2×C411yescharacteristic
74#6489C2×C411yescharacteristic
75#6489C2311yescharacteristic
76#6489C2311yescharacteristic
77C2×C4⋊Dic7C2×C411yescharacteristic
78C2×C4⋊Dic7C2×C411yescharacteristic
79C2×C4×Dic7—21no—
80C2×C4×Dic7C2312yes—
81C2×#1136C2312yes—
82C22×C7⋊C8—42no—
83C22×C7⋊C8C2×C411yescharacteristic
84C22×C7⋊C8C2×C411yescharacteristic
85C22×C7⋊C8C2×C411yescharacteristic
86C22×C7⋊C8C2×C411yescharacteristic
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2311yescharacteristic
89C7×C22⋊C8—21no—
90C7×C22⋊C8—21no—
91C7×C22⋊C8—22no—
92C7×C22⋊C8—22no—
93C7×C22⋊C8D411yescharacteristic
94C7×C22⋊C8D411yescharacteristic
95C7×C22⋊C8D412yes—
96#6411—24no—
97#6411—42no—
98#6409C2×C411yescharacteristic
99#6409C2×C411yescharacteristic
100#6409C2×C411yescharacteristic
101#6409C2×C411yescharacteristic
102#6409C2×C411yescharacteristic
103#6409C2×C411yescharacteristic
104#6409C2×C411yescharacteristic
105#6409C2×C411yescharacteristic
106#6406C2×C411yescharacteristic
107#6406C2×C411yescharacteristic
108#6406C2×C411yescharacteristic
109#6406C2×C411yescharacteristic
110#6406C2×C411yescharacteristic
111#6406C2×C411yescharacteristic
112#6406C2×C411yescharacteristic
113#6406C2×C411yescharacteristic
114C7⋊#95—21no—
115C22⋊(C7⋊C8)—21no—
116C22⋊(C7⋊C8)—21no—
117#6396—42no—
118#6389—22no—
119#6389—42no—
120Dic7⋊C8—22no—
121Dic7⋊C8—22no—
122C8×Dic7—22no—
123#2924—71no—
124C2×#416—71no—
125#2880D711yescharacteristic
126#1938—72no—
127#1929—72no—
128#1866—72no—
129#1826—71no—
130#1810—71no—
131#1789—71no—
132#1765—72no—
133#1752—71no—
134C24×C7—21no—
135C24×C7#4111yescharacteristic
136C7×#41—22no—
137C7×#41—22no—
138C7×#41—22no—
139C7×#41—22no—
140C7×#41—22no—
141C7×#41—44no—
142C7×#41—41no—
143C7×#41—41no—
144C7×#41—44no—
145C7×#41#4114yes—
146C2×C7×Q8—21no—
147C2×C7×Q8—22no—
148C2×C7×Q8—41no—
149C2×C7×Q8C22×C411yescharacteristic
150C2×C7×Q8C22×C411yescharacteristic
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4—21no—
157C2×C7×D4—41no—
158C2×C7×D4—44no—
159C2×C7×D4C22×C411yescharacteristic
160C2×C7×D4C22×C411yescharacteristic
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—44no—
169C22×C4×C7C22×C411yescharacteristic
170C22×C4×C7C22×C411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7#4111yescharacteristic
174C22×C4×C7C2411yescharacteristic
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—22no—
178C7×M16—22no—
179C7×M16—22no—
180C7×M16—44no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—44no—
185C2×C7×C8C2×D412yes—
186C2×C7×C8C2×D412yes—
187C22⋊Dic7—21no—
188C22⋊Dic7—21no—
189C22⋊Dic7—21no—
190C22⋊Dic7—21no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7C22×C411yescharacteristic
195C4⋊Dic7C22×C411yescharacteristic
196C4⋊Dic7C22×C411yescharacteristic
197C4⋊Dic7C22×C411yescharacteristic
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—22no—
203C4×Dic7—22no—
204#1136—22no—
205#1136—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—41no—
210C2×C7⋊C8—41no—
211C2×C7⋊C8—41no—
212C2×C7⋊C8—41no—
213C2×C7⋊C8C22×C411yescharacteristic
214C2×C7⋊C8C22×C411yescharacteristic
215C2×C7⋊C8C22×C411yescharacteristic
216C2×C7⋊C8C22×C411yescharacteristic
217C22×#41C2×D711yescharacteristic
218C2×#131C2×D712yes—
219C23×C8—71no—
220C2×#117—71no—
221#433—142no—
222#432—141no—
223#431—141no—
224C2×#108—72no—
225#424—72no—
226C2×#104—72no—
227C2×#104—72no—
228#416—71no—
229#416—71no—
230#416—71no—
231#416—71no—
232#416—71no—
233#416—71no—
234#416—71no—
235#416—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#102—71no—
239C2×#102—71no—
240#408—22no—
241#408C2×D712yes—
242#407—21no—
243#407—71no—
244#407C2×D711yescharacteristic
245C2×C22⋊C8—21no—
246C2×C22⋊C8C2×D711yescharacteristic
247#405—72no—
248C2×#97—72no—
249C4×C22⋊C4—71no—
250#341—72no—
251#339—72no—
252#339—72no—
253#336—72no—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7—22no—
257C23×C7—21no—
258C23×C7—41no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7—81no—
262C23×C7C4×D411yescharacteristic
263C23×C7C4×D411yescharacteristic
264C23×C7C2×#4111yescharacteristic
265C7×Q8—21no—
266C7×Q8—21no—
267C7×Q8—21no—
268C7×Q8—42no—
269C7×Q8#11712yes—
270C7×D4—24no—
271C7×D4—21no—
272C7×D4—21no—
273C7×D4—21no—
274C7×D4—42no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—42no—
278C7×D4#11712yes—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—24no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7C4×D411yescharacteristic
294C2×C4×C7C4×D411yescharacteristic
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4112yes—
298C2×C4×C7H(5,𝔽2)12yes—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—81no—
308C7×C8—22no—
309C7×C8—22no—
310C7×C8—42no—
311C7⋊C8—21no—
312C7⋊C8—21no—
313C7⋊C8—21no—
314C7⋊C8—21no—
315C7⋊C8—42no—
316C2×#41—21no—
317C2×#41—21no—
318C2×#41—24no—
319C2×#41—42no—
320C2×#41C4×D711yescharacteristic
321C2×#41C4×D711yescharacteristic
322C2×#41C22×D711yescharacteristic
323C2×#41C22×D711yescharacteristic
324C22×Q8C4×D711yescharacteristic
325C22×D4—21no—
326C22×D4C4×D711yescharacteristic
327C23×C4—21no—
328C23×C4C22×D711yescharacteristic
329#131—24no—
330#131—42no—
331C2×M16—22no—
332C2×M16—72no—
333C2×M16C22×D712yes—
334C22×C8—22no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C22×C8—282no—
342C22×C8C22×D712yes—
343#117—71no—
344#117—71no—
345#117—71no—
346#117—71no—
347#117—141no—
348#117—141no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C42—72no—
354C2×C42—141no—
355#108—282no—
356C4⋊C8—142no—
357C4⋊C8—142no—
358#104—144no—
359#104—282no—
360#103—71no—
361#103—71no—
362#103—71no—
363#103—71no—
364#103—71no—
365#103—71no—
366#103—71no—
367#103—71no—
368#102—71no—
369#102—71no—
370#102—71no—
371#102—71no—
372#102—71no—
373#102—71no—
374#102—71no—
375#102—71no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380C22⋊C8—141no—
381C22⋊C8—141no—
382C22⋊C8C22⋊D711yescharacteristic
383C22⋊C8C22⋊D711yescharacteristic
384C22⋊C8C22⋊D712yes—
385#97—142no—
386#97—282no—
387C4×C8—142no—
388#95—141no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—22no—
395C22×C7—21no—
396C22×C7—21no—
397C22×C7—44no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—84no—
404C22×C7#44412yes—
405C22×C7C2×C4×D411yescharacteristic
406C22×C7#55112yes—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—22no—
410C4×C7—41no—
411C4×C7C2×#11711yescharacteristic
412C4×C7#51811yescharacteristic
413C4×C7#54612yes—
414Dic7—41no—
415Dic7—41no—
416Dic7—41no—
417Dic7—41no—
418Dic7—82no—
419C24—21no—
420C24#115711yescharacteristic
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—22no—
426#41—44no—
427#41—41no—
428#41—41no—
429#41—44no—
430#41#115914yes—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—41no—
434C2×Q8C2×C4×D711yescharacteristic
435C2×Q8C2×C4×D711yescharacteristic
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—44no—
444C2×D4C2×C4×D711yescharacteristic
445C2×D4C2×C4×D711yescharacteristic
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—44no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4C2×C4×D711yescharacteristic
458C22×C4C2×C4×D711yescharacteristic
459C22×C4#115711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463M16—22no—
464M16—22no—
465M16—44no—
466M16—142no—
467M16—142no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C2×C8—71no—
473C2×C8—71no—
474C2×C8—71no—
475C2×C8—71no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8—142no—
479C2×C8—281no—
480C2×C8—281no—
481C2×C8—281no—
482C2×C8—281no—
483C2×C8C2×C22⋊D712yes—
484C2×C8C2×C22⋊D712yes—
485C4⋊C4—71no—
486C4⋊C4—71no—
487C4⋊C4—71no—
488C4⋊C4—71no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C22⋊C4—141no—
492C22⋊C4—141no—
493C22⋊C4—141no—
494C22⋊C4—141no—
495C22⋊C4—282no—
496C42—141no—
497C42—141no—
498C42—141no—
499C42—141no—
500C42—142no—
501C42—142no—
502C2×C7—22no—
503C2×C7—22no—
504C2×C7—41no—
505C2×C7—42no—
506C2×C7—81no—
507C2×C7#234211yescharacteristic
508C2×C7C2×#44411yescharacteristic
509C2×C7#295711yescharacteristic
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—81no—
518C23C4×C22⋊D711yescharacteristic
519C23C4×C22⋊D711yescharacteristic
520C23C2×#115711yescharacteristic
521Q8—21no—
522Q8—21no—
523Q8—21no—
524Q8—42no—
525Q8#644012yes—
526D4—24no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4#644012yes—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—561no—
558C2×C4C4×C22⋊D711yescharacteristic
559C2×C4C4×C22⋊D711yescharacteristic
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C2×#115912yes—
562C2×C4#654812yes—
563C2×C4C22×C22⋊D711yescharacteristic
564C8—22no—
565C8—22no—
566C8—42no—
567C8—141no—
568C8—141no—
569C8—141no—
570C8—141no—
571C8—282no—
572C7#2258611yescharacteristic, complemented, Sylow
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—44no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—84no—
588C22#8991712yes—
589C22#9018612yes—
590C22C2×C4×C22⋊D711yescharacteristic
591C4—21no—
592C4—21no—
593C4—22no—
594C4—41no—
595C4—281no—
596C4—281no—
597C4—281no—
598C4—281no—
599C4—562no—
600C4C2×#644011yescharacteristic
601C4#9017711yescharacteristic
602C4#9049712yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—42no—
607C2—81no—
608C2#1173954311yescharacteristic
609C2C2×#8991711yescharacteristic
610C2#1174125711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2880C2
socleC22×C7C2×C4×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenter#2880D7
hypocenterC7#22586
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #2880

Complements

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

NH ≅ G/Ntype
3. #11736029C2semidirect
5. #11729771C2semidirect
6. #11729763C2semidirect
7. #11727752C2semidirect
8. #11727750C2semidirect
9. #11727736C2semidirect
10. #11727729C2semidirect
11. #11727702C2semidirect
24. #90005C4semidirect
25. #90005C4semidirect
26. #90005C4semidirect
27. #90005C4semidirect
32. C2×#6409C4semidirect
33. C2×#6409C4semidirect
34. C2×#6406C4semidirect
35. C2×#6406C4semidirect
38. C2×#6396C22semidirect
39. #89874C22semidirect
40. #89873C22semidirect
41. C2×#6389C22semidirect
45. #89353C22semidirect
46. #89350C22semidirect
47. #89350C22semidirect
48. #89345C22semidirect
572. C7#22586semidirect

Maximal subgroups

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

subgroupindex
C7×#28802
C2×#900052
#117360292
#117298922
#117297712
#117297632
#117277522
#117277502
#117277362
#117277292
#117277022
#225867