Finite Group Databasebeta

← #13371142

Subgroups — #13371142

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
1C2×#89885C211yesmaximal, characteristic
2#11736076C211yesmaximal, characteristic
3#11736066C211yesmaximal, characteristic, complemented
4C7×#1810C211yesmaximal, characteristic, complemented
5#11729721C211yesmaximal, characteristic, complemented
6#11728476C212yesmaximal, complemented
7#11728453C212yesmaximal, complemented
8#11727745C211yesmaximal, characteristic
9#11727729C211yesmaximal, characteristic, complemented
10#11726681C212yesmaximal, complemented
11#11726673C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C2×#6489C2211yescharacteristic
15C2×#6470C2212yes—
16C2×C7×#108C2212yescomplemented
17C7×#407C2211yescharacteristic
18C7×#407C2211yescharacteristic
19C4×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2211yescharacteristic
21C2×#6399C2212yes—
22C2×#6399C2212yes—
23#89901C2212yes—
24#89894C2212yescomplemented
25#89891C2212yescomplemented
26#89885C411yescharacteristic, complemented
27#89885C411yescharacteristic, complemented
28#89885C411yescharacteristic, complemented
29#89885C411yescharacteristic, complemented
30#89885C2211yescharacteristic
31#89885C2211yescharacteristic
32#89885C2211yescharacteristic
33#89885C2211yescharacteristic
34C2×#6395C411yescharacteristic, complemented
35C2×#6395C411yescharacteristic, complemented
36C2×#6390C411yescharacteristic, complemented
37C2×#6390C411yescharacteristic, complemented
38C2×#6389C2212yescomplemented
39#89499—22no—
40#89498—21no—
41D7×C22⋊C8—21no—
42#89492—22no—
43#89491—21no—
44#89490—21no—
45C7×#341C2212yescomplemented
46#89350C2212yescomplemented
47#89350C2212yescomplemented
48#89345C2212yescomplemented
49#22586—71nomaximal, Sylow
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—21no—
52C2×#1157—21no—
53C2×#1157—21no—
54C2×#1157—24no—
55C2×#1157—42no—
56C2×#1157C2×C411yescharacteristic
57C2×#1157C2×C411yescharacteristic
58C2×#1157C2311yescharacteristic
59C2×#1157C2311yescharacteristic
60C22×D28C2×C411yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2×C411yescharacteristic
63C2×C7×M16C2312yes—
64C2×C7×M16C2312yes—
65C22×C7×C8—42no—
66C22×C7×C8C2×C411yescharacteristic
67C22×C7×C8C2×C411yescharacteristic
68C22×C7×C8C2×C411yescharacteristic
69C22×C7×C8C2×C411yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74#6489—21no—
75#6489—21no—
76#6489C2×C411yescharacteristic
77#6489C2×C411yescharacteristic
78#6489C2311yescharacteristic
79#6489C2311yescharacteristic
80C2×C4⋊Dic7C2×C411yescharacteristic
81C2×C4⋊Dic7C2×C411yescharacteristic
82C2×C4×Dic7—21no—
83C2×C4×Dic7C2312yes—
84C22×C7⋊C8C2312yes—
85#6470—24no—
86#6470—42no—
87C2×#1131—22no—
88C2×C8×D7—22no—
89C7×#108—42no—
90C7×C22⋊C8—21no—
91C7×C22⋊C8—21no—
92C7×C22⋊C8—21no—
93C7×C22⋊C8—21no—
94C7⋊#95—21no—
95C22⋊(C7⋊C8)D411yescharacteristic
96C22⋊(C7⋊C8)D411yescharacteristic
97C22⋊(C7⋊C8)D412yes—
98#6399—24no—
99#6399—42no—
100#6395C2×C411yescharacteristic
101#6395C2×C411yescharacteristic
102#6395C2×C411yescharacteristic
103#6395C2×C411yescharacteristic
104#6395C2×C411yescharacteristic
105#6395C2×C411yescharacteristic
106#6395C2×C411yescharacteristic
107#6395C2×C411yescharacteristic
108(C2×D7)⋊C8—22no—
109(C2×D7)⋊C8—22no—
110#6390C2×C411yescharacteristic
111#6390C2×C411yescharacteristic
112#6390C2×C411yescharacteristic
113#6390C2×C411yescharacteristic
114#6390C2×C411yescharacteristic
115#6390C2×C411yescharacteristic
116#6390C2×C411yescharacteristic
117#6390C2×C411yescharacteristic
118#6389—22no—
119#6389—42no—
120Dic7⋊C8—22no—
121Dic7⋊C8—22no—
122C8×Dic7—22no—
123#2924—71no—
124C2×#416—71no—
125#2880—71no—
126#1938—72no—
127#1929—72no—
128#1866—72no—
129#1826—71no—
130#1810D711yescharacteristic, complemented
131#1789—71no—
132#1765—72no—
133#1752—71no—
134C24×C7#4111yescharacteristic
135C23×D7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7C22×C411yescharacteristic
139C22×C4×C7C22×C411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7#4111yescharacteristic
143C22×C4×C7C2411yescharacteristic
144C2×C22⋊D7—21no—
145C2×C22⋊D7—21no—
146C2×C22⋊D7—21no—
147C2×C22⋊D7—21no—
148C2×C22⋊D7—44no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—22no—
153#1157—22no—
154#1157—22no—
155#1157—22no—
156#1157—22no—
157#1157—22no—
158#1157—41no—
159#1157—41no—
160#1157—44no—
161#1157—44no—
162#1157#4114yes—
163C2×D28—22no—
164C2×D28—21no—
165C2×D28—41no—
166C2×D28C22×C411yescharacteristic
167C2×D28C22×C411yescharacteristic
168C2×C4×D7—21no—
169C2×C4×D7—21no—
170C2×C4×D7—21no—
171C2×C4×D7—21no—
172C2×C4×D7—44no—
173C2×Dic14—22no—
174C2×Dic14—21no—
175C2×Dic14—41no—
176C2×Dic14C22×C411yescharacteristic
177C2×Dic14C22×C411yescharacteristic
178C7×M16—22no—
179C7×M16—22no—
180C7×M16—22no—
181C7×M16—22no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—41no—
187C2×C7×C8—41no—
188C2×C7×C8—41no—
189C2×C7×C8—41no—
190C2×C7×C8C22×C411yescharacteristic
191C2×C7×C8C22×C411yescharacteristic
192C2×C7×C8C22×C411yescharacteristic
193C2×C7×C8C22×C411yescharacteristic
194C22⋊Dic7—21no—
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197C22⋊Dic7—21no—
198C22⋊Dic7—42no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7C22×C411yescharacteristic
202C4⋊Dic7C22×C411yescharacteristic
203C4⋊Dic7C22×C411yescharacteristic
204C4⋊Dic7C22×C411yescharacteristic
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—22no—
210C4×Dic7—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8C2×D412yes—
214C2×C7⋊C8C2×D412yes—
215#1131—44no—
216C8×D7—44no—
217C22×#41—71no—
218C2×#131—72no—
219C23×C8C2×D711yescharacteristic
220C2×#117—71no—
221#433—142no—
222#432—141no—
223#431—141no—
224C2×#108C2×D712yescomplemented
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—72no—
241#408—142no—
242#407—141no—
243#407C2×D711yescharacteristic
244#407C2×D711yescharacteristic
245C2×C22⋊C8—71no—
246C2×C22⋊C8—141no—
247#405—72no—
248C2×#97—72no—
249C4×C22⋊C4—71no—
250#341C2×D712yescomplemented
251#339—72no—
252#339—72no—
253#336—72no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—41no—
257C23×C7C4×D411yescharacteristic
258C23×C7C4×D411yescharacteristic
259C23×C7C2×#4111yescharacteristic
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—42no—
263C22×D7—42no—
264C22×D7—81no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7C4×D411yescharacteristic
273C2×C4×C7C4×D411yescharacteristic
274C2×C4×C7C23×C411yescharacteristic
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7C2×#4112yes—
277C2×C4×C7H(5,𝔽2)12yes—
278C22⋊D7—24no—
279C22⋊D7—42no—
280C22⋊D7—42no—
281C22⋊D7—42no—
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—81no—
294D28—21no—
295D28—21no—
296D28—21no—
297D28—42no—
298D28#11712yes—
299C4×D7—24no—
300C4×D7—42no—
301C4×D7—42no—
302C4×D7—42no—
303Dic14—21no—
304Dic14—21no—
305Dic14—21no—
306Dic14—42no—
307Dic14#11712yes—
308C7×C8—21no—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—42no—
313C7×C8—42no—
314C7⋊C8—22no—
315C7⋊C8—22no—
316C2×#41—71no—
317C2×#41—71no—
318C2×#41—71no—
319C2×#41—71no—
320C2×#41—144no—
321C2×#41—141no—
322C2×#41—141no—
323C2×#41—282no—
324C22×Q8—71no—
325C22×D4—71no—
326C22×D4—141no—
327C23×C4—141no—
328C23×C4C22×D711yescharacteristic
329#131—144no—
330#131—282no—
331C2×M16—142no—
332C2×M16C22×D712yes—
333C2×M16C22×D712yes—
334C22×C8—42no—
335C22×C8—72no—
336C22×C8—142no—
337C22×C8C4×D711yescharacteristic
338C22×C8C4×D711yescharacteristic
339C22×C8C4×D711yescharacteristic
340C22×C8C4×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
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—42no—
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—21no—
379C22⋊C8—21no—
380C22⋊C8—71no—
381C22⋊C8—71no—
382C22⋊C8—72no—
383C22⋊C8—142no—
384C22⋊C8—142no—
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—21no—
395C22×C7—21no—
396C22×C7—44no—
397C22×C7#44412yes—
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#55112yes—
400C2×D7—22no—
401C2×D7—41no—
402C2×D7—41no—
403C2×D7—41no—
404C2×D7—41no—
405C2×D7—41no—
406C2×D7—84no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7C2×#11711yescharacteristic
410C4×C7#51811yescharacteristic
411C4×C7#54612yes—
412Dic7—22no—
413Dic7—41no—
414Dic7—41no—
415Dic7—41no—
416Dic7—41no—
417Dic7—41no—
418Dic7—82no—
419C24—141no—
420C24#116111yescharacteristic
421#41—74no—
422#41—142no—
423#41—142no—
424#41—142no—
425#41—142no—
426#41—142no—
427#41—284no—
428#41—281no—
429#41—281no—
430#41—284no—
431C2×Q8—71no—
432C2×Q8—71no—
433C2×Q8—142no—
434C2×Q8—141no—
435C2×Q8—281no—
436C2×D4—71no—
437C2×D4—71no—
438C2×D4—142no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4—281no—
445C2×D4—284no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—142no—
456C22×C4—284no—
457C22×C4C2×C4×D711yescharacteristic
458C22×C4C2×C4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4D4×D711yescharacteristic
461C22×C4#116111yescharacteristic
462C22×C4C23×D711yescharacteristic
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—284no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—72no—
477C2×C8—72no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8—284no—
481C2×C8C2×C4×D711yescharacteristic
482C2×C8C2×C4×D711yescharacteristic
483C2×C8C2×C4×D711yescharacteristic
484C2×C8C2×C4×D711yescharacteristic
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#234211yescharacteristic
506C2×C7C2×#44411yescharacteristic
507C2×C7#295711yescharacteristic
508D7—42no—
509D7—81no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—141no—
514C23—141no—
515C23—282no—
516C23—282no—
517C23—561no—
518C23#645611yescharacteristic
519C23#645611yescharacteristic
520C23C2×#116111yescharacteristic
521Q8—72no—
522Q8—141no—
523Q8—141no—
524Q8—141no—
525Q8—282no—
526D4—72no—
527D4—144no—
528D4—141no—
529D4—141no—
530D4—141no—
531D4—282no—
532D4—282no—
533D4—282no—
534D4—282no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—141no—
543C2×C4—144no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—561no—
558C2×C4#645611yescharacteristic
559C2×C4#645611yescharacteristic
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C2×D4×D711yescharacteristic
562C2×C4#654812yes—
563C2×C4D7×#4112yes—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—42no—
569C8—42no—
570C8—142no—
571C8—142no—
572C7#2258611yescharacteristic, complemented, Sylow
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—142no—
582C22—281no—
583C22—281no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22—564no—
588C22#8966612yes—
589C22C2×#645611yescharacteristic
590C22#9032812yes—
591C4—21no—
592C4—21no—
593C4—142no—
594C4—281no—
595C4—281no—
596C4—281no—
597C4—281no—
598C4—281no—
599C4—562no—
600C4#9017711yescharacteristic
601C4D7×#11711yescharacteristic
602C4#9030112yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—282no—
607C2—561no—
608C2#1173145411yescharacteristic
609C2C2×#8966611yescharacteristic
610C2#1173669411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1810C2
socleC22×C7C2×C4×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1810D7
hypocenterC7#22586
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
3. #11736066C2semidirect
4. C7×#1810C2semidirect
5. #11729721C2semidirect
6. #11728476C2semidirect
7. #11728453C2semidirect
9. #11727729C2semidirect
10. #11726681C2semidirect
11. #11726673C2semidirect
16. C2×C7×#108C22semidirect
24. #89894C22semidirect
25. #89891C22semidirect
26. #89885C4semidirect
27. #89885C4semidirect
28. #89885C4semidirect
29. #89885C4semidirect
34. C2×#6395C4semidirect
35. C2×#6395C4semidirect
36. C2×#6390C4semidirect
37. C2×#6390C4semidirect
38. C2×#6389C22semidirect
45. C7×#341C22semidirect
46. #89350C22semidirect
47. #89350C22semidirect
48. #89345C22semidirect
130. #1810D7semidirect
224. C2×#108C2×D7semidirect
250. #341C2×D7semidirect
572. C7#22586semidirect

Maximal subgroups

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

subgroupindex
C2×#898852
#117360762
#117360662
C7×#18102
#117297212
#117284762
#117284532
#117277452
#117277292
#117266812
#117266732
#225867