Finite Group Databasebeta

← #13371540

Subgroups — #13371540

Order 1792.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739685C211yesmaximal, characteristic
2C2×#89767C211yesmaximal, characteristic
3#11731323C211yesmaximal, characteristic
4C7×#1825C211yesmaximal, characteristic
5#11727703C211yesmaximal, characteristic
6#11727679C211yesmaximal, characteristic
7#11727267C211yesmaximal, characteristic
8#11726614C211yesmaximal, characteristic, complemented
9#11726613C211yesmaximal, characteristic, complemented
10#11726575C211yesmaximal, characteristic, complemented
11#11726569C211yesmaximal, characteristic, complemented
12#11726543C211yesmaximal, characteristic, complemented
13#11726542C211yesmaximal, characteristic, complemented
14#11726540C211yesmaximal, characteristic, complemented
15#11726538C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6440—21no—
20C2×C7×C22⋊C8C2211yescharacteristic
21C7×#380C2211yescharacteristic
22#89984C2211yescharacteristic
23C4×C22⋊Dic7C2211yescharacteristic
24C2×#6389C2211yescharacteristic
25C2×C8×Dic7C2211yescharacteristic
26#89767C418yescomplemented
27#89757C2211yescharacteristic
28#89757C2211yescharacteristic
29C4⋊C4×Dic7C2211yescharacteristic
30C4⋊C4×Dic7C2211yescharacteristic
31C4⋊C4×Dic7C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33#89745C2211yescharacteristic
34#89745C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89723—21no—
42#89723—21no—
43#89715—21no—
44#89709—21no—
45C4×C4⋊Dic7—21no—
46#89426—21no—
47#89426—21no—
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22622—71nomaximal, Sylow
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C7×#117—22no—
66C7×#117C2×C414yes—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C42×C7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7D411yescharacteristic
76C2×C22⋊Dic7D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2×C414yes—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C2×#1136—22no—
95C2×#1136—22no—
96C2×#1136C2×C414yes—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99#6440—44no—
100C7×C22⋊C8—22no—
101C7×#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113#6389—22no—
114C8×Dic7—22no—
115#6382—22no—
116#6382C2×C414yes—
117#6382C2×C414yes—
118#6381—22no—
119#6381C2×C414yes—
120#6381C2×C414yes—
121C2×#428—71no—
122#2347—71no—
123#2344—71no—
124#1943—71no—
125#1942—71no—
126#1921—71no—
127#1920—71no—
128#1919—71no—
129#1918—71no—
130#1825D711yescharacteristic
131#1824—71no—
132#1809—71no—
133#1776—71no—
134C4×#106—71no—
135#1753—71no—
136C24×C7C2×D411yescharacteristic
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C22×C412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7#4112yes—
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—24no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—42no—
155C22×Dic7—42no—
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4C22×C412yes—
173C7×C4⋊C4C22×C412yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—42no—
176C42×C7—24no—
177C22⋊Dic7—22no—
178C22⋊Dic7—41no—
179C22⋊Dic7—81no—
180C4⋊Dic7—24no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—42no—
188C4⋊Dic7—42no—
189C4⋊Dic7C22×C412yes—
190C4⋊Dic7C22×C412yes—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193C4×Dic7—21no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—42no—
198C4×Dic7—42no—
199C4×Dic7—81no—
200C4×Dic7—81no—
201#1136—28no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8C22×C412yes—
205C2×C7⋊C8C22×C412yes—
206C2×C7⋊C8#4112yes—
207C2×C7⋊C8#4112yes—
208C22×M16—71no—
209C2×#117—141no—
210C2×#117C2×D711yescharacteristic
211C22×C4⋊C4—71no—
212#428—78no—
213C2×#107—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×C22⋊C8C2×D711yescharacteristic
220C2×#97—71no—
221C2×C4×C8—71no—
222#389—71no—
223#389—71no—
224#389—71no—
225#389—71no—
226#388—141no—
227#388—141no—
228#387—141no—
229#387—141no—
230#383—141no—
231#381—141no—
232#380—71no—
233#380C2×D711yescharacteristic
234C4×C4⋊C4—71no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—141no—
239C4×C22⋊C4—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340C2×D711yescharacteristic
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#340C2×D711yescharacteristic
250#336—71no—
251#336—71no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7C4×D412yes—
259C23×C7C22×D411yescharacteristic
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7C4×D412yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—48no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7AGL(1,ℤ/8)12yes—
310C7×C8—41no—
311C7×C8—41no—
312C7⋊C8—22no—
313C7⋊C8#11714yescomplemented
314C23×C4—141no—
315C23×C4C22×D711yescharacteristic
316C2×M16—74no—
317C2×M16—142no—
318C2×M16—142no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8C22×D711yescharacteristic
322C22×C8C22×D711yescharacteristic
323#117—22no—
324#117—284no—
325#117C4×D714yes—
326C2×C4⋊C4—74no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—71no—
344C2×C22⋊C4—71no—
345C2×C22⋊C4—141no—
346C2×C42—21no—
347C2×C42—71no—
348C2×C42—71no—
349C2×C42—141no—
350C2×C42—141no—
351C2×C42—141no—
352C2×C42—141no—
353#107—74no—
354#107—74no—
355#107—142no—
356#106—74no—
357#106—74no—
358#106—142no—
359C22⋊C8—22no—
360#97—142no—
361C4×C8—142no—
362#95—21no—
363#95—71no—
364#95—71no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7C2×C4×D411yescharacteristic
382C22×C7#54711yescharacteristic
383C22×C7#54711yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57811yescharacteristic
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—81no—
392C4×C7—81no—
393C4×C7C2×#11711yescharacteristic
394C4×C7#51911yescharacteristic
395C4×C7#55211yescharacteristic
396C4×C7#55211yescharacteristic
397Dic7—22no—
398Dic7—41no—
399Dic7—41no—
400Dic7—41no—
401Dic7—42no—
402Dic7—42no—
403Dic7—81no—
404Dic7—81no—
405C24D4×D711yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—144no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—282no—
424C22×C4—282no—
425C22×C4C2×C4×D712yes—
426C22×C4D4×D711yescharacteristic
427C22×C4#115912yes—
428C22×C4C23×D711yescharacteristic
429M16—148no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—72no—
437C2×C8—72no—
438C2×C8—72no—
439C2×C8—72no—
440C2×C8—141no—
441C2×C8—141no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—72no—
449C4⋊C4—72no—
450C4⋊C4—144no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—282no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4C2×C4×D712yes—
462C4⋊C4C2×C4×D712yes—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—142no—
466C22⋊C4—281no—
467C22⋊C4—561no—
468C42—24no—
469C42—141no—
470C42—141no—
471C42—141no—
472C42—141no—
473C42—282no—
474C42—282no—
475C42—561no—
476C42—561no—
477C2×C7—42no—
478C2×C7#234411yescharacteristic
479C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
480C2×C7#294711yescharacteristic
481C2×C7#334511yescharacteristic
482C2×C7#334911yescharacteristic
483C2×C7#335111yescharacteristic
484C2×C7#335311yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23(C22⋊D7)⋊C412yes—
492C23C2×D4×D711yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—72no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—288no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4(C22⋊D7)⋊C412yes—
537C2×C4C22×C4×D711yescharacteristic
538C2×C4C2×#115911yescharacteristic
539C2×C4D7×#4111yescharacteristic
540C2×C4D7×#4111yescharacteristic
541C2×C4#655411yescharacteristic
542C2×C4#655411yescharacteristic
543C8—41no—
544C8—41no—
545C8—74no—
546C8—142no—
547C7#2262211yescharacteristic, complemented, Sylow
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×(C22⋊D7)⋊C411yescharacteristic
556C22#9034611yescharacteristic
557C22#9034611yescharacteristic
558C22#9046111yescharacteristic
559C22#9046211yescharacteristic
560C22D7×AGL(1,ℤ/8)11yescharacteristic
561C22#9046811yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—142no—
568C4—281no—
569C4—281no—
570C4—281no—
571C4—282no—
572C4—282no—
573C4—561no—
574C4—561no—
575C4#9020711yescharacteristic
576C4D7×#11711yescharacteristic
577C4#9034111yescharacteristic
578C4#9034111yescharacteristic
579C2—42no—
580C2#1173150511yescharacteristic
581C2#1173626611yescharacteristic
582C2#1173631111yescharacteristic
583C2#1173881211yescharacteristic
584C2#1173883611yescharacteristic
585C2#1173895511yescharacteristic
586C2#1173898411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1825C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1825D7
hypocenterC7#22622
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
8. #11726614C2semidirect
9. #11726613C2semidirect
10. #11726575C2semidirect
11. #11726569C2semidirect
12. #11726543C2semidirect
13. #11726542C2semidirect
14. #11726540C2semidirect
15. #11726538C2semidirect
26. #89767C4semidirect
313. C7⋊C8#117semidirect
547. C7#22622semidirect

Maximal subgroups

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

subgroupindex
#117396852
C2×#897672
#117313232
C7×#18252
#117277032
#117276792
#117272672
#117266142
#117266132
#117265752
#117265692
#117265432
#117265422
#117265402
#117265382
#226227