Finite Group Databasebeta

← #13371578

Subgroups — #13371578

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
1C7×#2346C211yesmaximal, characteristic
2C2×#89767C211yesmaximal, characteristic
3#11730901C211yesmaximal, characteristic
4#11729329C211yesmaximal, characteristic
5#11727375C211yesmaximal, characteristic, complemented
6#11727372C211yesmaximal, characteristic, complemented
7#11727356C211yesmaximal, characteristic, complemented
8#11727355C211yesmaximal, characteristic, complemented
9#11727267C211yesmaximal, characteristic
10#11727256C211yesmaximal, characteristic
11#11727235C211yesmaximal, characteristic, complemented
12C4×#6381C211yesmaximal, characteristic, complemented
13#11727215C211yesmaximal, characteristic, complemented
14#11727212C211yesmaximal, characteristic, complemented
15#11727202C211yesmaximal, characteristic
16C2×C7×#117C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6440—21no—
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#388—21no—
23C7×#388—21no—
24C7×#382—21no—
25C7×#380C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C22⋊C4C2211yescharacteristic
29C7×#376—21no—
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32#89767C418yescomplemented
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89715—21no—
40C4×C4⋊Dic7C2211yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×#6377C2211yescharacteristic
45C2×C4×C7⋊C8C2211yescharacteristic
46#89425—21no—
47#89425—21no—
48#89324C2211yescharacteristic
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22624—71nomaximal, Sylow
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C7×#117—22no—
64C7×#117C2×C414yes—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
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×C7×C22⋊C4D411yescharacteristic
74C2×C7×C22⋊C4D411yescharacteristic
75C2×C42×C7—21no—
76C2×C42×C7—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7C2×C414yes—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136C2×C414yes—
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99#6440—44no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95D411yescharacteristic
106C7×#95D411yescharacteristic
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C22⋊(C7⋊C8)—22no—
113#6382—22no—
114#6382C2×C414yes—
115#6382C2×C414yes—
116#6381—22no—
117#6381C2×C414yes—
118#6381C2×C414yes—
119#6377—22no—
120C4×C7⋊C8—22no—
121C2×#428—71no—
122#2346D711yescharacteristic
123#2344—71no—
124#1944—71no—
125#1941—71no—
126#1921—71no—
127#1920—71no—
128#1919—71no—
129#1918—71no—
130#1825—71no—
131#1824—71no—
132#1809—71no—
133#1776—71no—
134C4×#107—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×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7#4112yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—24no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—42no—
159C22×Dic7—42no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—81no—
167C7×C4⋊C4—81no—
168C7×C4⋊C4C22×C412yes—
169C7×C4⋊C4C22×C412yes—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—41no—
173C7×C22⋊C4—81no—
174C42×C7—24no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—42no—
180C42×C7—42no—
181C22⋊Dic7—42no—
182C4⋊Dic7—24no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7C22×C412yes—
190C4⋊Dic7C22×C412yes—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193C4×Dic7—42no—
194C4×Dic7—42no—
195#1136—28no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
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⋊C8—71no—
220C2×#97—71no—
221C2×C4×C8—71no—
222#389—71no—
223#389—71no—
224#389C2×D711yescharacteristic
225#389C2×D711yescharacteristic
226#388—21no—
227#388—21no—
228#387—141no—
229#387—141no—
230#382—21no—
231#381—141no—
232#380—71no—
233#380C2×D711yescharacteristic
234C4×C4⋊C4—71no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4C2×D711yescharacteristic
237C4×C4⋊C4C2×D711yescharacteristic
238C4×C22⋊C4C2×D711yescharacteristic
239#376—21no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
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—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7C4×D412yes—
290C2×C4×C7AGL(1,ℤ/8)12yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—48no—
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×Dic7—41no—
310C7⋊C8—22no—
311C7⋊C8—41no—
312C7⋊C8—41no—
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×C8—71no—
322C22×C8—71no—
323#117—22no—
324#117—284no—
325#117C4×D714yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—74no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—142no—
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—141no—
344C2×C22⋊C4C22⋊D711yescharacteristic
345C2×C22⋊C4C22⋊D711yescharacteristic
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—141no—
350C2×C42—141no—
351C2×C42C22×D711yescharacteristic
352C2×C42C22×D711yescharacteristic
353#107—74no—
354#107—74no—
355#107—142no—
356#106—74no—
357#106—74no—
358#106—142no—
359C22⋊C8—142no—
360#97—142no—
361C4×C8—142no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95C22⋊D711yescharacteristic
373#95C22⋊D711yescharacteristic
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—22no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—42no—
394C4×C7—42no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7C2×#11711yescharacteristic
398C4×C7#51911yescharacteristic
399C4×C7#55211yescharacteristic
400C4×C7#55211yescharacteristic
401Dic7—42no—
402Dic7—42no—
403Dic7—81no—
404Dic7—81no—
405C24C2×C22⋊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—144no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—282no—
420C22×C4—282no—
421C22×C4C2×C4×D712yes—
422C22×C4#115712yes—
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C23×D711yescharacteristic
429M16—148no—
430C2×C8—72no—
431C2×C8—72no—
432C2×C8—72no—
433C2×C8—72no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
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—81no—
451C4⋊C4—81no—
452C4⋊C4—144no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4C2×C4×D712yes—
462C4⋊C4C2×C4×D712yes—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—41no—
466C22⋊C4—81no—
467C22⋊C4—282no—
468C42—24no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—42no—
474C42—42no—
475C42—282no—
476C42—282no—
477C2×C7—42no—
478C2×C7#234411yescharacteristic
479C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
480C2×C7#294711yescharacteristic
481C2×C7#334311yescharacteristic
482C2×C7#334811yescharacteristic
483C2×C7#335211yescharacteristic
484C2×C7#335411yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23C4×C22⋊D712yes—
492C23C22×C22⋊D711yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—142no—
523C2×C4—142no—
524C2×C4—288no—
525C2×C4—281no—
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×C4C4×C22⋊D712yes—
536C2×C4#651212yes—
537C2×C4C22×C4×D711yescharacteristic
538C2×C4C2×#115711yescharacteristic
539C2×C4D7×#4111yescharacteristic
540C2×C4D7×#4111yescharacteristic
541C2×C4#655411yescharacteristic
542C2×C4#655411yescharacteristic
543C8—74no—
544C8—142no—
545C8—281no—
546C8—281no—
547C7#2262411yescharacteristic, complemented, Sylow
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D711yescharacteristic
556C22C2×#651211yescharacteristic
557C22#9051311yescharacteristic
558C22#9051511yescharacteristic
559C22#9051511yescharacteristic
560C22#9052711yescharacteristic
561C22#9052911yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—282no—
572C4—282no—
573C4—561no—
574C4—561no—
575C4#9020711yescharacteristic
576C4D7×#11711yescharacteristic
577C4#9022211yescharacteristic
578C4#9022211yescharacteristic
579C2—42no—
580C2#1173968311yescharacteristic
581C2C4×#651211yescharacteristic
582C2#1174001611yescharacteristic
583C2#1174103711yescharacteristic
584C2#1174104211yescharacteristic
585C2#1174109411yescharacteristic
586C2#1174109511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2346C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#2346D7
hypocenterC7#22624
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2346

Complements

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

NH ≅ G/Ntype
5. #11727375C2semidirect
6. #11727372C2semidirect
7. #11727356C2semidirect
8. #11727355C2semidirect
11. #11727235C2semidirect
12. C4×#6381C2semidirect
13. #11727215C2semidirect
14. #11727212C2semidirect
32. #89767C4semidirect
313. C7⋊C8#117semidirect
547. C7#22624semidirect

Maximal subgroups

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

subgroupindex
C7×#23462
C2×#897672
#117309012
#117293292
#117273752
#117273722
#117273562
#117273552
#117272672
#117272562
#117272352
C4×#63812
#117272152
#117272122
#117272022
#226247