Finite Group Databasebeta

← #13365955

Subgroups — #13365955

Order 1792.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89619C211yesmaximal, characteristic
2C4×#6433C211yesmaximal, characteristic, complemented
3#11730744C211yesmaximal, characteristic
4C7×#1767C211yesmaximal, characteristic, complemented
5#11728234C211yesmaximal, characteristic
6#11728215C211yesmaximal, characteristic, complemented
7#11728209C211yesmaximal, characteristic, complemented
8#11727574C211yesmaximal, characteristic, complemented
9#11727573C211yesmaximal, characteristic, complemented
10#11727572C211yesmaximal, characteristic
11#11727572C211yesmaximal, characteristic
12#11726380C211yesmaximal, characteristic
13#11726379C211yesmaximal, characteristic
14#11726376C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C2×C4×D28C2211yescharacteristic
17C2×#6435—21no—
18C2×#6433C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×#97C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C43×C7C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32#89712—21no—
33#89712—21no—
34#89711C2211yescharacteristic
35#89711C2211yescharacteristic
36C4×(C2×D7)⋊C4—21no—
37#89708C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89705—21no—
42#89705—21no—
43C4×Dic7⋊C4—21no—
44C4×Dic7⋊C4—21no—
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49#89619C418yescomplemented
50C7×#336C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#22120—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C42×C7D411yescharacteristic
68C2×C42×C7D411yescharacteristic
69C2×C42×C7D411yescharacteristic
70C2×C42×C7D411yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C4×D28—22no—
96C4×D28C2×C414yes—
97#6435—44no—
98#6433—22no—
99#6433C2×C414yes—
100C7×C4⋊C8—22no—
101C7×C4⋊C8C2×C414yes—
102C7×#97—41no—
103C7×#97—41no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113#6395—41no—
114#6395—41no—
115#6395C2×C414yes—
116#6395C2×C414yes—
117#6392—41no—
118#6392C2×C414yes—
119#6391—41no—
120#6391C2×C414yes—
121C2×#478—71no—
122C4×#125—71no—
123#2293—71no—
124#1989—71no—
125#1949—71no—
126#1944—71no—
127#1943—71no—
128#1936—71no—
129#1936—71no—
130#1926—71no—
131#1925—71no—
132#1919—71no—
133#1918—71no—
134#1804—71no—
135#1767D711yescharacteristic, complemented
136C23×D7—21no—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2411yescharacteristic
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28C22×C412yes—
157C2×C4×D7—24no—
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8C22×C412yes—
171C2×C7×C8C22×C412yes—
172C42×C7—22no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—41no—
178C42×C7—41no—
179C42×C7—41no—
180C42×C7C22×C412yes—
181C42×C7#4112yes—
182(C2×D7)⋊C4—24no—
183(C2×D7)⋊C4—42no—
184(C2×D7)⋊C4—42no—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4—42no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7C22×C412yes—
199C4⋊Dic7C22×C412yes—
200C4⋊Dic7C22×C412yes—
201Dic7⋊C4—24no—
202Dic7⋊C4—24no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C4×Dic7—42no—
208C4×Dic7—42no—
209C4×Dic7—42no—
210C2×#125—71no—
211C2×C4×D4—71no—
212C2×#117—141no—
213#478—78no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×C4⋊C8C2×D711yescharacteristic
219C2×C4⋊C8C2×D711yescharacteristic
220C2×#102—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#97C2×D711yescharacteristic
225C2×#97C2×D711yescharacteristic
226#390—71no—
227#390—71no—
228#389—71no—
229#389—71no—
230#389—71no—
231#389—71no—
232#388—141no—
233#388—141no—
234#387—141no—
235#387—141no—
236#382—71no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—141no—
241C4×C4⋊C4—141no—
242C4×C22⋊C4—141no—
243C43C2×D711yescharacteristic
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#336C2×D711yescharacteristic
253#336C2×D711yescharacteristic
254C23×C7C22×D411yescharacteristic
255C22×D7—22no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
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×C7C4×D412yes—
278C2×C4×C7C4×D412yes—
279C2×C4×C7AGL(1,ℤ/8)12yes—
280C2×C4×C7C23×C411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307D28—22no—
308D28—22no—
309D28—41no—
310D28—41no—
311C4×D7—48no—
312C7×C8—22no—
313C7×C8—22no—
314C7×C8—41no—
315C7×C8—41no—
316C22×D4—71no—
317C23×C4—141no—
318C22×C8C22×D711yescharacteristic
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322#125—74no—
323#125—142no—
324C4×D4—74no—
325C4×D4—142no—
326#117—284no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C42—141no—
346C2×C42—141no—
347C2×C42—141no—
348C2×C42D2811yescharacteristic
349C2×C42D2811yescharacteristic
350C2×C42D2811yescharacteristic
351C2×C42D2811yescharacteristic
352C2×C42C22×D711yescharacteristic
353C2×C42C22×D711yescharacteristic
354C2×C42C22×D711yescharacteristic
355#107—74no—
356#107—281no—
357#106—74no—
358#106—281no—
359C4⋊C8—22no—
360C4⋊C8C4×D714yes—
361#102—74no—
362#102—74no—
363#102—281no—
364#102—281no—
365#97—41no—
366#97—41no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376C22×C7C2×C4×D411yescharacteristic
377C22×C7#53611yescharacteristic
378C22×C7#53711yescharacteristic
379C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C7#57511yescharacteristic
381C22×C7#57511yescharacteristic
382C22×C7#57511yescharacteristic
383C2×D7—42no—
384C2×D7—42no—
385C2×D7—42no—
386C2×D7—81no—
387C2×D7—81no—
388C2×D7—81no—
389C2×D7—81no—
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—22no—
393C4×C7—41no—
394C4×C7—41no—
395C4×C7—41no—
396C4×C7#51911yescharacteristic
397C4×C7#52011yescharacteristic
398C4×C7#54111yescharacteristic
399C4×C7#54211yescharacteristic
400Dic7—42no—
401Dic7—42no—
402Dic7—42no—
403Dic7—81no—
404Dic7—81no—
405Dic7—81no—
406Dic7—81no—
407C24—141no—
408C2×D4—72no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—141no—
412C2×D4—141no—
413C2×D4—141no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—144no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—282no—
423C22×C4—282no—
424C22×C4C2×D2811yescharacteristic
425C22×C4C2×D2811yescharacteristic
426C22×C4C2×D2811yescharacteristic
427C22×C4C2×D2811yescharacteristic
428C22×C4C2×D2811yescharacteristic
429C22×C4C2×D2811yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8C2×C4×D712yes—
442C2×C8C2×C4×D712yes—
443C4⋊C4—72no—
444C4⋊C4—72no—
445C4⋊C4—72no—
446C4⋊C4—144no—
447C4⋊C4—144no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
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⋊C4—282no—
462C4⋊C4—282no—
463C22⋊C4—144no—
464C22⋊C4—282no—
465C22⋊C4—282no—
466C22⋊C4—282no—
467C22⋊C4—282no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—282no—
477C42—282no—
478C42—282no—
479C42C2×C4×D712yes—
480C42#115712yes—
481C2×C7#232311yescharacteristic
482C2×C7#294311yescharacteristic
483C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
484C2×C7#323411yescharacteristic
485C2×C7#324011yescharacteristic
486C2×C7#325111yescharacteristic
487C2×C7#325611yescharacteristic
488D7—82no—
489C23—142no—
490C23—281no—
491C23—281no—
492C23—281no—
493C23—281no—
494C23—281no—
495C23—281no—
496C23C22×D2811yescharacteristic
497D4—142no—
498D4—142no—
499D4—281no—
500D4—281no—
501C2×C4—21no—
502C2×C4—21no—
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—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—142no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—288no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
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×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4C4×D2812yes—
539C2×C4C4×D2812yes—
540C2×C4#647112yes—
541C2×C4C22×C4×D711yescharacteristic
542C2×C4C2×#115711yescharacteristic
543C2×C4#654811yescharacteristic
544C2×C4#655111yescharacteristic
545C2×C4#655411yescharacteristic
546C2×C4#655411yescharacteristic
547C8—22no—
548C8—22no—
549C8—41no—
550C8—41no—
551C7#2212011yescharacteristic, complemented, Sylow
552C22—282no—
553C22—282no—
554C22—282no—
555C22—561no—
556C22—561no—
557C22—561no—
558C22—561no—
559C22C2×C4×D2811yescharacteristic
560C22#9020111yescharacteristic
561C22#9021711yescharacteristic
562C22C2×#647111yescharacteristic
563C22#9043911yescharacteristic
564C22#9043911yescharacteristic
565C22#9043911yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—282no—
573C4—282no—
574C4—282no—
575C4—561no—
576C4—561no—
577C4—561no—
578C4—561no—
579C4#9023711yescharacteristic
580C4#9025211yescharacteristic
581C4#9026311yescharacteristic
582C4#9027411yescharacteristic
583C2—562no—
584C2#1173120711yescharacteristic
585C2#1173593911yescharacteristic
586C2C4×#647111yescharacteristic
587C2#1173650711yescharacteristic
588C2#1173651111yescharacteristic
589C2#1173676311yescharacteristic
590C2#1173676911yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1767C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1767D7
hypocenterC7#22120
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C4×#6433C2semidirect
4. C7×#1767C2semidirect
6. #11728215C2semidirect
7. #11728209C2semidirect
8. #11727574C2semidirect
9. #11727573C2semidirect
14. #11726376C2semidirect
15. #11726375C2semidirect
49. #89619C4semidirect
135. #1767D7semidirect
551. C7#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#896192
C4×#64332
#117307442
C7×#17672
#117282342
#117282152
#117282092
#117275742
#117275732
#117275722
#117263802
#117263792
#117263762
#117263752
#221207