Finite Group Databasebeta

← #13416713

Subgroups — #13416713

Order 1792.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2441C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic, complemented
3C2×#89767C211yesmaximal, characteristic, complemented
4#11730912C211yesmaximal, characteristic
5#11729493C211yesmaximal, characteristic, complemented
6#11729340C211yesmaximal, characteristic
7#11727384C211yesmaximal, characteristic, complemented
8#11727380C211yesmaximal, characteristic, complemented
9#11727378C211yesmaximal, characteristic, complemented
10#11727354C211yesmaximal, characteristic, complemented
11#11727341C211yesmaximal, characteristic, complemented
12#11727322C211yesmaximal, characteristic, complemented
13#11727271C211yesmaximal, characteristic
14#11727259C211yesmaximal, characteristic, complemented
15#11727204C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×#6493C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×(C22⋊D7)⋊C4—21no—
22C2×C4×D28C2211yescharacteristic
23C7×#399—21no—
24C7×#396—21no—
25C7×#389—21no—
26C7×#389C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#387—21no—
29C7×#384C2211yescharacteristic
30C7×#380C2211yescharacteristic
31C4×C7×C22⋊C4C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33#89838—24no—
34#89767—24no—
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89730—21no—
46#89711C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89442—21no—
52#89440—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26366—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C22×C22⋊D7—21no—
64C23×Dic7—21no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C7×#125—22no—
68C7×#125—41no—
69C7×#117—22no—
70C7×#117—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4D411yescharacteristic
84C2×C7×C22⋊C4D411yescharacteristic
85C2×C42×C7—21no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88#6493—22no—
89#6493Q814yes—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×C4⋊Dic7—22no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7—41no—
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×Dic7⋊C4—21no—
101C2×C4×Dic7—21no—
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136—22no—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109(C22⋊D7)⋊C4—44no—
110C4×D28—22no—
111C4×D28—41no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117C7⋊#95—21no—
118C7⋊#95—21no—
119C22⋊(C7⋊C8)—22no—
120#6383—22no—
121#6383—22no—
122#6383—22no—
123#6383—41no—
124#6382—22no—
125#6382—22no—
126#6382—22no—
127#6382—41no—
128#6381—22no—
129#6381D414yescomplemented
130C4⋊(C7⋊C8)—22no—
131C4⋊(C7⋊C8)—22no—
132C2×#478—71no—
133C2×#428—71no—
134#2441D711yescharacteristic, complemented
135#2361—71no—
136#2094—71no—
137#2088—71no—
138#2067—71no—
139#2059—71no—
140#1948—71no—
141#1918—71no—
142#1895—71no—
143#1875—71no—
144#1864—71no—
145#1853—71no—
146#1851—71no—
147C24×C7C2×D411yescharacteristic
148C23×D7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×Q812yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—24no—
166C2×C22⋊D7—42no—
167C2×C22⋊D7—42no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—42no—
172C22×Dic7—42no—
173C22×Dic7—81no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28—21no—
177C2×D28—21no—
178C2×D28—21no—
179C2×D28C2×Q812yes—
180C2×C4×D7—42no—
181C2×C4×D7—42no—
182C2×C4×D7—81no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4—42no—
194C7×C4⋊C4—81no—
195C7×C4⋊C4—81no—
196C7×C4⋊C4C2×D412yes—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—41no—
199C7×C22⋊C4—42no—
200C7×C22⋊C4—81no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7—21no—
205C42×C7—42no—
206(C2×D7)⋊C4—24no—
207(C2×D7)⋊C4—42no—
208(C2×D7)⋊C4—42no—
209(C2×D7)⋊C4—81no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—42no—
215C4⋊Dic7C2×D412yes—
216C4⋊Dic7C2×Q812yes—
217Dic7⋊C4—42no—
218C4×Dic7—42no—
219#1136—28no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8C2×D412yes—
231C2×C7⋊C8#4112yes—
232C22×M16—71no—
233C2×#125C2×D711yescharacteristic
234C2×#121—71no—
235C2×C4×D4—71no—
236C2×C4×D4—141no—
237C2×#117C2×D711yescharacteristic
238C22×C4⋊C4—71no—
239#478—144no—
240#428—144no—
241C2×#107—71no—
242C2×#107—71no—
243C2×#106—71no—
244C2×#106—71no—
245C2×#106—71no—
246C2×#106—71no—
247C2×C4⋊C8—71no—
248C2×C4⋊C8—71no—
249C2×#102—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×#102—71no—
253C2×C22⋊C8—71no—
254#399—21no—
255#396—21no—
256#390—71no—
257#389—21no—
258#389—71no—
259#389C2×D711yescharacteristic
260#389C2×D711yescharacteristic
261#388—141no—
262#387—21no—
263#386—141no—
264#385—141no—
265#384C2×D711yescharacteristic
266#380C2×D711yescharacteristic
267C4×C4⋊C4—71no—
268C4×C22⋊C4C2×D711yescharacteristic
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#336—71no—
276#336—71no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7—21no—
281C23×C7—21no—
282C23×C7—21no—
283C23×C7C22⋊Q812yes—
284C23×C7C22×D411yescharacteristic
285C22×D7—22no—
286C22×D7—41no—
287C22×D7—41no—
288C22×D7—41no—
289C22×D7—41no—
290C22×D7—41no—
291C22×D7—41no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7C22⋊Q812yes—
322C2×C4×C7AGL(1,ℤ/8)12yes—
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×Q811yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7H(5,𝔽2)11yescharacteristic
328C2×C4×C72−1+411yescharacteristic
329C22⋊D7—48no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—84no—
340D28—22no—
341D28—22no—
342D28—41no—
343D28—41no—
344C4×D7—84no—
345C7⋊C8—22no—
346C7⋊C8—22no—
347C7⋊C8—41no—
348C7⋊C8—41no—
349C22×D4—71no—
350C22×D4—141no—
351C23×C4—141no—
352C23×C4—141no—
353C23×C4C22×D711yescharacteristic
354C2×M16—142no—
355C2×M16—142no—
356C2×M16—142no—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8—71no—
360C22×C8—71no—
361#125—22no—
362#125—41no—
363#121—74no—
364#121—142no—
365C4×D4—142no—
366C4×D4—281no—
367C4×D4—284no—
368#117—22no—
369#117—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—71no—
379C2×C4⋊C4—142no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—281no—
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4C22⋊D711yescharacteristic
394C2×C22⋊C4C22⋊D711yescharacteristic
395C2×C42—21no—
396C2×C42—141no—
397C2×C42C22×D711yescharacteristic
398C2×C42C22×D711yescharacteristic
399#107—74no—
400#107—142no—
401#106—142no—
402#106—142no—
403#106—142no—
404#106—281no—
405C4⋊C8—142no—
406C4⋊C8—142no—
407#102—142no—
408#102—142no—
409#102—142no—
410#102—281no—
411C22⋊C8—142no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—141no—
416#95—141no—
417#95C22⋊D711yescharacteristic
418#95C22⋊D711yescharacteristic
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—41no—
423C22×C7—41no—
424C22×C7—41no—
425C22×C7—41no—
426C22×C7C2×C22⋊Q811yescharacteristic
427C22×C7D4211yescharacteristic
428C22×C7#54711yescharacteristic
429C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
430C22×C7#57511yescharacteristic
431C22×C7#57711yescharacteristic
432C22×C7#57711yescharacteristic
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—42no—
436C2×D7—81no—
437C2×D7—81no—
438C2×D7—81no—
439C2×D7—81no—
440C4×C7—22no—
441C4×C7—22no—
442C4×C7—41no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—81no—
447C4×C7—81no—
448C4×C7—81no—
449C4×C7#52511yescharacteristic
450C4×C7#53911yescharacteristic
451C4×C7D4×Q811yescharacteristic
452C4×C7#55411yescharacteristic
453Dic7—42no—
454Dic7—42no—
455Dic7—81no—
456C24—141no—
457C24C2×C22⋊D711yescharacteristic
458C2×D4—72no—
459C2×D4—144no—
460C2×D4—141no—
461C2×D4—141no—
462C2×D4—141no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—282no—
466C2×D4—282no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—561no—
483C22×C4—561no—
484C22×C4Q8×D712yes—
485C22×C4#116112yes—
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C23×D711yescharacteristic
492M16—148no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—72no—
517C4⋊C4—72no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—141no—
521C4⋊C4—141no—
522C4⋊C4—141no—
523C4⋊C4—141no—
524C4⋊C4—282no—
525C4⋊C4—282no—
526C4⋊C4D4×D712yes—
527C22⋊C4—22no—
528C22⋊C4—41no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C22⋊C4—144no—
532C22⋊C4—282no—
533C22⋊C4—282no—
534C22⋊C4—561no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—282no—
541C2×C7—42no—
542C2×C7#244911yescharacteristic
543C2×C7#307711yescharacteristic
544C2×C7#308411yescharacteristic
545C2×C7#327111yescharacteristic
546C2×C7#327211yescharacteristic
547C2×C7#334511yescharacteristic
548C2×C7#335111yescharacteristic
549D7—82no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—142no—
557C23—281no—
558C23—281no—
559C23—281no—
560C23—281no—
561C23—281no—
562C23—281no—
563C23#650912yes—
564C23C22×C22⋊D711yescharacteristic
565D4—142no—
566D4—142no—
567D4—288no—
568D4—281no—
569D4—281no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—564no—
609C2×C4—564no—
610C2×C4#650912yes—
611C2×C4#651212yes—
612C2×C4C2×D4×D711yescharacteristic
613C2×C4#654811yescharacteristic
614C2×C4C2×Q8×D711yescharacteristic
615C2×C4C2×#116111yescharacteristic
616C2×C4D7×#4111yescharacteristic
617C2×C4#655411yescharacteristic
618C8—142no—
619C8—142no—
620C8—281no—
621C8—281no—
622C7#2636611yescharacteristic, complemented, Sylow
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—282no—
631C22—282no—
632C22—282no—
633C22—561no—
634C22—561no—
635C22—561no—
636C22—561no—
637C22D4×C22⋊D711yescharacteristic
638C22C2×#650911yescharacteristic
639C22C2×#651211yescharacteristic
640C22#9051311yescharacteristic
641C22#9051511yescharacteristic
642C22#9052711yescharacteristic
643C22#9052811yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—282no—
654C4—282no—
655C4—561no—
656C4#9031811yescharacteristic
657C4#9032311yescharacteristic
658C4#9032711yescharacteristic
659C4#9033611yescharacteristic
660C2—42no—
661C2—562no—
662C2#1174020911yescharacteristic
663C2#1174079511yescharacteristic
664C2#1174092111yescharacteristic
665C2#1174096811yescharacteristic
666C2#1174102811yescharacteristic
667C2#1174104111yescharacteristic
668C2#1174105311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2441C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2441D7
hypocenterC7#26366
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C7×#2441C2semidirect
2. C2×#89838C2semidirect
3. C2×#89767C2semidirect
5. #11729493C2semidirect
7. #11727384C2semidirect
8. #11727380C2semidirect
9. #11727378C2semidirect
10. #11727354C2semidirect
11. #11727341C2semidirect
12. #11727322C2semidirect
14. #11727259C2semidirect
15. #11727204C2semidirect
129. #6381D4semidirect
134. #2441D7semidirect
622. C7#26366semidirect

Maximal subgroups

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

subgroupindex
C7×#24412
C2×#898382
C2×#897672
#117309122
#117294932
#117293402
#117273842
#117273802
#117273782
#117273542
#117273412
#117273222
#117272712
#117272592
#117272042
#263667