Finite Group Databasebeta

← #13375609

Subgroups — #13375609

Order 1792.

898 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89642C211yesmaximal, characteristic
2#11733647C211yesmaximal, characteristic
3#11731102C211yesmaximal, characteristic, complemented
4C7×#1919C211yesmaximal, characteristic, complemented
5#11728314C211yesmaximal, characteristic, complemented
6#11727372C211yesmaximal, characteristic, complemented
7C4×#6383C211yesmaximal, characteristic
8#11726649C211yesmaximal, characteristic
9#11726571C211yesmaximal, characteristic, complemented
10#11726557C211yesmaximal, characteristic, complemented
11#11726550C211yesmaximal, characteristic, complemented
12#11726534C211yesmaximal, characteristic
13#11726534C211yesmaximal, characteristic
14#11726425C211yesmaximal, characteristic
15#11726405C211yesmaximal, characteristic, complemented
16C2×#6456C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×D7×C4⋊C4—21no—
19C2×#6451C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×C8×Dic7C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89761—21no—
31#89757C2211yescharacteristic
32#89757C2211yescharacteristic
33#89748C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C4×Dic7⋊C4—21no—
41C4×Dic7⋊C4C2211yescharacteristic
42#89701—21no—
43#89700C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89642C418yescomplemented
46#89440—21no—
47#89439—21no—
48#89419—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53C7×#336C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22963—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×Dic14—21no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4D411yescharacteristic
88C2×Dic7⋊C4D411yescharacteristic
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6456—22no—
96#6456—22no—
97#6456—41no—
98#6456C2×C414yes—
99D7×C4⋊C4—44no—
100#6451—22no—
101#6451C2×C414yescomplemented
102C4×D28—22no—
103C4×D28—41no—
104C7×#106C2×C414yescomplemented
105C7×#95—21no—
106C7×#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95D411yescharacteristic
113C7⋊#95D411yescharacteristic
114#6395—22no—
115#6395C2×C414yes—
116#6391—41no—
117Dic7⋊C8C2×C414yescomplemented
118C8×Dic7—22no—
119C8×Dic7—41no—
120#6383C2×C414yes—
121#6383Q814yescomplemented
122#6382—41no—
123#6382C2×C414yescomplemented
124C4×C7⋊C8—41no—
125C2×#476—71no—
126#2369—71no—
127D4×C4⋊C4—71no—
128#1990—71no—
129#1948—71no—
130#1941—71no—
131#1941—71no—
132#1919—71no—
133#1919D711yescharacteristic, complemented
134#1864—71no—
135#1864—71no—
136C8×C4⋊C4—71no—
137C4×#102—71no—
138C4×#102—71no—
139C4×#102—71no—
140C23×D7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C2×D28—21no—
156C2×D28—21no—
157C2×D28—21no—
158C2×D28C22×C412yes—
159C2×D28C2×Q812yes—
160C2×D28#4112yes—
161C2×C4×D7—24no—
162C2×C4×D7—42no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C2×C4×D7—81no—
169C2×C4×D7—81no—
170C2×Dic14—24no—
171C2×Dic14—81no—
172C2×Dic14—81no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C22×C412yes—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4C22×C412yes—
186C7×C4⋊C4C22×C412yes—
187C7×C4⋊C4C2×Q812yes—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—81no—
191(C2×D7)⋊C4—24no—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—42no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—42no—
200C4⋊Dic7—81no—
201C4⋊Dic7C22×C412yes—
202Dic7⋊C4—24no—
203Dic7⋊C4—22no—
204Dic7⋊C4—42no—
205Dic7⋊C4—81no—
206Dic7⋊C4—81no—
207Dic7⋊C4SD1614yes—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—42no—
211C4×Dic7—81no—
212C4×Dic7C22×C412yes—
213C4×Dic7#4112yes—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8C22×C412yes—
219C2×C7⋊C8C2×Q812yes—
220C2×C4⋊Q8—71no—
221C2×C4×D4—71no—
222C2×C4×D4—71no—
223C2×C4×D4—71no—
224C22×C4⋊C4—141no—
225#476—78no—
226C2×#106—71no—
227C2×#106—71no—
228C2×#106—71no—
229C2×#106C2×D711yescharacteristic
230C2×C4⋊C8—71no—
231C2×#102—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×C4×C8—71no—
236C2×C4×C8—71no—
237C2×C4×C8—71no—
238#391—141no—
239#391—141no—
240#389—71no—
241#389—71no—
242#389—71no—
243#389—141no—
244#389C2×D711yescharacteristic
245#385—141no—
246#385—141no—
247#384—71no—
248#380—141no—
249C4×C4⋊C4—71no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—141no—
253C4×C4⋊C4C2×D711yescharacteristic
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261#340C2×D711yescharacteristic
262#336—71no—
263#336C2×D711yescharacteristic
264C23×C7C22×D411yescharacteristic
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
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×C7C2×SD1612yes—
290C2×C4×C7C23×C411yescharacteristic
291C2×C4×C7C22×Q811yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—22no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7C4×D412yes—
313C2×Dic7C4×D412yes—
314C2×Dic7C2×SD1612yes—
315C2×Dic7C2×SD1612yes—
316C2×Dic7#13512yes—
317D28—22no—
318D28—22no—
319D28—22no—
320D28C4×Q814yes—
321C4×D7—48no—
322C4×D7—84no—
323C4×D7—84no—
324Dic14—84no—
325C7×C8—22no—
326C7×C8—41no—
327C7⋊C8—22no—
328C7⋊C8—22no—
329C22×Q8—141no—
330C22×D4—71no—
331C23×C4—141no—
332C23×C4—141no—
333C23×C4—141no—
334C22×C8—71no—
335C22×C8—71no—
336C22×C8C22×D711yescharacteristic
337C22×C8C22×D711yescharacteristic
338C4⋊Q8—74no—
339C4⋊Q8—142no—
340C4×D4—74no—
341C4×D4—142no—
342C4×D4—142no—
343C4×D4—142no—
344C4×D4—281no—
345C4×D4—281no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—284no—
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4—141no—
366C2×C42—21no—
367C2×C42—71no—
368C2×C42—71no—
369C2×C42—141no—
370C2×C42—141no—
371C2×C42C22×D711yescharacteristic
372#106—74no—
373#106—281no—
374#106—281no—
375#106C4×D714yescomplemented
376C4⋊C8—74no—
377#102—74no—
378#102—74no—
379#102—74no—
380#102—142no—
381C4×C8—142no—
382C4×C8—281no—
383C4×C8—281no—
384#95—21no—
385#95—21no—
386#95—71no—
387#95—71no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393C22×C7C2×C4×D411yescharacteristic
394C22×C7#54611yescharacteristic
395C22×C7D4×Q811yescharacteristic
396C22×C7C22×SD1611yescharacteristic
397C22×C7C2×#13511yescharacteristic
398C22×C7#57611yescharacteristic
399C22×C7#57711yescharacteristic
400C2×D7—24no—
401C2×D7—42no—
402C2×D7—42no—
403C2×D7—42no—
404C2×D7—42no—
405C2×D7—42no—
406C2×D7—42no—
407C4×C7—22no—
408C4×C7—42no—
409C4×C7—42no—
410C4×C7—42no—
411C4×C7—41no—
412C4×C7—81no—
413C4×C7C2×C4×Q811yescharacteristic
414C4×C7#52011yescharacteristic
415C4×C7#55211yescharacteristic
416C4×C7#55411yescharacteristic
417Dic7—22no—
418Dic7—22no—
419Dic7—42no—
420Dic7—41no—
421Dic7—81no—
422Dic7—81no—
423Dic7C4×SD1614yes—
424C24—141no—
425C2×Q8—144no—
426C2×Q8—561no—
427C2×Q8—561no—
428C2×D4—72no—
429C2×D4—72no—
430C2×D4—72no—
431C2×D4—141no—
432C2×D4—141no—
433C2×D4—141no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—144no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—282no—
447C22×C4—282no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4—561no—
453C22×C4—561no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—72no—
463C2×C8—72no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8C2×C4×D712yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—74no—
476C4⋊C4—72no—
477C4⋊C4—142no—
478C4⋊C4—144no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—561no—
487C4⋊C4—561no—
488C4⋊C4—561no—
489C4⋊C4C2×C4×D712yes—
490C4⋊C4C2×C4×D712yes—
491C4⋊C4Q8×D712yes—
492C22⋊C4—144no—
493C22⋊C4—282no—
494C22⋊C4—282no—
495C42—21no—
496C42—21no—
497C42—72no—
498C42—72no—
499C42—81no—
500C42—141no—
501C42—141no—
502C42—282no—
503C42—561no—
504C2×C7#236011yescharacteristic
505C2×C7C2×C4×SD1611yescharacteristic
506C2×C7#294611yescharacteristic
507C2×C7#329511yescharacteristic
508C2×C7#330311yescharacteristic
509C2×C7Q8×SD1611yescharacteristic
510C2×C7#338011yescharacteristic
511D7—44no—
512C23—142no—
513C23—142no—
514C23—142no—
515C23—281no—
516C23—281no—
517C23—281no—
518C23—281no—
519C23C2×D4×D711yescharacteristic
520Q8—564no—
521D4—74no—
522D4—142no—
523D4—142no—
524D4—142no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—72no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—72no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—142no—
554C2×C4—141no—
555C2×C4—288no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—564no—
565C2×C4—564no—
566C2×C4D7×SD1612yes—
567C2×C4C22×C4×D711yescharacteristic
568C2×C4C2×Q8×D711yescharacteristic
569C2×C4D7×#4111yescharacteristic
570C2×C4D7×#4111yescharacteristic
571C2×C4#655311yescharacteristic
572C2×C4#655411yescharacteristic
573C8—22no—
574C8—41no—
575C8—142no—
576C8—142no—
577C7#2296311yescharacteristic, complemented, Sylow
578C22—144no—
579C22—282no—
580C22—282no—
581C22—282no—
582C22—282no—
583C22—282no—
584C22—282no—
585C22C4×D4×D711yescharacteristic
586C22#9032311yescharacteristic
587C22#9034511yescharacteristic
588C22C2×D7×SD1611yescharacteristic
589C22D7×#13511yescharacteristic
590C22#9046511yescharacteristic
591C22#9046911yescharacteristic
592C4—22no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—74no—
598C4—81no—
599C4—142no—
600C4—142no—
601C4—281no—
602C4—282no—
603C4—561no—
604C4—561no—
605C4#9021411yescharacteristic
606C4C4×Q8×D711yescharacteristic
607C4#9038111yescharacteristic
608C4#9039911yescharacteristic
609C2—284no—
610C2#1173241011yescharacteristic
611C2C4×D7×SD1611yescharacteristic
612C2#1173722011yescharacteristic
613C2#1173848411yescharacteristic
614C2#1173860111yescharacteristic
615C2#1173878711yescharacteristic
616C2#1173899011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1919C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1919D7
hypocenterC7#22963
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
3. #11731102C2semidirect
4. C7×#1919C2semidirect
5. #11728314C2semidirect
6. #11727372C2semidirect
9. #11726571C2semidirect
10. #11726557C2semidirect
11. #11726550C2semidirect
15. #11726405C2semidirect
45. #89642C4semidirect
101. #6451C2×C4semidirect
104. C7×#106C2×C4semidirect
117. Dic7⋊C8C2×C4semidirect
121. #6383Q8semidirect
123. #6382C2×C4semidirect
133. #1919D7semidirect
375. #106C4×D7semidirect
577. C7#22963semidirect

Maximal subgroups

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

subgroupindex
C2×#896422
#117336472
#117311022
C7×#19192
#117283142
#117273722
C4×#63832
#117266492
#117265712
#117265572
#117265502
#117265342
#117264252
#117264052
#229637