Finite Group Databasebeta

← #13375598

Subgroups — #13375598

Order 1792.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89662C211yesmaximal, characteristic
2#11733647C211yesmaximal, characteristic
3#11731097C211yesmaximal, characteristic, complemented
4C7×#1918C211yesmaximal, characteristic, complemented
5#11728366C211yesmaximal, characteristic, complemented
6#11727373C211yesmaximal, characteristic, complemented
7C4×#6383C211yesmaximal, characteristic
8#11726648C211yesmaximal, characteristic
9#11726569C211yesmaximal, characteristic, complemented
10#11726557C211yesmaximal, characteristic, complemented
11#11726549C211yesmaximal, characteristic, complemented
12#11726534C211yesmaximal, characteristic
13#11726534C211yesmaximal, characteristic
14#11726424C211yesmaximal, characteristic
15#11726405C211yesmaximal, characteristic, complemented
16C2×#6456C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×D7×C4⋊C4—21no—
19C2×#6451C2211yescharacteristic
20C2×C4×D28C2211yescharacteristic
21C2×C4×Dic14—21no—
22C2×C7×#107C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31#89761—21no—
32#89757C2211yescharacteristic
33#89757C2211yescharacteristic
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89705—21no—
42C4×Dic7⋊C4C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89662C418yescomplemented
46#89440—21no—
47#89439—21no—
48#89427—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#22962—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—
104C4×Dic14—44no—
105C7×#107C2×C414yescomplemented
106C7×#95—21no—
107C7×#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115#6395—22no—
116#6395C2×C414yes—
117#6392—41no—
118Dic7⋊C8C2×C414yescomplemented
119C8×Dic7—22no—
120C8×Dic7—41no—
121#6383C2×C414yes—
122#6383Q814yescomplemented
123#6381—41no—
124#6381C2×C414yescomplemented
125C4×C7⋊C8—41no—
126C2×#474—71no—
127#2365—71no—
128D4×C4⋊C4—71no—
129#1990—71no—
130#1947—71no—
131#1942—71no—
132#1942—71no—
133#1918—71no—
134#1918D711yescharacteristic, complemented
135#1862—71no—
136#1862—71no—
137C8×C4⋊C4—71no—
138C4×#102—71no—
139C4×#102—71no—
140C4×#102—71no—
141C23×D7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C2×D28—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28C22×C412yes—
160C2×D28C2×Q812yes—
161C2×D28#4112yes—
162C2×C4×D7—24no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—81no—
170C2×C4×D7—81no—
171C2×Dic14—24no—
172C2×Dic14—42no—
173C2×Dic14—42no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C22×C412yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4C22×C412yes—
187C7×C4⋊C4C22×C412yes—
188C7×C4⋊C4C2×Q812yes—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—42no—
192(C2×D7)⋊C4—24no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—42no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7—42no—
202C4⋊Dic7C22×C412yes—
203Dic7⋊C4—24no—
204Dic7⋊C4—22no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—81no—
208Dic7⋊C4D814yes—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C4×Dic7C22×C412yes—
214C4×Dic7#4112yes—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C22×C412yes—
220C2×C7⋊C8C2×Q812yes—
221C2×C4⋊Q8—71no—
222C2×C4×Q8—141no—
223C2×C4×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4—71no—
226C22×C4⋊C4—141no—
227#474—78no—
228C2×#107—71no—
229C2×#107—71no—
230C2×#107—71no—
231C2×#107C2×D711yescharacteristic
232C2×C4⋊C8—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—71no—
237C2×C4×C8—71no—
238C2×C4×C8—71no—
239C2×C4×C8—71no—
240#391—141no—
241#389—71no—
242#389—71no—
243#389—71no—
244#389C2×D711yescharacteristic
245#387—141no—
246#387—141no—
247#385—141no—
248#385—141no—
249#384—71no—
250#380—141no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4C2×D711yescharacteristic
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340C2×D711yescharacteristic
261#340C2×D711yescharacteristic
262#340C2×D711yescharacteristic
263#336—71no—
264#336C2×D711yescharacteristic
265C23×C7C22×D411yescharacteristic
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
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×C7—41no—
290C2×C4×C7C2×D812yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—22no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7C4×D412yes—
314C2×Dic7C4×D412yes—
315C2×Dic7C2×D812yes—
316C2×Dic7C2×D812yes—
317C2×Dic7#13512yes—
318D28—22no—
319D28—22no—
320D28—22no—
321D28C4×Q814yescomplemented
322C4×D7—48no—
323C4×D7—84no—
324C4×D7—84no—
325Dic14—48no—
326C7×C8—22no—
327C7×C8—41no—
328C7⋊C8—22no—
329C7⋊C8—22no—
330C22×Q8—141no—
331C22×D4—71no—
332C23×C4—141no—
333C23×C4—141no—
334C23×C4—141no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8C22×D711yescharacteristic
338C22×C8C22×D711yescharacteristic
339C4⋊Q8—74no—
340C4⋊Q8—142no—
341C4×Q8—284no—
342C4×D4—74no—
343C4×D4—142no—
344C4×D4—142no—
345C4×D4—142no—
346C4×D4—281no—
347C4×D4—281no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—284no—
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C22⋊C4—141no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C42—21no—
369C2×C42—71no—
370C2×C42—71no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42C22×D711yescharacteristic
374#107—74no—
375#107—281no—
376#107—281no—
377#107C4×D714yescomplemented
378C4⋊C8—74no—
379#102—74no—
380#102—74no—
381#102—74no—
382#102—142no—
383C4×C8—142no—
384C4×C8—281no—
385C4×C8—281no—
386#95—21no—
387#95—21no—
388#95—71no—
389#95—71no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395C22×C7C2×C4×D411yescharacteristic
396C22×C7#54611yescharacteristic
397C22×C7D4×Q811yescharacteristic
398C22×C7C22×D811yescharacteristic
399C22×C7C2×#13511yescharacteristic
400C22×C7#57711yescharacteristic
401C22×C7#57811yescharacteristic
402C2×D7—24no—
403C2×D7—42no—
404C2×D7—42no—
405C2×D7—42no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—42no—
409C4×C7—22no—
410C4×C7—42no—
411C4×C7—42no—
412C4×C7—42no—
413C4×C7—41no—
414C4×C7—81no—
415C4×C7C2×C4×Q811yescharacteristic
416C4×C7#52011yescharacteristic
417C4×C7#55211yescharacteristic
418C4×C7#55411yescharacteristic
419Dic7—22no—
420Dic7—22no—
421Dic7—42no—
422Dic7—41no—
423Dic7—81no—
424Dic7—81no—
425Dic7C4×D814yes—
426C24—141no—
427C2×Q8—144no—
428C2×Q8—282no—
429C2×Q8—282no—
430C2×D4—72no—
431C2×D4—72no—
432C2×D4—72no—
433C2×D4—141no—
434C2×D4—141no—
435C2×D4—141no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—144no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—561no—
455C22×C4—561no—
456C22×C4D4×D711yescharacteristic
457C22×C4D4×D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—72no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8C2×C4×D712yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—74no—
478C4⋊C4—72no—
479C4⋊C4—142no—
480C4⋊C4—144no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—561no—
491C4⋊C4C2×C4×D712yes—
492C4⋊C4C2×C4×D712yes—
493C4⋊C4Q8×D712yes—
494C22⋊C4—144no—
495C22⋊C4—282no—
496C22⋊C4—282no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—72no—
501C42—72no—
502C42—141no—
503C42—141no—
504C42—282no—
505C42—282no—
506C2×C7#236011yescharacteristic
507C2×C7C2×C4×D811yescharacteristic
508C2×C7#294711yescharacteristic
509C2×C7#329411yescharacteristic
510C2×C7#330211yescharacteristic
511C2×C7Q8×D811yescharacteristic
512C2×C7#338111yescharacteristic
513D7—44no—
514C23—142no—
515C23—142no—
516C23—142no—
517C23—281no—
518C23—281no—
519C23—281no—
520C23—281no—
521C23C2×D4×D711yescharacteristic
522Q8—288no—
523D4—74no—
524D4—142no—
525D4—142no—
526D4—142no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—72no—
547C2×C4—72no—
548C2×C4—72no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—142no—
556C2×C4—141no—
557C2×C4—288no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—564no—
567C2×C4—564no—
568C2×C4D7×D812yes—
569C2×C4C22×C4×D711yescharacteristic
570C2×C4C2×Q8×D711yescharacteristic
571C2×C4D7×#4111yescharacteristic
572C2×C4D7×#4111yescharacteristic
573C2×C4#655311yescharacteristic
574C2×C4#655411yescharacteristic
575C8—22no—
576C8—41no—
577C8—142no—
578C8—142no—
579C7#2296211yescharacteristic, complemented, Sylow
580C22—144no—
581C22—282no—
582C22—282no—
583C22—282no—
584C22—282no—
585C22—282no—
586C22—282no—
587C22C4×D4×D711yescharacteristic
588C22#9032311yescharacteristic
589C22#9034511yescharacteristic
590C22C2×D7×D811yescharacteristic
591C22D7×#13511yescharacteristic
592C22#9046911yescharacteristic
593C22#9047011yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—74no—
600C4—81no—
601C4—142no—
602C4—142no—
603C4—281no—
604C4—282no—
605C4—561no—
606C4—561no—
607C4#9021411yescharacteristic
608C4C4×Q8×D711yescharacteristic
609C4#9038111yescharacteristic
610C4#9039911yescharacteristic
611C2—284no—
612C2#1173241011yescharacteristic
613C2C4×D7×D811yescharacteristic
614C2#1173727511yescharacteristic
615C2#1173840211yescharacteristic
616C2#1173866611yescharacteristic
617C2#1173874711yescharacteristic
618C2#1173898911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1918C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1918D7
hypocenterC7#22962
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
3. #11731097C2semidirect
4. C7×#1918C2semidirect
5. #11728366C2semidirect
6. #11727373C2semidirect
9. #11726569C2semidirect
10. #11726557C2semidirect
11. #11726549C2semidirect
15. #11726405C2semidirect
45. #89662C4semidirect
101. #6451C2×C4semidirect
105. C7×#107C2×C4semidirect
118. Dic7⋊C8C2×C4semidirect
122. #6383Q8semidirect
124. #6381C2×C4semidirect
134. #1918D7semidirect
321. D28C4×Q8semidirect
377. #107C4×D7semidirect
579. C7#22962semidirect

Maximal subgroups

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

subgroupindex
C2×#896622
#117336472
#117310972
C7×#19182
#117283662
#117273732
C4×#63832
#117266482
#117265692
#117265572
#117265492
#117265342
#117264242
#117264052
#229627