Finite Group Databasebeta

← #13375596

Subgroups — #13375596

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
1D4×Dic7⋊C4C211yesmaximal, characteristic
2C2×#89531C211yesmaximal, characteristic
3#11731097C211yesmaximal, characteristic, complemented
4C4×C7×#102C211yesmaximal, characteristic
5#11729360C211yesmaximal, characteristic
6#11727922C211yesmaximal, characteristic
7Dic7×#102C211yesmaximal, characteristic
8Dic7×#102C211yesmaximal, characteristic
9C8×Dic7⋊C4C211yesmaximal, characteristic, complemented
10#11727524C211yesmaximal, characteristic, complemented
11#11727212C211yesmaximal, characteristic, complemented
12#11726613C211yesmaximal, characteristic, complemented
13#11726568C211yesmaximal, characteristic, complemented
14#11726557C211yesmaximal, characteristic, complemented
15#11726549C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C22×Dic7⋊C4—21no—
20C2×#6451C2211yescharacteristic
21C2×C4×Dic14—21no—
22C2×C7×#102C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28#89982—21no—
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34#89757C2211yescharacteristic
35#89748C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38#89745C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89718—21no—
42#89714—21no—
43#89705—21no—
44C4×Dic7⋊C4C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89531C418yescomplemented
48#89427—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22962—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×Dic7—21no—
64C23×Dic7—21no—
65C22×Dic14—21no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C4×C7×D4—22no—
69C4×C7×D4—41no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78D4×Dic7—22no—
79D4×Dic7—22no—
80D4×Dic7—41no—
81D4×Dic7C2×C414yes—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—44no—
94C2×Dic7⋊C4D411yescharacteristic
95C2×Dic7⋊C4D411yescharacteristic
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7C2311yescharacteristic
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6451—22no—
103#6451C2×C414yescomplemented
104C4×Dic14—44no—
105C7×#102C2×C414yes—
106C7×#102Q814yescomplemented
107C4×C7×C8—41no—
108C7×#95—21no—
109#6406—22no—
110#6406C2×C414yes—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6392—41no—
120#6392C2×C414yescomplemented
121Dic7⋊C8C2×C414yescomplemented
122C8×Dic7—22no—
123C8×Dic7—41no—
124#6381—41no—
125#6381C2×C414yescomplemented
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#1918—71no—
135#1862—71no—
136#1862—71no—
137C8×C4⋊C4—71no—
138C4×#102—71no—
139C4×#102—71no—
140C4×#102D711yescharacteristic
141C24×C7—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4C22×C412yes—
146C2×C7×D4C2×Q812yes—
147C2×C7×D4#4112yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—81no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—24no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—42no—
163C22×Dic7—42no—
164C22×Dic7—42no—
165C22×Dic7—42no—
166C22×Dic7—81no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
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×C8C22×C412yes—
179C2×C7×C8C2×Q812yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4C22×C412yes—
185C7×C4⋊C4C2×Q812yes—
186C7×C22⋊C4—42no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—42no—
190C22⋊Dic7—24no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—42no—
199C4⋊Dic7C22×C412yes—
200C4⋊Dic7C22×C412yes—
201Dic7⋊C4—24no—
202Dic7⋊C4—22no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
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⋊C8—21no—
220C2×C7⋊C8C22×C412yes—
221C2×C4⋊Q8—71no—
222C2×C4×Q8—141no—
223C2×C4×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4C2×D711yescharacteristic
226C22×C4⋊C4—141no—
227#474—78no—
228C2×#107—71no—
229C2×#107—71no—
230C2×#107—71no—
231C2×#107—71no—
232C2×C4⋊C8—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#102C2×D711yescharacteristic
236C2×#102C2×D711yescharacteristic
237C2×C4×C8—71no—
238C2×C4×C8—71no—
239C2×C4×C8C2×D711yescharacteristic
240#391—141no—
241#389—71no—
242#389—71no—
243#389—71no—
244#389—71no—
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#340—71no—
261#340C2×D711yescharacteristic
262#340C2×D711yescharacteristic
263#336—71no—
264#336—71no—
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7C22×D411yescharacteristic
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—22no—
276C7×D4C4×Q814yescomplemented
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
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×C7—41no—
291C2×C4×C7—84no—
292C2×C4×C7C2×D812yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7H(5,𝔽2)11yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—48no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—84no—
320C2×Dic7C4×D412yes—
321C2×Dic7C4×D412yes—
322C2×Dic7C2×D812yes—
323C2×Dic7C2×D812yes—
324C2×Dic7#13512yes—
325Dic14—48no—
326C7×C8—22no—
327C7×C8—22no—
328C7⋊C8—22no—
329C7⋊C8—41no—
330C22×Q8—141no—
331C22×D4C22×D711yescharacteristic
332C23×C4—21no—
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—22no—
343C4×D4—41no—
344C4×D4—74no—
345C4×D4—142no—
346C4×D4—142no—
347C4×D4—281no—
348C2×C4⋊C4—71no—
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—141no—
361C2×C4⋊C4—284no—
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C22⋊C4—21no—
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—74no—
376#107—281no—
377#107—281no—
378C4⋊C8—74no—
379#102—74no—
380#102—142no—
381#102Dic1414yescomplemented
382#102C4×D714yes—
383C4×C8—41no—
384C4×C8—142no—
385C4×C8—281no—
386#95—21no—
387#95—71no—
388#95—71no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395C22×C7—24no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7C2×C4×D411yescharacteristic
403C22×C7#54611yescharacteristic
404C22×C7D4×Q811yescharacteristic
405C22×C7C22×D811yescharacteristic
406C22×C7C2×#13511yescharacteristic
407C22×C7#57711yescharacteristic
408C22×C7#57811yescharacteristic
409C4×C7—22no—
410C4×C7—42no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—81no—
414C4×C7C2×C4×Q811yescharacteristic
415C4×C7#52011yescharacteristic
416C4×C7#55211yescharacteristic
417C4×C7#55411yescharacteristic
418Dic7—22no—
419Dic7—22no—
420Dic7—42no—
421Dic7—42no—
422Dic7—41no—
423Dic7—81no—
424Dic7—81no—
425Dic7C4×D814yes—
426C24—21no—
427C2×Q8—144no—
428C2×Q8—282no—
429C2×Q8—282no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×Dic1412yes—
434C2×D4C2×C4×D712yes—
435C2×D4#115712yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—81no—
446C22×C4—144no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—282no—
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—72no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8C2×Dic1412yes—
470C2×C8C2×C4×D712yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—74no—
476C4⋊C4—72no—
477C4⋊C4—72no—
478C4⋊C4—142no—
479C4⋊C4—144no—
480C4⋊C4—141no—
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—282no—
491C4⋊C4—561no—
492C4⋊C4C2×Dic1412yes—
493C4⋊C4C2×C4×D712yes—
494C22⋊C4—42no—
495C22⋊C4—144no—
496C22⋊C4—282no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—72no—
501C42—72no—
502C42—141no—
503C42—141no—
504C42—282no—
505C42—282no—
506C2×C7—44no—
507C2×C7#236011yescharacteristic
508C2×C7C2×C4×D811yescharacteristic
509C2×C7#294711yescharacteristic
510C2×C7#329411yescharacteristic
511C2×C7#330211yescharacteristic
512C2×C7Q8×D811yescharacteristic
513C2×C7#338111yescharacteristic
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D711yescharacteristic
522Q8—288no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4C4×Dic1414yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—72no—
542C2×C4—72no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—84no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—288no—
557C2×C4—281no—
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—281no—
567C2×C4—564no—
568C2×C4D7×D812yes—
569C2×C4C22×Dic1411yescharacteristic
570C2×C4C22×C4×D711yescharacteristic
571C2×C4C2×#115711yescharacteristic
572C2×C4#654811yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4#655411yescharacteristic
575C8—22no—
576C8—22no—
577C8—142no—
578C8—281no—
579C7#2296211yescharacteristic, complemented, Sylow
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22#9018111yescharacteristic
588C22D4×Dic1411yescharacteristic
589C22C4×D4×D711yescharacteristic
590C22C2×D7×D811yescharacteristic
591C22#9045311yescharacteristic
592C22D7×#13511yescharacteristic
593C22#9046211yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—74no—
599C4—81no—
600C4—142no—
601C4—142no—
602C4—282no—
603C4—282no—
604C4—281no—
605C4—561no—
606C4—561no—
607C4C2×C4×Dic1411yescharacteristic
608C4#9022211yescharacteristic
609C4#9023011yescharacteristic
610C4#9023711yescharacteristic
611C2—44no—
612C2#1173091111yescharacteristic
613C2#1173620211yescharacteristic
614C2#1173629411yescharacteristic
615C2D8×Dic1411yescharacteristic
616C2C4×D7×D811yescharacteristic
617C2#1173707811yescharacteristic
618C2#1173709311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#102C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC4×#102D7
hypocenterC7#22962
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #11731097C2semidirect
9. C8×Dic7⋊C4C2semidirect
10. #11727524C2semidirect
11. #11727212C2semidirect
12. #11726613C2semidirect
13. #11726568C2semidirect
14. #11726557C2semidirect
15. #11726549C2semidirect
47. #89531C4semidirect
103. #6451C2×C4semidirect
106. C7×#102Q8semidirect
120. #6392C2×C4semidirect
121. Dic7⋊C8C2×C4semidirect
125. #6381C2×C4semidirect
276. C7×D4C4×Q8semidirect
381. #102Dic14semidirect
526. D4C4×Dic14semidirect
579. C7#22962semidirect

Maximal subgroups

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

subgroupindex
D4×Dic7⋊C42
C2×#895312
#117310972
C4×C7×#1022
#117293602
#117279222
Dic7×#1022
C8×Dic7⋊C42
#117275242
#117272122
#117266132
#117265682
#117265572
#117265492
#229627