Finite Group Databasebeta

← #13276419

Subgroups — #13276419

Order 1792.

1630 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
2C2×#89723C212yesmaximal
3#11729881C211yesmaximal, characteristic
4#11729881C211yesmaximal, characteristic
5#11729300C212yesmaximal
6#11729289C211yesmaximal, characteristic, complemented
7#11729289C211yesmaximal, characteristic, complemented
8#11729288C211yesmaximal, characteristic, complemented
9C4×C7⋊#95C211yesmaximal, characteristic, complemented
10#11727166C212yesmaximal, complemented
11#11727161C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C22×C42×C7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2212yes—
19C22×C4⋊Dic7C2212yes—
20C22×C4⋊Dic7C2212yes—
21C22×C4×Dic7C2212yes—
22C4×C7×C22⋊C4C418yescomplemented
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36#89723—28no—
37#13948—71nomaximal, Sylow
38C25×C7C2311yescharacteristic
39C23×C4×C7—22no—
40C23×C4×C7—22no—
41C23×C4×C7—42no—
42C23×C4×C7—42no—
43C23×C4×C7C2×C414yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2312yes—
52C23×Dic7C2312yes—
53C23×Dic7C2312yes—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4C2×C414yes—
57C2×C7×C22⋊C4C2×C414yes—
58C2×C7×C22⋊C4D414yes—
59C2×C7×C22⋊C4Q814yes—
60C2×C42×C7—42no—
61C2×C42×C7—42no—
62C2×C42×C7C2×C414yes—
63C2×C42×C7C2×C414yes—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—42no—
69C2×C22⋊Dic7—42no—
70C2×C4⋊Dic7—24no—
71C2×C4⋊Dic7—24no—
72C2×C4⋊Dic7—24no—
73C2×C4⋊Dic7—24no—
74C2×C4⋊Dic7—42no—
75C2×C4⋊Dic7—42no—
76C2×C4×Dic7—24no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—42no—
79C7×#95C2×C414yes—
80C7×#95C2×C414yes—
81C7⋊#95—24no—
82C7⋊#95—24no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—24no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95—24no—
89C7⋊#95—42no—
90C7⋊#95—42no—
91C7⋊#95—42no—
92C2×#388—72no—
93C2×C4×C22⋊C4D711yescharacteristic
94#1442—72no—
95#1439—72no—
96#1436—72no—
97C4⋊#95—71no—
98C4⋊#95—71no—
99#1434—71no—
100C22⋊#95—71no—
101C22⋊#95—71no—
102C4×#95—71no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—21no—
106C24×C7—21no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7C22×C412yes—
112C24×C7C2×D412yes—
113C24×C7C2×Q812yes—
114C24×C7C2411yescharacteristic
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—28no—
132C22×C4×C7—44no—
133C22×C4×C7—44no—
134C22×C4×C7—48no—
135C22×C4×C7—48no—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C22×C412yes—
141C22×C4×C7C22×C412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×Q812yes—
145C22×C4×C7C2×Q812yes—
146C22×C4×C7#4112yes—
147C22×C4×C7#4112yes—
148C22×C4×C7#4112yes—
149C22×C4×C7#4112yes—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7#4114yes—
177C22×Dic7#4114yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4C4⋊C4116yescomplemented
180C42×C7—44no—
181C42×C7—44no—
182C22⋊Dic7—48no—
183C22⋊Dic7—48no—
184C4⋊Dic7—48no—
185C4⋊Dic7—48no—
186C4×Dic7—48no—
187C24×C4C2×D711yescharacteristic
188C22×C4⋊C4—72no—
189C22×C4⋊C4—72no—
190C22×C22⋊C4—72no—
191C22×C22⋊C4—72no—
192C22×C22⋊C4C2×D711yescharacteristic
193C22×C22⋊C4C2×D711yescharacteristic
194C22×C42—72no—
195C22×C42C2×D711yescharacteristic
196C22×C42C2×D711yescharacteristic
197#388—148no—
198C4×C22⋊C4Dic718yescomplemented
199C2×#95—71no—
200C2×#95—71no—
201C2×#95—72no—
202C2×#95—71no—
203C2×#95—71no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95C2×D711yescharacteristic
211C2×#95C2×D711yescharacteristic
212C23×C7—22no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7C2×C4⋊C414yes—
235C23×C7C23×C411yescharacteristic
236C23×C7C22×D411yescharacteristic
237C23×C7C22×Q811yescharacteristic
238C23×C7C2×#4111yescharacteristic
239C23×C7C2×#4111yescharacteristic
240C23×C7C2×#4111yescharacteristic
241C23×C7C2×#4111yescharacteristic
242C23×C7C2×#4112yes—
243C23×C7C2×#4112yes—
244C23×C7H(5,𝔽2)12yes—
245C23×C7H(5,𝔽2)12yes—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—416no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7C2×C4⋊C414yes—
274C2×C4×C7C2×C4⋊C414yes—
275C2×C4×C7#11714yes—
276C2×C4×C7#11714yes—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7—24no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C25C22×D711yescharacteristic
306C23×C4—22no—
307C23×C4—22no—
308C23×C4—42no—
309C23×C4—42no—
310C23×C4—72no—
311C23×C4—72no—
312C23×C4—72no—
313C23×C4—72no—
314C23×C4C2×Dic714yes—
315C23×C4C22×D711yescharacteristic
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C23×C4C22×D711yescharacteristic
319C23×C4C22×D711yescharacteristic
320C23×C4C22×D711yescharacteristic
321C2×C4⋊C4—144no—
322C2×C4⋊C4—144no—
323C2×C4⋊C4—144no—
324C2×C4⋊C4—144no—
325C2×C4⋊C4—282no—
326C2×C4⋊C4—282no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—144no—
330C2×C22⋊C4—144no—
331C2×C22⋊C4—144no—
332C2×C22⋊C4—144no—
333C2×C22⋊C4—282no—
334C2×C22⋊C4—282no—
335C2×C22⋊C4Dic1414yes—
336C2×C22⋊C4D2814yes—
337C2×C22⋊C4C2×Dic714yes—
338C2×C22⋊C4C2×Dic714yes—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—144no—
342C2×C42—144no—
343C2×C42—282no—
344C2×C42C2×Dic714yes—
345C2×C42C2×Dic714yes—
346#95—144no—
347#95—144no—
348#95—144no—
349#95—142no—
350#95—142no—
351#95—144no—
352#95—142no—
353#95—142no—
354#95—282no—
355#95—282no—
356#95—282no—
357#95C2×Dic714yes—
358#95C2×Dic714yes—
359C22×C7—24no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7—42no—
363C22×C7—44no—
364C22×C7—44no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7C22×C4⋊C411yescharacteristic
371C22×C7C2×#11711yescharacteristic
372C22×C7C2×#11711yescharacteristic
373C22×C7C4×#4111yescharacteristic
374C22×C7C4×#4111yescharacteristic
375C22×C7#51811yescharacteristic
376C22×C7#51811yescharacteristic
377C22×C7#52912yes—
378C22×C7#52912yes—
379C22×C7#53211yescharacteristic
380C22×C7#53211yescharacteristic
381C22×C7#53311yescharacteristic
382C22×C7#53311yescharacteristic
383C22×C7#53812yes—
384C22×C7#53812yes—
385C22×C7#54612yes—
386C22×C7#54612yes—
387C22×C7#55112yes—
388C22×C7#55312yes—
389C22×C7#55312yes—
390C22×C7#55412yes—
391C22×C7#55412yes—
392C22×C7#55912yes—
393C4×C7—24no—
394C4×C7—24no—
395C4×C7—24no—
396C4×C7—24no—
397C4×C7—42no—
398C4×C7—42no—
399Dic7—44no—
400Dic7—44no—
401Dic7—82no—
402Dic7—82no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24C2×Dic1412yes—
412C24C2×D2812yes—
413C24C22×Dic712yes—
414C24C23×D711yescharacteristic
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—74no—
437C22×C4—74no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4C2×Dic1412yes—
465C22×C4C2×Dic1412yes—
466C22×C4C2×D2812yes—
467C22×C4C2×D2812yes—
468C22×C4#115712yes—
469C22×C4#115712yes—
470C22×C4#115712yes—
471C22×C4#115712yes—
472C22×C4C22×Dic712yes—
473C22×C4C22×Dic712yes—
474C22×C4C22×Dic712yes—
475C22×C4C22×Dic712yes—
476C22×C4C22×Dic712yes—
477C22×C4C22×Dic712yes—
478C4⋊C4—288no—
479C4⋊C4—288no—
480C22⋊C4—28no—
481C22⋊C4—288no—
482C22⋊C4—288no—
483C22⋊C4C4⋊Dic7116yescomplemented
484C42—44no—
485C42—44no—
486C42—288no—
487C2×C7—44no—
488C2×C7#228511yescharacteristic
489C2×C7#229811yescharacteristic
490C2×C7#231211yescharacteristic
491C2×C7#234911yescharacteristic
492C2×C7#234911yescharacteristic
493C2×C7#235311yescharacteristic
494C2×C7#235711yescharacteristic
495C2×C7#255712yes—
496C2×C7#255912yes—
497C2×C7#258712yes—
498C2×C7#258912yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C2×C4⋊Dic714yes—
522C23C22×Dic1411yescharacteristic
523C23C22×D2811yescharacteristic
524C23C2×#115711yescharacteristic
525C23C2×#115711yescharacteristic
526C23C2×#115711yescharacteristic
527C23C2×#115711yescharacteristic
528C23#654812yes—
529C23#654812yes—
530C23D7×#4112yes—
531C23D7×#4112yes—
532C23C23×Dic711yescharacteristic
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—144no—
561C2×C4—144no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—144no—
565C2×C4—144no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4—282no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4—282no—
580C2×C4—282no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4C2×C4⋊Dic714yes—
589C2×C4C2×C4⋊Dic714yes—
590C2×C4#648914yes—
591C2×C4#648914yes—
592C7#1394811yescharacteristic, complemented, Sylow
593C22—24no—
594C22—24no—
595C22—24no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22#9016111yescharacteristic
605C22#9016111yescharacteristic
606C22#9016811yescharacteristic
607C22#9016811yescharacteristic
608C22#9018512yes—
609C22#9018512yes—
610C22#9021912yes—
611C22#9021912yes—
612C22#9023012yes—
613C22#9023012yes—
614C22#9023112yes—
615C22#9024512yes—
616C22#9024512yes—
617C22#9024712yes—
618C22#9024912yes—
619C22#9025012yes—
620C22C22×C4⋊Dic711yescharacteristic
621C22C2×#648911yescharacteristic
622C22C2×#648911yescharacteristic
623C22#9048911yescharacteristic
624C22#9048911yescharacteristic
625C22#41×Dic711yescharacteristic
626C22#41×Dic711yescharacteristic
627C4—24no—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—284no—
634C4—284no—
635C4—562no—
636C4—562no—
637C2—44no—
638C2#1173099312yes—
639C2#1173099412yes—
640C2#1173104712yes—
641C2#1173104912yes—
642C2#1173950711yescharacteristic
643C2#1173953811yescharacteristic
644C2#1173967411yescharacteristic
645C2#1173977811yescharacteristic
646C2#1173977811yescharacteristic
647C2#1173978111yescharacteristic
648C2#1173978311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C4×C7×C22⋊C4C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×C4×C22⋊C4D7
hypocenterC7#13948
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #11729289C2semidirect
7. #11729289C2semidirect
8. #11729288C2semidirect
9. C4×C7⋊#95C2semidirect
10. #11727166C2semidirect
11. #11727161C2semidirect
22. C4×C7×C22⋊C4C4semidirect
179. C7×C22⋊C4C4⋊C4semidirect
198. C4×C22⋊C4Dic7semidirect
483. C22⋊C4C4⋊Dic7semidirect
592. C7#13948semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×C22⋊C42
C2×#897232
#117298812
#117293002
#117292892
#117292882
C4×C7⋊#952
#117271662
#117271612
#139487