Finite Group Databasebeta

← #13371066

Subgroups — #13371066

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×C22⋊C4C211yesmaximal, characteristic
2C2×#89997C211yesmaximal, characteristic
3#11730905C211yesmaximal, characteristic
4#11729886C211yesmaximal, characteristic
5#11729385C211yesmaximal, characteristic, complemented
6C4×#6409C211yesmaximal, characteristic, complemented
7#11729383C211yesmaximal, characteristic, complemented
8#11729382C211yesmaximal, characteristic, complemented
9#11727429C211yesmaximal, characteristic, complemented
10#11727428C211yesmaximal, characteristic, complemented
11#11727421C211yesmaximal, characteristic, complemented
12#11727420C211yesmaximal, characteristic, complemented
13#11727267C211yesmaximal, characteristic
14#11727257C211yesmaximal, characteristic
15#11727202C211yesmaximal, characteristic
16C23×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C7×#391—21no—
22C7×#391—21no—
23C7×#391C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#380—21no—
26C7×#380C2211yescharacteristic
27C4×C7×C22⋊C4—21no—
28C4×C7×C22⋊C4C2211yescharacteristic
29#89997C418yescomplemented
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36C2×#6409C2211yescharacteristic
37C2×#6409C2211yescharacteristic
38#89984C2211yescharacteristic
39C2×C22⋊(C7⋊C8)C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89717—21no—
43#89717—21no—
44C4×C4⋊Dic7C2211yescharacteristic
45C4×C4⋊Dic7C2211yescharacteristic
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48C2×#6377C2211yescharacteristic
49C2×C4×C7⋊C8C2211yescharacteristic
50#89423—21no—
51#89421—21no—
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22581—71nomaximal, Sylow
61C22×C7×Q8—21no—
62C22×C7×Q8—22no—
63C22×C7×Q8—22no—
64C22×C7×Q8—22no—
65C22×C7×Q8—41no—
66C22×C7×Q8—41no—
67C22×C7×Q8C2×C414yes—
68C22×C7×Q8C2311yescharacteristic
69C22×C7×Q8C2311yescharacteristic
70C23×C4×C7—21no—
71C23×C4×C7C2311yescharacteristic
72C4×C7×Q8—22no—
73C4×C7×Q8—22no—
74C4×C7×Q8—41no—
75C4×C7×Q8—41no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C42×C7—21no—
84C2×C42×C7—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—21no—
89#6489—22no—
90#6489C2×C414yes—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×#1136—41no—
101C2×#1136—41no—
102C2×#1136C2×C414yes—
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114#6409—22no—
115#6409—22no—
116#6409—41no—
117#6409—41no—
118#6409C2×C414yes—
119#6409C2×C414yes—
120#6409C2×C414yes—
121#6409C2×C414yes—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C22⋊(C7⋊C8)—22no—
128#6382—41no—
129#6381—41no—
130#6377—41no—
131C4×C7⋊C8—41no—
132C2×#419—71no—
133#2347—71no—
134Q8×C22⋊C4D711yescharacteristic
135#1935—71no—
136#1933—71no—
137#1928—71no—
138#1927—71no—
139#1865—71no—
140#1863—71no—
141#1825—71no—
142#1809—71no—
143#1787—71no—
144#1763—71no—
145C4×#103—71no—
146#1753—71no—
147C24×C7C2×D411yescharacteristic
148C2×C7×Q8—24no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—28no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—42no—
159C2×C7×Q8—42no—
160C2×C7×Q8—44no—
161C2×C7×Q8—44no—
162C2×C7×Q8—44no—
163C2×C7×Q8C22×C412yes—
164C2×C7×Q8C22×C412yes—
165C2×C7×Q8#4112yes—
166C2×C7×Q8#4112yes—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—24no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7C22×C412yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7#4112yes—
185C22×C4×C7C2411yescharacteristic
186C22×Dic7—21no—
187C22×Dic7—21no—
188C22×Dic7—21no—
189C22×Dic7—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4—42no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—41no—
198C7×C22⋊C4—42no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—42no—
204C42×C7—42no—
205C22⋊Dic7—24no—
206C22⋊Dic7—81no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7C22×C412yes—
214C4⋊Dic7C22×C412yes—
215Dic7⋊C4—42no—
216Dic7⋊C4—42no—
217C4×Dic7—24no—
218C4×Dic7—81no—
219C4×Dic7—81no—
220#1136—44no—
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⋊C8—21no—
231C2×C7⋊C8C22×C412yes—
232C2×C7⋊C8C22×C412yes—
233C23×Q8C2×D711yescharacteristic
234C22×M16—71no—
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×Q8C2×D711yescharacteristic
237C2×#117—71no—
238C2×#107—71no—
239C2×#106—71no—
240#419—78no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×C22⋊C8—71no—
250C2×#97—71no—
251C2×C4×C8—71no—
252#391—21no—
253#391—21no—
254#391C2×D711yescharacteristic
255#391C2×D711yescharacteristic
256#389—71no—
257#389—71no—
258#388—141no—
259#388—141no—
260#383—141no—
261#380—21no—
262#380—71no—
263#380C2×D711yescharacteristic
264C4×C4⋊C4—71no—
265C4×C4⋊C4—71no—
266C4×C4⋊C4—141no—
267C4×C22⋊C4—21no—
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×C7C4×D412yes—
284C23×C7C22×D411yescharacteristic
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—22no—
288C7×Q8—22no—
289C7×Q8—22no—
290C7×Q8—48no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×Q8#11714yes—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—48no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7C4×D412yes—
325C2×C4×C7#13712yes—
326C2×C4×C7C23×C411yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7C2×#4111yescharacteristic
330C2×C4×C7H(5,𝔽2)11yescharacteristic
331C2×C4×C7H(5,𝔽2)11yescharacteristic
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C7⋊C8—22no—
345C7⋊C8—22no—
346C7⋊C8—41no—
347C7⋊C8—41no—
348C22×Q8—22no—
349C22×Q8—21no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—41no—
353C22×Q8—41no—
354C22×Q8C4×D714yes—
355C22×Q8C22×D711yescharacteristic
356C22×Q8C22×D711yescharacteristic
357C23×C4—21no—
358C23×C4C22×D711yescharacteristic
359C2×M16—74no—
360C2×M16—281no—
361C2×M16—281no—
362C22×C8—71no—
363C22×C8—71no—
364C22×C8—71no—
365C22×C8—71no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—41no—
369C4×Q8—41no—
370#117—74no—
371#117—142no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4C22⋊D711yescharacteristic
386C2×C22⋊C4C22⋊D711yescharacteristic
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—141no—
390C2×C42—141no—
391C2×C42—141no—
392C2×C42C22×D711yescharacteristic
393C2×C42C22×D711yescharacteristic
394#107—281no—
395#106—281no—
396#103—74no—
397#103—74no—
398#103—74no—
399#103—74no—
400#103—142no—
401#103—142no—
402#103—281no—
403#103—281no—
404C22⋊C8—142no—
405#97—281no—
406C4×C8—281no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—141no—
413#95—141no—
414#95—141no—
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×C4×D411yescharacteristic
427C22×C7#54611yescharacteristic
428C22×C7#54611yescharacteristic
429C22×C7C2×#13711yescharacteristic
430C22×C7#57511yescharacteristic
431C22×C7#57711yescharacteristic
432C22×C7#57811yescharacteristic
433C4×C7—22no—
434C4×C7—24no—
435C4×C7—22no—
436C4×C7—41no—
437C4×C7—42no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—81no—
441C4×C7—81no—
442C4×C7C2×#11711yescharacteristic
443C4×C7#51811yescharacteristic
444C4×C7#55111yescharacteristic
445C4×C7#55111yescharacteristic
446Dic7—42no—
447Dic7—42no—
448Dic7—81no—
449Dic7—81no—
450C24C2×C22⋊D711yescharacteristic
451C2×Q8—24no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—28no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8C2×C4×D712yes—
467C2×Q8C2×C4×D712yes—
468C2×Q8#115912yes—
469C2×Q8#115912yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4C2×C4×D712yes—
486C22×C4#115712yes—
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C23×D711yescharacteristic
493M16—284no—
494C2×C8—72no—
495C2×C8—72no—
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—
505C2×C8—141no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—72no—
513C4⋊C4—72no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—282no—
521C4⋊C4—282no—
522C22⋊C4—22no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—144no—
526C22⋊C4—561no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—42no—
532C42—42no—
533C42—144no—
534C42—561no—
535C42—561no—
536C2×C7—42no—
537C2×C7#234211yescharacteristic
538C2×C7C4×#13711yescharacteristic
539C2×C7#294711yescharacteristic
540C2×C7#331311yescharacteristic
541C2×C7#332011yescharacteristic
542C2×C7#332611yescharacteristic
543C2×C7#332811yescharacteristic
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23C4×C22⋊D712yes—
551C23C22×C22⋊D711yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—48no—
558Q8—41no—
559Q8—41no—
560Q8#644014yes—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4C4×C22⋊D712yes—
604C2×C4#650512yes—
605C2×C4C22×C4×D711yescharacteristic
606C2×C4C2×#115711yescharacteristic
607C2×C4C2×#115911yescharacteristic
608C2×C4C2×#115911yescharacteristic
609C2×C4#654811yescharacteristic
610C2×C4#654811yescharacteristic
611C8—142no—
612C8—142no—
613C8—281no—
614C8—281no—
615C7#2258111yescharacteristic, complemented, Sylow
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D711yescharacteristic
624C22#9049711yescharacteristic
625C22#9049711yescharacteristic
626C22C2×#650511yescharacteristic
627C22#9051311yescharacteristic
628C22#9052811yescharacteristic
629C22#9052911yescharacteristic
630C4—22no—
631C4—22no—
632C4—24no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—282no—
640C4—282no—
641C4—561no—
642C4—561no—
643C4C2×#644011yescharacteristic
644C4#9017711yescharacteristic
645C4#9018611yescharacteristic
646C4#9018611yescharacteristic
647C2—42no—
648C2#1173954311yescharacteristic
649C2C4×#650511yescharacteristic
650C2#1173999211yescharacteristic
651C2#1174098211yescharacteristic
652C2#1174098511yescharacteristic
653C2#1174099811yescharacteristic
654C2#1174099911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×Q8×C22⋊C4C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterQ8×C22⋊C4D7
hypocenterC7#22581
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. #11729385C2semidirect
6. C4×#6409C2semidirect
7. #11729383C2semidirect
8. #11729382C2semidirect
9. #11727429C2semidirect
10. #11727428C2semidirect
11. #11727421C2semidirect
12. #11727420C2semidirect
29. #89997C4semidirect
615. C7#22581semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×C22⋊C42
C2×#899972
#117309052
#117298862
#117293852
C4×#64092
#117293832
#117293822
#117274292
#117274282
#117274212
#117274202
#117272672
#117272572
#117272022
#225817