Finite Group Databasebeta

← #13321641

Subgroups — #13321641

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3506C211yesmaximal, characteristic
2#11739781C211yesmaximal, characteristic, complemented
3#11739747C211yesmaximal, characteristic, complemented
4#11729591C211yesmaximal, characteristic, complemented
5#11729590C211yesmaximal, characteristic, complemented
6#11729567C211yesmaximal, characteristic, complemented
7#11729566C211yesmaximal, characteristic, complemented
8#11729464C212yesmaximal, complemented
9#11729463C212yesmaximal, complemented
10#11729462C212yesmaximal, complemented
11#11729461C212yesmaximal, complemented
12C7×#559—22no—
13C7×#556C412yescomplemented
14C7×#555C412yescomplemented
15C7×#553—22no—
16C7×#550C412yescomplemented
17C7×#547C412yescomplemented
18C7×#538—21no—
19C7×#538—21no—
20C7×#535—21no—
21C7×#533C2211yescharacteristic
22C7×#532C2211yescharacteristic
23C7×#525C2212yes—
24C7×#525C2212yes—
25C7×#520—21no—
26C2×C4×C7×D4C2211yescharacteristic
27#89992—21no—
28#89984C2211yescharacteristic
29#89983C2211yescharacteristic
30C4×C22⋊Dic7C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32#89979C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34#89783—22no—
35#89783—22no—
36#89753—21no—
37#89723—21no—
38C22⋊C4×Dic7—21no—
39#89709C2211yescharacteristic
40#89704C2211yescharacteristic
41#89698C2211yescharacteristic
42#89698C2211yescharacteristic
43#89333C2212yes—
44#89333C2212yes—
45#89333C2212yes—
46#89333C2212yes—
47#89332C2212yes—
48#89332C2212yes—
49#89332C2212yes—
50#89332C2212yes—
51#17629—71nomaximal, Sylow
52C2×C7×#41—21no—
53C2×C7×#41C2×C412yes—
54C2×C7×#41C2×C412yescomplemented
55C22×C7×D4D411yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C23×Dic7—21no—
59C7×C4⋊Q8C2×C411yescharacteristic, complemented
60C7×C4⋊Q8C2×C411yescharacteristic, complemented
61C7×C4⋊Q8C2×C411yescharacteristic, complemented
62C7×#127—22no—
63C7×#127C2×C411yescharacteristic, complemented
64C7×#126—22no—
65C7×#126—22no—
66C7×#125—21no—
67C7×#124—22no—
68C7×#124—22no—
69C7×#124—22no—
70C7×#124—21no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×C22⋊Q8—21no—
76C7×C22⋊Q8C2×C412yes—
77C7×C22⋊Q8C2×C412yescomplemented
78C7×C22⋊Q8C2312yes—
79C7×#121—22no—
80C7×#121—22no—
81C7×#121—21no—
82C7×#121—22no—
83C7×#121—22no—
84C7×#121C2×C412yes—
85C7×#121C2×C412yescomplemented
86C7×#121C2312yes—
87C7×#120—22no—
88C7×#120—22no—
89C4×C7×Q8—22no—
90C4×C7×Q8—21no—
91C4×C7×Q8C2×C412yescomplemented
92C4×C7×D4—22no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4—22no—
97C4×C7×D4—21no—
98C4×C7×D4C2×C412yes—
99C4×C7×D4C2×C412yes—
100C4×C7×D4C2×C412yescomplemented
101C4×C7×D4D412yes—
102C4×C7×D4D412yes—
103C7×#117—21no—
104C7×#117—22no—
105C7×#117C2×C412yes—
106C7×#117C2×C412yescomplemented
107C2×C7×C4⋊C4—22no—
108C2×C7×C4⋊C4D411yescharacteristic
109C2×C7×C22⋊C4D411yescharacteristic
110C2×C7×C22⋊C4D411yescharacteristic
111C2×C42×C7C2311yescharacteristic
112C2×C22⋊Dic7—21no—
113C2×C22⋊Dic7—21no—
114C2×C22⋊Dic7—21no—
115C2×C4⋊Dic7—21no—
116C2×C4×Dic7—21no—
117C2×C4×Dic7—21no—
118C2×#1136—22no—
119C22×C7⋊C8—22no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95C2311yescharacteristic
126C7⋊#95C2311yescharacteristic
127C7⋊#95C2311yescharacteristic
128C7⋊#95C2311yescharacteristic
129C22⋊(C7⋊C8)—22no—
130C22⋊(C7⋊C8)—22no—
131C22⋊(C7⋊C8)C2312yes—
132C22⋊(C7⋊C8)C2312yes—
133C4⋊(C7⋊C8)—21no—
134C4⋊(C7⋊C8)—21no—
135#6377—21no—
136#6377—21no—
137#3506D711yescharacteristic
138#2973—71no—
139#2353—71no—
140#1516—72no—
141#1515—71no—
142#1514—71no—
143#1513—72no—
144#1506—71no—
145#1505—71no—
146#1504—72no—
147#1501—72no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C7×#41—42no—
151C7×#41—42no—
152C7×#41—82no—
153C7×#41—82no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8C22×C412yes—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—41no—
165C2×C7×D4—41no—
166C2×C7×D4C22⋊C412yes—
167C2×C7×D4C22⋊C412yes—
168C2×C7×D4C22×C412yes—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—81no—
173C22×C4×C7—81no—
174C22×C4×C7C22×C412yes—
175C22×C4×C7C22×C412yes—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2411yescharacteristic
183C22×Dic7—21no—
184C22×Dic7—21no—
185C22×Dic7—21no—
186C22×Dic7—21no—
187C22×Dic7—42no—
188C22×Dic7—42no—
189C22×Dic7—81no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4C22⋊C412yes—
199C7×C4⋊C4C22⋊C412yes—
200C7×C4⋊C4C22×C412yes—
201C7×C4⋊C4C22×C412yes—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4C22⋊C412yes—
212C7×C22⋊C4C22⋊C412yes—
213C7×C22⋊C4C22⋊C412yes—
214C7×C22⋊C4C22⋊C412yes—
215C42×C7—22no—
216C42×C7—22no—
217C42×C7—22no—
218C42×C7C22×C411yescharacteristic
219C42×C7C22×C411yescharacteristic
220C42×C7C2×D411yescharacteristic
221C42×C7C2×D411yescharacteristic
222C22⋊Dic7—42no—
223C22⋊Dic7—42no—
224C22⋊Dic7—81no—
225C4⋊Dic7—81no—
226C4×Dic7—42no—
227C4×Dic7—81no—
228#1136—82no—
229C2×C7⋊C8—22no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—82no—
232#559—22no—
233#556Dic712yescomplemented
234#555Dic712yescomplemented
235#553—22no—
236#550Dic712yescomplemented
237#547Dic712yescomplemented
238#538—21no—
239#538—21no—
240#535—21no—
241#533C2×D711yescharacteristic
242#532C2×D711yescharacteristic
243#525C2×D712yes—
244#525C2×D712yes—
245#520—21no—
246C2×C4×D4C2×D711yescharacteristic
247#435—142no—
248#435—142no—
249#432—71no—
250#432—71no—
251#407—72no—
252C2×C22⋊C8—72no—
253#388—141no—
254#387—141no—
255#386—141no—
256#383—71no—
257#381—71no—
258#380—71no—
259C4×C22⋊C4—71no—
260C4×C22⋊C4—141no—
261#376—71no—
262C2×#95—71no—
263#328—72no—
264#328—72no—
265#328—72no—
266#328—72no—
267#327—72no—
268#327—72no—
269#327—72no—
270#327—72no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—22no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C2×C22⋊C412yes—
281C23×C7C2×C22⋊C412yes—
282C23×C7C22×D411yescharacteristic
283C7×Q8—42no—
284C7×Q8—42no—
285C7×Q8—81no—
286C7×D4—42no—
287C7×D4—42no—
288C7×D4—42no—
289C7×D4—42no—
290C7×D4—42no—
291C7×D4—42no—
292C7×D4—81no—
293C7×D4—82no—
294C7×D4—82no—
295C7×D4—82no—
296C7×D4—82no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—44no—
311C2×C4×C7—82no—
312C2×C4×C7—82no—
313C2×C4×C7C2×C22⋊C411yescharacteristic
314C2×C4×C7C2×C22⋊C412yes—
315C2×C4×C7C2×C22⋊C412yes—
316C2×C4×C7C2×C22⋊C411yescharacteristic
317C2×C4×C7C2×C22⋊C411yescharacteristic
318C2×C4×C7C2×C22⋊C411yescharacteristic
319C2×C4×C7C23×C411yescharacteristic
320C2×C4×C7C22×D411yescharacteristic
321C2×Dic7—22no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—84no—
334C7⋊C8—42no—
335C7⋊C8—42no—
336C2×#41—21no—
337C2×#41C2×Dic712yes—
338C2×#41C2×Dic712yescomplemented
339C22×D4C22⋊D711yescharacteristic
340C23×C4—141no—
341C23×C4C22×D711yescharacteristic
342C23×C4C22×D711yescharacteristic
343C2×M16—142no—
344C22×C8—142no—
345C4⋊Q8C2×Dic711yescharacteristic, complemented
346C4⋊Q8C2×Dic711yescharacteristic, complemented
347C4⋊Q8C2×Dic711yescharacteristic, complemented
348#127—22no—
349#127C2×Dic711yescharacteristic, complemented
350#126—22no—
351#126—22no—
352#125—21no—
353#124—22no—
354#124—22no—
355#124—22no—
356#124—21no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361C22⋊Q8—21no—
362C22⋊Q8C2×Dic712yes—
363C22⋊Q8C2×Dic712yescomplemented
364C22⋊Q8C22×D712yes—
365#121—22no—
366#121—22no—
367#121—21no—
368#121—22no—
369#121—22no—
370#121C2×Dic712yes—
371#121C2×Dic712yescomplemented
372#121C22×D712yes—
373#120—22no—
374#120—22no—
375C4×Q8—22no—
376C4×Q8—21no—
377C4×Q8C2×Dic712yescomplemented
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4C2×Dic712yes—
385C4×D4C2×Dic712yes—
386C4×D4C2×Dic712yescomplemented
387C4×D4C22⋊D712yes—
388C4×D4C22⋊D712yes—
389#117—21no—
390#117—22no—
391#117C2×Dic712yes—
392#117C2×Dic712yescomplemented
393C2×C4⋊C4—22no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4C22⋊D711yescharacteristic
396C2×C22⋊C4—141no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4C22⋊D711yescharacteristic
400C2×C22⋊C4C22⋊D711yescharacteristic
401C2×C42—141no—
402C2×C42—141no—
403C2×C42C22×D711yescharacteristic
404C4⋊C8—141no—
405C4⋊C8—141no—
406C22⋊C8—72no—
407C22⋊C8—72no—
408C22⋊C8—142no—
409C22⋊C8—142no—
410#97—141no—
411#97—141no—
412#95—71no—
413#95—71no—
414#95—71no—
415#95—71no—
416#95—141no—
417#95—141no—
418#95—141no—
419#95—141no—
420#95—141no—
421C22×C7—21no—
422C22×C7—21no—
423C22×C7—22no—
424C22×C7—22no—
425C22×C7—22no—
426C22×C7—22no—
427C22×C7—22no—
428C22×C7—22no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—42no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—81no—
435C22×C7—81no—
436C22×C7C22×C22⋊C411yescharacteristic
437C22×C7#57511yescharacteristic
438C22×C7#57511yescharacteristic
439C4×C7—21no—
440C4×C7—21no—
441C4×C7—22no—
442C4×C7—22no—
443C4×C7—21no—
444C4×C7—21no—
445C4×C7—21no—
446C4×C7—21no—
447C4×C7—42no—
448C4×C7—42no—
449C4×C7—42no—
450Dic7—42no—
451Dic7—81no—
452Dic7—81no—
453Dic7—81no—
454C24C2×C22⋊D711yescharacteristic
455C24C2×C22⋊D711yescharacteristic
456#41—42no—
457#41—42no—
458#41—82no—
459#41—82no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8C22×Dic712yes—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4C22⋊Dic712yes—
473C2×D4C22⋊Dic712yes—
474C2×D4C22×Dic712yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—282no—
485C22×C4—282no—
486C22×C4—561no—
487C22×C4C22×Dic712yes—
488C22×C4C22×Dic712yes—
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C2×C22⋊D711yescharacteristic
493C22×C4C2×C22⋊D712yes—
494C22×C4C2×C22⋊D712yes—
495C22×C4C23×D711yescharacteristic
496M16—562no—
497C2×C8—142no—
498C2×C8—142no—
499C2×C8—562no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—561no—
509C4⋊C4C22⋊Dic712yes—
510C4⋊C4C22⋊Dic712yes—
511C4⋊C4C22×Dic712yes—
512C4⋊C4C22×Dic712yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—282no—
523C22⋊C4—282no—
524C22⋊C4—561no—
525C22⋊C4C22⋊Dic712yes—
526C22⋊C4C22⋊Dic712yes—
527C22⋊C4C22⋊Dic712yes—
528C22⋊C4C22⋊Dic712yes—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—282no—
533C42—561no—
534C42C22×Dic711yescharacteristic
535C42C22×Dic711yescharacteristic
536C42C2×C22⋊D711yescharacteristic
537C42C2×C22⋊D711yescharacteristic
538C2×C7—21no—
539C2×C7—21no—
540C2×C7—42no—
541C2×C7—42no—
542C2×C7—82no—
543C2×C7#288311yescharacteristic
544C2×C7#289611yescharacteristic
545C2×C7#290111yescharacteristic
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×C22⋊Dic712yes—
556C23C2×C22⋊Dic712yes—
557C23C22×C22⋊D711yescharacteristic
558Q8—42no—
559Q8—42no—
560Q8—81no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—81no—
568D4—82no—
569D4—82no—
570D4—82no—
571D4—82no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—142no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—564no—
601C2×C4C2×C22⋊Dic711yescharacteristic
602C2×C4C2×C22⋊Dic712yes—
603C2×C4C2×C22⋊Dic712yes—
604C2×C4C2×C22⋊Dic711yescharacteristic
605C2×C4C2×C22⋊Dic711yescharacteristic
606C2×C4C2×C22⋊Dic711yescharacteristic
607C2×C4C23×Dic711yescharacteristic
608C2×C4C22×C22⋊D711yescharacteristic
609C8—282no—
610C8—282no—
611C7#1762911yescharacteristic, complemented, Sylow
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22#9051311yescharacteristic
628C22#9051311yescharacteristic
629C22C22×C22⋊Dic711yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—282no—
642C4—561no—
643C4—561no—
644C4—561no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—82no—
650C2#1174151711yescharacteristic
651C2#1174155211yescharacteristic
652C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#3506C2
socleC22×C7C22×C22⋊C4
cosocleC22×C4×C7C24
hypercenter#3506D7
hypocenterC7#17629
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
2. #11739781C2semidirect
3. #11739747C2semidirect
4. #11729591C2semidirect
5. #11729590C2semidirect
6. #11729567C2semidirect
7. #11729566C2semidirect
8. #11729464C2semidirect
9. #11729463C2semidirect
10. #11729462C2semidirect
11. #11729461C2semidirect
13. C7×#556C4semidirect
14. C7×#555C4semidirect
16. C7×#550C4semidirect
17. C7×#547C4semidirect
54. C2×C7×#41C2×C4semidirect
59. C7×C4⋊Q8C2×C4semidirect
60. C7×C4⋊Q8C2×C4semidirect
61. C7×C4⋊Q8C2×C4semidirect
63. C7×#127C2×C4semidirect
77. C7×C22⋊Q8C2×C4semidirect
85. C7×#121C2×C4semidirect
91. C4×C7×Q8C2×C4semidirect
100. C4×C7×D4C2×C4semidirect
106. C7×#117C2×C4semidirect
233. #556Dic7semidirect
234. #555Dic7semidirect
236. #550Dic7semidirect
237. #547Dic7semidirect
338. C2×#41C2×Dic7semidirect
345. C4⋊Q8C2×Dic7semidirect
346. C4⋊Q8C2×Dic7semidirect
347. C4⋊Q8C2×Dic7semidirect
349. #127C2×Dic7semidirect
363. C22⋊Q8C2×Dic7semidirect
371. #121C2×Dic7semidirect
377. C4×Q8C2×Dic7semidirect
386. C4×D4C2×Dic7semidirect
392. #117C2×Dic7semidirect
611. C7#17629semidirect

Maximal subgroups

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

subgroupindex
C7×#35062
#117397812
#117397472
#117295912
#117295902
#117295672
#117295662
#117294642
#117294632
#117294622
#117294612
#176297