Finite Group Databasebeta

← #13358696

Subgroups — #13358696

Order 1792.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739589C211yesmaximal, characteristic
2C2×#89492C211yesmaximal, characteristic
3#11731313C211yesmaximal, characteristic
4C7×#1852C211yesmaximal, characteristic
5#11729310C211yesmaximal, characteristic
6#11728370C211yesmaximal, characteristic, complemented
7#11728370C211yesmaximal, characteristic, complemented
8#11728359C211yesmaximal, characteristic, complemented
9Dic7×#107C211yesmaximal, characteristic, complemented
10#11727683C211yesmaximal, characteristic
11#11727260C211yesmaximal, characteristic
12#11726550C211yesmaximal, characteristic, complemented
13#11726549C211yesmaximal, characteristic, complemented
14#11726540C211yesmaximal, characteristic, complemented
15#11726538C211yesmaximal, characteristic, complemented
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C2×C7×#107C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C7×#380C2211yescharacteristic
23#89984C2211yescharacteristic
24C4×C22⋊Dic7—21no—
25C4×C22⋊Dic7—21no—
26C2×C22⋊(C7⋊C8)C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89749C2211yescharacteristic
34#89748C2211yescharacteristic
35C4⋊C4×Dic7—21no—
36C4⋊C4×Dic7—21no—
37C4⋊C4×Dic7C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42C4⋊C4×Dic7C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47#89717—21no—
48#89716—21no—
49#89492C418yescomplemented
50#89426—21no—
51#89425—21no—
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21367—71nomaximal, Sylow
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7D411yescharacteristic
76C2×C22⋊Dic7D411yescharacteristic
77#6489—22no—
78#6489—22no—
79#6489C2×C414yes—
80#6489Q814yes—
81C2×C4⋊Dic7D411yescharacteristic
82C2×C4⋊Dic7D411yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7C2311yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C7×#107—22no—
101C7×#107—22no—
102C7×C22⋊C8C2×C414yes—
103C7×#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C22⋊(C7⋊C8)C2×C414yes—
114#6392—22no—
115#6391—22no—
116#6389C2×C414yescomplemented
117Dic7⋊C8C2×C414yescomplemented
118Dic7⋊C8C2×C414yescomplemented
119C8×Dic7C2×C414yescomplemented
120#6382—22no—
121#6382—22no—
122#6382—22no—
123#6381—22no—
124C2×#433—71no—
125#2314—71no—
126#2274—71no—
127#1949—71no—
128#1949—71no—
129#1948—71no—
130#1947—71no—
131#1854—71no—
132#1854—71no—
133#1853—71no—
134#1852D711yescharacteristic
135#1776—71no—
136#1776—71no—
137C4×#107—71no—
138C4×#106—71no—
139C24×C7C2×D411yescharacteristic
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—81no—
150C22×C4×C7—81no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×Q812yes—
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8C22×C412yes—
168C2×C7×C8C22×C412yes—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4#4112yes—
177C7×C4⋊C4#4112yes—
178C7×C22⋊C4—81no—
179C22⋊Dic7—24no—
180C22⋊Dic7—81no—
181C22⋊Dic7C4⋊C414yes—
182C22⋊Dic7C4⋊C414yes—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7C4⋊C414yes—
190C4⋊Dic7C4⋊C414yes—
191C4⋊Dic7#4112yes—
192C4⋊Dic7#4112yes—
193Dic7⋊C4—81no—
194Dic7⋊C4—81no—
195C4×Dic7—24no—
196C4×Dic7—81no—
197C4×Dic7—81no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C4×Dic7—81no—
201C4×Dic7—81no—
202C4×Dic7C22×C412yes—
203C4×Dic7C22×C412yes—
204C4×Dic7C2×Q812yes—
205C4×Dic7C2×Q812yes—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8C22×C412yes—
211C2×C7⋊C8C22×C412yes—
212C2×#117—71no—
213C2×#117—71no—
214C22×C4⋊C4C2×D711yescharacteristic
215#433—78no—
216C2×#107—71no—
217C2×#107—71no—
218C2×#107C2×D711yescharacteristic
219C2×#107C2×D711yescharacteristic
220C2×#106—71no—
221C2×#106—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226C2×C22⋊C8—71no—
227C2×C22⋊C8C2×D711yescharacteristic
228C2×#97—71no—
229C2×C4×C8—71no—
230#388—141no—
231#388—141no—
232#387—141no—
233#387—141no—
234#384—71no—
235#382—71no—
236#380—71no—
237#380C2×D711yescharacteristic
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—141no—
245C4×C4⋊C4—141no—
246C4×C22⋊C4—141no—
247C4×C22⋊C4—141no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340C2×D711yescharacteristic
255#340C2×D711yescharacteristic
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7C2×C4⋊C412yes—
263C23×C7C22×D411yescharacteristic
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—84no—
283C2×C4×C7C2×C4⋊C412yes—
284C2×C4×C7C23×C411yescharacteristic
285C2×C4×C7C22×Q811yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7C2×C4⋊C412yes—
311C2×Dic7C2×C4⋊C412yes—
312C2×Dic7C2×C4⋊C412yes—
313C2×Dic7C2×C4⋊C412yes—
314C7×C8—22no—
315C7×C8—22no—
316C7⋊C8—22no—
317C7⋊C8—22no—
318C23×C4—21no—
319C23×C4C22×D711yescharacteristic
320C22×C8—71no—
321C22×C8—71no—
322C22×C8C22×D711yescharacteristic
323C22×C8C22×D711yescharacteristic
324#117—74no—
325#117—74no—
326#117—142no—
327#117—142no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—71no—
345C2×C22⋊C4—71no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—141no—
351C2×C42—141no—
352C2×C42—141no—
353C2×C42—141no—
354C2×C42—141no—
355C2×C42—141no—
356C2×C42—141no—
357#107—22no—
358#107—22no—
359#107—142no—
360#107—142no—
361#106—142no—
362#106—142no—
363#106—142no—
364#106—142no—
365C4⋊C8—74no—
366C4⋊C8—74no—
367C22⋊C8—74no—
368C22⋊C8C4×D714yes—
369#97—74no—
370C4×C8—74no—
371#95—21no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—22no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7—41no—
388C22×C7C22×C4⋊C411yescharacteristic
389C22×C7#52511yescharacteristic
390C22×C7#52511yescharacteristic
391C22×C7#57611yescharacteristic
392C22×C7#57711yescharacteristic
393C22×C7#57711yescharacteristic
394C22×C7#57811yescharacteristic
395C4×C7—22no—
396C4×C7—42no—
397C4×C7—42no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7C4×#4111yescharacteristic
401C4×C7C4×#4111yescharacteristic
402C4×C7#53311yescharacteristic
403C4×C7#53311yescharacteristic
404Dic7—22no—
405Dic7—22no—
406Dic7—22no—
407Dic7—22no—
408Dic7—42no—
409Dic7—42no—
410Dic7—81no—
411Dic7—81no—
412C24D4×D711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—71no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4C2×C4×D712yes—
433C22×C4D4×D711yescharacteristic
434C22×C4Q8×D712yes—
435C22×C4C23×D711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—72no—
441C2×C8—72no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8C2×C4×D712yes—
447C2×C8C2×C4×D712yes—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—74no—
456C4⋊C4—74no—
457C4⋊C4—72no—
458C4⋊C4—72no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—561no—
466C4⋊C4—561no—
467C4⋊C4#115912yes—
468C4⋊C4#115912yes—
469C22⋊C4—74no—
470C22⋊C4—74no—
471C22⋊C4—81no—
472C22⋊C4—144no—
473C22⋊C4—561no—
474C42—72no—
475C42—72no—
476C42—72no—
477C42—72no—
478C42—144no—
479C42—561no—
480C42—561no—
481C42—561no—
482C42—561no—
483C42—561no—
484C42—561no—
485C2×C7—42no—
486C2×C7#229611yescharacteristic
487C2×C7#291211yescharacteristic
488C2×C7#291311yescharacteristic
489C2×C7#311811yescharacteristic
490C2×C7#311911yescharacteristic
491C2×C7#312011yescharacteristic
492C2×C7#312111yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23D7×C4⋊C412yes—
500C23C2×D4×D711yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—72no—
520C2×C4—72no—
521C2×C4—72no—
522C2×C4—72no—
523C2×C4—84no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4D7×C4⋊C412yes—
545C2×C4C22×C4×D711yescharacteristic
546C2×C4C2×#115911yescharacteristic
547C2×C4C2×#115911yescharacteristic
548C2×C4C2×Q8×D711yescharacteristic
549C2×C4D7×#4111yescharacteristic
550C2×C4D7×#4111yescharacteristic
551C8—22no—
552C8—22no—
553C8—142no—
554C8—142no—
555C7#2136711yescharacteristic, complemented, Sylow
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×D7×C4⋊C411yescharacteristic
564C22#9027711yescharacteristic
565C22#9034711yescharacteristic
566C22#9044811yescharacteristic
567C22#9045711yescharacteristic
568C22#9046611yescharacteristic
569C22#9046911yescharacteristic
570C4—22no—
571C4—42no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—142no—
576C4—142no—
577C4—142no—
578C4—142no—
579C4—282no—
580C4—282no—
581C4—561no—
582C4—561no—
583C4C4×#115911yescharacteristic
584C4C4×#115911yescharacteristic
585C4#9031311yescharacteristic
586C4#9031311yescharacteristic
587C2—42no—
588C2#1173182311yescharacteristic
589C2#1173659111yescharacteristic
590C2#1173665411yescharacteristic
591C2#1173743711yescharacteristic
592C2#1173762111yescharacteristic
593C2#1173871711yescharacteristic
594C2#1173890111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1852C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1852D7
hypocenterC7#21367
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
6. #11728370C2semidirect
7. #11728370C2semidirect
8. #11728359C2semidirect
9. Dic7×#107C2semidirect
12. #11726550C2semidirect
13. #11726549C2semidirect
14. #11726540C2semidirect
15. #11726538C2semidirect
49. #89492C4semidirect
116. #6389C2×C4semidirect
117. Dic7⋊C8C2×C4semidirect
118. Dic7⋊C8C2×C4semidirect
119. C8×Dic7C2×C4semidirect
555. C7#21367semidirect

Maximal subgroups

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

subgroupindex
#117395892
C2×#894922
#117313132
C7×#18522
#117293102
#117283702
#117283592
Dic7×#1072
#117276832
#117272602
#117265502
#117265492
#117265402
#117265382
#213677