Finite Group Databasebeta

← #13364413

Subgroups — #13364413

Order 1792.

762 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
1C2×#89481C211yesmaximal, characteristic
2#11731803C211yesmaximal, characteristic
3#11731670C211yesmaximal, characteristic
4C7×#1836C211yesmaximal, characteristic, complemented
5#11727570C211yesmaximal, characteristic, complemented
6#11727328C211yesmaximal, characteristic, complemented
7#11727303C211yesmaximal, characteristic, complemented
8#11726631C211yesmaximal, characteristic
9#11726630C211yesmaximal, characteristic
10#11726629C211yesmaximal, characteristic
11#11726628C211yesmaximal, characteristic
12#11726552C211yesmaximal, characteristic, complemented
13#11726551C211yesmaximal, characteristic, complemented
14#11726550C211yesmaximal, characteristic, complemented
15#11726549C211yesmaximal, characteristic, complemented
16C2×#6455C2211yescharacteristic
17C2×D7×C4⋊C4C2211yescharacteristic
18C2×#6435C2211yescharacteristic
19C2×C7×#97C2211yescharacteristic
20C7×#384C2211yescharacteristic
21C7×#382C2211yescharacteristic
22C2×(C2×D7)⋊C8C2211yescharacteristic
23C2×(C2×D7)⋊C8C2211yescharacteristic
24C2×#6392C2211yescharacteristic
25C2×#6392C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30#89762—21no—
31#89762—21no—
32#89759C2211yescharacteristic
33#89759C2211yescharacteristic
34#89750—21no—
35#89750—21no—
36#89749C2211yescharacteristic
37#89748C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89708C2211yescharacteristic
45#89707C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89481C418yescomplemented
48#89440—21no—
49#89440—21no—
50#89419—21no—
51#89419—21no—
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21992—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4D411yescharacteristic
76C2×(C2×D7)⋊C4D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4D411yescharacteristic
87C2×Dic7⋊C4D411yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94#6455—41no—
95#6455—41no—
96D7×C4⋊C4—41no—
97D7×C4⋊C4—41no—
98#6435C2×C414yes—
99#6435Q814yes—
100C7×#97C2×C414yescomplemented
101C7×#95—21no—
102C7×#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111(C2×D7)⋊C8C2×C414yes—
112(C2×D7)⋊C8C2×C414yes—
113#6392—41no—
114#6392—41no—
115#6391—41no—
116#6391—41no—
117Dic7⋊C8C2×C414yescomplemented
118Dic7⋊C8C2×C414yescomplemented
119#6382—41no—
120#6382—41no—
121#6381—41no—
122#6381—41no—
123C4×C7⋊C8C2×C414yescomplemented
124C2×#433—71no—
125#2317—71no—
126#2314—71no—
127#1949—71no—
128#1949—71no—
129#1948—71no—
130#1947—71no—
131#1854—71no—
132#1854—71no—
133#1853—71no—
134#1852—71no—
135#1847—71no—
136#1846—71no—
137#1836D711yescharacteristic, complemented
138#1834—71no—
139C23×D7C2×D411yescharacteristic
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C2×C4×D7—21no—
155C2×C4×D7—21no—
156C2×C4×D7—21no—
157C2×C4×D7—21no—
158C2×C4×D7—42no—
159C2×C4×D7—81no—
160C2×C4×D7—81no—
161C2×C4×D7C22×C412yes—
162C2×C4×D7C2×Q812yes—
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—21no—
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—21no—
177C7×C4⋊C4—81no—
178C7×C4⋊C4—81no—
179C42×C7C22×C412yes—
180C42×C7C2×Q812yes—
181(C2×D7)⋊C4—42no—
182(C2×D7)⋊C4—81no—
183(C2×D7)⋊C4—81no—
184(C2×D7)⋊C4C4⋊C414yes—
185(C2×D7)⋊C4C4⋊C414yes—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—81no—
195C4⋊Dic7—81no—
196Dic7⋊C4—42no—
197Dic7⋊C4—81no—
198Dic7⋊C4—81no—
199Dic7⋊C4C4⋊C414yes—
200Dic7⋊C4C4⋊C414yes—
201C4×Dic7—42no—
202C4×Dic7—81no—
203C4×Dic7—81no—
204C4×Dic7C22×C412yes—
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⋊C4—71no—
215#433—78no—
216C2×#107—71no—
217C2×#107—71no—
218C2×#107—71no—
219C2×#107—71no—
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⋊C8—71no—
228C2×#97C2×D711yescharacteristic
229C2×C4×C8—71no—
230#389—141no—
231#389—141no—
232#388—141no—
233#388—141no—
234#387—141no—
235#387—141no—
236#385—141no—
237#385—141no—
238#384—71no—
239#384—71no—
240#384C2×D711yescharacteristic
241#382—71no—
242#382—71no—
243#382C2×D711yescharacteristic
244#380—71no—
245#380—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340C2×D711yescharacteristic
253#340C2×D711yescharacteristic
254#340C2×D711yescharacteristic
255#340C2×D711yescharacteristic
256C23×C7C22×D411yescharacteristic
257C22×D7—21no—
258C22×D7—21no—
259C22×D7—21no—
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—21no—
263C22×D7C2×C4⋊C412yes—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
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×C7C2×C4⋊C412yes—
281C2×C4×C7C2×C4⋊C412yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C22×Q811yescharacteristic
284C2×C4×C7H(5,𝔽2)11yescharacteristic
285C2×C4×C7H(5,𝔽2)11yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C2×C4×C72−1+411yescharacteristic
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7C2×C4⋊C412yes—
307C2×Dic7C2×C4⋊C412yes—
308C2×Dic7C2×C4⋊C412yes—
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—22no—
313C4×D7—84no—
314C7×C8—22no—
315C7×C8—22no—
316C7⋊C8—22no—
317C7⋊C8—22no—
318C23×C4—71no—
319C23×C4—141no—
320C22×C8—71no—
321C22×C8—71no—
322C22×C8C22×D711yescharacteristic
323C22×C8C22×D711yescharacteristic
324#117—74no—
325#117—74no—
326#117—281no—
327#117—281no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—281no—
342C2×C4⋊C4—281no—
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C22⋊C4—71no—
348C2×C22⋊C4—71no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C42—71no—
353C2×C42—141no—
354C2×C42—141no—
355C2×C42—141no—
356C2×C42C22×D711yescharacteristic
357#107—281no—
358#107—281no—
359#107—281no—
360#107—281no—
361#106—281no—
362#106—281no—
363#106—281no—
364#106—281no—
365C4⋊C8—74no—
366C4⋊C8—74no—
367C22⋊C8—74no—
368C22⋊C8—74no—
369#97C4×D714yescomplemented
370C4×C8—74no—
371#95—21no—
372#95—21no—
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×C7C22×C4⋊C411yescharacteristic
382C22×C7#53511yescharacteristic
383C22×C7#53611yescharacteristic
384C22×C7#57611yescharacteristic
385C22×C7#57711yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57811yescharacteristic
388C2×D7—22no—
389C2×D7—22no—
390C2×D7—22no—
391C2×D7—41no—
392C2×D7—41no—
393C2×D7—41no—
394C2×D7—41no—
395C4×C7—22no—
396C4×C7—22no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7—81no—
401C4×C7#52011yescharacteristic
402C4×C7#52011yescharacteristic
403C4×C7#54411yescharacteristic
404C4×C7#54411yescharacteristic
405Dic7—22no—
406Dic7—22no—
407Dic7—22no—
408Dic7—81no—
409Dic7—81no—
410Dic7—81no—
411Dic7—81no—
412C24—71no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—72no—
421C22×C4—72no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—282no—
431C22×C4—561no—
432C22×C4—561no—
433C22×C4D4×D711yescharacteristic
434C22×C4D4×D711yescharacteristic
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—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—74no—
457C4⋊C4—74no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—561no—
470C4⋊C4—561no—
471C4⋊C4—561no—
472C4⋊C4—561no—
473C22⋊C4—74no—
474C22⋊C4—74no—
475C22⋊C4—282no—
476C22⋊C4—561no—
477C22⋊C4—561no—
478C42—72no—
479C42—72no—
480C42—282no—
481C42—561no—
482C42—561no—
483C42C2×C4×D712yes—
484C42Q8×D712yes—
485C2×C7#231711yescharacteristic
486C2×C7#291211yescharacteristic
487C2×C7#291311yescharacteristic
488C2×C7#322411yescharacteristic
489C2×C7#322511yescharacteristic
490C2×C7#324111yescharacteristic
491C2×C7#324211yescharacteristic
492D7—42no—
493C23—72no—
494C23—141no—
495C23—141no—
496C23—141no—
497C23—141no—
498C23—141no—
499C23—141no—
500C23C2×D4×D711yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
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—72no—
518C2×C4—72no—
519C2×C4—72no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
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—564no—
543C2×C4D7×C4⋊C412yes—
544C2×C4D7×C4⋊C412yes—
545C2×C4C22×C4×D711yescharacteristic
546C2×C4C2×Q8×D711yescharacteristic
547C2×C4#655311yescharacteristic
548C2×C4#655311yescharacteristic
549C2×C4#655411yescharacteristic
550C2×C4#655411yescharacteristic
551C8—22no—
552C8—22no—
553C8—142no—
554C8—142no—
555C7#2199211yescharacteristic, complemented, Sylow
556C22—142no—
557C22—142no—
558C22—142no—
559C22—281no—
560C22—281no—
561C22—281no—
562C22—281no—
563C22C2×D7×C4⋊C411yescharacteristic
564C22#9036511yescharacteristic
565C22#9036611yescharacteristic
566C22#9046511yescharacteristic
567C22#9046611yescharacteristic
568C22#9046911yescharacteristic
569C22#9047011yescharacteristic
570C4—22no—
571C4—22no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—142no—
577C4—142no—
578C4—142no—
579C4—561no—
580C4—561no—
581C4—561no—
582C4—561no—
583C4#9021411yescharacteristic
584C4#9021411yescharacteristic
585C4#9038011yescharacteristic
586C4#9038011yescharacteristic
587C2—282no—
588C2#1173167011yescharacteristic
589C2#1173665411yescharacteristic
590C2#1173665511yescharacteristic
591C2#1173913311yescharacteristic
592C2#1173913411yescharacteristic
593C2#1173916811yescharacteristic
594C2#1173919211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1836C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1836D7
hypocenterC7#21992
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. C7×#1836C2semidirect
5. #11727570C2semidirect
6. #11727328C2semidirect
7. #11727303C2semidirect
12. #11726552C2semidirect
13. #11726551C2semidirect
14. #11726550C2semidirect
15. #11726549C2semidirect
47. #89481C4semidirect
100. C7×#97C2×C4semidirect
117. Dic7⋊C8C2×C4semidirect
118. Dic7⋊C8C2×C4semidirect
123. C4×C7⋊C8C2×C4semidirect
137. #1836D7semidirect
369. #97C4×D7semidirect
555. C7#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#894812
#117318032
#117316702
C7×#18362
#117275702
#117273282
#117273032
#117266312
#117266302
#117266292
#117266282
#117265522
#117265512
#117265502
#117265492
#219927