Finite Group Databasebeta

← #13364329

Subgroups — #13364329

Order 1792.

850 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739669C211yesmaximal, characteristic
2C2×#89490C211yesmaximal, characteristic
3#11731309C211yesmaximal, characteristic
4C7×#1854C211yesmaximal, characteristic
5#11729330C211yesmaximal, characteristic
6#11728370C211yesmaximal, characteristic, complemented
7#11728360C211yesmaximal, characteristic, complemented
8#11728318C211yesmaximal, characteristic, complemented
9#11728308C211yesmaximal, characteristic, complemented
10#11727683C211yesmaximal, characteristic
11#11727260C211yesmaximal, characteristic
12#11726601C211yesmaximal, characteristic, complemented
13#11726601C211yesmaximal, characteristic, complemented
14#11726552C211yesmaximal, characteristic, complemented
15#11726551C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C2×#6440—21no—
20C2×C7×#107C2211yescharacteristic
21C2×C7×#106C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C7×#380C2211yescharacteristic
24#89984C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89749C2211yescharacteristic
33#89749C2211yescharacteristic
34#89748C2211yescharacteristic
35#89748C2211yescharacteristic
36#89748C2211yescharacteristic
37#89748C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89714—21no—
45#89714—21no—
46#89714—21no—
47C4×Dic7⋊C4—21no—
48C4×Dic7⋊C4—21no—
49#89490C418yescomplemented
50#89422—21no—
51#89421—21no—
52C7×#340C2211yescharacteristic
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21984—71nomaximal, Sylow
61C23×C4×C7C2311yescharacteristic
62C23×Dic7D411yescharacteristic
63C23×Dic7D411yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×#117—41no—
67C7×#117—41no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77#6489—41no—
78#6489—41no—
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⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7C2×C414yes—
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7D411yescharacteristic
97C2×C4×Dic7Q814yes—
98C2×C4×Dic7C2311yescharacteristic
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6440—44no—
103C7×#107—41no—
104C7×#106—41no—
105C7×C22⋊C8C2×C414yes—
106C7×#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C22⋊(C7⋊C8)C2×C414yes—
115#6392—41no—
116#6391—41no—
117Dic7⋊C8C2×C414yes—
118Dic7⋊C8C2×C414yes—
119C8×Dic7C2×C414yes—
120C8×Dic7C2×C414yes—
121#6382—41no—
122#6382—41no—
123#6381—41no—
124#6381—41no—
125C2×#431—71no—
126#2285—71no—
127#2285—71no—
128#1949—71no—
129#1949—71no—
130#1949—71no—
131#1949—71no—
132#1854—71no—
133#1854—71no—
134#1854—71no—
135#1854D711yescharacteristic
136#1845—71no—
137#1844—71no—
138#1834—71no—
139#1834—71no—
140C24×C7C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×Q812yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7C4⋊C414yes—
162C22×Dic7C4⋊C414yes—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8C22×C412yes—
172C2×C7×C8C22×C412yes—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—42no—
183C42×C7—42no—
184C22⋊Dic7—42no—
185C22⋊Dic7—42no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—42no—
201Dic7⋊C4—42no—
202C4×Dic7—42no—
203C4×Dic7—42no—
204C4×Dic7—42no—
205C4×Dic7C4⋊C414yes—
206C4×Dic7C4⋊C414yes—
207C4×Dic7C22×C412yes—
208C4×Dic7C22×C412yes—
209C4×Dic7C2×Q812yes—
210C4×Dic7C2×Q812yes—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8C22×C412yes—
216C2×C7⋊C8C22×C412yes—
217C2×#117—71no—
218C2×#117—141no—
219C2×#117C2×D711yescharacteristic
220C22×C42—71no—
221#431—78no—
222C2×#107—71no—
223C2×#107—71no—
224C2×#107—71no—
225C2×#107C2×D711yescharacteristic
226C2×#106—71no—
227C2×#106—71no—
228C2×#106—71no—
229C2×#106C2×D711yescharacteristic
230C2×C4⋊C8—71no—
231C2×C4⋊C8—71no—
232C2×C22⋊C8—71no—
233C2×C22⋊C8C2×D711yescharacteristic
234C2×C4×C8—71no—
235C2×C4×C8—71no—
236#384—71no—
237#384—71no—
238#384—71no—
239#384—71no—
240#382—71no—
241#382—71no—
242#382—141no—
243#380—71no—
244#380—141no—
245#380—141no—
246#380—141no—
247#380C2×D711yescharacteristic
248C4×C4⋊C4—71no—
249C4×C4⋊C4—71no—
250C4×C4⋊C4—141no—
251C4×C4⋊C4—141no—
252C4×C4⋊C4—141no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7C2×C4⋊C412yes—
268C23×C7C22×D411yescharacteristic
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
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—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7C2×C4⋊C412yes—
288C2×C4×C7C23×C411yescharacteristic
289C2×C4×C7C22×Q811yescharacteristic
290C2×C4×C7H(5,𝔽2)11yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C72−1+411yescharacteristic
293C2×C4×C72−1+411yescharacteristic
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—28no—
299C2×Dic7—28no—
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×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7C2×C4⋊C412yes—
313C2×Dic7C2×C4⋊C412yes—
314C2×Dic7C2×C4⋊C412yes—
315C2×Dic7C2×C4⋊C412yes—
316C2×Dic7#13512yes—
317C2×Dic7#13512yes—
318C2×Dic7AGL(1,ℤ/8)12yes—
319C2×Dic7#13712yes—
320C7×C8—22no—
321C7×C8—22no—
322C7⋊C8—22no—
323C7⋊C8—22no—
324C23×C4—71no—
325C23×C4—71no—
326C23×C4C22×D711yescharacteristic
327C22×C8—71no—
328C22×C8—71no—
329C22×C8C22×D711yescharacteristic
330C22×C8C22×D711yescharacteristic
331#117—41no—
332#117—41no—
333#117—284no—
334#117—281no—
335#117—281no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—141no—
355C2×C22⋊C4—141no—
356C2×C42—21no—
357C2×C42—74no—
358C2×C42—71no—
359C2×C42—71no—
360C2×C42—74no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—141no—
364C2×C42—141no—
365C2×C42—141no—
366#107—41no—
367#107—281no—
368#107—281no—
369#107—281no—
370#106—41no—
371#106—281no—
372#106—281no—
373#106—281no—
374C4⋊C8—74no—
375C4⋊C8—74no—
376C22⋊C8—74no—
377C22⋊C8C4×D714yes—
378C4×C8—74no—
379C4×C8—74no—
380#95—21no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388C22×C7—22no—
389C22×C7—22no—
390C22×C7—22no—
391C22×C7—41no—
392C22×C7—41no—
393C22×C7—41no—
394C22×C7—41no—
395C22×C7C22×C4⋊C411yescharacteristic
396C22×C7#53511yescharacteristic
397C22×C7#53611yescharacteristic
398C22×C7C2×#13511yescharacteristic
399C22×C7C2×#13511yescharacteristic
400C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
401C22×C7C2×#13711yescharacteristic
402C4×C7—22no—
403C4×C7—81no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7—81no—
407C4×C7#52011yescharacteristic
408C4×C7#52011yescharacteristic
409C4×C7#54411yescharacteristic
410C4×C7#54411yescharacteristic
411Dic7—22no—
412Dic7—22no—
413Dic7—81no—
414Dic7—81no—
415Dic7—81no—
416Dic7—81no—
417Dic7#42714yes—
418Dic7#42814yes—
419C24D4×D711yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—74no—
431C22×C4—74no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4C2×C4×D712yes—
443C22×C4D4×D711yescharacteristic
444C22×C4Q8×D712yes—
445C22×C4C23×D711yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—72no—
451C2×C8—72no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8C2×C4×D712yes—
457C2×C8C2×C4×D712yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4—282no—
479C4⋊C4—282no—
480C4⋊C4—282no—
481C4⋊C4—282no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—282no—
485C22⋊C4—282no—
486C42—42no—
487C42—74no—
488C42—74no—
489C42—72no—
490C42—72no—
491C42—72no—
492C42—72no—
493C42—282no—
494C42—282no—
495C42—282no—
496C2×C7—42no—
497C2×C7#231711yescharacteristic
498C2×C7C2×#42711yescharacteristic
499C2×C7C2×#42811yescharacteristic
500C2×C7#320711yescharacteristic
501C2×C7#321011yescharacteristic
502C2×C7#322811yescharacteristic
503C2×C7#323111yescharacteristic
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23D7×C4⋊C412yes—
511C23C2×D4×D711yescharacteristic
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—72no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—72no—
537C2×C4—72no—
538C2×C4—148no—
539C2×C4—148no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4D7×C4⋊C412yes—
557C2×C4C22×C4×D711yescharacteristic
558C2×C4#654811yescharacteristic
559C2×C4#654811yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4#655411yescharacteristic
562C2×C4#655411yescharacteristic
563C8—22no—
564C8—22no—
565C8—142no—
566C8—142no—
567C7#2198411yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×D7×C4⋊C411yescharacteristic
576C22#9028311yescharacteristic
577C22#9033911yescharacteristic
578C22D7×#13511yescharacteristic
579C22D7×#13511yescharacteristic
580C22D7×AGL(1,ℤ/8)11yescharacteristic
581C22D7×#13711yescharacteristic
582C4—22no—
583C4—74no—
584C4—74no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—142no—
590C4—142no—
591C4—561no—
592C4—561no—
593C4—561no—
594C4—561no—
595C4#9023711yescharacteristic
596C4#9023711yescharacteristic
597C4#9031411yescharacteristic
598C4#9031411yescharacteristic
599C2—42no—
600C2#1173182711yescharacteristic
601C2D7×#42711yescharacteristic
602C2D7×#42811yescharacteristic
603C2#1173752111yescharacteristic
604C2#1173766511yescharacteristic
605C2#1173880111yescharacteristic
606C2#1173894511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1854C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1854D7
hypocenterC7#21984
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
6. #11728370C2semidirect
7. #11728360C2semidirect
8. #11728318C2semidirect
9. #11728308C2semidirect
12. #11726601C2semidirect
13. #11726601C2semidirect
14. #11726552C2semidirect
15. #11726551C2semidirect
49. #89490C4semidirect
567. C7#21984semidirect

Maximal subgroups

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

subgroupindex
#117396692
C2×#894902
#117313092
C7×#18542
#117293302
#117283702
#117283602
#117283182
#117283082
#117276832
#117272602
#117266012
#117265522
#117265512
#219847