Finite Group Databasebeta

← #13412652

Subgroups — #13412652

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739616C211yesmaximal, characteristic
2C2×#89736C211yesmaximal, characteristic
3C2×#89628C211yesmaximal, characteristic, complemented
4#11731345C211yesmaximal, characteristic
5C7×#1853C211yesmaximal, characteristic
6#11728321C211yesmaximal, characteristic, complemented
7#11728320C211yesmaximal, characteristic, complemented
8#11728308C211yesmaximal, characteristic, complemented
9#11727703C211yesmaximal, characteristic
10#11727683C211yesmaximal, characteristic
11#11727267C211yesmaximal, characteristic
12#11726614C211yesmaximal, characteristic, complemented
13#11726613C211yesmaximal, characteristic, complemented
14#11726588C211yesmaximal, characteristic, complemented
15#11726581C211yesmaximal, characteristic, complemented
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C2×#6487—21no—
19C22×#1136C2211yescharacteristic
20C2×#6453C2211yescharacteristic
21C2×#6451C2211yescharacteristic
22C2×C7×#106C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24C2×C7×C22⋊C8C2211yescharacteristic
25C7×#380C2211yescharacteristic
26#89984C2211yescharacteristic
27#89982—21no—
28C4×C22⋊Dic7C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6389C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89757C2211yescharacteristic
34#89757C2211yescharacteristic
35#89756—21no—
36#89751—21no—
37#89748C2211yescharacteristic
38#89748C2211yescharacteristic
39#89745C2211yescharacteristic
40#89745C2211yescharacteristic
41#89736—24no—
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48#89721—21no—
49#89720—21no—
50#89628—24no—
51#89427—21no—
52#89419—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#26078—71nomaximal, Sylow
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C22×Dic14—21no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—41no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7D411yescharacteristic
78C2×C22⋊Dic7D411yescharacteristic
79#6489—22no—
80#6489—41no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88#6487—44no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7C2311yescharacteristic
99C2×C4×Dic7C2311yescharacteristic
100C2×#1136—22no—
101C2×#1136—22no—
102C2×#1136Q814yescomplemented
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6453—22no—
106#6453—41no—
107#6451—22no—
108#6451—41no—
109C7×#106—22no—
110C7×#106—41no—
111C7×C22⋊C8Q814yescomplemented
112C7×#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95D411yescharacteristic
119C7⋊#95D411yescharacteristic
120#6392—22no—
121#6391—41no—
122#6389—22no—
123C8×Dic7—22no—
124#6382—22no—
125#6382—22no—
126#6382—22no—
127#6381—22no—
128#6381—22no—
129#6381D414yes—
130C2×#501—71no—
131C2×#428—71no—
132#2441—71no—
133#2440—71no—
134#2084—71no—
135#2083—71no—
136#2059—71no—
137#2058—71no—
138#1943—71no—
139#1942—71no—
140#1854—71no—
141#1853D711yescharacteristic
142#1844—71no—
143#1809—71no—
144#1753—71no—
145C24×C7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—81no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×Q812yes—
159C22×C4×C7C2×Q812yes—
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C2×Dic14—42no—
170C2×Dic14—42no—
171C2×Dic14—81no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8C2×Q812yes—
177C2×C7×C8C2×Q812yes—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4C2×D412yes—
187C7×C22⋊C4—81no—
188C22⋊Dic7—22no—
189C22⋊Dic7—22no—
190C22⋊Dic7—42no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—42no—
200C4⋊Dic7—42no—
201C4⋊Dic7—81no—
202C4⋊Dic7C2×D412yes—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—81no—
210Dic7⋊C4—81no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7—21no—
214C4×Dic7—21no—
215C4×Dic7—42no—
216#1136—28no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C2×D412yes—
220C2×C7⋊C8C2×Q812yes—
221C2×C7⋊C8C2×Q812yes—
222C2×C7⋊C8#4112yes—
223C22×M16—71no—
224C2×C4⋊Q8—71no—
225C2×#125—71no—
226C2×C22⋊Q8—141no—
227C2×#117—71no—
228C22×C4⋊C4C2×D711yescharacteristic
229#501—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#106—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106C2×D711yescharacteristic
240C2×#106C2×D711yescharacteristic
241C2×C22⋊C8C2×D711yescharacteristic
242C2×#97—71no—
243C2×C4×C8—71no—
244#399—141no—
245#398—141no—
246#396—141no—
247#395—141no—
248#389—71no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—141no—
253#387—141no—
254#385—141no—
255#384—71no—
256#384—71no—
257#380—71no—
258#380C2×D711yescharacteristic
259C4×C22⋊C4—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340C2×D711yescharacteristic
265#340C2×D711yescharacteristic
266#336—71no—
267#336—71no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7C4⋊Q812yes—
275C23×C7C22×D411yescharacteristic
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—84no—
295C2×C4×C7C4⋊Q812yes—
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C22×Q811yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7H(5,𝔽2)11yescharacteristic
301C2×C4×C72−1+411yescharacteristic
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—22no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7AGL(1,ℤ/8)12yes—
325C2×Dic7#13712yes—
326Dic14—84no—
327C7×C8—22no—
328C7×C8—22no—
329C7⋊C8—22no—
330C7⋊C8C22⋊Q814yescomplemented
331C22×Q8—141no—
332C23×C4—21no—
333C23×C4C22×D711yescharacteristic
334C2×M16—74no—
335C2×M16—142no—
336C2×M16—142no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C4⋊Q8—142no—
342C4⋊Q8—281no—
343#125—142no—
344#125—281no—
345C22⋊Q8—284no—
346#117—142no—
347#117—281no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—41no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—71no—
372C2×C22⋊C4—71no—
373C2×C22⋊C4—141no—
374C2×C22⋊C4—141no—
375C2×C42—71no—
376C2×C42—71no—
377C2×C42—141no—
378#107—74no—
379#107—142no—
380#107—142no—
381#107—142no—
382#106—22no—
383#106—41no—
384#106—142no—
385#106—142no—
386#106—142no—
387#106—281no—
388C22⋊C8Dic1414yescomplemented
389#97—142no—
390C4×C8—142no—
391#95—21no—
392#95—71no—
393#95—71no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398#95—141no—
399C22×C7—22no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7C2×C4⋊Q811yescharacteristic
407C22×C7D4211yescharacteristic
408C22×C7#54711yescharacteristic
409C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
410C22×C7C2×#13711yescharacteristic
411C22×C7#57711yescharacteristic
412C22×C7#57711yescharacteristic
413C4×C7—22no—
414C4×C7—42no—
415C4×C7—81no—
416C4×C7—81no—
417C4×C7—81no—
418C4×C7C2×C22⋊Q811yescharacteristic
419C4×C7#54411yescharacteristic
420C4×C7D4×Q811yescharacteristic
421C4×C7#55411yescharacteristic
422Dic7—22no—
423Dic7—22no—
424Dic7—41no—
425Dic7—42no—
426Dic7—41no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430C24D4×D711yescharacteristic
431C2×Q8—282no—
432C2×Q8—282no—
433C2×Q8—561no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—81no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4C2×Dic1412yes—
454C22×C4D4×D711yescharacteristic
455C22×C4Q8×D712yes—
456C22×C4C23×D711yescharacteristic
457M16—148no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—72no—
463C2×C8—72no—
464C2×C8—72no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8C2×Dic1412yes—
469C2×C8C2×Dic1412yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—72no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C4⋊C4D4×D712yes—
498C22⋊C4—81no—
499C22⋊C4—142no—
500C22⋊C4—142no—
501C22⋊C4—282no—
502C22⋊C4—282no—
503C42—141no—
504C42—141no—
505C42—141no—
506C42—141no—
507C42—282no—
508C2×C7—42no—
509C2×C7#244011yescharacteristic
510C2×C7#316311yescharacteristic
511C2×C7#316511yescharacteristic
512C2×C7#327211yescharacteristic
513C2×C7#327811yescharacteristic
514C2×C7#334411yescharacteristic
515C2×C7#334511yescharacteristic
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23#645112yes—
523C23C2×D4×D711yescharacteristic
524Q8—564no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—84no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—142no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4#645112yes—
569C2×C4C22×Dic1411yescharacteristic
570C2×C4C2×D4×D711yescharacteristic
571C2×C4#654811yescharacteristic
572C2×C4C2×Q8×D711yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4#655411yescharacteristic
575C8—22no—
576C8—22no—
577C8—74no—
578C8—142no—
579C7#2607811yescharacteristic, complemented, Sylow
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#645111yescharacteristic
588C22#9028111yescharacteristic
589C22#9034611yescharacteristic
590C22#9045311yescharacteristic
591C22#9045311yescharacteristic
592C22D7×AGL(1,ℤ/8)11yescharacteristic
593C22D7×#13711yescharacteristic
594C4—22no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—142no—
600C4—142no—
601C4—281no—
602C4—282no—
603C4—281no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4D4×Dic1411yescharacteristic
608C4#9023011yescharacteristic
609C4#9031411yescharacteristic
610C4D7×C22⋊Q811yescharacteristic
611C2—42no—
612C2#1173186111yescharacteristic
613C2#1173655511yescharacteristic
614C2#1173664811yescharacteristic
615C2#1173751111yescharacteristic
616C2#1173761211yescharacteristic
617C2#1173878011yescharacteristic
618C2#1173888311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1853C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1853D7
hypocenterC7#26078
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. C2×#89628C2semidirect
6. #11728321C2semidirect
7. #11728320C2semidirect
8. #11728308C2semidirect
12. #11726614C2semidirect
13. #11726613C2semidirect
14. #11726588C2semidirect
15. #11726581C2semidirect
102. C2×#1136Q8semidirect
111. C7×C22⋊C8Q8semidirect
330. C7⋊C8C22⋊Q8semidirect
388. C22⋊C8Dic14semidirect
579. C7#26078semidirect

Maximal subgroups

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

subgroupindex
#117396162
C2×#897362
C2×#896282
#117313452
C7×#18532
#117283212
#117283202
#117283082
#117277032
#117276832
#117272672
#117266142
#117266132
#117265882
#117265812
#260787