Finite Group Databasebeta

← #14220032

Subgroups — #14220032

Order 1792.

1930 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
1C4×#6551C211yesmaximal, characteristic
2C7×Q8×M16C211yesmaximal, characteristic, complemented
3#11739882C211yesmaximal, characteristic, complemented
4#11737408C213yesmaximal, complemented
5D7×#446C213yesmaximal, complemented
6#11737390C213yesmaximal, complemented
7#11737388C213yesmaximal, complemented
8#11737366C211yesmaximal, characteristic, complemented
9Q8×#1131C211yesmaximal, characteristic, complemented
10#11737351C211yesmaximal, characteristic, complemented
11C8×#1161C211yesmaximal, characteristic, complemented
12#11736496C213yesmaximal, complemented
13#11736495C213yesmaximal, complemented
14#11735969C213yesmaximal, complemented
15#11735968C213yesmaximal, complemented
16C2×#6551C411yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18#90503C411yescharacteristic
19#90265C413yes—
20#90265C411yescharacteristic
21C4×#1161C2211yescharacteristic
22C4×#1161C2213yes—
23#90263C413yes—
24#90263C411yescharacteristic
25C4×Q8×D7C2211yescharacteristic
26C4×Q8×D7C2213yes—
27#90195C413yes—
28#90195C413yes—
29C4×#1157C2213yes—
30C4×#1157C2213yes—
31C7×#446C2213yes—
32C7×#446C2213yes—
33C7×C8×Q8C2211yescharacteristic
34C7×C8×Q8C2211yescharacteristic
35C7×#423C2213yes—
36C4×C7×M16C2213yes—
37#89900C2211yescharacteristic
38#89900C2213yes—
39#89899C2213yes—
40#89899C2213yes—
41#89899C2216yes—
42#89796C2213yes—
43#89796C2213yes—
44Q8×C7⋊C8C2211yescharacteristic
45Q8×C7⋊C8C2211yescharacteristic
46#89698C2213yes—
47#89697C2213yes—
48#89609C2213yes—
49#89608C2216yes—
50#89607C2213yes—
51#89607C2216yes—
52#89606C2213yes—
53#89606C2213yes—
54#89605C2213yes—
55D7×C4⋊C8C2213yes—
56D7×C4⋊C8C2216yes—
57#89603C2213yes—
58#89603C2213yes—
59#89600C2216yes—
60#89478C2213yes—
61#89478C2213yes—
62#89477C2216yes—
63D7×#97C2216yes—
64#89473C2213yes—
65#89473C2213yes—
66C4×#1131C2213yes—
67C8×D28C2213yes—
68C8×D28C2213yes—
69#89457C2213yes—
70#89457C2213yes—
71C4×C8×D7C2213yes—
72#89451C2213yes—
73#89451C2213yes—
74#60343—71nomaximal, Sylow
75C22×C7×Q8C2×C411yescharacteristic
76#6551—28no—
77C2×#1161C2×C413yes—
78C2×#1161C2×C411yescharacteristic
79C2×Q8×D7C2×C411yescharacteristic
80C2×Q8×D7C2×C413yes—
81C2×#1157C2×C413yes—
82C2×#1157C2×C413yes—
83C2×C7×M16C2311yescharacteristic
84C2×C7×M16C2313yes—
85C4×C7×Q8C2×C413yes—
86C4×C7×Q8C2×C411yescharacteristic
87C4×C7×Q8C2311yescharacteristic
88C4×C7×Q8C2313yes—
89C2×C7×C4⋊C4C2×C413yes—
90C2×C42×C7C2313yes—
91Q8×Dic7C2×C413yes—
92Q8×Dic7C2×C411yescharacteristic
93Q8×Dic7C2×C413yes—
94Q8×Dic7C2×C411yescharacteristic
95C4×C22⋊D7C2×C413yes—
96C4×C22⋊D7C2×C413yes—
97C4×C22⋊D7C2×C416yes—
98C4×C22⋊D7C2313yes—
99C4×C22⋊D7C2311yescharacteristic
100#6489C2×C413yes—
101#6489C2×C413yes—
102C2×#1131C2311yescharacteristic
103C2×#1131C2313yes—
104C2×C8×D7C2313yes—
105C2×C8×D7C2313yes—
106C2×C8×D7C2316yes—
107#6456C2×C416yes—
108#6456C2×C413yes—
109#6456C2×C413yes—
110#6455C2×C413yes—
111#6455C2×C416yes—
112#6455C2×C413yes—
113D7×C4⋊C4C2×C413yes—
114D7×C4⋊C4C2×C416yes—
115D7×C4⋊C4C2×C413yes—
116#6450C2×C413yes—
117#6450C2×C413yes—
118#6450C2×C416yes—
119C4×D28C2×C413yes—
120C4×D28C2×C413yes—
121C4×D28C2313yes—
122C4×D28C2313yes—
123#6435C2×C413yes—
124#6435C2×C416yes—
125#6435C2×C413yes—
126C42×D7C2313yes—
127C42×D7C2316yes—
128C42×D7C2313yes—
129C4×Dic14C2×C413yes—
130C4×Dic14C2×C413yes—
131C4×Dic14C2313yes—
132C4×Dic14C2313yes—
133C7×C4⋊C8C2316yes—
134C7×C4⋊C8C2313yes—
135C7×C4⋊C8C2313yes—
136C7×#97C2313yes—
137C7×#97C2313yes—
138C4×C7×C8C2313yes—
139C4×C7×C8C2313yes—
140C22⋊(C7⋊C8)C2311yescharacteristic
141C22⋊(C7⋊C8)C2313yes—
142(C2×D7)⋊C8C2316yes—
143(C2×D7)⋊C8C2313yes—
144(C2×D7)⋊C8C2313yes—
145(C2×D7)⋊C8C2313yes—
146(C2×D7)⋊C8C2311yescharacteristic
147#6389C2316yes—
148#6389C2313yes—
149#6389C2313yes—
150Dic7⋊C8C2313yes—
151Dic7⋊C8C2316yes—
152Dic7⋊C8C2313yes—
153Dic7⋊C8C2313yes—
154Dic7⋊C8C2311yescharacteristic
155C8×Dic7C2313yes—
156C8×Dic7C2311yescharacteristic
157C4⋊(C7⋊C8)C2313yes—
158C4⋊(C7⋊C8)C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4⋊(C7⋊C8)C2313yes—
161#6377C2316yes—
162C4×C7⋊C8C2313yes—
163C4×C7⋊C8C2313yes—
164C4×2−1+4—71no—
165#2990—73no—
166#2990—73no—
167#2989—73no—
168#2988—71no—
169#2984—71no—
170#2984—71no—
171#2983—73no—
172#2970—73no—
173#2970—73no—
174#2965—73no—
175C8×#41—71no—
176Q8×M16—71no—
177Q8×M16D711yescharacteristic, complemented
178C2×#446—73no—
179C2×C7×Q8—24no—
180C2×C7×Q8C22×C413yes—
181C2×C7×Q8C22×C411yescharacteristic
182C22×C4×C7C22×C413yes—
183C22×C4×C7C22×C413yes—
184C22×C4×C7C2411yescharacteristic
185C2×C22⋊D7C22×C413yes—
186C2×C22⋊D7C22×C411yescharacteristic
187#1161—212no—
188#1161M1618yescomplemented
189Q8×D7—24no—
190Q8×D7—212no—
191#1157—212no—
192#1157—212no—
193C2×D28C22×C413yes—
194C2×D28C22×C413yes—
195C2×C4×D7C22×C416yes—
196C2×C4×D7C22×C413yes—
197C2×C4×D7C22×C413yes—
198C2×C4×D7C22×C413yes—
199C2×C4×D7C22×C416yes—
200C2×C4×D7C22×C413yes—
201C2×C4×D7C2413yes—
202C2×C4×D7C2411yescharacteristic
203C2×Dic14C22×C413yes—
204C2×Dic14C22×C413yes—
205C7×M16—22no—
206C7×M16—43no—
207C2×C7×C8C2411yescharacteristic
208C2×C7×C8C2413yes—
209C2×C7×C8C2413yes—
210C2×C7×C8C2411yescharacteristic
211C7×C4⋊C4C22×C416yes—
212C7×C4⋊C4C22×C413yes—
213C7×C4⋊C4C22×C413yes—
214C42×C7C22×C413yes—
215C42×C7C22×C413yes—
216C42×C7C2413yes—
217C42×C7C2413yes—
218C22⋊Dic7C22×C413yes—
219C22⋊Dic7C22×C411yescharacteristic
220(C2×D7)⋊C4C22×C411yescharacteristic
221(C2×D7)⋊C4C22×C413yes—
222(C2×D7)⋊C4C22×C413yes—
223(C2×D7)⋊C4C22×C416yes—
224(C2×D7)⋊C4C22×C413yes—
225C4⋊Dic7C22×C413yes—
226C4⋊Dic7C22×C413yes—
227C4⋊Dic7C22×C413yes—
228C4⋊Dic7C22×C413yes—
229Dic7⋊C4C22×C413yes—
230Dic7⋊C4C22×C411yescharacteristic
231Dic7⋊C4C22×C413yes—
232Dic7⋊C4C22×C416yes—
233Dic7⋊C4C22×C413yes—
234C4×Dic7C22×C416yes—
235C4×Dic7C22×C413yes—
236C4×Dic7C22×C413yes—
237C4×Dic7C22×C416yes—
238C4×Dic7C22×C413yes—
239C4×Dic7C22×C413yes—
240C4×Dic7C2411yescharacteristic
241C4×Dic7C2413yes—
242C2×C7⋊C8C2411yescharacteristic
243C2×C7⋊C8C2413yes—
244C2×C7⋊C8C2413yes—
245C2×C7⋊C8C2411yescharacteristic
246#1131—22no—
247#1131—43no—
248C8×D7—26no—
249C8×D7—43no—
250C8×D7—46no—
251C2×2−1+4—71no—
252#520—71no—
253#520—73no—
254#520—73no—
255#520—73no—
256#519—71no—
257#519—73no—
258#519—71no—
259C4×#41—71no—
260C4×#41—73no—
261C4×#41—73no—
262C4×#41—73no—
263C2×C4×Q8—73no—
264C2×C4×Q8—71no—
265C2×C4×Q8C2×D711yescharacteristic
266#446—73no—
267#446—73no—
268#446—76no—
269#446—73no—
270#446—73no—
271#446—73no—
272#446—73no—
273#446—73no—
274#446—73no—
275#446C2×D713yes—
276#446C2×D713yes—
277C8×Q8—71no—
278C8×Q8—71no—
279C8×Q8C2×D711yescharacteristic
280C8×Q8C2×D711yescharacteristic
281#436—71no—
282#436—73no—
283#435—73no—
284#435—73no—
285#435—76no—
286#435—76no—
287#435—73no—
288#435—73no—
289C8×D4—73no—
290C8×D4—73no—
291C8×D4—73no—
292C8×D4—73no—
293#432—73no—
294#432—76no—
295#432—73no—
296#432—76no—
297#431—73no—
298#431—73no—
299#431—73no—
300#431—73no—
301#423—73no—
302#423C2×D713yes—
303C2×C4⋊C8—73no—
304C2×C4⋊C8—76no—
305C4×M16—73no—
306C4×M16C2×D713yes—
307C2×#97—76no—
308C2×C4×C8—73no—
309C23×C7C23×C411yescharacteristic
310C22×D7C23×C411yescharacteristic
311C22×D7C23×C413yes—
312C7×Q8—26no—
313C7×Q8C2×M1614yes—
314C2×C4×C7—26no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—43no—
318C2×C4×C7C23×C413yes—
319C2×C4×C7C23×C413yes—
320C2×C4×C7C23×C413yes—
321C2×C4×C7C23×C411yescharacteristic
322C2×C4×C7C23×C413yes—
323C2×C4×C7C2511yescharacteristic
324C22⋊D7—26no—
325C22⋊D7—22no—
326C2×Dic7C23×C413yes—
327C2×Dic7C23×C411yescharacteristic
328C2×Dic7C23×C413yes—
329C2×Dic7C23×C411yescharacteristic
330C2×Dic7C23×C413yes—
331C2×Dic7C23×C411yescharacteristic
332D28—26no—
333D28C2×M16112yes—
334C4×D7—212no—
335C4×D7—26no—
336C4×D7—23no—
337C4×D7—21no—
338C4×D7—23no—
339C4×D7—21no—
340C4×D7—43no—
341C4×D7—46no—
342C4×D7—43no—
343C4×D7C2×M16112yes—
344Dic14—26no—
345Dic14—26no—
346C7×C8—21no—
347C7×C8—23no—
348C7×C8—23no—
349C7×C82−1+412yescomplemented
350C7⋊C8—23no—
351C7⋊C8—21no—
352C7⋊C8—21no—
353C7⋊C8—23no—
3542−1+4—148no—
355C2×#41—71no—
356C2×#41—73no—
357C2×#41—73no—
358C2×#41—73no—
359C22×Q8—73no—
360C22×Q8—71no—
361C22×Q8C4×D711yescharacteristic
362C2×M16—71no—
363C2×M16—73no—
364C2×M16C22×D711yescharacteristic
365C2×M16C22×D713yes—
366C22×C8—73no—
367C22×C8—73no—
368C22×C8—76no—
369C4×Q8—73no—
370C4×Q8—71no—
371C4×Q8—73no—
372C4×Q8—71no—
373C4×Q8—76no—
374C4×Q8—73no—
375C4×Q8—73no—
376C4×Q8—73no—
377C4×Q8—73no—
378C4×Q8—73no—
379C4×Q8—73no—
380C4×Q8C4×D713yes—
381C4×Q8C4×D711yescharacteristic
382C4×Q8C22×D711yescharacteristic
383C4×Q8C22×D713yes—
384C4×D4—73no—
385C4×D4—73no—
386C4×D4—76no—
387C4×D4—73no—
388C4×D4—71no—
389C4×D4—73no—
390C4×D4—73no—
391C4×D4—76no—
392C4×D4—73no—
393C4×D4—73no—
394C4×D4—73no—
395C4×D4—73no—
396#117—73no—
397#117—73no—
398#117—73no—
399#117—73no—
400#117—76no—
401#117—73no—
402#117—73no—
403#117—76no—
404C2×C4⋊C4—73no—
405C2×C4⋊C4—76no—
406C2×C4⋊C4—73no—
407C2×C4⋊C4C4×D713yes—
408C2×C42—73no—
409C2×C42—76no—
410C2×C42—73no—
411C2×C42C22×D713yes—
412C4⋊C8—73no—
413C4⋊C8—73no—
414C4⋊C8—76no—
415C4⋊C8—71no—
416C4⋊C8—73no—
417C4⋊C8—73no—
418C4⋊C8—73no—
419C4⋊C8—73no—
420C4⋊C8—73no—
421C4⋊C8C22×D713yes—
422C4⋊C8C22×D716yes—
423C4⋊C8C22×D713yes—
424C22⋊C8—71no—
425C22⋊C8—73no—
426C22⋊C8—71no—
427C22⋊C8—73no—
428C22⋊C8—76no—
429C22⋊C8—73no—
430C22⋊C8—73no—
431#97—73no—
432#97—76no—
433#97—73no—
434#97—76no—
435#97C22×D713yes—
436#97C22×D713yes—
437C4×C8—71no—
438C4×C8—73no—
439C4×C8—73no—
440C4×C8—73no—
441C4×C8C22×D713yes—
442C4×C8C22×D713yes—
443C22×C7—21no—
444C22×C7—21no—
445C22×C7—21no—
446C22×C7C24×C411yescharacteristic
447C2×D7—23no—
448C2×D7—23no—
449C2×D7—21no—
450C2×D7—21no—
451C2×D7—21no—
452C2×D7C22×M1616yes—
453C4×C7—21no—
454C4×C7—23no—
455C4×C7—23no—
456C4×C7—23no—
457C4×C7C22×M1616yes—
458C4×C7C2×2−1+411yescharacteristic
459C4×C7#58511yescharacteristic
460Dic7—23no—
461Dic7—21no—
462Dic7—23no—
463Dic7—21no—
464Dic7—23no—
465Dic7C22×M1612yes—
466#41—78no—
467#41—1412no—
468#41—1412no—
469#41—1412no—
470C2×Q8—24no—
471C2×Q8—73no—
472C2×Q8—73no—
473C2×Q8—1412no—
474C2×Q8—144no—
475C2×Q8C2×C4×D713yes—
476C2×Q8C2×C4×D711yescharacteristic
477C2×D4—71no—
478C2×D4—73no—
479C2×D4—73no—
480C2×D4—73no—
481C22×C4—73no—
482C22×C4—76no—
483C22×C4—73no—
484C22×C4—73no—
485C22×C4—71no—
486C22×C4—73no—
487C22×C4—76no—
488C22×C4—73no—
489C22×C4C2×C4×D713yes—
490C22×C4C2×C4×D713yes—
491C22×C4C23×D711yescharacteristic
492M16—22no—
493M16—43no—
494M16—142no—
495M16—283no—
496C2×C8—71no—
497C2×C8—71no—
498C2×C8—73no—
499C2×C8—73no—
500C2×C8—146no—
501C2×C8—286no—
502C2×C8—283no—
503C2×C8C23×D713yes—
504C2×C8C23×D711yescharacteristic
505C2×C8C23×D713yes—
506C2×C8C23×D711yescharacteristic
507C4⋊C4—73no—
508C4⋊C4—73no—
509C4⋊C4—76no—
510C4⋊C4—71no—
511C4⋊C4—73no—
512C4⋊C4—73no—
513C4⋊C4—73no—
514C4⋊C4—73no—
515C4⋊C4—73no—
516C4⋊C4C2×C4×D716yes—
517C4⋊C4C2×C4×D713yes—
518C4⋊C4C2×C4×D713yes—
519C22⋊C4—73no—
520C22⋊C4—71no—
521C22⋊C4—76no—
522C22⋊C4—73no—
523C22⋊C4—73no—
524C22⋊C4—73no—
525C22⋊C4—71no—
526C42—73no—
527C42—76no—
528C42—73no—
529C42—73no—
530C42—76no—
531C42—73no—
532C42—71no—
533C42—73no—
534C42C2×C4×D713yes—
535C42C2×C4×D713yes—
536C42C23×D713yes—
537C42C23×D713yes—
538C2×C7—41no—
539C2×C7C4×2−1+411yescharacteristic
540C2×C7C23×M1611yescharacteristic
541C2×C7#357311yescharacteristic
542D7—26no—
543D7—41no—
544C23—73no—
545C23—71no—
546C23C22×C4×D711yescharacteristic
547Q8—26no—
548Q8—146no—
549Q8—146no—
550Q8D7×M1614yes—
551D4—712no—
552D4—146no—
553D4—142no—
554D4—146no—
555C2×C4—26no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—43no—
559C2×C4—71no—
560C2×C4—712no—
561C2×C4—73no—
562C2×C4—71no—
563C2×C4—71no—
564C2×C4—73no—
565C2×C4—73no—
566C2×C4—1412no—
567C2×C4—146no—
568C2×C4—143no—
569C2×C4—141no—
570C2×C4—143no—
571C2×C4—141no—
572C2×C4—283no—
573C2×C4—286no—
574C2×C4—283no—
575C2×C4C22×C4×D713yes—
576C2×C4C22×C4×D713yes—
577C2×C4C22×C4×D713yes—
578C2×C4C22×C4×D711yescharacteristic
579C2×C4C22×C4×D713yes—
580C2×C4C24×D711yescharacteristic
581C8—21no—
582C8—23no—
583C8—23no—
584C8—143no—
585C8—141no—
586C8—141no—
587C8—143no—
588C8#655112yescomplemented
589C7#6034311yescharacteristic, complemented, Sylow
590C22—21no—
591C22—21no—
592C22—21no—
593C22—76no—
594C22—143no—
595C22—143no—
596C22—141no—
597C22—141no—
598C22—141no—
599C22C23×C4×D711yescharacteristic
600C4—21no—
601C4—23no—
602C4—23no—
603C4—23no—
604C4—72no—
605C4—143no—
606C4—141no—
607C4—143no—
608C4—141no—
609C4—143no—
610C4C2×D7×M1616yes—
611C4C2×#655111yescharacteristic
612C4#9061611yescharacteristic
613C2—41no—
614C2—146no—
615C2—281no—
616C2C4×#655111yescharacteristic
617C2C22×D7×M1611yescharacteristic
618C2#1174423411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×Q8×M16C2
prime radicalC2×#6551C4
cosocleC2×C4×C7C25
hypercenterQ8×M16D7
hypocenterC7#60343
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. C7×Q8×M16C2semidirect
3. #11739882C2semidirect
4. #11737408C2semidirect
5. D7×#446C2semidirect
6. #11737390C2semidirect
7. #11737388C2semidirect
8. #11737366C2semidirect
9. Q8×#1131C2semidirect
10. #11737351C2semidirect
11. C8×#1161C2semidirect
12. #11736496C2semidirect
13. #11736495C2semidirect
14. #11735969C2semidirect
15. #11735968C2semidirect
177. Q8×M16D7semidirect
188. #1161M16semidirect
349. C7×C82−1+4semidirect
588. C8#6551semidirect
589. C7#60343semidirect

Maximal subgroups

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

subgroupindex
C4×#65512
C7×Q8×M162
#117398822
#117374082
D7×#4462
#117373902
#117373882
#117373662
Q8×#11312
#117373512
C8×#11612
#117364962
#117364952
#117359692
#117359682
#603437