Finite Group Databasebeta

← #14219862

Subgroups — #14219862

Order 1792.

1926 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×#6554C211yesmaximal, characteristic
2C7×#2971C211yesmaximal, characteristic, complemented
3#11739968C211yesmaximal, characteristic, complemented
4#11737366C211yesmaximal, characteristic, complemented
5#11737365C211yesmaximal, characteristic, complemented
6Q8×#1131C211yesmaximal, characteristic, complemented
7C8×Q8×D7C211yesmaximal, characteristic, complemented
8#11736969C213yesmaximal, complemented
9#11736968C213yesmaximal, complemented
10#11736967C213yesmaximal, complemented
11#11736962C213yesmaximal, complemented
12#11736784C213yesmaximal, complemented
13#11736760C213yesmaximal, complemented
14#11735968C213yesmaximal, complemented
15M16×Dic14C213yesmaximal, complemented
16C2×#6554C411yescharacteristic
17C4×C7×#41C2211yescharacteristic
18#90518C411yescharacteristic
19#90263C411yescharacteristic
20#90263C411yescharacteristic
21C4×Q8×D7C2211yescharacteristic
22C4×Q8×D7C2211yescharacteristic
23#90237C413yes—
24#90237C413yes—
25C4×#1159C2213yes—
26C4×#1159C2213yes—
27#90214C413yes—
28#90207C413yes—
29C4×#1157C2213yes—
30C2×C4×Dic14C2213yes—
31C7×C8×Q8C2211yescharacteristic
32C7×C8×Q8C2211yescharacteristic
33C7×#436C2213yes—
34C7×#436C2213yes—
35C7×#432C2213yes—
36C4×C7×M16C2213yes—
37#89900C2213yes—
38#89900C2213yes—
39#89899C2213yes—
40#89899C2213yes—
41#89890C2216yes—
42M16×Dic7C2216yes—
43Q8×C7⋊C8C2211yescharacteristic
44Q8×C7⋊C8C2211yescharacteristic
45#89784C2213yes—
46#89784C2213yes—
47#89697C2213yes—
48#89696C2213yes—
49#89609C2213yes—
50#89607C2213yes—
51#89605C2213yes—
52D7×C4⋊C8C2213yes—
53#89603C2213yes—
54#89603C2213yes—
55#89602C2213yes—
56#89602C2213yes—
57#89501C2216yes—
58#89500C2216yes—
59#89491C2216yes—
60#89490C2216yes—
61#89478C2211yescharacteristic
62#89478C2213yes—
63#89473C2213yes—
64#89473C2216yes—
65#89473C2213yes—
66C4×#1131C2213yes—
67#89457C2213yes—
68#89457C2213yes—
69C4×C8×D7C2213yes—
70#89451C2213yes—
71#89451C2213yes—
72C8×Dic14C2213yes—
73C8×Dic14C2213yes—
74#60333—71nomaximal, Sylow
75C2×C7×#41C2×C411yescharacteristic
76#6554—28no—
77C2×Q8×D7C2×C411yescharacteristic
78C2×Q8×D7C2×C411yescharacteristic
79C2×#1159C2×C413yes—
80C2×#1159C2×C413yes—
81C2×#1157C2×C413yes—
82C22×Dic14C2×C413yes—
83C2×C7×M16C2313yes—
84C2×C7×M16C2313yes—
85C4×C7×Q8C2×C411yescharacteristic
86C4×C7×Q8C2311yescharacteristic
87C4×C7×D4C2×C413yes—
88C4×C7×D4C2313yes—
89C7×#117C2×C413yes—
90C2×C42×C7C2313yes—
91Q8×Dic7C2×C411yescharacteristic
92Q8×Dic7C2×C411yescharacteristic
93D4×Dic7C2×C413yes—
94D4×Dic7C2×C413yes—
95C4×C22⋊D7C2×C413yes—
96C4×C22⋊D7C2×C413yes—
97C4×C22⋊D7C2313yes—
98C4×C22⋊D7C2313yes—
99#6489C2×C413yes—
100C2×C4⋊Dic7C2×C413yes—
101C2×Dic7⋊C4C2×C416yes—
102C2×C4×Dic7C2316yes—
103C2×#1136C2316yes—
104C2×#1131C2311yescharacteristic
105C2×#1131C2313yes—
106C2×C8×D7C2311yescharacteristic
107C2×C8×D7C2313yes—
108#6455C2×C413yes—
109#6455C2×C413yes—
110D7×C4⋊C4C2×C413yes—
111D7×C4⋊C4C2×C413yes—
112#6450C2×C413yes—
113#6450C2×C413yes—
114#6450C2×C413yes—
115#6450C2×C413yes—
116(C22⋊D7)⋊C4C2×C416yes—
117(C22⋊D7)⋊C4C2×C416yes—
118#6440C2×C416yes—
119#6440C2×C416yes—
120C4×D28C2×C413yes—
121C4×D28C2311yescharacteristic
122#6435C2×C413yes—
123#6435C2×C413yes—
124C42×D7C2313yes—
125C42×D7C2313yes—
126C4×Dic14C2×C416yes—
127C4×Dic14C2×C413yes—
128C4×Dic14C2×C413yes—
129C4×Dic14C2×C413yes—
130C4×Dic14C2313yes—
131C4×Dic14C2313yes—
132C4×Dic14C2313yes—
133C7×C4⋊C8C2313yes—
134C7×C4⋊C8C2313yes—
135C7×C22⋊C8C2316yes—
136C7×#97C2311yescharacteristic
137C7×#97C2313yes—
138C4×C7×C8C2313yes—
139C4×C7×C8C2313yes—
140C22⋊(C7⋊C8)C2313yes—
141C22⋊(C7⋊C8)C2313yes—
142(C2×D7)⋊C8C2313yes—
143(C2×D7)⋊C8C2313yes—
144(C2×D7)⋊C8C2311yescharacteristic
145(C2×D7)⋊C8C2311yescharacteristic
146#6389C2313yes—
147#6389C2316yes—
148#6389C2313yes—
149Dic7⋊C8C2313yes—
150Dic7⋊C8C2316yes—
151Dic7⋊C8C2313yes—
152Dic7⋊C8C2316yes—
153Dic7⋊C8C2313yes—
154Dic7⋊C8C2313yes—
155C8×Dic7C2313yes—
156C8×Dic7C2316yes—
157C8×Dic7C2313yes—
158C4⋊(C7⋊C8)C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4⋊(C7⋊C8)C2313yes—
161C4⋊(C7⋊C8)C2311yescharacteristic
162C4×C7⋊C8C2313yes—
163C4×C7⋊C8C2313yes—
164C4×2−1+4—71no—
165#2991—73no—
166#2991—71no—
167#2989—73no—
168#2989—73no—
169#2984—71no—
170#2984—73no—
171#2982—71no—
172#2971—73no—
173#2971D711yescharacteristic, complemented
174#2965—73no—
175#2965—73no—
176Q8×M16—71no—
177Q8×M16—73no—
178C2×C8×Q8—71no—
179C7×#41—24no—
180C2×C7×Q8C22×C411yescharacteristic
181C2×C7×D4C22×C413yes—
182C22×C4×C7C22×C413yes—
183C22×C4×C7C22×C413yes—
184C22×C4×C7C2413yes—
185C2×C22⋊D7C22×C413yes—
186C2×C22⋊D7C22×C413yes—
187C22×Dic7C22×C416yes—
188C22×Dic7C22×C416yes—
189Q8×D7—24no—
190Q8×D7M1618yescomplemented
191#1159—212no—
192#1159—212no—
193#1157—212no—
194C2×D28C22×C411yescharacteristic
195C2×C4×D7C22×C413yes—
196C2×C4×D7C22×C413yes—
197C2×C4×D7C22×C413yes—
198C2×C4×D7C22×C413yes—
199C2×C4×D7C2411yescharacteristic
200C2×C4×D7C2411yescharacteristic
201C2×Dic14—212no—
202C2×Dic14C22×C413yes—
203C2×Dic14C22×C413yes—
204C2×Dic14C22×C413yes—
205C7×M16—26no—
206C7×M16—43no—
207C2×C7×C8C2411yescharacteristic
208C2×C7×C8C2413yes—
209C2×C7×C8C2413yes—
210C2×C7×C8C2411yescharacteristic
211C7×C4⋊C4C22×C413yes—
212C7×C4⋊C4C22×C413yes—
213C7×C22⋊C4C22×C416yes—
214C42×C7C22×C413yes—
215C42×C7C22×C413yes—
216C42×C7C2413yes—
217C42×C7C2411yescharacteristic
218C22⋊Dic7C22×C413yes—
219C22⋊Dic7C22×C413yes—
220(C2×D7)⋊C4C22×C411yescharacteristic
221(C2×D7)⋊C4C22×C411yescharacteristic
222(C2×D7)⋊C4C22×C413yes—
223(C2×D7)⋊C4C22×C413yes—
224C4⋊Dic7C22×C411yescharacteristic
225C4⋊Dic7C22×C413yes—
226C4⋊Dic7C22×C413yes—
227C4⋊Dic7C22×C413yes—
228Dic7⋊C4C22×C413yes—
229Dic7⋊C4C22×C413yes—
230Dic7⋊C4C22×C416yes—
231Dic7⋊C4C22×C416yes—
232Dic7⋊C4C22×C413yes—
233Dic7⋊C4C22×C413yes—
234C4×Dic7C22×C416yes—
235C4×Dic7C22×C416yes—
236C4×Dic7C22×C413yes—
237C4×Dic7C22×C413yes—
238C4×Dic7C22×C413yes—
239C4×Dic7C22×C413yes—
240C4×Dic7C2413yes—
241C4×Dic7C2413yes—
242#1136—46no—
243C2×C7⋊C8C2411yescharacteristic
244C2×C7⋊C8C2411yescharacteristic
245C2×C7⋊C8C2413yes—
246C2×C7⋊C8C2413yes—
247#1131—22no—
248#1131—43no—
249C8×D7—22no—
250C8×D7—43no—
251C2×2−1+4—71no—
252#520—71no—
253#520—73no—
254#520—73no—
255#520—73no—
256#519—71no—
257#519—71no—
258#519—73no—
259C4×#41—73no—
260C4×#41—73no—
261C4×#41—73no—
262C4×#41C2×D711yescharacteristic
263C2×C4×Q8—71no—
264C2×C4×Q8—71no—
265C2×C4×Q8—73no—
266#446—73no—
267#446—73no—
268#446—73no—
269#446—76no—
270#446—73no—
271#446—73no—
272#446—73no—
273C8×Q8—71no—
274C8×Q8—71no—
275C8×Q8—73no—
276C8×Q8—73no—
277C8×Q8—73no—
278C8×Q8—73no—
279C8×Q8C2×D711yescharacteristic
280C8×Q8C2×D711yescharacteristic
281#436—73no—
282#436—73no—
283#436—73no—
284#436—73no—
285#436—76no—
286#436C2×D713yes—
287#436C2×D713yes—
288#435—73no—
289#435—73no—
290#435—76no—
291#435—71no—
292#435—73no—
293#432—73no—
294#432—76no—
295#432C2×D713yes—
296#431—73no—
297#431—73no—
298#431—76no—
299#431—73no—
300#431—73no—
301#423—76no—
302#423—73no—
303#423—73no—
304C2×C4⋊C8—73no—
305C4×M16—76no—
306C4×M16—73no—
307C4×M16C2×D713yes—
308C2×C4×C8—73no—
309C23×C7C23×C413yes—
310C22×D7C23×C411yescharacteristic
311C22×D7C23×C411yescharacteristic
312C7×Q8C2×M1614yes—
313C7×D4—26no—
314C2×C4×C7—26no—
315C2×C4×C7—23no—
316C2×C4×C7—23no—
317C2×C4×C7—43no—
318C2×C4×C7C23×C413yes—
319C2×C4×C7C23×C413yes—
320C2×C4×C7C23×C411yescharacteristic
321C2×C4×C7C23×C411yescharacteristic
322C2×C4×C7C23×C413yes—
323C2×C4×C7C2511yescharacteristic
324C22⋊D7—26no—
325C22⋊D7—26no—
326C2×Dic7—212no—
327C2×Dic7—46no—
328C2×Dic7C23×C411yescharacteristic
329C2×Dic7C23×C411yescharacteristic
330C2×Dic7C23×C413yes—
331C2×Dic7C23×C413yes—
332C2×Dic7C23×C413yes—
333C2×Dic7C23×C413yes—
334D28—22no—
335C4×D7—26no—
336C4×D7—21no—
337C4×D7—21no—
338C4×D7—21no—
339C4×D7—21no—
340C4×D7—43no—
341C4×D7—43no—
342C4×D7C2×M16112yes—
343Dic14—26no—
344Dic14—26no—
345Dic14C2×M16112yes—
346C7×C8—23no—
347C7×C8—23no—
348C7×C8—21no—
349C7×C82−1+412yescomplemented
350C7⋊C8—21no—
351C7⋊C8—21no—
352C7⋊C8—23no—
353C7⋊C8—23no—
3542−1+4—148no—
355C2×#41—73no—
356C2×#41—73no—
357C2×#41—73no—
358C2×#41C4×D711yescharacteristic
359C22×Q8—71no—
360C22×Q8—71no—
361C22×Q8—73no—
362C2×M16—76no—
363C2×M16—71no—
364C2×M16—73no—
365C2×M16C22×D713yes—
366C2×M16C22×D713yes—
367C22×C8—71no—
368C22×C8—73no—
369C4×Q8—71no—
370C4×Q8—71no—
371C4×Q8—73no—
372C4×Q8—73no—
373C4×Q8—73no—
374C4×Q8—73no—
375C4×Q8—73no—
376C4×Q8—73no—
377C4×Q8—76no—
378C4×Q8—73no—
379C4×Q8—73no—
380C4×Q8—73no—
381C4×Q8—73no—
382C4×Q8C4×D711yescharacteristic
383C4×Q8C22×D711yescharacteristic
384C4×D4—73no—
385C4×D4—73no—
386C4×D4—73no—
387C4×D4—73no—
388C4×D4—73no—
389C4×D4—73no—
390C4×D4—76no—
391C4×D4—76no—
392C4×D4—73no—
393C4×D4—71no—
394C4×D4C4×D713yes—
395C4×D4C22×D713yes—
396#117—73no—
397#117—73no—
398#117—73no—
399#117—76no—
400#117—76no—
401#117—73no—
402#117—73no—
403#117C4×D713yes—
404C2×C4⋊C4—76no—
405C2×C4⋊C4—73no—
406C2×C4⋊C4—73no—
407C2×C4⋊C4—73no—
408C2×C42—76no—
409C2×C42—73no—
410C2×C42—73no—
411C2×C42C22×D713yes—
412C4⋊C8—76no—
413C4⋊C8—76no—
414C4⋊C8—73no—
415C4⋊C8—73no—
416C4⋊C8—73no—
417C4⋊C8—73no—
418C4⋊C8—73no—
419C4⋊C8—73no—
420C4⋊C8—73no—
421C4⋊C8—71no—
422C4⋊C8C22×D713yes—
423C4⋊C8C22×D713yes—
424C22⋊C8—73no—
425C22⋊C8—73no—
426C22⋊C8—71no—
427C22⋊C8—73no—
428C22⋊C8—71no—
429C22⋊C8—73no—
430C22⋊C8C22×D716yes—
431#97—73no—
432#97—73no—
433#97—76no—
434#97C22×D711yescharacteristic
435#97C22×D713yes—
436C4×C8—73no—
437C4×C8—73no—
438C4×C8—76no—
439C4×C8—73no—
440C4×C8—73no—
441C4×C8C22×D713yes—
442C4×C8C22×D713yes—
443C22×C7—23no—
444C22×C7—23no—
445C22×C7—23no—
446C22×C7C24×C411yescharacteristic
447C2×D7—21no—
448C2×D7—21no—
449C2×D7—21no—
450C2×D7—21no—
451C2×D7—21no—
452C2×D7C22×M1612yes—
453C4×C7—23no—
454C4×C7—21no—
455C4×C7—23no—
456C4×C7—21no—
457C4×C7C22×M1616yes—
458C4×C7C2×2−1+411yescharacteristic
459C4×C7#58511yescharacteristic
460Dic7—21no—
461Dic7—21no—
462Dic7—23no—
463Dic7—23no—
464Dic7—23no—
465Dic7C22×M1616yes—
466#41—24no—
467#41—1412no—
468#41—1412no—
469#41—1412no—
470C2×Q8—78no—
471C2×Q8—73no—
472C2×Q8—73no—
473C2×Q8—73no—
474C2×Q8—1412no—
475C2×Q8—144no—
476C2×Q8C2×C4×D711yescharacteristic
477C2×D4—73no—
478C2×D4—73no—
479C2×D4—71no—
480C2×D4C2×C4×D713yes—
481C22×C4—73no—
482C22×C4—76no—
483C22×C4—76no—
484C22×C4—73no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—73no—
488C22×C4—73no—
489C22×C4C2×C4×D713yes—
490C22×C4C2×C4×D713yes—
491C22×C4C23×D713yes—
492M16—26no—
493M16—43no—
494M16—142no—
495M16—283no—
496M16—286no—
497C2×C8—71no—
498C2×C8—71no—
499C2×C8—73no—
500C2×C8—73no—
501C2×C8—142no—
502C2×C8—283no—
503C2×C8C23×D713yes—
504C2×C8C23×D713yes—
505C2×C8C23×D711yescharacteristic
506C2×C8C23×D711yescharacteristic
507C4⋊C4—76no—
508C4⋊C4—73no—
509C4⋊C4—76no—
510C4⋊C4—73no—
511C4⋊C4—73no—
512C4⋊C4—73no—
513C4⋊C4—73no—
514C4⋊C4—73no—
515C4⋊C4—73no—
516C4⋊C4—71no—
517C4⋊C4C2×C4×D713yes—
518C4⋊C4C2×C4×D713yes—
519C22⋊C4—73no—
520C22⋊C4—73no—
521C22⋊C4—71no—
522C22⋊C4—71no—
523C22⋊C4—73no—
524C22⋊C4—73no—
525C22⋊C4C2×C4×D716yes—
526C42—73no—
527C42—76no—
528C42—73no—
529C42—73no—
530C42—73no—
531C42—76no—
532C42—73no—
533C42—73no—
534C42C2×C4×D713yes—
535C42C2×C4×D713yes—
536C42C23×D713yes—
537C42C23×D711yescharacteristic
538C2×C7—43no—
539C2×C7C4×2−1+411yescharacteristic
540C2×C7C23×M1611yescharacteristic
541C2×C7#357311yescharacteristic
542D7—22no—
543D7—41no—
544C23—71no—
545C23—71no—
546C23C22×C4×D713yes—
547Q8—712no—
548Q8—146no—
549Q8—146no—
550Q8D7×M1614yes—
551D4—26no—
552D4—142no—
553D4—146no—
554D4—146no—
555C2×C4—26no—
556C2×C4—23no—
557C2×C4—23no—
558C2×C4—43no—
559C2×C4—73no—
560C2×C4—712no—
561C2×C4—71no—
562C2×C4—71no—
563C2×C4—73no—
564C2×C4—73no—
565C2×C4—73no—
566C2×C4—1412no—
567C2×C4—146no—
568C2×C4—141no—
569C2×C4—141no—
570C2×C4—141no—
571C2×C4—141no—
572C2×C4—283no—
573C2×C4—283no—
574C2×C4—286no—
575C2×C4C22×C4×D713yes—
576C2×C4C22×C4×D713yes—
577C2×C4C22×C4×D711yescharacteristic
578C2×C4C22×C4×D711yescharacteristic
579C2×C4C22×C4×D713yes—
580C2×C4C24×D711yescharacteristic
581C8—23no—
582C8—23no—
583C8—21no—
584C8—141no—
585C8—141no—
586C8—143no—
587C8—143no—
588C8#655412yescomplemented
589C7#6033311yescharacteristic, complemented, Sylow
590C22—23no—
591C22—23no—
592C22—23no—
593C22—72no—
594C22—141no—
595C22—141no—
596C22—141no—
597C22—141no—
598C22—141no—
599C22C23×C4×D711yescharacteristic
600C4—23no—
601C4—21no—
602C4—21no—
603C4—23no—
604C4—76no—
605C4—141no—
606C4—141no—
607C4—143no—
608C4—143no—
609C4—143no—
610C4C2×D7×M1616yes—
611C4C2×#655411yescharacteristic
612C4#9061611yescharacteristic
613C2—43no—
614C2—142no—
615C2—281no—
616C2C4×#655411yescharacteristic
617C2C22×D7×M1611yescharacteristic
618C2#1174423411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2971C2
prime radicalC2×#6554C4
cosocleC2×C4×C7C25
hypercenter#2971D7
hypocenterC7#60333
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. C7×#2971C2semidirect
3. #11739968C2semidirect
4. #11737366C2semidirect
5. #11737365C2semidirect
6. Q8×#1131C2semidirect
7. C8×Q8×D7C2semidirect
8. #11736969C2semidirect
9. #11736968C2semidirect
10. #11736967C2semidirect
11. #11736962C2semidirect
12. #11736784C2semidirect
13. #11736760C2semidirect
14. #11735968C2semidirect
15. M16×Dic14C2semidirect
173. #2971D7semidirect
190. Q8×D7M16semidirect
349. C7×C82−1+4semidirect
588. C8#6554semidirect
589. C7#60333semidirect

Maximal subgroups

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

subgroupindex
C4×#65542
C7×#29712
#117399682
#117373662
#117373652
Q8×#11312
C8×Q8×D72
#117369692
#117369682
#117369672
#117369622
#117367842
#117367602
#117359682
M16×Dic142
#603337