Finite Group Databasebeta

← #13513948

Subgroups — #13513948

Order 1792.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90213C211yesmaximal, characteristic
2C7×#2314C211yesmaximal, characteristic, complemented
3#11739676C211yesmaximal, characteristic, complemented
4#11733280C214yesmaximal, complemented
5#11732921C214yesmaximal, complemented
6#11732751C214yesmaximal, complemented
7#11732409C214yesmaximal, complemented
8#11731803C211yesmaximal, characteristic, complemented
9#11731763C211yesmaximal, characteristic, complemented
10#11731669C211yesmaximal, characteristic, complemented
11D7×#382C211yesmaximal, characteristic, complemented
12#11731448C212yesmaximal
13#11731308C212yesmaximal, complemented
14#11731197C212yesmaximal, complemented
15#11731195C212yesmaximal, complemented
16C22×#1157C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19#90213C4116yescomplemented
20C2×#6456C2212yes—
21C2×#6455C2212yes—
22C2×D7×C4⋊C4C2212yes—
23C2×#6450C2212yes—
24C2×(C22⋊D7)⋊C4C2212yes—
25C2×(C22⋊D7)⋊C4C2212yes—
26C2×#6435C2211yescharacteristic
27C2×#6435C2211yescharacteristic
28C2×C42×D7C2211yescharacteristic
29C2×C42×D7C2211yescharacteristic
30C7×#388C2214yes—
31C7×#387C2214yes—
32C7×#384C2211yescharacteristic
33C7×#382C2211yescharacteristic
34C7×#380C2212yes—
35C4×C7×C4⋊C4C2212yes—
36#89984C2212yes—
37#89762C2214yes—
38#89759C2214yes—
39#89759C2212yes—
40#89759C2212yes—
41#89753C2214yes—
42#89750C2214yes—
43#89749C2212yes—
44#89749C2212yes—
45C4⋊C4×Dic7C2214yes—
46C4⋊C4×Dic7C2212yes—
47C4⋊C4×Dic7C2212yes—
48#89729C2214yes—
49#89728C2214yes—
50#89723C2214yes—
51#89715C2214yes—
52C22⋊C4×Dic7C2214yes—
53C4×(C2×D7)⋊C4C2214yes—
54C4×(C2×D7)⋊C4C2214yes—
55#89708C2211yescharacteristic
56#89707C2211yescharacteristic
57C4×C4⋊Dic7C2212yes—
58#89704C2214yes—
59C4×Dic7⋊C4C2214yes—
60#89442C2214yes—
61#89438C2214yes—
62#89437C2214yes—
63#89436C2214yes—
64D7×#95C2214yes—
65#89427C2214yes—
66#89423C2214yes—
67#89423C2214yes—
68#89422C2214yes—
69#89421C2214yes—
70#89420C2214yes—
71#89418C2214yes—
72#34189—71nomaximal, Sylow
73C23×C4×C7C2311yescharacteristic
74C22×C22⋊D7C2311yescharacteristic
75C22×C22⋊D7C2311yescharacteristic
76C23×Dic7C2312yes—
77C2×#1157—24no—
78C2×#1157—24no—
79C2×#1157C2×C418yes—
80C22×D28C2311yescharacteristic
81C22×C4×D7C2311yescharacteristic
82C22×C4×D7C2314yes—
83C22×C4×D7C2312yes—
84C22×C4×D7C2312yes—
85C22×C4×D7C2311yescharacteristic
86C22×Dic14C2311yescharacteristic
87C7×#117C2×C418yescomplemented
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2314yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C22⋊C4C2314yes—
93C2×C7×C22⋊C4C2312yes—
94C2×C42×C7C2314yes—
95C2×C42×C7C2311yescharacteristic
96C2×C42×C7C2311yescharacteristic
97C2×C22⋊Dic7C2314yes—
98C2×(C2×D7)⋊C4C2314yes—
99C2×(C2×D7)⋊C4C2314yes—
100C2×(C2×D7)⋊C4C2314yes—
101C2×(C2×D7)⋊C4C2312yes—
102C2×(C2×D7)⋊C4C2312yes—
103C2×C4⋊Dic7C2312yes—
104C2×C4⋊Dic7C2312yes—
105C2×C4⋊Dic7C2314yes—
106C2×Dic7⋊C4C2314yes—
107C2×Dic7⋊C4C2312yes—
108C2×Dic7⋊C4C2312yes—
109C2×C4×Dic7C2×C418yescomplemented
110C2×C4×Dic7C2314yes—
111C2×C4×Dic7C2314yes—
112C2×C4×Dic7C2314yes—
113C2×C4×Dic7C2314yes—
114C2×C4×Dic7C2311yescharacteristic
115C2×C4×Dic7C2312yes—
116C2×C4×Dic7C2312yes—
117C2×C4×Dic7C2311yescharacteristic
118C2×C4×Dic7C2312yes—
119#6456C2×C4116yes—
120#6455—28no—
121D7×C4⋊C4—28no—
122#6450C2×C4116yescomplemented
123(C22⋊D7)⋊C4C2×C4116yes—
124(C22⋊D7)⋊C4C2×C4116yescomplemented
125#6435C2×C418yes—
126#6435C2×C418yescomplemented
127C42×D7C2×C418yes—
128C42×D7C2×C418yescomplemented
129C7×#95C2314yes—
130C7×#95C2314yes—
131C7×#95C2314yes—
132C7⋊#95C2314yes—
133C7⋊#95C2314yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137C7⋊#95C2314yes—
138C7⋊#95C2314yes—
139C7⋊#95C2314yes—
140C7⋊#95C2314yes—
141C7⋊#95C2314yes—
142C7⋊#95C2314yes—
143C2×C4×#41—71no—
144#2357—74no—
145#2356—74no—
146#2353—74no—
147#2353—74no—
148#2314—71no—
149#2314—71no—
150#2314—71no—
151#2314D711yescharacteristic, complemented
152#2311—72no—
153#2285—71no—
154#2283—72no—
155#2283—72no—
156C2×#382—71no—
157#2274—72no—
158C24×C7C2411yescharacteristic
159C23×D7C2411yescharacteristic
160C23×D7C2411yescharacteristic
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7C22×C414yes—
164C22×C4×C7C2414yes—
165C22×C4×C7C2414yes—
166C22×C4×C7C2411yescharacteristic
167C22×C4×C7C2412yes—
168C22×C4×C7C2412yes—
169C22×C4×C7C2411yescharacteristic
170C2×C22⋊D7—22no—
171C2×C22⋊D7—22no—
172C2×C22⋊D7—22no—
173C2×C22⋊D7—22no—
174C2×C22⋊D7C22×C414yes—
175C2×C22⋊D7C22×C414yes—
176C22×Dic7—42no—
177C22×Dic7—42no—
178C22×Dic7C22×C418yes—
179C22×Dic7C2414yes—
180C22×Dic7C2414yes—
181C22×Dic7C2412yes—
182C22×Dic7C2411yescharacteristic
183C22×Dic7C2412yes—
184C22×Dic7C2411yescharacteristic
185#1157—232no—
186C2×D28—22no—
187C2×D28—22no—
188C2×D28C22×C414yes—
189C2×C4×D7—22no—
190C2×C4×D7—22no—
191C2×C4×D7—28no—
192C2×C4×D7—42no—
193C2×C4×D7—42no—
194C2×C4×D7—42no—
195C2×C4×D7—42no—
196C2×C4×D7—44no—
197C2×C4×D7—44no—
198C2×C4×D7C22×C414yes—
199C2×C4×D7C22×C418yes—
200C2×C4×D7C22×C418yes—
201C2×C4×D7C22×C414yes—
202C2×C4×D7#4114yes—
203C2×C4×D7#4114yes—
204C2×Dic14—22no—
205C2×Dic14—22no—
206C2×Dic14C22×C414yes—
207C7×C4⋊C4—24no—
208C7×C4⋊C4—24no—
209C7×C4⋊C4—44no—
210C7×C4⋊C4C22×C418yes—
211C7×C22⋊C4—44no—
212C7×C22⋊C4C22×C418yes—
213C42×C7—44no—
214C42×C7C22×C414yes—
215C42×C7C22×C414yes—
216C22⋊Dic7—44no—
217(C2×D7)⋊C4—28no—
218(C2×D7)⋊C4—44no—
219(C2×D7)⋊C4—44no—
220(C2×D7)⋊C4C22×C418yes—
221(C2×D7)⋊C4C22×C418yes—
222C4⋊Dic7—24no—
223C4⋊Dic7—24no—
224C4⋊Dic7—44no—
225Dic7⋊C4—28no—
226Dic7⋊C4C22×C418yes—
227Dic7⋊C4C22×C418yes—
228C4×Dic7—28no—
229C4×Dic7—44no—
230C4×Dic7—44no—
231C4×Dic7—44no—
232C4×Dic7C22×C414yes—
233C4×Dic7C22×C418yes—
234C4×Dic7C22×C418yes—
235C4×Dic7C22×C414yes—
236C4×Dic7C22×C418yes—
237C22×#41—71no—
238C4×#41—716no—
239C2×C4×Q8—72no—
240C2×C4×D4—72no—
241C2×C4×D4—72no—
242C2×C4×D4—72no—
243C2×#117—72no—
244C2×#117—71no—
245C2×#117—71no—
246C2×#117C2×D711yescharacteristic
247C22×C4⋊C4—72no—
248C22×C42—71no—
249C22×C42—71no—
250C22×C42—71no—
251#391—74no—
252#390—74no—
253#388—74no—
254#388—74no—
255#388—74no—
256#388C2×D714yes—
257#387—74no—
258#387—74no—
259#387—74no—
260#387C2×D714yes—
261#386—74no—
262#386—74no—
263#384—71no—
264#384C2×D711yescharacteristic
265#383—74no—
266#383—74no—
267#382—72no—
268#382—72no—
269#382—71no—
270#382—74no—
271#382C2×D711yescharacteristic
272#381—74no—
273#381—74no—
274#380—72no—
275#380—72no—
276#380—72no—
277#380—74no—
278#380C2×D712yes—
279C4×C4⋊C4—72no—
280C4×C4⋊C4—72no—
281C4×C4⋊C4—74no—
282C4×C4⋊C4—72no—
283C4×C4⋊C4—74no—
284C4×C4⋊C4—74no—
285C4×C4⋊C4C2×D712yes—
286C4×C22⋊C4—74no—
287C4×C22⋊C4—74no—
288C4×C22⋊C4—74no—
289C4×C22⋊C4—74no—
290#376—74no—
291#376—74no—
292C2×#95—74no—
293C23×C7—21no—
294C23×C7—21no—
295C23×C7—22no—
296C23×C7—22no—
297C23×C7C23×C412yes—
298C23×C7C2511yescharacteristic
299C22×D7—22no—
300C22×D7—22no—
301C22×D7—21no—
302C22×D7—21no—
303C22×D7—22no—
304C22×D7—22no—
305C22×D7C23×C412yes—
306C22×D7C23×C412yes—
307C22×D7C2×#4112yes—
308C22×D7C2×#4112yes—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—28no—
316C2×C4×C7—24no—
317C2×C4×C7—24no—
318C2×C4×C7—24no—
319C2×C4×C7—24no—
320C2×C4×C7—24no—
321C2×C4×C7—24no—
322C2×C4×C7C23×C412yes—
323C2×C4×C7C23×C414yes—
324C2×C4×C7C23×C414yes—
325C2×C4×C7C23×C412yes—
326C2×C4×C7C2×#4112yes—
327C2×C4×C7C2×#4112yes—
328C22⋊D7—28no—
329C22⋊D7—28no—
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—22no—
335C2×Dic7—22no—
336C2×Dic7—24no—
337C2×Dic7—24no—
338C2×Dic7—24no—
339C2×Dic7—24no—
340C2×Dic7—24no—
341C2×Dic7—24no—
342C2×Dic7—48no—
343C2×Dic7C23×C414yes—
344C2×Dic7C23×C414yes—
345C2×Dic7C23×C412yes—
346C2×Dic7C23×C412yes—
347C2×Dic7C2×#4112yes—
348C2×Dic7C2×#4112yes—
349D28—28no—
350C4×D7—28no—
351C4×D7—48no—
352C4×D7—48no—
353C4×D7—88no—
354C4×D7#117116yes—
355Dic14—28no—
356C2×#41—78no—
357C2×#41—144no—
358C2×#41—144no—
359C22×Q8—71no—
360C22×D4—71no—
361C22×D4—71no—
362C22×D4—71no—
363C23×C4—71no—
364C23×C4—72no—
365C23×C4—71no—
366C23×C4—72no—
367C23×C4—72no—
368C23×C4—74no—
369C23×C4C22×D711yescharacteristic
370C4×Q8—716no—
371C4×D4—716no—
372C4×D4—716no—
373C4×D4—716no—
374#117—78no—
375#117—78no—
376#117—148no—
377#117C4×D718yescomplemented
378C2×C4⋊C4—72no—
379C2×C4⋊C4—72no—
380C2×C4⋊C4—74no—
381C2×C4⋊C4—72no—
382C2×C4⋊C4—72no—
383C2×C4⋊C4—74no—
384C2×C4⋊C4—148no—
385C2×C4⋊C4C22×D712yes—
386C2×C4⋊C4C22×D712yes—
387C2×C4⋊C4C22×D714yes—
388C2×C4⋊C4C22×D712yes—
389C2×C22⋊C4—72no—
390C2×C22⋊C4—72no—
391C2×C22⋊C4—74no—
392C2×C22⋊C4—74no—
393C2×C22⋊C4—74no—
394C2×C22⋊C4—74no—
395C2×C22⋊C4C22×D714yes—
396C2×C22⋊C4C22×D712yes—
397C2×C42—78no—
398C2×C42—71no—
399C2×C42—72no—
400C2×C42—72no—
401C2×C42—72no—
402C2×C42—71no—
403C2×C42—78no—
404C2×C42—78no—
405C2×C42—74no—
406C2×C42—74no—
407C2×C42—74no—
408C2×C42—74no—
409C2×C42C22×D714yes—
410C2×C42C22×D711yescharacteristic
411C2×C42C22×D711yescharacteristic
412#95—74no—
413#95—74no—
414#95—74no—
415#95—74no—
416#95—74no—
417#95—74no—
418#95—74no—
419#95—74no—
420#95—74no—
421#95—74no—
422#95—74no—
423#95C22×D714yes—
424#95C22×D714yes—
425#95C22×D714yes—
426C22×C7—22no—
427C22×C7—24no—
428C22×C7—42no—
429C22×C7—41no—
430C22×C7—41no—
431C22×C7C24×C411yescharacteristic
432C22×C7C22×#4111yescharacteristic
433C22×C7C22×#4111yescharacteristic
434C22×C7#58512yes—
435C22×C7#58512yes—
436C2×D7—24no—
437C2×D7—24no—
438C2×D7—22no—
439C2×D7—22no—
440C2×D7—22no—
441C2×D7—24no—
442C2×D7—42no—
443C2×D7—41no—
444C2×D7—41no—
445C2×D7C2×#11714yes—
446C4×C7—22no—
447C4×C7—24no—
448C4×C7—24no—
449C4×C7—44no—
450C4×C7—44no—
451C4×C7C2×#11714yes—
452Dic7—22no—
453Dic7—24no—
454Dic7—24no—
455Dic7—44no—
456Dic7—44no—
457Dic7C2×#11714yes—
458C24—71no—
459C24—71no—
460C24C23×D711yescharacteristic
461#41—1432no—
462C2×Q8—74no—
463C2×Q8—142no—
464C2×Q8—142no—
465C2×D4—74no—
466C2×D4—74no—
467C2×D4—74no—
468C2×D4—142no—
469C2×D4—142no—
470C2×D4—142no—
471C2×D4—142no—
472C2×D4—142no—
473C2×D4—142no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—71no—
477C22×C4—74no—
478C22×C4—74no—
479C22×C4—74no—
480C22×C4—72no—
481C22×C4—71no—
482C22×C4—72no—
483C22×C4—78no—
484C22×C4—74no—
485C22×C4—78no—
486C22×C4—78no—
487C22×C4—74no—
488C22×C4—74no—
489C22×C4—148no—
490C22×C4—142no—
491C22×C4—142no—
492C22×C4—282no—
493C22×C4—282no—
494C22×C4—282no—
495C22×C4—282no—
496C22×C4—284no—
497C22×C4—284no—
498C22×C4—282no—
499C22×C4—282no—
500C22×C4C2×C4×D714yes—
501C22×C4C23×D714yes—
502C22×C4C23×D714yes—
503C22×C4C23×D711yescharacteristic
504C22×C4C23×D712yes—
505C22×C4C23×D712yes—
506C22×C4C23×D711yescharacteristic
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—44no—
510C4⋊C4—78no—
511C4⋊C4—78no—
512C4⋊C4—148no—
513C4⋊C4—144no—
514C4⋊C4—144no—
515C4⋊C4—284no—
516C4⋊C4C2×C4×D718yes—
517C22⋊C4—44no—
518C22⋊C4—78no—
519C22⋊C4—78no—
520C22⋊C4—148no—
521C22⋊C4—284no—
522C22⋊C4—284no—
523C22⋊C4—284no—
524C22⋊C4C2×C4×D718yes—
525C42—44no—
526C42—74no—
527C42—74no—
528C42—78no—
529C42—78no—
530C42—78no—
531C42—148no—
532C42—284no—
533C42—284no—
534C42—284no—
535C42C2×C4×D714yes—
536C42C2×C4×D714yes—
537C2×C7—42no—
538C2×C7C22×#11711yescharacteristic
539C2×C7#342812yes—
540C2×C7#351012yes—
541C2×C7#351211yescharacteristic
542C2×C7#351211yescharacteristic
543D7—24no—
544D7—42no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—72no—
550C23—72no—
551C23—72no—
552C23—72no—
553C23—142no—
554C23—142no—
555C23—141no—
556C23—141no—
557C23—142no—
558C23—142no—
559C23C22×C4×D712yes—
560C23C24×D711yescharacteristic
561Q8—148no—
562D4—148no—
563D4—148no—
564D4—148no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—72no—
579C2×C4—72no—
580C2×C4—72no—
581C2×C4—716no—
582C2×C4—74no—
583C2×C4—74no—
584C2×C4—72no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—148no—
592C2×C4—144no—
593C2×C4—144no—
594C2×C4—144no—
595C2×C4—144no—
596C2×C4—144no—
597C2×C4—144no—
598C2×C4—288no—
599C2×C4—288no—
600C2×C4—288no—
601C2×C4—568no—
602C2×C4C22×C4×D712yes—
603C2×C4C22×C4×D714yes—
604C2×C4C22×C4×D714yes—
605C2×C4C22×C4×D712yes—
606C2×C4D7×#4112yes—
607C2×C4D7×#4112yes—
608C7#3418911yescharacteristic, complemented, Sylow
609C22—22no—
610C22—24no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—74no—
615C22—142no—
616C22—142no—
617C22—144no—
618C22—144no—
619C22—142no—
620C22—144no—
621C22—281no—
622C22—281no—
623C22—282no—
624C22C23×C4×D711yescharacteristic
625C22C2×D7×#4111yescharacteristic
626C22C2×D7×#4111yescharacteristic
627C22#9061812yes—
628C22#9062012yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—74no—
635C4—142no—
636C4—144no—
637C4—144no—
638C4—284no—
639C4—284no—
640C4D7×#11714yes—
641C2—42no—
642C2—144no—
643C2—282no—
644C2C2×D7×#11711yescharacteristic
645C2#1174295312yes—
646C2#1174353711yescharacteristic
647C2#1174356511yescharacteristic
648C2#1174362012yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2314C2
socle, cosocleC23×C7C25
prime radicalC22×#1157C22
hypercenter#2314D7
hypocenterC7#34189
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. C7×#2314C2semidirect
3. #11739676C2semidirect
4. #11733280C2semidirect
5. #11732921C2semidirect
6. #11732751C2semidirect
7. #11732409C2semidirect
8. #11731803C2semidirect
9. #11731763C2semidirect
10. #11731669C2semidirect
11. D7×#382C2semidirect
13. #11731308C2semidirect
14. #11731197C2semidirect
15. #11731195C2semidirect
19. #90213C4semidirect
87. C7×#117C2×C4semidirect
109. C2×C4×Dic7C2×C4semidirect
122. #6450C2×C4semidirect
124. (C22⋊D7)⋊C4C2×C4semidirect
126. #6435C2×C4semidirect
128. C42×D7C2×C4semidirect
151. #2314D7semidirect
377. #117C4×D7semidirect
608. C7#34189semidirect

Maximal subgroups

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

subgroupindex
C2×#902132
C7×#23142
#117396762
#117332802
#117329212
#117327512
#117324092
#117318032
#117317632
#117316692
D7×#3822
#117314482
#117313082
#117311972
#117311952
#341897