Finite Group Databasebeta

← #13275106

Subgroups — #13275106

Order 1792.

1862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C2×#89418C212yesmaximal
3C2×#89421C211yesmaximal, characteristic
4C2×#89420C211yesmaximal, characteristic
5C7×#1434C211yesmaximal, characteristic
6#11729288C211yesmaximal, characteristic
7#11727161C212yesmaximal
8#11727160C212yesmaximal
9Dic7⋊#95C212yesmaximal
10#11727156C211yesmaximal, characteristic
11#11727156C211yesmaximal, characteristic
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C22×Dic7⋊C4C2211yescharacteristic
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2212yes—
18C22×C4×Dic7C2211yescharacteristic
19C22×C4×Dic7C2212yes—
20C22×C4×Dic7C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C2×C4×Dic14—24no—
23C2×C4×Dic14—24no—
24C2×C4×Dic14C418yescomplemented
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89421—24no—
39#89420—24no—
40#89418C4116yescomplemented
41#13786—71nomaximal, Sylow
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2312yes—
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2311yescharacteristic
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14C2×C414yes—
57C22×Dic14C2×C418yes—
58C2×C42×C7—22no—
59C2×C42×C7—22no—
60C2×C42×C7C2×C414yes—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7—22no—
63C2×C4⋊Dic7C2×C414yes—
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—22no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4—24no—
70C2×Dic7⋊C4C2×C414yes—
71C2×Dic7⋊C4C2×C414yes—
72C2×Dic7⋊C4C2×C418yes—
73C2×C4×Dic7—22no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—24no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—42no—
79C2×C4×Dic7—42no—
80C2×C4×Dic7C2×C414yes—
81C2×C4×Dic7C2×C418yes—
82C2×C4×Dic7C2×C414yes—
83C2×C4×Dic7D414yes—
84C2×C4×Dic7D414yes—
85C4×Dic14—232no—
86C7×#95—22no—
87C7×#95—22no—
88C7×#95—42no—
89C7×#95C2×C418yes—
90C7⋊#95—22no—
91C7⋊#95—22no—
92C7⋊#95—24no—
93C7⋊#95—42no—
94C7⋊#95—42no—
95C7⋊#95—42no—
96C7⋊#95C2×C418yes—
97C7⋊#95C2×C418yes—
98C7⋊#95C2×C418yes—
99C22×C4×Q8—71no—
100C2×#391—72no—
101C2×C4×C4⋊C4—71no—
102C2×#376—71no—
103#1436—72no—
104#1436—72no—
105C4⋊#95—72no—
106#1434—71no—
107#1434—71no—
108#1434—71no—
109#1434D711yescharacteristic
110C24×C7C2411yescharacteristic
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7C22×C414yes—
135C22×C4×C7C22×C414yes—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C22×C414yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7C22×C414yes—
168C22×Dic7C22×C414yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C22×C414yes—
171C22×Dic7C22×C412yes—
172C22×Dic7C22×C412yes—
173C22×Dic7C22×C412yes—
174C22×Dic7C22×C414yes—
175C22×Dic7C2×D412yes—
176C22×Dic7C2×D412yes—
177C22×Dic7C2×D412yes—
178C22×Dic7C2×D412yes—
179C22×Dic7#4114yes—
180C2×Dic14—28no—
181C2×Dic14—28no—
182C2×Dic14—216no—
183C2×Dic14—216no—
184C2×Dic14C42116yescomplemented
185C42×C7—28no—
186C4⋊Dic7—28no—
187Dic7⋊C4—28no—
188Dic7⋊C4—28no—
189Dic7⋊C4—216no—
190C4×Dic7—28no—
191C4×Dic7—216no—
192C4×Dic7—48no—
193C4×Dic7C22⋊C4116yes—
194C23×Q8—71no—
195C2×C4×Q8—78no—
196C2×C4×Q8—144no—
197C2×C4×Q8—144no—
198C22×C4⋊C4—71no—
199C22×C4⋊C4—71no—
200C22×C4⋊C4—71no—
201C22×C4⋊C4—72no—
202C22×C42—71no—
203C22×C42—71no—
204C22×C42—72no—
205C22×C42—72no—
206C22×C42C2×D711yescharacteristic
207#391—716no—
208C4×C4⋊C4—144no—
209#376—144no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95—71no—
214C2×#95—71no—
215C2×#95—72no—
216C2×#95—72no—
217C2×#95—72no—
218C2×#95—72no—
219C2×#95C2×D711yescharacteristic
220C2×#95C2×D711yescharacteristic
221C2×#95C2×D712yes—
222C2×#95C2×D712yes—
223C23×C7C23×C411yescharacteristic
224C23×C7C23×C412yes—
225C23×C7C22×D411yescharacteristic
226C23×C7C22×D411yescharacteristic
227C23×C7C2×#4112yes—
228C23×C7H(5,𝔽2)11yescharacteristic
229C23×C7H(5,𝔽2)11yescharacteristic
230C23×C72−1+411yescharacteristic
231C23×C72−1+412yes—
232C23×C72−1+412yes—
233C23×C72−1+411yescharacteristic
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—42no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7C2×C4214yes—
264C2×C4×C7C2×C22⋊C414yes—
265C2×Dic7—22no—
266C2×Dic7—22no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—42no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7C2×C4214yes—
297C2×Dic7C2×C4214yes—
298C2×Dic7C2×C22⋊C414yes—
299C2×Dic7C2×C22⋊C414yes—
300C2×Dic7#11714yes—
301C2×Dic7C4×D418yes—
302C2×Dic7C4×D418yes—
303Dic14—232no—
304C22×Q8—74no—
305C22×Q8—78no—
306C22×Q8—142no—
307C22×Q8—142no—
308C22×Q8—144no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—72no—
314C23×C4—72no—
315C23×C4C22×D711yescharacteristic
316C23×C4C22×D712yes—
317C23×C4C22×D712yes—
318C23×C4C22×D711yescharacteristic
319C23×C4C22×D711yescharacteristic
320C4×Q8—1432no—
321C2×C4⋊C4—74no—
322C2×C4⋊C4—74no—
323C2×C4⋊C4—74no—
324C2×C4⋊C4—78no—
325C2×C4⋊C4—142no—
326C2×C4⋊C4—142no—
327C2×C4⋊C4—142no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—144no—
332C2×C4⋊C4—144no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—74no—
336C2×C42—74no—
337C2×C42—74no—
338C2×C42—74no—
339C2×C42—78no—
340C2×C42—144no—
341C2×C42—144no—
342C2×C42—142no—
343C2×C42—142no—
344C2×C42—144no—
345C2×C42—282no—
346C2×C42—282no—
347C2×C42C4×D714yes—
348#95—22no—
349#95—22no—
350#95—42no—
351#95—78no—
352#95—78no—
353#95—78no—
354#95—144no—
355#95—142no—
356#95—142no—
357#95—282no—
358#95—282no—
359#95—282no—
360#95C4×D718yes—
361C22×C7C22×C4211yescharacteristic
362C22×C7C22×C22⋊C411yescharacteristic
363C22×C7C2×#11711yescharacteristic
364C22×C7C2×C4×D412yes—
365C22×C7C2×C4×D412yes—
366C22×C7#51911yescharacteristic
367C22×C7#51911yescharacteristic
368C22×C7#51912yes—
369C22×C7#51912yes—
370C22×C7#52011yescharacteristic
371C22×C7#52011yescharacteristic
372C22×C7#52012yes—
373C22×C7#52012yes—
374C22×C7#53511yescharacteristic
375C22×C7#53611yescharacteristic
376C22×C7#53612yes—
377C22×C7#53612yes—
378C22×C7#53711yescharacteristic
379C22×C7#53711yescharacteristic
380C22×C7#54112yes—
381C22×C7#54112yes—
382C22×C7#54212yes—
383C22×C7#54212yes—
384C4×C7—24no—
385C4×C7—24no—
386C4×C7—42no—
387C4×C7—44no—
388C4×C7—44no—
389Dic7—24no—
390Dic7—24no—
391Dic7—42no—
392Dic7—44no—
393Dic7—44no—
394Dic7C4×C22⋊C418yes—
395C24C23×D711yescharacteristic
396C2×Q8—716no—
397C2×Q8—148no—
398C2×Q8—148no—
399C2×Q8—1416no—
400C2×Q8—1416no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—72no—
425C22×C4—74no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—74no—
429C22×C4—74no—
430C22×C4—72no—
431C22×C4—72no—
432C22×C4—72no—
433C22×C4—72no—
434C22×C4—74no—
435C22×C4—74no—
436C22×C4—72no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—142no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—142no—
450C22×C4—141no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4C2×C4×D714yes—
463C22×C4C2×C4×D714yes—
464C22×C4C2×C4×D712yes—
465C22×C4C2×C4×D714yes—
466C22×C4C2×C4×D712yes—
467C22×C4C2×C4×D712yes—
468C22×C4D4×D712yes—
469C22×C4D4×D712yes—
470C4⋊C4—1416no—
471C4⋊C4—148no—
472C4⋊C4—148no—
473C4⋊C4—148no—
474C42—28no—
475C42—716no—
476C42—1416no—
477C42—148no—
478C42—288no—
479C2×C7C2×C4×C22⋊C411yescharacteristic
480C2×C7#227611yescharacteristic
481C2×C7#227611yescharacteristic
482C2×C7#231011yescharacteristic
483C2×C7#231111yescharacteristic
484C2×C7#232011yescharacteristic
485C2×C7#232011yescharacteristic
486C2×C7#232312yes—
487C2×C7#232312yes—
488C2×C7#232612yes—
489C2×C7#232712yes—
490C23C22×C4×D711yescharacteristic
491C23C22×C4×D712yes—
492C23C2×D4×D711yescharacteristic
493C23C2×D4×D711yescharacteristic
494C23#654811yescharacteristic
495C23#655111yescharacteristic
496C23D7×#4112yes—
497C23#655311yescharacteristic
498C23#655412yes—
499C23#655411yescharacteristic
500C23#655412yes—
501Q8—1432no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—74no—
532C2×C4—74no—
533C2×C4—74no—
534C2×C4—78no—
535C2×C4—78no—
536C2×C4—74no—
537C2×C4—74no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—142no—
553C2×C4—144no—
554C2×C4—144no—
555C2×C4—144no—
556C2×C4—144no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—282no—
560C2×C4—282no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—282no—
565C2×C4—281no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4C42×D714yes—
570C2×C4D7×C22⋊C414yes—
571C7#1378611yescharacteristic, complemented, Sylow
572C22C2×C42×D711yescharacteristic
573C22C2×D7×C22⋊C411yescharacteristic
574C22#9019511yescharacteristic
575C22#9020711yescharacteristic
576C22#9020711yescharacteristic
577C22#9020712yes—
578C22D7×#11711yescharacteristic
579C22#9021412yes—
580C22#9021411yescharacteristic
581C22C4×D4×D712yes—
582C22C4×D4×D712yes—
583C22#9023712yes—
584C22#9026312yes—
585C22#9029811yescharacteristic
586C22#9029911yescharacteristic
587C22#9031911yescharacteristic
588C22#9032011yescharacteristic
589C22#9033912yes—
590C22#9033912yes—
591C22#9036012yes—
592C22#9038712yes—
593C22#9039212yes—
594C22#9040112yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—42no—
600C4—78no—
601C4—144no—
602C4—144no—
603C4—284no—
604C4—284no—
605C4—282no—
606C2#1173082811yescharacteristic
607C2C4×D7×C22⋊C411yescharacteristic
608C2#1173106111yescharacteristic
609C2#1173144811yescharacteristic
610C2#1173149811yescharacteristic
611C2#1173159611yescharacteristic
612C2#1173170511yescharacteristic
613C2#1173239212yes—
614C2#1173239312yes—
615C2#1173290512yes—
616C2#1173290712yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC22×C7C22×C42
FittingC7×#1434C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1434D7
hypocenterC7#13786
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
24. C2×C4×Dic14C4semidirect
40. #89418C4semidirect
184. C2×Dic14C42semidirect
571. C7#13786semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C2×#894182
C2×#894212
C2×#894202
C7×#14342
#117292882
#117271612
#117271602
Dic7⋊#952
#117271562
#137867