Finite Group Databasebeta

← #13410301

Subgroups — #13410301

Order 1792.

1292 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C2×#89854C211yesmaximal, characteristic, complemented
3#11739828C211yesmaximal, characteristic, complemented
4C2×#89781C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal, complemented
6#11729889C212yesmaximal, complemented
7#11729532C211yesmaximal, characteristic, complemented
8#11729449C212yesmaximal, complemented
9#11729448C212yesmaximal, complemented
10#11729369C211yesmaximal, characteristic, complemented
11#11729363C211yesmaximal, characteristic, complemented
12C22×C7×#41C2212yes—
13C7×#547—24no—
14C7×#547—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×#121—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22C2×#6433C2211yescharacteristic
23C2×#6409C2212yes—
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27#89994—21no—
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2212yes—
31#89984C2212yes—
32#89984C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89854—24no—
36#89781—24no—
37#89754—21no—
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89717—22no—
41C4×C4⋊Dic7C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#25892—71nomaximal, Sylow
46C2×C7×#41—24no—
47C2×C7×#41—24no—
48C2×C7×#41—24no—
49C2×C7×#41—24no—
50C2×C7×#41—42no—
51C2×C7×#41—42no—
52C22×C7×Q8C2312yes—
53C22×C7×D4—22no—
54C22×C7×D4D411yescharacteristic
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7—21no—
58C23×C4×C7C2312yes—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8D414yes—
61C7×#123—216no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—24no—
64C7×#121—28no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C4×C7×D4—22no—
68C4×C7×D4D414yes—
69C2×C7×C4⋊C4—28no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4D412yes—
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—22no—
80C2×C22⋊Dic7—22no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90#6433—22no—
91#6433—22no—
92#6409—24no—
93#6409—24no—
94#6409—42no—
95#6409—42no—
96#6406—22no—
97#6406—22no—
98#6406—24no—
99C7⋊#95—21no—
100C7⋊#95—22no—
101C7⋊#95—22no—
102C22⋊(C7⋊C8)—42no—
103C22⋊(C7⋊C8)—42no—
104#6382—22no—
105#6381—22no—
106C4⋊(C7⋊C8)—22no—
107#6377—22no—
108C2×#547D711yescharacteristic
109C2×#491—71no—
110C2×#478—71no—
111#2562—71no—
112#2036—72no—
113#2035—72no—
114#1981—71no—
115#1955—71no—
116#1788—72no—
117#1788—72no—
118#1762—71no—
119C24×C7—21no—
120C24×C7C2×D412yes—
121C7×#41—48no—
122C7×#41—48no—
123C7×#41—48no—
124C7×#41—48no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8C2×D414yes—
131C2×C7×D4—28no—
132C2×C7×D4—22no—
133C2×C7×D4—21no—
134C2×C7×D4—22no—
135C2×C7×D4—42no—
136C2×C7×D4—44no—
137C2×C7×D4—44no—
138C2×C7×D4C2×D412yes—
139C2×C7×D4#4114yes—
140C22×C4×C7—28no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—44no—
151C22×C4×C7—44no—
152C22×C4×C7—44no—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—22no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4C2×D412yes—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—24no—
169C42×C7C2×D412yes—
170C42×C7C2×D412yes—
171C22⋊Dic7—44no—
172C22⋊Dic7—82no—
173C4⋊Dic7—21no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179Dic7⋊C4—42no—
180Dic7⋊C4—42no—
181C4×Dic7—81no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C22×#41C2×D712yes—
190#547—24no—
191#547—24no—
192C2×C4⋊Q8C2×D711yescharacteristic
193C2×#125—71no—
194C2×#123—22no—
195C2×C22⋊Q8C2×D712yes—
196C2×#121—21no—
197C2×C4×D4C2×D711yescharacteristic
198C2×C4×D4C2×D711yescharacteristic
199C22×C4⋊C4—21no—
200#491—144no—
201#478—144no—
202C2×#107—71no—
203C2×#106—71no—
204C2×C4⋊C8—71no—
205C2×#103—72no—
206C2×#103—72no—
207C2×#103—72no—
208C2×#103—72no—
209C2×#102—71no—
210C2×#102—71no—
211C2×#102—72no—
212C2×C22⋊C8—72no—
213C2×C22⋊C8—72no—
214C2×#97—71no—
215#400—141no—
216#394—141no—
217#388—142no—
218#380—72no—
219#380—72no—
220C4×C4⋊C4—71no—
221#340—72no—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—42no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7C22×D411yescharacteristic
235C7×Q8—24no—
236C7×Q8—24no—
237C7×Q8—42no—
238C7×Q8—42no—
239C7×D4—22no—
240C7×D4—24no—
241C7×D4—44no—
242C7×D4—416no—
243C7×D4#12414yes—
244C2×C4×C7—22no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—48no—
253C2×C4×C7—416no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7#12012yes—
262C2×C4×C7#12012yes—
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4112yes—
266C2×C4×C7H(5,𝔽2)12yes—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C7⋊C8—41no—
276C7⋊C8—41no—
277C7⋊C8—42no—
278C2×#41—24no—
279C2×#41—24no—
280C2×#41—24no—
281C2×#41—24no—
282C2×#41—42no—
283C2×#41—42no—
284C22×Q8C22×D712yes—
285C22×D4—22no—
286C22×D4C22⋊D711yescharacteristic
287C22×D4C22×D711yescharacteristic
288C23×C4—21no—
289C23×C4—22no—
290C23×C4C22×D712yes—
291C22×C8—72no—
292C22×C8—71no—
293C22×C8—71no—
294C4⋊Q8—22no—
295C4⋊Q8C22⋊D714yes—
296#125—142no—
297#125—142no—
298#123—216no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301#121—28no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4C22⋊D714yes—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—72no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4C22⋊D711yescharacteristic
316C2×C4⋊C4C22×D711yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—142no—
320C2×C22⋊C4—142no—
321C2×C22⋊C4C22⋊D712yes—
322C2×C42—141no—
323C2×C42C22×D711yescharacteristic
324#107—142no—
325#106—142no—
326C4⋊C8—142no—
327#103—144no—
328#103—144no—
329#103—282no—
330#103—282no—
331#102—144no—
332#102—142no—
333#102—142no—
334C22⋊C8—282no—
335C22⋊C8—282no—
336#97—142no—
337#95—141no—
338#95—142no—
339#95—142no—
340C22×C7—24no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—44no—
344C22×C7—44no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7C2×#12011yescharacteristic
353C22×C7#54612yes—
354C22×C7#57512yes—
355C22×C7#57711yescharacteristic
356C22×C7#57811yescharacteristic
357C4×C7—22no—
358C4×C7—22no—
359C4×C7—42no—
360C4×C7—44no—
361C4×C7—42no—
362C4×C7C2×#12411yescharacteristic
363C4×C7#53511yescharacteristic
364C4×C7#54612yes—
365Dic7—81no—
366Dic7—81no—
367Dic7—82no—
368C24—21no—
369C24C2×C22⋊D712yes—
370#41—48no—
371#41—48no—
372#41—48no—
373#41—48no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C2×C22⋊D714yes—
380C2×D4—28no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4#115914yes—
388C2×D4C2×C22⋊D712yes—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—142no—
403C22×C4—141no—
404C22×C4—141no—
405C22×C4C2×C22⋊D712yes—
406C22×C4C2×C22⋊D711yescharacteristic
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C23×D711yescharacteristic
409C2×C8—142no—
410C2×C8—142no—
411C2×C8—142no—
412C2×C8—142no—
413C2×C8—142no—
414C2×C8—141no—
415C2×C8—141no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—142no—
422C4⋊C4—142no—
423C4⋊C4—141no—
424C4⋊C4—141no—
425C4⋊C4—141no—
426C4⋊C4—141no—
427C4⋊C4—282no—
428C4⋊C4—282no—
429C4⋊C4C2×C22⋊D712yes—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—284no—
434C22⋊C4—562no—
435C42—561no—
436C42C2×C22⋊D712yes—
437C42C2×C22⋊D712yes—
438C2×C7—44no—
439C2×C7—82no—
440C2×C7#243511yescharacteristic
441C2×C7#300511yescharacteristic
442C2×C7#301311yescharacteristic
443C2×C7#330712yes—
444C2×C7#331512yes—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23C22×C22⋊D711yescharacteristic
458Q8—24no—
459Q8—24no—
460Q8—42no—
461Q8—42no—
462D4—22no—
463D4—24no—
464D4—44no—
465D4—416no—
466D4#649914yes—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—282no—
488C2×C4—281no—
489C2×C4—281no—
490C2×C4—281no—
491C2×C4—281no—
492C2×C4#651612yes—
493C2×C4#651612yes—
494C2×C4C2×#115912yes—
495C2×C4#654812yes—
496C2×C4C22×C22⋊D711yescharacteristic
497C2×C4C22×C22⋊D711yescharacteristic
498C8—281no—
499C8—281no—
500C8—282no—
501C7#2589211yescharacteristic, complemented, Sylow
502C22—24no—
503C22—44no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22#9049712yes—
515C22#9051312yes—
516C22#9052811yescharacteristic
517C22#9052911yescharacteristic
518C22C2×#651611yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—562no—
525C4—561no—
526C4—561no—
527C4C2×#649911yescharacteristic
528C4#9049611yescharacteristic
529C4#9049712yes—
530C2—44no—
531C2—82no—
532C2#1174095712yes—
533C2#1174095812yes—
534C2#1174149511yescharacteristic
535C2#1174152611yescharacteristic
536C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#547C2
socleC23×C7C22×D4
hypercenterC2×#547D7
hypocenterC7#25892
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#89854C2semidirect
3. #11739828C2semidirect
4. C2×#89781C2semidirect
5. #11729889C2semidirect
6. #11729889C2semidirect
7. #11729532C2semidirect
8. #11729449C2semidirect
9. #11729448C2semidirect
10. #11729369C2semidirect
11. #11729363C2semidirect
501. C7#25892semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×#898542
#117398282
C2×#897812
#117298892
#117295322
#117294492
#117294482
#117293692
#117293632
#258927