Finite Group Databasebeta

← #13405666

Subgroups — #13405666

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89602C211yesmaximal, characteristic
2C2×#89563C211yesmaximal, characteristic
3#11733993C211yesmaximal, characteristic
4#11733505C211yesmaximal, characteristic
5C7×#2066C211yesmaximal, characteristic
6#11728317C211yesmaximal, characteristic
7#11728314C211yesmaximal, characteristic
8#11728218C211yesmaximal, characteristic
9#11728015C211yesmaximal, characteristic
10#11728008C211yesmaximal, characteristic
11Dic7×#103C211yesmaximal, characteristic
12#11727428C211yesmaximal, characteristic
13#11727372C211yesmaximal, characteristic
14#11726563C211yesmaximal, characteristic
15#11726557C211yesmaximal, characteristic
16C2×Q8×Dic7C2211yescharacteristic
17C2×#6451—21no—
18C2×#6451C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C7×#106C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#389C2211yescharacteristic
26#90002C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89756—21no—
37#89748C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39#89745C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89602—24no—
46#89563—24no—
47#89433—21no—
48#89429—21no—
49#89429—21no—
50#89419—21no—
51#89418—21no—
52#89418—21no—
53C7×#340C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25472—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×Dic14—21no—
64C22×Dic14D411yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72Q8×Dic7—22no—
73Q8×Dic7—22no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7D411yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4D411yescharacteristic
89C2×Dic7⋊C4D411yescharacteristic
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6451—22no—
97#6451—22no—
98#6451—44no—
99C4×Dic14—22no—
100C4×Dic14—22no—
101C7×#106—22no—
102C7×C4⋊C8—22no—
103C7×#103—22no—
104C7×#103—22no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108#6409—22no—
109#6409Q814yescomplemented
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118#6391—22no—
119Dic7⋊C8—22no—
120Dic7⋊C8D414yes—
121C8×Dic7—22no—
122C8×Dic7—22no—
123#6382—22no—
124#6382—22no—
125C4×C7⋊C8—22no—
126C2×#475—71no—
127C2×C8×Q8—71no—
128#2582—71no—
129#2546—71no—
130#2088—71no—
131#2088—71no—
132#2066—71no—
133#2066D711yescharacteristic
134#1990—71no—
135#1948—71no—
136#1941—71no—
137#1941—71no—
138#1933—71no—
139#1933—71no—
140C4×#103—71no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8C2×Q812yes—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—42no—
165C2×Dic14—42no—
166C2×Dic14—42no—
167C2×Dic14SD1614yes—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8C2×D412yes—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4#4112yes—
181C42×C7—21no—
182C42×C7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—22no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—42no—
192C4⋊Dic7—42no—
193C4⋊Dic7—81no—
194C4⋊Dic7C2×Q812yes—
195Dic7⋊C4—22no—
196Dic7⋊C4—22no—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—42no—
201Dic7⋊C4—81no—
202Dic7⋊C4—81no—
203C4×Dic7—21no—
204C4×Dic7—21no—
205C4×Dic7—42no—
206C4×Dic7—42no—
207C4×Dic7C2×D412yes—
208C4×Dic7#4112yes—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8C2×D412yes—
214C2×C7⋊C8C2×Q812yes—
215C2×C4⋊Q8—71no—
216C2×C4⋊Q8—141no—
217C2×C4×Q8—71no—
218C2×C4×Q8—71no—
219#475—144no—
220C8×Q8—144no—
221C2×#106—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×#106C2×D711yescharacteristic
225C2×C4⋊C8—71no—
226C2×C4⋊C8—71no—
227C2×C4⋊C8C2×D711yescharacteristic
228C2×#103—71no—
229C2×#103—71no—
230C2×#103C2×D711yescharacteristic
231C2×#103C2×D711yescharacteristic
232C2×C4×C8—71no—
233C2×C4×C8—71no—
234C2×C4×C8—71no—
235#398—141no—
236#398—141no—
237#398—141no—
238#398—141no—
239#391—71no—
240#391—141no—
241#391—141no—
242#391C2×D711yescharacteristic
243#389—71no—
244#389—71no—
245#389—71no—
246#389—71no—
247#389—71no—
248#389—141no—
249#389C2×D711yescharacteristic
250#384—71no—
251C4×C4⋊C4—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260C23×C7C22×D411yescharacteristic
261C7×Q8—22no—
262C7×Q8—22no—
263C7×Q8—41no—
264C7×Q8—41no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7C2×SD1612yes—
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×Q811yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C72−1+411yescharacteristic
288C2×C4×C72−1+411yescharacteristic
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—22no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7#12112yes—
309C2×Dic7#12112yes—
310C2×Dic7C2×SD1612yes—
311C2×Dic7C2×SD1612yes—
312C2×Dic7#13512yes—
313Dic14—28no—
314Dic14—84no—
315C7×C8—22no—
316C7×C8—41no—
317C7⋊C8—22no—
318C7⋊C8—22no—
319C22×Q8—71no—
320C22×Q8—141no—
321C22×Q8C22×D711yescharacteristic
322C22×C8—71no—
323C22×C8—71no—
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C4⋊Q8—142no—
327C4⋊Q8—142no—
328C4⋊Q8—284no—
329C4×Q8—142no—
330C4×Q8—142no—
331C4×Q8—142no—
332C4×Q8—142no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42C22×D711yescharacteristic
358#106—22no—
359#106—142no—
360#106—142no—
361#106—142no—
362C4⋊C8—22no—
363C4⋊C8—74no—
364C4⋊C8—142no—
365#103—22no—
366#103—22no—
367#103—74no—
368#103—142no—
369C4×C8—142no—
370C4×C8—142no—
371C4×C8—142no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383C22×C7C2×#12111yescharacteristic
384C22×C7#54811yescharacteristic
385C22×C7D4×Q811yescharacteristic
386C22×C7C22×SD1611yescharacteristic
387C22×C7C2×#13511yescharacteristic
388C22×C7#57711yescharacteristic
389C22×C7#57811yescharacteristic
390C4×C7—22no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—41no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7#53311yescharacteristic
397C4×C7#54111yescharacteristic
398C4×C7#54711yescharacteristic
399C4×C7D4×Q811yescharacteristic
400Dic7—22no—
401Dic7—22no—
402Dic7—42no—
403Dic7—41no—
404Dic7—81no—
405Dic7—81no—
406Dic7—81no—
407Dic7#46514yes—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—74no—
414C2×Q8—142no—
415C2×Q8—142no—
416C2×Q8—282no—
417C2×Q8—282no—
418C2×Q8—282no—
419C2×Q8Q8×D712yes—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—71no—
425C22×C4—71no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4D4×D711yescharacteristic
433C22×C4D4×D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—72no—
441C2×C8—72no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8D4×D712yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—72no—
454C4⋊C4—142no—
455C4⋊C4—142no—
456C4⋊C4—142no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4—282no—
467C4⋊C4—282no—
468C4⋊C4—282no—
469C4⋊C4—282no—
470C4⋊C4—561no—
471C4⋊C4—561no—
472C4⋊C4—561no—
473C4⋊C4#115912yes—
474C42—21no—
475C42—21no—
476C42—72no—
477C42—72no—
478C42—141no—
479C42—141no—
480C42—282no—
481C42—282no—
482C2×C7#242911yescharacteristic
483C2×C7C2×#46511yescharacteristic
484C2×C7#306311yescharacteristic
485C2×C7#335911yescharacteristic
486C2×C7#336511yescharacteristic
487C2×C7Q8×SD1611yescharacteristic
488C2×C7#338311yescharacteristic
489C23C2×D4×D711yescharacteristic
490Q8—22no—
491Q8—22no—
492Q8—41no—
493Q8—41no—
494Q8—148no—
495Q8—564no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—72no—
514C2×C4—72no—
515C2×C4—72no—
516C2×C4—72no—
517C2×C4—72no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—142no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4D7×SD1612yes—
538C2×C4C2×D4×D711yescharacteristic
539C2×C4C2×#115911yescharacteristic
540C2×C4C2×Q8×D711yescharacteristic
541C2×C4D7×#4111yescharacteristic
542C2×C4#655411yescharacteristic
543C2×C4#655411yescharacteristic
544C8—22no—
545C8—41no—
546C8—142no—
547C8—142no—
548C7#2547211yescharacteristic, complemented, Sylow
549C22#9028911yescharacteristic
550C22D7×#12111yescharacteristic
551C22#9032311yescharacteristic
552C22C2×D7×SD1611yescharacteristic
553C22D7×#13511yescharacteristic
554C22#9046911yescharacteristic
555C22#9047011yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—74no—
561C4—81no—
562C4—81no—
563C4—142no—
564C4—142no—
565C4—282no—
566C4—281no—
567C4—561no—
568C4—561no—
569C4—561no—
570C4#9030211yescharacteristic
571C4#9031311yescharacteristic
572C4#9032311yescharacteristic
573C4#9039211yescharacteristic
574C2#1173473211yescharacteristic
575C2#1173785911yescharacteristic
576C2#1173802611yescharacteristic
577C2D7×#46511yescharacteristic
578C2#1173835211yescharacteristic
579C2#1173848411yescharacteristic
580C2#1173866711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2066C2
socle, prime radicalC23×C7C22×D4
hypercenter#2066D7
hypocenterC7#25472
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2066

Complements

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

NH ≅ G/Ntype
109. #6409Q8semidirect
548. C7#25472semidirect

Maximal subgroups

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

subgroupindex
C2×#896022
C2×#895632
#117339932
#117335052
C7×#20662
#117283172
#117283142
#117282182
#117280152
#117280082
Dic7×#1032
#117274282
#117273722
#117265632
#117265572
#254727