Finite Group Databasebeta

← #13416756

Subgroups — #13416756

Order 1792.

788 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×Q16C211yesmaximal, characteristic
2#11739915C211yesmaximal, characteristic
3#11739906C211yesmaximal, characteristic
4C2×#89563C211yesmaximal, characteristic
5#11729819C211yesmaximal, characteristic
6#11729819C211yesmaximal, characteristic
7#11729383C211yesmaximal, characteristic
8#11729382C211yesmaximal, characteristic
9#11728374C211yesmaximal, characteristic
10#11728371C211yesmaximal, characteristic
11#11728006C211yesmaximal, characteristic
12#11728001C211yesmaximal, characteristic
13#11727992C211yesmaximal, characteristic
14#11727538C211yesmaximal, characteristic
15#11727523C211yesmaximal, characteristic
16C22×C7×Q16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×#6451C2211yescharacteristic
22C4×C7×Q16—24no—
23C2×C7×#107C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27#90002C2211yescharacteristic
28#90002C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6391C2211yescharacteristic
34C2×#6391C2211yescharacteristic
35C2×Dic7⋊C8C2211yescharacteristic
36#89757C2211yescharacteristic
37#89756—21no—
38#89751—21no—
39#89749—21no—
40#89749C2211yescharacteristic
41#89745—21no—
42#89745C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89705—21no—
46C4×Dic7⋊C4—21no—
47C4×Dic7⋊C4C2211yescharacteristic
48#89701—21no—
49#89700—21no—
50#89700C2211yescharacteristic
51#89700C2211yescharacteristic
52#89563—24no—
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89319C2211yescharacteristic
61#26369—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×C7×Q8C2311yescharacteristic
64C22×Dic14—21no—
65C2×C7×Q16—22no—
66C2×C7×Q16—22no—
67C2×C7×Q16Q814yescomplemented
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×Q8—41no—
73C4×C7×Q8—41no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C42×C7—21no—
79C2×C42×C7—21no—
80C2×C42×C7C2311yescharacteristic
81Q8×Dic7—22no—
82Q8×Dic7—22no—
83Q8×Dic7—22no—
84Q8×Dic7—41no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4D411yescharacteristic
97C2×Dic7⋊C4D411yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7C2311yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104#6451—22no—
105#6451—41no—
106C7×#107—22no—
107C7×#103—22no—
108C7×#103D414yes—
109C4×C7×C8—22no—
110#6409—22no—
111#6409—22no—
112#6409—22no—
113#6409—41no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95D411yescharacteristic
125C7⋊#95D411yescharacteristic
126#6391—22no—
127#6391—22no—
128Dic7⋊C8—22no—
129#6382—22no—
130#6381—41no—
131C2×#475—71no—
132C2×C4×Q16D711yescharacteristic
133#2451—71no—
134#2351—71no—
135#2097—71no—
136#2085—71no—
137#2066—71no—
138#2057—71no—
139#1941—71no—
140#1916—71no—
141#1878—71no—
142#1878—71no—
143#1865—71no—
144#1865—71no—
145#1863—71no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8C2×D412yes—
155C2×C7×Q8C2×Q812yes—
156C2×C7×Q8C2×Q812yes—
157C2×C7×Q8#4112yes—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C22×Dic7C2×D411yescharacteristic
173C2×Dic14—42no—
174C2×Dic14—42no—
175C2×Dic14—81no—
176C7×Q16—28no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8C2×D412yes—
182C2×C7×C8C2×Q812yes—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4—42no—
188C7×C4⋊C4C2×D412yes—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—42no—
192C42×C7—42no—
193C4⋊Dic7—24no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—42no—
202C4⋊Dic7—81no—
203C4⋊Dic7#4112yes—
204Dic7⋊C4—22no—
205Dic7⋊C4—22no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—81no—
210C4×Dic7—24no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7—21no—
214C4×Dic7—21no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C22×Q16C2×D711yescharacteristic
223C2×C4⋊Q8—71no—
224C2×C4×Q8—71no—
225C2×C4×Q8—71no—
226C2×C4×Q8C2×D711yescharacteristic
227C2×C4×Q8C2×D711yescharacteristic
228#475—144no—
229C4×Q16—24no—
230C2×#107—71no—
231C2×#107C2×D711yescharacteristic
232C2×#106—71no—
233C2×#106—71no—
234C2×#106—71no—
235C2×C4⋊C8—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#103—71no—
239C2×#103—71no—
240C2×#103C2×D711yescharacteristic
241C2×#103C2×D711yescharacteristic
242C2×C4×C8C2×D711yescharacteristic
243#398—141no—
244#395—141no—
245#391—71no—
246#391—71no—
247#391—141no—
248#389—71no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—141no—
253#389—141no—
254#387—141no—
255#382—71no—
256#382—141no—
257C4×C4⋊C4—71no—
258C4×C4⋊C4—141no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#336—71no—
264#336—71no—
265#336—71no—
266#336—71no—
267C23×C7C22×D411yescharacteristic
268C7×Q8—22no—
269C7×Q8—22no—
270C7×Q8—22no—
271C7×Q8—22no—
272C7×Q8—22no—
273C7×Q8—41no—
274C7×Q8—41no—
275C7×Q8C22⋊Q814yes—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7#13712yes—
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×Q811yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—22no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7#13512yes—
324Dic14—84no—
325C7×C8—22no—
326C7×C8—22no—
327C7⋊C8—41no—
328C7⋊C8—41no—
329C22×Q8—141no—
330C22×Q8C22×D711yescharacteristic
331C22×Q8C22×D711yescharacteristic
332C2×Q16—22no—
333C2×Q16—22no—
334C2×Q16Dic1414yescomplemented
335C22×C8—71no—
336C22×C8—71no—
337C22×C8C22×D711yescharacteristic
338C22×C8C22×D711yescharacteristic
339C4⋊Q8—142no—
340C4⋊Q8—281no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—41no—
344C4×Q8—41no—
345C4×Q8—142no—
346C4×Q8—142no—
347C4×Q8—142no—
348C4×Q8—281no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—71no—
369C2×C42—71no—
370C2×C42—141no—
371C2×C42—141no—
372C2×C42C22×D711yescharacteristic
373#107—22no—
374#107—281no—
375#106—142no—
376#106—142no—
377#106—142no—
378C4⋊C8—142no—
379#103—22no—
380#103—142no—
381#103—142no—
382#103—142no—
383#103—281no—
384#103C22⋊D714yes—
385C4×C8—22no—
386#95—71no—
387#95—71no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398C22×C7#52511yescharacteristic
399C22×C7D4211yescharacteristic
400C22×C7D4×Q811yescharacteristic
401C22×C7C2×#13511yescharacteristic
402C22×C7C2×#13711yescharacteristic
403C22×C7#57711yescharacteristic
404C22×C7#57811yescharacteristic
405C4×C7—24no—
406C4×C7—22no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7—41no—
410C4×C7—81no—
411C4×C7C2×C22⋊Q811yescharacteristic
412C4×C7#53911yescharacteristic
413C4×C7#54711yescharacteristic
414C4×C7#55411yescharacteristic
415Dic7—22no—
416Dic7—41no—
417Dic7—42no—
418Dic7—41no—
419Dic7—41no—
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—282no—
432C2×Q8—282no—
433C2×Q8—561no—
434C2×Q8C2×Dic1412yes—
435C2×Q8C2×Dic1412yes—
436C2×Q8#115712yes—
437C2×Q8C2×C22⋊D712yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4D4×D711yescharacteristic
451C22×C4D4×D711yescharacteristic
452C22×C4C23×D711yescharacteristic
453Q16—28no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8C2×Dic1412yes—
465C2×C8C2×C22⋊D712yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—72no—
472C4⋊C4—142no—
473C4⋊C4—144no—
474C4⋊C4—142no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—282no—
483C4⋊C4—282no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—561no—
487C4⋊C4—561no—
488C4⋊C4C2×C22⋊D712yes—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42—144no—
494C42—141no—
495C42—141no—
496C42—141no—
497C42—141no—
498C42—282no—
499C2×C7#244911yescharacteristic
500C2×C7#310711yescharacteristic
501C2×C7#312111yescharacteristic
502C2×C7#327011yescharacteristic
503C2×C7#328211yescharacteristic
504C2×C7#338311yescharacteristic
505C2×C7#338711yescharacteristic
506C23C2×D4×D711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8—22no—
512Q8—41no—
513Q8—41no—
514Q8—564no—
515Q8#648714yes—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—72no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—142no—
545C2×C4—141no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4#648112yes—
558C2×C4C22×Dic1411yescharacteristic
559C2×C4C2×#115711yescharacteristic
560C2×C4#654811yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4#655411yescharacteristic
563C2×C4C22×C22⋊D711yescharacteristic
564C8—22no—
565C8—22no—
566C8—281no—
567C8—281no—
568C7#2636911yescharacteristic, complemented, Sylow
569C22D4×Dic1411yescharacteristic
570C22#9023811yescharacteristic
571C22C2×#648111yescharacteristic
572C22#9045711yescharacteristic
573C22D7×#13511yescharacteristic
574C22#9046111yescharacteristic
575C22D4×C22⋊D711yescharacteristic
576C4—22no—
577C4—24no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—81no—
582C4—142no—
583C4—281no—
584C4—282no—
585C4—281no—
586C4—281no—
587C4—561no—
588C4—561no—
589C4—561no—
590C4#9023011yescharacteristic
591C4#9025511yescharacteristic
592C4C2×#648711yescharacteristic
593C4#9051511yescharacteristic
594C2#1173715211yescharacteristic
595C2#1173715311yescharacteristic
596C2#1173719211yescharacteristic
597C2#1173719511yescharacteristic
598C2#1173980511yescharacteristic
599C2#1174133311yescharacteristic
600C2#1174136411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×Q16C2
socle, prime radicalC23×C7C22×D4
hypercenterC2×C4×Q16D7
hypocenterC7#26369
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
67. C2×C7×Q16Q8semidirect
334. C2×Q16Dic14semidirect
568. C7#26369semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×Q162
#117399152
#117399062
C2×#895632
#117298192
#117293832
#117293822
#117283742
#117283712
#117280062
#117280012
#117279922
#117275382
#117275232
#263697