Finite Group Databasebeta

← #13416792

Subgroups — #13416792

Order 1792.

780 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#89794C211yesmaximal, characteristic
3C2×C4×#1144C211yesmaximal, characteristic
4#11731121C211yesmaximal, characteristic
5#11729386C211yesmaximal, characteristic
6#11729386C211yesmaximal, characteristic
7#11729383C211yesmaximal, characteristic
8#11729383C211yesmaximal, characteristic
9#11727388C211yesmaximal, characteristic
10#11727385C211yesmaximal, characteristic
11#11727372C211yesmaximal, characteristic
12#11727360C211yesmaximal, characteristic
13#11727345C211yesmaximal, characteristic
14#11727323C211yesmaximal, characteristic
15#11727236C211yesmaximal, characteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C2×#6433C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#389—21no—
23C7×#389—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#384—21no—
27C7×#384C2211yescharacteristic
28C4×C7×C4⋊C4—21no—
29C4×C7×C4⋊C4C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C4×#1144—24no—
34#89794—24no—
35#89752—21no—
36#89744—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89701C2211yescharacteristic
46#89701C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89430—21no—
52#89424—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26371—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×Dic14C2311yescharacteristic
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—41no—
69C4×C7×Q8—41no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×#1144—22no—
89C2×#1144—22no—
90C2×#1144Q814yes—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104#6433—22no—
105#6433—41no—
106C4×Dic14—22no—
107C4×Dic14—41no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95D411yescharacteristic
111C7×#95D411yescharacteristic
112#6409—22no—
113#6409—22no—
114#6409D414yes—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121#6384—22no—
122#6384—22no—
123#6384—41no—
124#6382—22no—
125#6382—22no—
126#6382—41no—
127#6381—22no—
128#6381—22no—
129C4⋊(C7⋊C8)—22no—
130C4×C7⋊C8—22no—
131C2×#479—71no—
132C2×C4×Q16—71no—
133#2453—71no—
134Q8×C4⋊C4D711yescharacteristic
135#2096—71no—
136#2083—71no—
137#2066—71no—
138#2059—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×Q8C2×D412yes—
150C2×C7×Q8C2×Q812yes—
151C2×C7×Q8#4112yes—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14C2×Q812yes—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—81no—
185C7×C4⋊C4—81no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4#4112yes—
188C42×C7—24no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—42no—
196C42×C7—42no—
197#1144—28no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7C2×D412yes—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209C4×Dic7—42no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C2×D412yes—
221C2×C7⋊C8C2×Q812yes—
222C22×Q16—71no—
223C2×#125—71no—
224C2×C4×Q8—71no—
225C2×C4×Q8C2×D711yescharacteristic
226C2×C4×Q8C2×D711yescharacteristic
227C2×C4×Q8C2×D711yescharacteristic
228#479—144no—
229C4×Q16—144no—
230C2×#107—71no—
231C2×#107—71no—
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×#103—71no—
241C2×#103—71no—
242C2×C4×C8—71no—
243#400—141no—
244#398—141no—
245#391—71no—
246#391—71no—
247#389—21no—
248#389—21no—
249#389—71no—
250#389—71no—
251#389—141no—
252#389C2×D711yescharacteristic
253#389C2×D711yescharacteristic
254#387—141no—
255#384—21no—
256#384C2×D711yescharacteristic
257C4×C4⋊C4—21no—
258C4×C4⋊C4C2×D711yescharacteristic
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×Q8C22⋊Q814yes—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7#13512yes—
301C2×C4×C7#13712yes—
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×Q811yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7H(5,𝔽2)11yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320Dic14—22no—
321Dic14—22no—
322Dic14—41no—
323Dic14—41no—
324C7⋊C8—22no—
325C7⋊C8—22no—
326C7⋊C8—41no—
327C7⋊C8—41no—
328C22×Q8—71no—
329C22×Q8C22×D711yescharacteristic
330C2×Q16—74no—
331C2×Q16—142no—
332C2×Q16—142no—
333C22×C8—71no—
334C22×C8—71no—
335C22×C8—71no—
336C22×C8—71no—
337#125—142no—
338#125—281no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—41no—
344C4×Q8—41no—
345C4×Q8—142no—
346C4×Q8—281no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
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⋊C4C22⋊D711yescharacteristic
363C2×C4⋊C4C22⋊D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—141no—
371C2×C42C22×D711yescharacteristic
372C2×C42C22×D711yescharacteristic
373C2×C42C22×D711yescharacteristic
374#107—142no—
375#107—142no—
376#106—142no—
377#106—142no—
378#106—281no—
379C4⋊C8—142no—
380#103—74no—
381#103—142no—
382#103—142no—
383#103—142no—
384#103—142no—
385#103—281no—
386C4×C8—142no—
387#95—21no—
388#95—21no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95C22⋊D711yescharacteristic
396#95C22⋊D711yescharacteristic
397C22×C7#52511yescharacteristic
398C22×C7D4211yescharacteristic
399C22×C7D4×Q811yescharacteristic
400C22×C7C2×#13511yescharacteristic
401C22×C7C2×#13711yescharacteristic
402C22×C7#57711yescharacteristic
403C22×C7#57811yescharacteristic
404C4×C7—22no—
405C4×C7—24no—
406C4×C7—22no—
407C4×C7—41no—
408C4×C7—42no—
409C4×C7—41no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—81no—
413C4×C7—81no—
414C4×C7C2×C22⋊Q811yescharacteristic
415C4×C7#53911yescharacteristic
416C4×C7#54711yescharacteristic
417C4×C7#55411yescharacteristic
418Dic7—42no—
419Dic7—42no—
420Dic7—81no—
421Dic7—81no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—72no—
426C2×Q8—141no—
427C2×Q8—141no—
428C2×Q8—141no—
429C2×Q8—141no—
430C2×Q8—141no—
431C2×Q8#115712yes—
432C2×Q8D4×D712yes—
433C2×Q8Q8×D712yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C23×D711yescharacteristic
449Q16—148no—
450C2×C8—72no—
451C2×C8—72no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—72no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
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—282no—
487C4⋊C4#115712yes—
488C42—24no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—42no—
497C42—282no—
498C2×C7#244911yescharacteristic
499C2×C7#310811yescharacteristic
500C2×C7#312011yescharacteristic
501C2×C7#327011yescharacteristic
502C2×C7#328411yescharacteristic
503C2×C7#338311yescharacteristic
504C2×C7#338711yescharacteristic
505C23C22×C22⋊D711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—142no—
510Q8—142no—
511Q8—281no—
512Q8—281no—
513Q8#645914yes—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—281no—
545C2×C4—281no—
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#650512yes—
555C2×C4#651312yes—
556C2×C4C2×#115711yescharacteristic
557C2×C4C2×#115711yescharacteristic
558C2×C4C2×D4×D711yescharacteristic
559C2×C4#654811yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4#655111yescharacteristic
562C8—142no—
563C8—142no—
564C8—281no—
565C8—281no—
566C7#2637111yescharacteristic, complemented, Sylow
567C22#9048211yescharacteristic
568C22D4×C22⋊D711yescharacteristic
569C22C2×#650511yescharacteristic
570C22Q8×C22⋊D711yescharacteristic
571C22C2×#651311yescharacteristic
572C22#9052711yescharacteristic
573C22#9052911yescharacteristic
574C4—22no—
575C4—24no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—282no—
585C4—282no—
586C4—561no—
587C4—561no—
588C4#9019811yescharacteristic
589C4C2×#645911yescharacteristic
590C4#9020211yescharacteristic
591C4#9020611yescharacteristic
592C2#1173961811yescharacteristic
593C2#1173988811yescharacteristic
594C2#1174002011yescharacteristic
595C2#1174101011yescharacteristic
596C2#1174102411yescharacteristic
597C2#1174112711yescharacteristic
598C2#1174115411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×Q8×C4⋊C4C2
socle, prime radicalC23×C7C22×D4
hypercenterQ8×C4⋊C4D7
hypocenterC7#26371
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
566. C7#26371semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×C4⋊C42
C2×#897942
C2×C4×#11442
#117311212
#117293862
#117293832
#117273882
#117273852
#117273722
#117273602
#117273452
#117273232
#117272362
#263717