Finite Group Databasebeta

← #13405637

Subgroups — #13405637

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89531C211yesmaximal, characteristic
2C2×C8×Dic14C211yesmaximal, characteristic, complemented
3#11733791C211yesmaximal, characteristic
4#11733505C211yesmaximal, characteristic, complemented
5C7×#1934C211yesmaximal, characteristic
6#11727939C211yesmaximal, characteristic
7#11727934C211yesmaximal, characteristic
8Dic7×#102C211yesmaximal, characteristic
9#11727524C211yesmaximal, characteristic, complemented
10#11727407C211yesmaximal, characteristic
11#11727376C211yesmaximal, characteristic, complemented
12#11726613C211yesmaximal, characteristic, complemented
13#11726592C211yesmaximal, characteristic, complemented
14#11726557C211yesmaximal, characteristic, complemented
15#11726549C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C2×#6451—21no—
18C2×#6451C2211yescharacteristic
19C2×#6441—21no—
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×#102C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C7×#390C2211yescharacteristic
25C7×#389C2211yescharacteristic
26#89993C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35#89757C2211yescharacteristic
36#89757C2211yescharacteristic
37#89756—21no—
38#89748C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40#89745C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89720—21no—
44#89720—21no—
45#89714—21no—
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49#89531—24no—
50C8×Dic14—24no—
51#89429—21no—
52#89418—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25469—71nomaximal, Sylow
62C22×C7×D4C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14—21no—
65C22×Dic14D411yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78D4×Dic7—22no—
79D4×Dic7—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7D411yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4D411yescharacteristic
94C2×Dic7⋊C4D411yescharacteristic
95C2×C4×Dic7—21no—
96C2×C4×Dic7C2311yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100#6451—22no—
101#6451—22no—
102#6451—44no—
103#6441—44no—
104C4×Dic14—22no—
105C4×Dic14—22no—
106C7×#102—22no—
107C7×#102Q814yescomplemented
108C4×C7×C8—22no—
109C7×#95—21no—
110#6406—22no—
111#6406—22no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118#6392—22no—
119#6392—22no—
120Dic7⋊C8—22no—
121Dic7⋊C8D414yes—
122C8×Dic7—22no—
123C8×Dic7—22no—
124#6381—22no—
125#6381—22no—
126C4⋊(C7⋊C8)—22no—
127C2×#474—71no—
128C2×C8×Q8—71no—
129#2546—71no—
130#2538—71no—
131#2086—71no—
132#2086—71no—
133#2067—71no—
134#2067—71no—
135#1990—71no—
136#1947—71no—
137#1942—71no—
138#1942—71no—
139#1934—71no—
140#1934D711yescharacteristic
141C4×#102—71no—
142C24×C7—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4C2×Q812yes—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—42no—
160C22×Dic7—42no—
161C22×Dic7—42no—
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C2×Dic14—22no—
167C2×Dic14—22no—
168C2×Dic14—42no—
169C2×Dic14—42no—
170C2×Dic14—42no—
171C2×Dic14D814yes—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8C2×D412yes—
177C2×C7×C8C2×Q812yes—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4C2×Q812yes—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—81no—
187C42×C7—21no—
188C42×C7—21no—
189C22⋊Dic7—42no—
190C22⋊Dic7—42no—
191C22⋊Dic7—81no—
192C4⋊Dic7—21no—
193C4⋊Dic7—22no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7—42no—
202C4⋊Dic7#4112yes—
203Dic7⋊C4—22no—
204Dic7⋊C4—22no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—81no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7—42no—
214C4×Dic7C2×D412yes—
215C4×Dic7#4112yes—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8C2×D412yes—
222C2×C4⋊Q8—71no—
223C2×C4⋊Q8—141no—
224C2×C22⋊Q8—141no—
225C2×C4×Q8—71no—
226C2×C4×D4—71no—
227#474—144no—
228C8×Q8—144no—
229C2×#107—71no—
230C2×#107—71no—
231C2×#107—71no—
232C2×#107—71no—
233C2×C4⋊C8—71no—
234C2×C4⋊C8—71no—
235C2×C4⋊C8—71no—
236C2×#102—71no—
237C2×#102—71no—
238C2×#102C2×D711yescharacteristic
239C2×#102C2×D711yescharacteristic
240C2×C4×C8—71no—
241C2×C4×C8—71no—
242C2×C4×C8C2×D711yescharacteristic
243#398—141no—
244#398—141no—
245#396—141no—
246#396—141no—
247#391—141no—
248#390—71no—
249#390C2×D711yescharacteristic
250#389—71no—
251#389—71no—
252#389—71no—
253#389—71no—
254#389—71no—
255#389C2×D711yescharacteristic
256#384—71no—
257#380—141no—
258C4×C4⋊C4—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340C2×D711yescharacteristic
266#340C2×D711yescharacteristic
267C23×C7—22no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7C22×D411yescharacteristic
275C7×D4—22no—
276C7×D4—22no—
277C7×D4—41no—
278C7×D4—41no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—41no—
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×C7C2×D812yes—
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×Q811yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—22no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—41no—
309C2×Dic7—41no—
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—84no—
320C2×Dic7#12112yes—
321C2×Dic7#12112yes—
322C2×Dic7C2×D812yes—
323C2×Dic7C2×D812yes—
324C2×Dic7#13512yes—
325Dic14—28no—
326Dic14—84no—
327C7×C8—22no—
328C7×C8—22no—
329C7⋊C8—22no—
330C7⋊C8—41no—
331C22×Q8—71no—
332C22×Q8—141no—
333C22×D4C22×D711yescharacteristic
334C23×C4—141no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8C22×D711yescharacteristic
338C22×C8C22×D711yescharacteristic
339C4⋊Q8—142no—
340C4⋊Q8—142no—
341C4⋊Q8—284no—
342C22⋊Q8—284no—
343C4×Q8—142no—
344C4×Q8—142no—
345C4×D4—142no—
346C4×D4—142no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
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⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—71no—
372C2×C42—71no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375#107—142no—
376#107—142no—
377#107—142no—
378#107—142no—
379C4⋊C8—74no—
380C4⋊C8—142no—
381C4⋊C8—142no—
382#102—22no—
383#102—142no—
384#102—142no—
385#102Dic1414yescomplemented
386C4×C8—22no—
387C4×C8—142no—
388C4×C8—142no—
389#95—21no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—81no—
400C22×C7—81no—
401C22×C7—81no—
402C22×C7—81no—
403C22×C7C2×#12111yescharacteristic
404C22×C7#54811yescharacteristic
405C22×C7D4×Q811yescharacteristic
406C22×C7C22×D811yescharacteristic
407C22×C7C2×#13511yescharacteristic
408C22×C7#57711yescharacteristic
409C22×C7#57811yescharacteristic
410C4×C7—22no—
411C4×C7—42no—
412C4×C7—41no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7#53311yescharacteristic
416C4×C7#54111yescharacteristic
417C4×C7#54711yescharacteristic
418C4×C7D4×Q811yescharacteristic
419Dic7—22no—
420Dic7—22no—
421Dic7—42no—
422Dic7—41no—
423Dic7—81no—
424Dic7—81no—
425Dic7—81no—
426Dic7#46614yes—
427C24—21no—
428C2×Q8—74no—
429C2×Q8—142no—
430C2×Q8—142no—
431C2×Q8—282no—
432C2×Q8—282no—
433C2×Q8—282no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×Dic1412yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—72no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8C2×Dic1412yes—
468C2×C8D4×D712yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—72no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C4⋊C4—142no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—561no—
494C4⋊C4C2×Dic1412yes—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—282no—
498C22⋊C4—282no—
499C22⋊C4—561no—
500C42—21no—
501C42—21no—
502C42—72no—
503C42—72no—
504C42—141no—
505C42—141no—
506C42—282no—
507C2×C7—82no—
508C2×C7#242911yescharacteristic
509C2×C7C2×#46611yescharacteristic
510C2×C7#306311yescharacteristic
511C2×C7#335811yescharacteristic
512C2×C7#336411yescharacteristic
513C2×C7Q8×D811yescharacteristic
514C2×C7#338111yescharacteristic
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D711yescharacteristic
523Q8—148no—
524Q8—564no—
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—72no—
547C2×C4—72no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—142no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—564no—
568C2×C4D7×D812yes—
569C2×C4C22×Dic1411yescharacteristic
570C2×C4C2×D4×D711yescharacteristic
571C2×C4D7×#4111yescharacteristic
572C2×C4D7×#4111yescharacteristic
573C2×C4#655411yescharacteristic
574C2×C4#655411yescharacteristic
575C8—22no—
576C8—22no—
577C8—142no—
578C8—281no—
579C7#2546911yescharacteristic, complemented, Sylow
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22D4×Dic1411yescharacteristic
588C22D7×#12111yescharacteristic
589C22#9036811yescharacteristic
590C22C2×D7×D811yescharacteristic
591C22#9045311yescharacteristic
592C22D7×#13511yescharacteristic
593C22#9046211yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—74no—
598C4—81no—
599C4—81no—
600C4—142no—
601C4—142no—
602C4—282no—
603C4—281no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4#9020811yescharacteristic
608C4D4×Dic1411yescharacteristic
609C4#9034611yescharacteristic
610C4#9039211yescharacteristic
611C2—82no—
612C2#1173294511yescharacteristic
613C2D8×Dic1411yescharacteristic
614C2#1173707811yescharacteristic
615C2#1173810711yescharacteristic
616C2D7×#46611yescharacteristic
617C2#1173902911yescharacteristic
618C2#1173907911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1934C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#1934D7
hypocenterC7#25469
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic14C2semidirect
4. #11733505C2semidirect
9. #11727524C2semidirect
11. #11727376C2semidirect
12. #11726613C2semidirect
13. #11726592C2semidirect
14. #11726557C2semidirect
15. #11726549C2semidirect
107. C7×#102Q8semidirect
385. #102Dic14semidirect
579. C7#25469semidirect

Maximal subgroups

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

subgroupindex
C2×#895312
C2×C8×Dic142
#117337912
#117335052
C7×#19342
#117279392
#117279342
Dic7×#1022
#117275242
#117274072
#117273762
#117266132
#117265922
#117265572
#117265492
#254697