Finite Group Databasebeta

← #13271326

Subgroups — #13271326

Order 1792.

910 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2435C211yesmaximal, characteristic
2C22×#6408C211yesmaximal, characteristic
3#11729884C211yesmaximal, characteristic
4C2×#89336C212yesmaximal, complemented
5#11725447C212yesmaximal, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#124C411yescharacteristic, complemented
8C2×C7×#124C2211yescharacteristic
9C2×C7×C22⋊Q8C411yescharacteristic, complemented
10C22×C22⋊Dic7—21no—
11C7×#394—21no—
12C7×#394—21no—
13C7×#390C411yescharacteristic, complemented
14C7×#390C2211yescharacteristic
15C7×#379—21no—
16C7×#379—21no—
17C4×C7×C22⋊C4C411yescharacteristic, complemented
18#90019—21no—
19C2×#6408—21no—
20C2×#6408—24no—
21C2×#6408—21no—
22C2×#6408C2212yes—
23C2×#6408C2212yes—
24#89336—24no—
25#13090—71nomaximal, Sylow
26C25×C7D411yescharacteristic
27C25×C7D411yescharacteristic
28C22×C7×Q8C2×C411yescharacteristic, complemented
29C22×C7×D4—22no—
30C22×C7×D4—22no—
31C22×C7×D4—41no—
32C22×C7×D4—41no—
33C22×C7×D4—42no—
34C22×C7×D4C2×C411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C22×C7×D4C2311yescharacteristic
38C23×C4×C7C2×C411yescharacteristic
39C23×Dic7—41no—
40C7×#124—22no—
41C7×#124—22no—
42C7×#124—42no—
43C7×C22⋊Q8—41no—
44C7×C22⋊Q8—41no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—21no—
57C2×C42×C7C2×C411yescharacteristic, complemented
58C2×C42×C7C2×C411yescharacteristic, complemented
59C2×C22⋊Dic7—24no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—41no—
63C2×C22⋊Dic7—42no—
64C2×C22⋊Dic7—41no—
65C2×C22⋊Dic7—41no—
66C7×#95—21no—
67#6408—24no—
68#6408—22no—
69#6408—22no—
70#6408—22no—
71#6408—42no—
72#6408D414yes—
73C22×#99—71no—
74#2435D711yescharacteristic
75C2×#354—72no—
76#1781—71no—
77#1393—72no—
78C24×C7—21no—
79C24×C7—22no—
80C24×C7—21no—
81C24×C7—21no—
82C24×C7—21no—
83C24×C7—21no—
84C24×C7—21no—
85C24×C7—22no—
86C24×C7—41no—
87C24×C7—42no—
88C24×C7—42no—
89C24×C7—81no—
90C24×C7—81no—
91C24×C7C22⋊C411yescharacteristic
92C24×C7C22⋊C411yescharacteristic
93C24×C7C22⋊C411yescharacteristic
94C24×C7C2×D411yescharacteristic
95C24×C7C2×D411yescharacteristic
96C2×C7×Q8—22no—
97C2×C7×Q8—41no—
98C2×C7×Q8—41no—
99C2×C7×D4—22no—
100C2×C7×D4—22no—
101C2×C7×D4—21no—
102C2×C7×D4—22no—
103C2×C7×D4—21no—
104C2×C7×D4—22no—
105C2×C7×D4—41no—
106C2×C7×D4—41no—
107C2×C7×D4—41no—
108C2×C7×D4—41no—
109C2×C7×D4—41no—
110C2×C7×D4—41no—
111C2×C7×D4—44no—
112C2×C7×D4—44no—
113C2×C7×D4—44no—
114C2×C7×D4—44no—
115C2×C7×D4C2×D412yes—
116C2×C7×D4#4112yes—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—41no—
125C22×C4×C7C22⋊C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×Dic7—44no—
128C22×Dic7—42no—
129C22×Dic7—42no—
130C22×Dic7—82no—
131C22×Dic7—82no—
132C7×C4⋊C4—41no—
133C7×C4⋊C4—42no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4—42no—
136C7×C22⋊C4—41no—
137C7×C22⋊C4—41no—
138C7×C22⋊C4—42no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—42no—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—81no—
143C7×C22⋊C4—81no—
144C7×C22⋊C4—81no—
145C7×C22⋊C4—81no—
146C42×C7—22no—
147C42×C7—41no—
148C22⋊Dic7—24no—
149C22⋊Dic7—44no—
150C22⋊Dic7—42no—
151C22⋊Dic7—42no—
152C22⋊Dic7—42no—
153C22⋊Dic7—82no—
154C22⋊Dic7—82no—
155C23×D4C2×D711yescharacteristic
156C2×#124Dic711yescharacteristic, complemented
157C2×#124C2×D711yescharacteristic
158C2×C22⋊Q8Dic711yescharacteristic, complemented
159C22×C22⋊C4—141no—
160C2×#99—72no—
161C2×#99—72no—
162C2×#99—144no—
163C2×#99—141no—
164C2×#99—141no—
165#394—21no—
166#394—21no—
167#390Dic711yescharacteristic, complemented
168#390C2×D711yescharacteristic
169#379—21no—
170#379—21no—
171#379—141no—
172C4×C22⋊C4Dic711yescharacteristic, complemented
173#354—144no—
174C23×C7—22no—
175C23×C7—22no—
176C23×C7—22no—
177C23×C7—22no—
178C23×C7—22no—
179C23×C7—21no—
180C23×C7—21no—
181C23×C7—21no—
182C23×C7—21no—
183C23×C7—21no—
184C23×C7—21no—
185C23×C7—21no—
186C23×C7—21no—
187C23×C7—21no—
188C23×C7—21no—
189C23×C7—21no—
190C23×C7—41no—
191C23×C7—42no—
192C23×C7—42no—
193C23×C7—42no—
194C23×C7—41no—
195C23×C7—41no—
196C23×C7—41no—
197C23×C7—41no—
198C23×C7—41no—
199C23×C7—41no—
200C23×C7—41no—
201C23×C7—41no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—81no—
206C23×C7—81no—
207C23×C7—81no—
208C23×C7—81no—
209C23×C7—84no—
210C23×C7—82no—
211C23×C7—82no—
212C23×C7—84no—
213C23×C7—84no—
214C23×C7#9911yescharacteristic
215C23×C7#9911yescharacteristic
216C23×C7#9912yes—
217C23×C7C2×C22⋊C411yescharacteristic
218C23×C7#12011yescharacteristic
219C23×C7#12112yes—
220C23×C7#12311yescharacteristic
221C7×Q8—82no—
222C7×D4—42no—
223C7×D4—42no—
224C7×D4—44no—
225C7×D4—44no—
226C7×D4—42no—
227C7×D4—42no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7C4×D412yes—
248C2×Dic7—44no—
249C2×Dic7—88no—
250C2×Dic7—82no—
251C2×Dic7—82no—
252C2×Dic7—82no—
253C2×Dic7—82no—
254C2×Dic7—82no—
255C25C22⋊D711yescharacteristic
256C25C22⋊D711yescharacteristic
257C22×Q8C2×Dic711yescharacteristic, complemented
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—41no—
261C22×D4—41no—
262C22×D4—42no—
263C22×D4C2×Dic711yescharacteristic
264C22×D4C22⋊D711yescharacteristic
265C22×D4C22⋊D711yescharacteristic
266C22×D4C22×D711yescharacteristic
267C23×C4—281no—
268C23×C4C2×Dic711yescharacteristic
269#124—22no—
270#124—22no—
271#124—42no—
272C22⋊Q8—41no—
273C22⋊Q8—41no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C2×Dic711yescharacteristic, complemented
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—144no—
287C2×C22⋊C4—142no—
288C2×C22⋊C4—142no—
289C2×C22⋊C4—281no—
290C2×C22⋊C4—282no—
291C2×C22⋊C4—281no—
292C2×C22⋊C4—281no—
293C2×C42C2×Dic711yescharacteristic, complemented
294C2×C42C2×Dic711yescharacteristic, complemented
295#99—74no—
296#99—144no—
297#99—142no—
298#99—142no—
299#99—142no—
300#99—282no—
301#95—21no—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—21no—
305C22×C7—21no—
306C22×C7—21no—
307C22×C7—21no—
308C22×C7—21no—
309C22×C7—22no—
310C22×C7—21no—
311C22×C7—22no—
312C22×C7—21no—
313C22×C7—42no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7—41no—
317C22×C7—42no—
318C22×C7—41no—
319C22×C7—44no—
320C22×C7—44no—
321C22×C7—42no—
322C22×C7—41no—
323C22×C7—41no—
324C22×C7—41no—
325C22×C7—41no—
326C22×C7—41no—
327C22×C7—41no—
328C22×C7—41no—
329C22×C7—41no—
330C22×C7—41no—
331C22×C7—81no—
332C22×C7—81no—
333C22×C7—81no—
334C22×C7—84no—
335C22×C7—81no—
336C22×C7—82no—
337C22×C7—82no—
338C22×C7—84no—
339C22×C7#35412yescomplemented
340C22×C7#38611yescharacteristic
341C22×C7C2×#9911yescharacteristic
342C22×C7C2×#9911yescharacteristic
343C4×C7—22no—
344C4×C7—41no—
345C4×C7—81no—
346C4×C7—81no—
347C4×C7—81no—
348C4×C7—81no—
349Dic7—84no—
350Dic7—162no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—41no—
360C24—42no—
361C24—42no—
362C24—81no—
363C24—81no—
364C24C22⋊Dic711yescharacteristic
365C24C22⋊Dic711yescharacteristic
366C24C22⋊Dic711yescharacteristic
367C24C2×C22⋊D711yescharacteristic
368C24C2×C22⋊D711yescharacteristic
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4D4×D712yes—
389C2×D4#115912yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—284no—
399C22×C4—282no—
400C22×C4—282no—
401C22×C4—562no—
402C22×C4—562no—
403C22×C4C22⋊Dic711yescharacteristic
404C22×C4C22×Dic711yescharacteristic
405C4⋊C4—41no—
406C4⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—42no—
415C22⋊C4—81no—
416C22⋊C4—81no—
417C22⋊C4—81no—
418C22⋊C4—81no—
419C22⋊C4—144no—
420C22⋊C4—284no—
421C22⋊C4—282no—
422C22⋊C4—282no—
423C22⋊C4—282no—
424C22⋊C4—562no—
425C22⋊C4—562no—
426C42—22no—
427C42—41no—
428C2×C7—21no—
429C2×C7—21no—
430C2×C7—22no—
431C2×C7—41no—
432C2×C7—42no—
433C2×C7—41no—
434C2×C7—42no—
435C2×C7—41no—
436C2×C7—81no—
437C2×C7#189811yescharacteristic
438C2×C7C2×#35411yescharacteristic
439C2×C7#212711yescharacteristic
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—84no—
476C23—82no—
477C23—82no—
478C23—84no—
479C23—84no—
480C23#640811yescharacteristic
481C23#640811yescharacteristic
482C23#640812yes—
483C23#649811yescharacteristic
484C23#650011yescharacteristic
485C23#650212yes—
486C23C2×C22⋊Dic711yescharacteristic
487Q8—82no—
488D4—42no—
489D4—42no—
490D4—44no—
491D4—44no—
492D4—42no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
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—284no—
514C2×C4—568no—
515C2×C4—562no—
516C2×C4—562no—
517C2×C4—562no—
518C2×C4—562no—
519C2×C4—562no—
520C2×C4D4×Dic712yes—
521C7#1309011yescharacteristic, complemented, Sylow
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—84no—
555C22—81no—
556C22—82no—
557C22—82no—
558C22—84no—
559C22#8933612yescomplemented
560C22C2×#640811yescharacteristic
561C22C2×#640811yescharacteristic
562C22#8999211yescharacteristic
563C4—22no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—564no—
570C4—1122no—
571C2—21no—
572C2—21no—
573C2—22no—
574C2—41no—
575C2—42no—
576C2—41no—
577C2—42no—
578C2—41no—
579C2—81no—
580C2#1172942411yescharacteristic
581C2C2×#8933611yescharacteristic
582C2#1172961411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2435C2
socleC22×C7C2×#99
prime radicalC23×C7×D4C22
cosocleC22×C7×D4C23
hypercenter#2435D7
hypocenterC7#13090
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2435

Complements

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

NH ≅ G/Ntype
4. C2×#89336C2semidirect
5. #11725447C2semidirect
7. C2×C7×#124C4semidirect
9. C2×C7×C22⋊Q8C4semidirect
13. C7×#390C4semidirect
17. C4×C7×C22⋊C4C4semidirect
28. C22×C7×Q8C2×C4semidirect
46. C2×C7×C4⋊C4C2×C4semidirect
57. C2×C42×C7C2×C4semidirect
58. C2×C42×C7C2×C4semidirect
156. C2×#124Dic7semidirect
158. C2×C22⋊Q8Dic7semidirect
167. #390Dic7semidirect
172. C4×C22⋊C4Dic7semidirect
257. C22×Q8C2×Dic7semidirect
275. C2×C4⋊C4C2×Dic7semidirect
293. C2×C42C2×Dic7semidirect
294. C2×C42C2×Dic7semidirect
339. C22×C7#354semidirect
521. C7#13090semidirect
559. C22#89336semidirect

Maximal subgroups

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

subgroupindex
C7×#24352
C22×#64082
#117298842
C2×#893362
#117254472
#130907