Finite Group Databasebeta

← #13271321

Subgroups — #13271321

Order 1792.

782 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2431C211yesmaximal, characteristic
2C2×#90012C211yesmaximal, characteristic, complemented
3#11729893C211yesmaximal, characteristic, complemented
4C2×#89340C212yesmaximal, complemented
5#11725447C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×#127C2211yescharacteristic
8C2×C7×#124C411yescharacteristic, complemented
9C2×C7×#121C411yescharacteristic, complemented
10C2×#6489—21no—
11C7×#392—21no—
12C7×#392—21no—
13C7×#391C411yescharacteristic, complemented
14C7×#390C2211yescharacteristic
15C7×#386—21no—
16C7×#386—21no—
17C4×C7×C22⋊C4C411yescharacteristic, complemented
18#90012—24no—
19C2×#6408—21no—
20C2×#6408—21no—
21C2×#6408C2212yes—
22C2×#6408C2212yes—
23#89983—21no—
24#89340—24no—
25#13089—71nomaximal, Sylow
26C2×C7×#41—22no—
27C2×C7×#41—22no—
28C2×C7×#41—41no—
29C2×C7×#41—41no—
30C2×C7×#41—42no—
31C22×C7×Q8C2×C411yescharacteristic
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4C2×C411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C22×C7×D4D411yescharacteristic
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7D411yescharacteristic
41C23×C4×C7D411yescharacteristic
42C7×#127—22no—
43C7×#127—22no—
44C7×#124—42no—
45C7×#121—41no—
46C7×#121—41no—
47C2×C7×C4⋊C4C2×C411yescharacteristic
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×C42×C7C2×C411yescharacteristic
55C2×C42×C7C2×C411yescharacteristic
56C2×C22⋊Dic7—22no—
57C2×C22⋊Dic7—22no—
58#6489—24no—
59#6489—42no—
60C2×C4⋊Dic7—41no—
61C2×C4×Dic7—41no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65#6408—24no—
66#6408—22no—
67#6408—22no—
68#6408—22no—
69#6408—42no—
70#6408D414yescomplemented
71C7⋊#95—41no—
72C7⋊#95—41no—
73C2×#410—71no—
74#2431D711yescharacteristic
75C2×#353—72no—
76#1782—71no—
77#1393—72no—
78C24×C7—21no—
79C24×C7C22⋊C411yescharacteristic
80C24×C7C2×D411yescharacteristic
81C24×C7C2×D411yescharacteristic
82C7×#41—44no—
83C7×#41—44no—
84C7×#41—44no—
85C7×#41—44no—
86C2×C7×Q8—22no—
87C2×C7×Q8—22no—
88C2×C7×Q8—41no—
89C2×C7×D4—21no—
90C2×C7×D4—22no—
91C2×C7×D4—21no—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—42no—
100C2×C7×D4—42no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—42no—
104C2×C7×D4—41no—
105C2×C7×D4—42no—
106C2×C7×D4—41no—
107C2×C7×D4—81no—
108C2×C7×D4—81no—
109C2×C7×D4—81no—
110C2×C7×D4—81no—
111C2×C7×D4C2×D412yes—
112C2×C7×D4#4112yes—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
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—22no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—41no—
128C22×C4×C7—41no—
129C22×C4×C7C22⋊C411yescharacteristic
130C22×C4×C7C22⋊C411yescharacteristic
131C22×C4×C7C22⋊C411yescharacteristic
132C22×C4×C7C22×C411yescharacteristic
133C22×Dic7—42no—
134C22×Dic7—42no—
135C7×C4⋊C4—41no—
136C7×C22⋊C4—42no—
137C7×C22⋊C4—42no—
138C7×C22⋊C4—41no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—42no—
141C7×C22⋊C4—81no—
142C7×C22⋊C4—81no—
143C42×C7—22no—
144C42×C7—41no—
145C22⋊Dic7—24no—
146C22⋊Dic7—42no—
147C22⋊Dic7—42no—
148C22⋊Dic7—42no—
149C4⋊Dic7—44no—
150C4×Dic7—44no—
151C22×#41C2×D711yescharacteristic
152C2×#127C2×D711yescharacteristic
153C2×#124Dic711yescharacteristic, complemented
154C2×#121Dic711yescharacteristic, complemented
155C2×#117—141no—
156#410—144no—
157C2×#99—72no—
158C2×#99—72no—
159C2×#99—141no—
160C2×#99—141no—
161#392—21no—
162#392—21no—
163#391Dic711yescharacteristic, complemented
164#390C2×D711yescharacteristic
165#386—21no—
166#386—21no—
167#381—141no—
168C4×C22⋊C4Dic711yescharacteristic, complemented
169#353—144no—
170C23×C7—21no—
171C23×C7—21no—
172C23×C7—21no—
173C23×C7—21no—
174C23×C7—21no—
175C23×C7—21no—
176C23×C7—21no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7C2×C22⋊C411yescharacteristic
191C23×C7#12011yescharacteristic
192C23×C7#12112yes—
193C23×C7#12311yescharacteristic
194C7×Q8—44no—
195C7×D4—42no—
196C7×D4—42no—
197C7×D4—44no—
198C7×D4—42no—
199C7×D4—42no—
200C7×D4—82no—
201C7×D4—82no—
202C7×D4—84no—
203C7×D4—82no—
204C7×D4—84no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—22no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7#9911yescharacteristic
232C2×C4×C7#9911yescharacteristic
233C2×C4×C7#9912yes—
234C2×C4×C7C4×D412yes—
235C2×Dic7—44no—
236C2×Dic7—82no—
237C2×Dic7—82no—
238C2×Dic7—82no—
239C2×Dic7—82no—
240C2×Dic7—82no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—41no—
244C2×#41—41no—
245C2×#41—42no—
246C22×Q8C2×Dic711yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×Dic711yescharacteristic
251C22×D4C22⋊D711yescharacteristic
252C22×D4C22⋊D711yescharacteristic
253C22×D4C22×D711yescharacteristic
254C23×C4C2×Dic711yescharacteristic
255C23×C4C22⋊D711yescharacteristic
256C23×C4C22⋊D711yescharacteristic
257#127—22no—
258#127—22no—
259#124—42no—
260#121—41no—
261#121—41no—
262#117—144no—
263#117—282no—
264C2×C4⋊C4—281no—
265C2×C4⋊C4C2×Dic711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—142no—
273C2×C22⋊C4—142no—
274C2×C42—281no—
275C2×C42C2×Dic711yescharacteristic
276C2×C42C2×Dic711yescharacteristic
277#99—74no—
278#99—144no—
279#99—142no—
280#99—142no—
281#99—142no—
282#99—282no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—281no—
287#95—281no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—21no—
291C22×C7—22no—
292C22×C7—21no—
293C22×C7—41no—
294C22×C7—42no—
295C22×C7—41no—
296C22×C7—41no—
297C22×C7—41no—
298C22×C7—41no—
299C22×C7—41no—
300C22×C7—41no—
301C22×C7—41no—
302C22×C7—41no—
303C22×C7—81no—
304C22×C7—81no—
305C22×C7—81no—
306C22×C7—81no—
307C22×C7—81no—
308C22×C7—81no—
309C22×C7—81no—
310C22×C7—81no—
311C22×C7—81no—
312C22×C7—81no—
313C22×C7—81no—
314C22×C7#38611yescharacteristic
315C22×C7C2×#9911yescharacteristic
316C22×C7C2×#9911yescharacteristic
317C4×C7—21no—
318C4×C7—21no—
319C4×C7—22no—
320C4×C7—22no—
321C4×C7—42no—
322C4×C7—41no—
323C4×C7—81no—
324C4×C7—81no—
325Dic7—84no—
326Dic7—162no—
327C24—21no—
328C24C22⋊Dic711yescharacteristic
329C24C2×C22⋊D711yescharacteristic
330C24C2×C22⋊D711yescharacteristic
331#41—44no—
332#41—44no—
333#41—44no—
334#41—44no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—81no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4D4×D712yes—
361C2×D4#115912yes—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—282no—
379C22×C4—282no—
380C22×C4C22⋊Dic711yescharacteristic
381C22×C4C22⋊Dic711yescharacteristic
382C22×C4C22⋊Dic711yescharacteristic
383C22×C4C22×Dic711yescharacteristic
384C4⋊C4—41no—
385C4⋊C4—284no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—42no—
391C22⋊C4—81no—
392C22⋊C4—81no—
393C22⋊C4—144no—
394C22⋊C4—282no—
395C22⋊C4—282no—
396C22⋊C4—282no—
397C42—22no—
398C42—41no—
399C42—284no—
400C2×C7—21no—
401C2×C7—21no—
402C2×C7—41no—
403C2×C7—42no—
404C2×C7—41no—
405C2×C7—81no—
406C2×C7—161no—
407C2×C7#189811yescharacteristic
408C2×C7#212611yescharacteristic
409C2×C7#212811yescharacteristic
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23#649811yescharacteristic
431C23#650011yescharacteristic
432C23#650212yes—
433C23C2×C22⋊Dic711yescharacteristic
434Q8—44no—
435D4—42no—
436D4—42no—
437D4—44no—
438D4—42no—
439D4—42no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—84no—
444D4—84no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—284no—
472C2×C4—562no—
473C2×C4—562no—
474C2×C4—562no—
475C2×C4—562no—
476C2×C4—562no—
477C2×C4#640812yes—
478C2×C4#640811yescharacteristic
479C2×C4#640811yescharacteristic
480C2×C4D4×Dic712yes—
481C7#1308911yescharacteristic, complemented, Sylow
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—42no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C2×#640811yescharacteristic
509C22C2×#640811yescharacteristic
510C22#8999211yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—81no—
518C4—81no—
519C4—564no—
520C4—1122no—
521C2—21no—
522C2—21no—
523C2—41no—
524C2—42no—
525C2—41no—
526C2—81no—
527C2—161no—
528C2#1172942411yescharacteristic
529C2#1172957311yescharacteristic
530C2#1172961511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×D4C22×D7
Fitting, prime radicalC7×#2431C2
socleC22×C7C2×#99
cosocleC22×C7×D4C23
hypercenter#2431D7
hypocenterC7#13089
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2431

Complements

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

NH ≅ G/Ntype
2. C2×#90012C2semidirect
3. #11729893C2semidirect
4. C2×#89340C2semidirect
5. #11725447C2semidirect
8. C2×C7×#124C4semidirect
9. C2×C7×#121C4semidirect
13. C7×#391C4semidirect
17. C4×C7×C22⋊C4C4semidirect
70. #6408D4semidirect
153. C2×#124Dic7semidirect
154. C2×#121Dic7semidirect
163. #391Dic7semidirect
168. C4×C22⋊C4Dic7semidirect
481. C7#13089semidirect

Maximal subgroups

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

subgroupindex
C7×#24312
C2×#900122
#117298932
C2×#893402
#117254472
#130897