Finite Group Databasebeta

← #13413126

Subgroups — #13413126

Order 1792.

734 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731215C211yesmaximal, characteristic
2#11731168C211yesmaximal, characteristic, complemented
3C7×#1939C211yesmaximal, characteristic, complemented
4#11728234C211yesmaximal, characteristic
5#11728234C211yesmaximal, characteristic
6#11728233C211yesmaximal, characteristic
7#11728218C211yesmaximal, characteristic, complemented
8#11728217C211yesmaximal, characteristic, complemented
9#11728209C211yesmaximal, characteristic, complemented
10#11727517C211yesmaximal, characteristic, complemented
11C4×#6395C211yesmaximal, characteristic
12#11726425C211yesmaximal, characteristic
13#11726424C211yesmaximal, characteristic
14#11726398C211yesmaximal, characteristic, complemented
15#11726396C211yesmaximal, characteristic, complemented
16C2×C4×D28C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30#89761—21no—
31#89761—21no—
32#89758—21no—
33#89756—21no—
34#89708C2211yescharacteristic
35#89708C2211yescharacteristic
36#89708C2211yescharacteristic
37#89707C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39#89700C2211yescharacteristic
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43#89439—21no—
44D7×#95—21no—
45#89427—21no—
46#89426—21no—
47C7×#336C2211yescharacteristic
48C7×#336C2211yescharacteristic
49#89346C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#26115—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×C4×Dic7—21no—
90C4×D28—22no—
91C4×D28—22no—
92C4×D28—22no—
93C4×D28—41no—
94C4×D28—41no—
95C4×D28Q814yes—
96C7×C4⋊C8—22no—
97C7×C4⋊C8—41no—
98C7×C4⋊C8—41no—
99C4×C7×C8—22no—
100C7×#95D411yescharacteristic
101C7×#95D411yescharacteristic
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113#6395—22no—
114#6395—22no—
115#6395—22no—
116#6395—22no—
117#6392—41no—
118#6391—41no—
119#2570—71no—
120#2349—71no—
121#2101—71no—
122#2099—71no—
123#2088—71no—
124#2087—71no—
125#1949—71no—
126#1939D711yescharacteristic, complemented
127#1864—71no—
128#1862—71no—
129#1835—71no—
130#1804—71no—
131#1804—71no—
132#1802—71no—
133C4×#102—71no—
134C23×D7—21no—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28C2×Q812yes—
153C2×D28#4112yes—
154C2×D28#4112yes—
155C2×C4×D7—24no—
156C2×C4×D7—42no—
157C2×C4×D7—42no—
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C4×D7—81no—
161C2×C4×D7—81no—
162C2×C4×D7—81no—
163C2×C4×D7—81no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—41no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7C2×Q812yes—
183C42×C7#4112yes—
184(C2×D7)⋊C4—24no—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4—42no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7—81no—
202C4⋊Dic7C2×Q812yes—
203Dic7⋊C4—42no—
204Dic7⋊C4—81no—
205Dic7⋊C4—81no—
206Dic7⋊C4—81no—
207C4×Dic7—81no—
208C2×C4×D4—71no—
209C2×C4×D4—71no—
210C2×C4×D4—71no—
211C2×#107—71no—
212C2×#106—71no—
213C2×C4⋊C8C2×D711yescharacteristic
214C2×C4⋊C8C2×D711yescharacteristic
215C2×C4⋊C8C2×D711yescharacteristic
216C2×#102—71no—
217C2×#102—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×C4×C8C2×D711yescharacteristic
221#400—141no—
222#398—141no—
223#389—71no—
224#389—71no—
225#389—71no—
226#389—71no—
227#387—141no—
228#387—141no—
229#385—141no—
230#385—141no—
231#384—71no—
232#382—71no—
233#382—71no—
234#382—71no—
235#380—141no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4C2×D711yescharacteristic
238C2×#95—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#336C2×D711yescharacteristic
250#336C2×D711yescharacteristic
251C23×C7C22×D411yescharacteristic
252C22×D7—22no—
253C22×D7—22no—
254C22×D7—22no—
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7C22⋊Q812yes—
275C2×C4×C7C22⋊Q812yes—
276C2×C4×C7#13512yes—
277C2×C4×C7C22×Q811yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7H(5,𝔽2)11yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×Dic7—22no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304D28—22no—
305D28—22no—
306D28—22no—
307D28#12514yes—
308C4×D7—84no—
309C4×D7—84no—
310C4×D7—84no—
311C7×C8—41no—
312C7×C8—41no—
313C7×C8—41no—
314C7×C8—41no—
315C22×D4—71no—
316C23×C4—141no—
317C23×C4—141no—
318C23×C4—141no—
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322C22×C8C22×D711yescharacteristic
323C4×D4—74no—
324C4×D4—142no—
325C4×D4—142no—
326C4×D4—142no—
327C4×D4—281no—
328C4×D4—281no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4D2811yescharacteristic
343C2×C4⋊C4D2811yescharacteristic
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C42—141no—
348C2×C42C22×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350C2×C42C22×D711yescharacteristic
351#107—281no—
352#106—281no—
353C4⋊C8—22no—
354C4⋊C8—41no—
355C4⋊C8—41no—
356#102—142no—
357#102—142no—
358#102—142no—
359#102—142no—
360C4×C8—22no—
361#95—141no—
362#95—141no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95D2811yescharacteristic
373#95D2811yescharacteristic
374C22×C7C2×C22⋊Q811yescharacteristic
375C22×C7#54611yescharacteristic
376C22×C7#54811yescharacteristic
377C22×C7C2×#13511yescharacteristic
378C22×C7#57511yescharacteristic
379C22×C7#57611yescharacteristic
380C22×C7#57711yescharacteristic
381C2×D7—24no—
382C2×D7—42no—
383C2×D7—42no—
384C2×D7—42no—
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—42no—
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—22no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7C2×#12511yescharacteristic
395C4×C7#54411yescharacteristic
396C4×C7#55211yescharacteristic
397C4×C7#55311yescharacteristic
398Dic7—42no—
399Dic7—81no—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24—141no—
406C2×D4—72no—
407C2×D4—72no—
408C2×D4—72no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—141no—
412C22×C4—141no—
413C22×C4—144no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—282no—
421C22×C4—282no—
422C22×C4—282no—
423C22×C4—282no—
424C22×C4—561no—
425C22×C4—561no—
426C22×C4—561no—
427C22×C4—561no—
428C22×C4C2×D2811yescharacteristic
429C22×C4C2×D2811yescharacteristic
430C22×C4C2×D2811yescharacteristic
431C22×C4C2×D2811yescharacteristic
432C22×C4C2×D2811yescharacteristic
433C22×C4C2×D2811yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C4⋊C4—22no—
448C4⋊C4—41no—
449C4⋊C4—72no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—282no—
464C4⋊C4—282no—
465C4⋊C4—561no—
466C4⋊C4—561no—
467C4⋊C4—561no—
468C4⋊C4—561no—
469C22⋊C4—144no—
470C22⋊C4—282no—
471C22⋊C4—282no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—561no—
477C42#115912yes—
478C42Q8×D712yes—
479C2×C7#244211yescharacteristic
480C2×C7#307611yescharacteristic
481C2×C7#308111yescharacteristic
482C2×C7#329611yescharacteristic
483C2×C7#332211yescharacteristic
484C2×C7#336111yescharacteristic
485C2×C7#337611yescharacteristic
486D7—44no—
487C23—142no—
488C23—142no—
489C23—142no—
490C23—281no—
491C23—281no—
492C23—281no—
493C23—281no—
494C23C22×D2811yescharacteristic
495D4—74no—
496D4—142no—
497D4—142no—
498D4—142no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—142no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—564no—
536C2×C4—564no—
537C2×C4—564no—
538C2×C4#646012yes—
539C2×C4#646012yes—
540C2×C4#646712yes—
541C2×C4C2×#115911yescharacteristic
542C2×C4#654811yescharacteristic
543C2×C4C2×Q8×D711yescharacteristic
544C2×C4#655111yescharacteristic
545C2×C4D7×#4111yescharacteristic
546C2×C4D7×#4111yescharacteristic
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C7#2611511yescharacteristic, complemented, Sylow
552C22—144no—
553C22—282no—
554C22—282no—
555C22—282no—
556C22—282no—
557C22—282no—
558C22—282no—
559C22C2×#646011yescharacteristic
560C22#9024511yescharacteristic
561C22#9026711yescharacteristic
562C22C2×#646711yescharacteristic
563C22#9043911yescharacteristic
564C22#9044211yescharacteristic
565C22#9044311yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—282no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4—561no—
578C4—561no—
579C4#9030511yescharacteristic
580C4#9033011yescharacteristic
581C4D7×#12511yescharacteristic
582C4#9042011yescharacteristic
583C2—284no—
584C2#1173344811yescharacteristic
585C2#1173647011yescharacteristic
586C2#1173653511yescharacteristic
587C2#1173688511yescharacteristic
588C2#1173699811yescharacteristic
589C2#1173737211yescharacteristic
590C2#1173741911yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1939C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1939D7
hypocenterC7#26115
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
2. #11731168C2semidirect
3. C7×#1939C2semidirect
7. #11728218C2semidirect
8. #11728217C2semidirect
9. #11728209C2semidirect
10. #11727517C2semidirect
14. #11726398C2semidirect
15. #11726396C2semidirect
126. #1939D7semidirect
551. C7#26115semidirect

Maximal subgroups

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

subgroupindex
#117312152
#117311682
C7×#19392
#117282342
#117282332
#117282182
#117282172
#117282092
#117275172
C4×#63952
#117264252
#117264242
#117263982
#117263962
#261157