Finite Group Databasebeta

← #13412839

Subgroups — #13412839

Order 1792.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2357C211yesmaximal, characteristic
2#11731168C211yesmaximal, characteristic, complemented
3C4×#6406C211yesmaximal, characteristic
4#11729359C211yesmaximal, characteristic
5#11727409C211yesmaximal, characteristic
6#11727408C211yesmaximal, characteristic
7#11727406C211yesmaximal, characteristic
8#11727406C211yesmaximal, characteristic
9#11727303C211yesmaximal, characteristic, complemented
10#11727294C211yesmaximal, characteristic, complemented
11#11727238C211yesmaximal, characteristic, complemented
12#11727222C211yesmaximal, characteristic, complemented
13#11727221C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727212C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C7×#390C2211yescharacteristic
18C7×#390C2211yescharacteristic
19C7×#388—21no—
20C7×#388—21no—
21C7×#382C2211yescharacteristic
22C7×#381—21no—
23C4×C7×C4⋊C4C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C22⋊C4—21no—
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31#89758—21no—
32#89756—21no—
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89708C2211yescharacteristic
40#89708C2211yescharacteristic
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4×C7⋊C8C2211yescharacteristic
49#89427—21no—
50#89426—21no—
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26094—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C4×C7×D4—22no—
63C4×C7×D4Q814yes—
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C7×#95—21no—
93C7×#95—21no—
94C7×#95—21no—
95C7×#95—21no—
96C7×#95D411yescharacteristic
97C7×#95D411yescharacteristic
98#6406—22no—
99#6406—22no—
100#6406—41no—
101#6406—41no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109#6382—41no—
110#6382—41no—
111#6382—41no—
112#6381—22no—
113#6381—22no—
114#6381—41no—
115C4⋊(C7⋊C8)—22no—
116C4⋊(C7⋊C8)—22no—
117C4⋊(C7⋊C8)—22no—
118C4×C7⋊C8—22no—
119#2570—71no—
120#2357D711yescharacteristic
121#2093—71no—
122#2092—71no—
123#2065—71no—
124#2065—71no—
125#1949—71no—
126#1949—71no—
127#1948—71no—
128#1939—71no—
129#1921—71no—
130#1918—71no—
131#1846—71no—
132#1804—71no—
133C4×#102—71no—
134C24×C7—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4C2×Q812yes—
141C22×C4×C7—21no—
142C22×C4×C7—24no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—81no—
146C22×C4×C7—81no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4C2×Q812yes—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—42no—
168C7×C22⋊C4—42no—
169C7×C22⋊C4—81no—
170C7×C22⋊C4—81no—
171C42×C7—21no—
172C42×C7—21no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—81no—
176C42×C7—81no—
177C42×C7C2×Q812yes—
178C42×C7#4112yes—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—42no—
188C4⋊Dic7—81no—
189Dic7⋊C4—42no—
190Dic7⋊C4—81no—
191C4×Dic7—81no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8#4112yes—
203C2×C7⋊C8#4112yes—
204C2×C4×D4C2×D711yescharacteristic
205C2×#107—71no—
206C2×#107—71no—
207C2×#107—71no—
208C2×#106—71no—
209C2×#106—71no—
210C2×#106—71no—
211C2×C4⋊C8—71no—
212C2×C4⋊C8—71no—
213C2×C4⋊C8—71no—
214C2×#102—71no—
215C2×#102—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×C4×C8—71no—
219#400—141no—
220#398—141no—
221#390C2×D711yescharacteristic
222#390C2×D711yescharacteristic
223#389—71no—
224#389—71no—
225#388—21no—
226#388—21no—
227#387—141no—
228#387—141no—
229#384—71no—
230#382—71no—
231#382—71no—
232#382C2×D711yescharacteristic
233#381—21no—
234C4×C4⋊C4—71no—
235C4×C4⋊C4C2×D711yescharacteristic
236C4×C4⋊C4C2×D711yescharacteristic
237C4×C4⋊C4C2×D711yescharacteristic
238C4×C22⋊C4—21no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#336—71no—
246#336—71no—
247C23×C7—22no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7C22×D411yescharacteristic
255C7×D4—22no—
256C7×D4—22no—
257C7×D4—41no—
258C7×D4—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—22no—
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×C7—21no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—84no—
284C2×C4×C7C22⋊Q812yes—
285C2×C4×C7C22⋊Q812yes—
286C2×C4×C7#13512yes—
287C2×C4×C7C22×Q811yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C72−1+411yescharacteristic
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—
304C2×Dic7—41no—
305C7⋊C8—22no—
306C7⋊C8—22no—
307C7⋊C8—41no—
308C7⋊C8—41no—
309C22×D4C22×D711yescharacteristic
310C23×C4—21no—
311C22×C8—71no—
312C22×C8—71no—
313C22×C8—71no—
314C22×C8—71no—
315C4×D4—22no—
316C4×D4Dic1414yes—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—71no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4C22⋊D711yescharacteristic
326C2×C4⋊C4C22⋊D711yescharacteristic
327C2×C4⋊C4C22×D711yescharacteristic
328C2×C4⋊C4C22×D711yescharacteristic
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—141no—
338C2×C42C22×D711yescharacteristic
339C2×C42C22×D711yescharacteristic
340C2×C42C22×D711yescharacteristic
341#107—142no—
342#107—142no—
343#107—281no—
344#106—281no—
345#106—281no—
346#106—281no—
347C4⋊C8—142no—
348C4⋊C8—142no—
349C4⋊C8—142no—
350#102—142no—
351#102—142no—
352#102—281no—
353#102—281no—
354C4×C8—142no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—141no—
360#95—141no—
361#95—141no—
362#95—141no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95C22⋊D711yescharacteristic
367#95C22⋊D711yescharacteristic
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7C2×C22⋊Q811yescharacteristic
376C22×C7#54711yescharacteristic
377C22×C7#55011yescharacteristic
378C22×C7C2×#13511yescharacteristic
379C22×C7#57511yescharacteristic
380C22×C7#57611yescharacteristic
381C22×C7#57711yescharacteristic
382C4×C7—22no—
383C4×C7—22no—
384C4×C7—22no—
385C4×C7—42no—
386C4×C7—41no—
387C4×C7—41no—
388C4×C7—41no—
389C4×C7—81no—
390C4×C7—81no—
391C4×C7#53311yescharacteristic
392C4×C7#54411yescharacteristic
393C4×C7#55211yescharacteristic
394C4×C7#55311yescharacteristic
395Dic7—81no—
396Dic7—81no—
397Dic7—81no—
398Dic7—81no—
399C24—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4C2×Dic1412yes—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4C2×C22⋊D711yescharacteristic
417C22×C4C2×C22⋊D711yescharacteristic
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C2×C22⋊D711yescharacteristic
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423C2×C8—72no—
424C2×C8—72no—
425C2×C8—141no—
426C2×C8—141no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—141no—
443C4⋊C4—141no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—282no—
451C4⋊C4—282no—
452C4⋊C4—561no—
453C4⋊C4—561no—
454C4⋊C4C2×Dic1412yes—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—81no—
465C42—81no—
466C42—561no—
467C42C2×Dic1412yes—
468C42#115712yes—
469C2×C7—82no—
470C2×C7#244111yescharacteristic
471C2×C7#307611yescharacteristic
472C2×C7#308111yescharacteristic
473C2×C7#333711yescharacteristic
474C2×C7#335511yescharacteristic
475C2×C7#339611yescharacteristic
476C2×C7#340111yescharacteristic
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C22×C22⋊D711yescharacteristic
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
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—84no—
514C2×C4—281no—
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#648712yes—
527C2×C4#648712yes—
528C2×C4#651312yes—
529C2×C4C22×Dic1411yescharacteristic
530C2×C4C2×#115711yescharacteristic
531C2×C4D7×#4111yescharacteristic
532C2×C4D7×#4111yescharacteristic
533C2×C4#655311yescharacteristic
534C2×C4#655411yescharacteristic
535C8—142no—
536C8—142no—
537C8—281no—
538C8—281no—
539C7#2609411yescharacteristic, complemented, Sylow
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#648711yescharacteristic
548C22C2×#651311yescharacteristic
549C22#9051311yescharacteristic
550C22#9051511yescharacteristic
551C22#9052511yescharacteristic
552C22#9052611yescharacteristic
553C22#9052711yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—561no—
564C4—561no—
565C4—561no—
566C4—561no—
567C4#9020811yescharacteristic
568C4#9021011yescharacteristic
569C4#9022011yescharacteristic
570C4#9022111yescharacteristic
571C2—82no—
572C2#1173969411yescharacteristic
573C2#1173995211yescharacteristic
574C2#1173998411yescharacteristic
575C2#1174105111yescharacteristic
576C2#1174105811yescharacteristic
577C2#1174115711yescharacteristic
578C2#1174116511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2357C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2357D7
hypocenterC7#26094
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #11731168C2semidirect
9. #11727303C2semidirect
10. #11727294C2semidirect
11. #11727238C2semidirect
12. #11727222C2semidirect
13. #11727221C2semidirect
14. #11727215C2semidirect
15. #11727212C2semidirect
539. C7#26094semidirect

Maximal subgroups

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

subgroupindex
C7×#23572
#117311682
C4×#64062
#117293592
#117274092
#117274082
#117274062
#117273032
#117272942
#117272382
#117272222
#117272212
#117272152
#117272122
#260947