Finite Group Databasebeta

← #13365979

Subgroups — #13365979

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739770C211yesmaximal, characteristic
2C2×#89781C211yesmaximal, characteristic
3#11731767C211yesmaximal, characteristic, complemented
4C7×#1802C211yesmaximal, characteristic
5#11728215C211yesmaximal, characteristic, complemented
6#11728196C211yesmaximal, characteristic, complemented
7#11727942C211yesmaximal, characteristic
8#11727940C211yesmaximal, characteristic
9#11727928C211yesmaximal, characteristic
10#11727928C211yesmaximal, characteristic
11#11727322C211yesmaximal, characteristic, complemented
12#11726614C211yesmaximal, characteristic, complemented
13#11726612C211yesmaximal, characteristic, complemented
14#11726573C211yesmaximal, characteristic, complemented
15#11726569C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×#6433C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C7×#384C2211yescharacteristic
22#89993C2211yescharacteristic
23#89993C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26#89984—21no—
27#89983—21no—
28C2×#6389C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89781C418yescomplemented
31#89757C2211yescharacteristic
32#89757C2211yescharacteristic
33#89750—21no—
34#89750—21no—
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39#89744—21no—
40#89744—21no—
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89723—21no—
46#89723—21no—
47C4×C4⋊Dic7C2211yescharacteristic
48#89704C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22122—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C4×C7×D4—22no—
66C4×C7×D4C2×C414yes—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C42×C7C2311yescharacteristic
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6433—22no—
96#6433C2×C414yes—
97C7×C4⋊C8—22no—
98C7×#102—41no—
99C7×#102—41no—
100#6406C2×C414yes—
101#6406C2×C414yes—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95D411yescharacteristic
110C7⋊#95D411yescharacteristic
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113#6389—41no—
114C8×Dic7—41no—
115#6382—41no—
116#6382—41no—
117#6382C2×C414yes—
118#6381C2×C414yes—
119C4⋊(C7⋊C8)C2×C414yes—
120C2×#478—71no—
121#2330—71no—
122#2319—71no—
123#1989—71no—
124#1948—71no—
125#1943—71no—
126#1941—71no—
127#1936—71no—
128#1934—71no—
129#1926—71no—
130#1926—71no—
131#1920—71no—
132#1918—71no—
133#1802D711yescharacteristic
134#1768—71no—
135C24×C7—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C22×C412yes—
142C22×C4×C7—21no—
143C22×C4×C7—24no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—81no—
147C22×C4×C7—81no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—81no—
171C7×C4⋊C4—81no—
172C7×C4⋊C4C22×C412yes—
173C7×C22⋊C4—24no—
174C42×C7C22×C412yes—
175C42×C7#4112yes—
176C22⋊Dic7—81no—
177C22⋊Dic7—81no—
178C22⋊Dic7—81no—
179C22⋊Dic7—81no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—81no—
187C4⋊Dic7—81no—
188C4⋊Dic7C22×C412yes—
189C4⋊Dic7C22×C412yes—
190Dic7⋊C4—24no—
191Dic7⋊C4—24no—
192Dic7⋊C4—81no—
193Dic7⋊C4—81no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C4×Dic7—81no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8C22×C412yes—
206C2×C7⋊C8C22×C412yes—
207C2×#125—71no—
208C2×C4×D4C2×D711yescharacteristic
209#478—78no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8C2×D711yescharacteristic
216C2×#102—71no—
217C2×#102—71no—
218C2×#102C2×D711yescharacteristic
219C2×#102C2×D711yescharacteristic
220C2×#97—71no—
221C2×C4×C8—71no—
222#390—71no—
223#390—71no—
224#389—71no—
225#389—71no—
226#389—71no—
227#389—71no—
228#389—141no—
229#389—141no—
230#388—141no—
231#388—141no—
232#387—141no—
233#387—141no—
234#384C2×D711yescharacteristic
235#381—141no—
236#380—141no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240#376—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#336—71no—
250#336—71no—
251C23×C7—22no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—41no—
262C7×D4—41no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—84no—
277C2×C4×C7C4×D412yes—
278C2×C4×C7C4×D412yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7H(5,𝔽2)11yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C7×C8—41no—
310C7×C8—41no—
311C7⋊C8—22no—
312C7⋊C8—22no—
313C22×D4C22×D711yescharacteristic
314C23×C4—21no—
315C22×C8—71no—
316C22×C8—71no—
317C22×C8C22×D711yescharacteristic
318C22×C8C22×D711yescharacteristic
319#125—74no—
320#125—142no—
321C4×D4—22no—
322C4×D4C4×D714yes—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C42—71no—
344C2×C42—71no—
345C2×C42—141no—
346C2×C42—141no—
347C2×C42—141no—
348C2×C42C22×D711yescharacteristic
349#107—74no—
350#106—74no—
351#106—281no—
352#106—281no—
353C4⋊C8—22no—
354C4⋊C8—74no—
355#102—41no—
356#102—41no—
357#102—74no—
358#102—74no—
359#97—281no—
360C4×C8—281no—
361#95—71no—
362#95—71no—
363#95—71no—
364#95—71no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7C2×C4×D411yescharacteristic
380C22×C7#53611yescharacteristic
381C22×C7#53711yescharacteristic
382C22×C7#57611yescharacteristic
383C22×C7#57711yescharacteristic
384C22×C7#57711yescharacteristic
385C22×C7#57711yescharacteristic
386C4×C7—22no—
387C4×C7—22no—
388C4×C7—42no—
389C4×C7—81no—
390C4×C7—81no—
391C4×C7#51911yescharacteristic
392C4×C7#52011yescharacteristic
393C4×C7#54111yescharacteristic
394C4×C7#54211yescharacteristic
395Dic7—42no—
396Dic7—42no—
397Dic7—41no—
398Dic7—41no—
399Dic7—41no—
400Dic7—41no—
401Dic7—81no—
402Dic7—81no—
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×C4×D712yes—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4D4×D711yescharacteristic
425C22×C4D4×D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—72no—
434C2×C8—72no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—72no—
445C4⋊C4—72no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—144no—
449C4⋊C4—144no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—561no—
457C4⋊C4—561no—
458C4⋊C4—561no—
459C4⋊C4—561no—
460C4⋊C4C2×C4×D712yes—
461C22⋊C4—24no—
462C22⋊C4—561no—
463C22⋊C4—561no—
464C22⋊C4—561no—
465C22⋊C4—561no—
466C42—141no—
467C42—141no—
468C42—141no—
469C42—141no—
470C42—561no—
471C42—561no—
472C42—561no—
473C42C2×C4×D712yes—
474C42#115912yes—
475C2×C7—82no—
476C2×C7#232311yescharacteristic
477C2×C7#294611yescharacteristic
478C2×C7#295211yescharacteristic
479C2×C7#324111yescharacteristic
480C2×C7#324411yescharacteristic
481C2×C7#325711yescharacteristic
482C2×C7#326011yescharacteristic
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×D4×D711yescharacteristic
491D4—22no—
492D4—22no—
493D4—41no—
494D4—41no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
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(C22⋊D7)⋊C412yes—
534C2×C4(C22⋊D7)⋊C412yes—
535C2×C4C22×C4×D711yescharacteristic
536C2×C4C2×#115911yescharacteristic
537C2×C4#654811yescharacteristic
538C2×C4#655111yescharacteristic
539C2×C4#655411yescharacteristic
540C2×C4#655411yescharacteristic
541C8—41no—
542C8—41no—
543C8—142no—
544C8—142no—
545C7#2212211yescharacteristic, complemented, Sylow
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×(C22⋊D7)⋊C411yescharacteristic
554C22#9029911yescharacteristic
555C22#9031911yescharacteristic
556C22#9044811yescharacteristic
557C22#9045311yescharacteristic
558C22#9046611yescharacteristic
559C22#9046911yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—281no—
566C4—281no—
567C4—282no—
568C4—281no—
569C4—281no—
570C4—282no—
571C4—561no—
572C4—561no—
573C4#9019511yescharacteristic
574C4#9020711yescharacteristic
575C4#9029411yescharacteristic
576C4#9031611yescharacteristic
577C2—82no—
578C2#1173131611yescharacteristic
579C2#1173617111yescharacteristic
580C2#1173631411yescharacteristic
581C2#1173779111yescharacteristic
582C2#1173788811yescharacteristic
583C2#1173841811yescharacteristic
584C2#1173851411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1802C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1802D7
hypocenterC7#22122
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. #11731767C2semidirect
5. #11728215C2semidirect
6. #11728196C2semidirect
11. #11727322C2semidirect
12. #11726614C2semidirect
13. #11726612C2semidirect
14. #11726573C2semidirect
15. #11726569C2semidirect
30. #89781C4semidirect
545. C7#22122semidirect

Maximal subgroups

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

subgroupindex
#117397702
C2×#897812
#117317672
C7×#18022
#117282152
#117281962
#117279422
#117279402
#117279282
#117273222
#117266142
#117266122
#117265732
#117265692
#221227