Finite Group Databasebeta

← #13365998

Subgroups — #13365998

Order 1792.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2319C211yesmaximal, characteristic
2C2×#89780C211yesmaximal, characteristic
3#11730836C211yesmaximal, characteristic, complemented
4#11729359C211yesmaximal, characteristic
5#11727399C211yesmaximal, characteristic
6#11727398C211yesmaximal, characteristic
7#11727391C211yesmaximal, characteristic
8#11727390C211yesmaximal, characteristic
9#11727375C211yesmaximal, characteristic, complemented
10#11727372C211yesmaximal, characteristic, complemented
11#11727356C211yesmaximal, characteristic, complemented
12#11727355C211yesmaximal, characteristic, complemented
13#11727244C211yesmaximal, characteristic, complemented
14#11727222C211yesmaximal, characteristic, complemented
15#11727190C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×#6432C2211yescharacteristic
18C7×#390C2211yescharacteristic
19C7×#390C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#388—21no—
23C7×#388—21no—
24C7×#381—21no—
25C7×#380—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#376C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89780C418yescomplemented
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89708C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×#6377C2211yescharacteristic
45C2×C4×C7⋊C8C2211yescharacteristic
46#89426—21no—
47#89426—21no—
48#89418—21no—
49#89418—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22123—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C22×Dic14—21no—
64C4×C7×D4—22no—
65C4×C7×D4C2×C414yes—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
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—
88C2×C4×Dic7—21no—
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93#6432—22no—
94#6432C2×C414yes—
95C7×#95—21no—
96C7×#95—21no—
97C7×#95—21no—
98C7×#95D411yescharacteristic
99C7×#95D411yescharacteristic
100C7×#95D411yescharacteristic
101C7×#95D411yescharacteristic
102#6406—41no—
103#6406—41no—
104#6406C2×C414yes—
105#6406C2×C414yes—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112#6382—41no—
113#6382C2×C414yes—
114#6382C2×C414yes—
115#6381—41no—
116C4⋊(C7⋊C8)—22no—
117C4⋊(C7⋊C8)C2×C414yes—
118#6377—41no—
119C4×C7⋊C8—41no—
120C2×#476—71no—
121#2331—71no—
122#2319D711yescharacteristic
123#1988—71no—
124#1949—71no—
125#1944—71no—
126#1941—71no—
127#1936—71no—
128#1934—71no—
129#1926—71no—
130#1925—71no—
131#1920—71no—
132#1919—71no—
133#1804—71no—
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×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C2×Dic14—24no—
160C2×Dic14—81no—
161C2×Dic14—81no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—81no—
168C7×C4⋊C4—81no—
169C7×C4⋊C4C22×C412yes—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—81no—
172C7×C22⋊C4—81no—
173C7×C22⋊C4—81no—
174C7×C22⋊C4—81no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—81no—
180C42×C7C22×C412yes—
181C42×C7#4112yes—
182C4⋊Dic7—24no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7C22×C412yes—
190C4⋊Dic7C22×C412yes—
191Dic7⋊C4—81no—
192Dic7⋊C4—81no—
193C4×Dic7—81no—
194C4×Dic7—81no—
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—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8C22×C412yes—
206C2×C7⋊C8C22×C412yes—
207C2×C4⋊Q8—71no—
208C2×C4×D4C2×D711yescharacteristic
209#476—78no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×#102—71no—
219C2×#102—71no—
220C2×#97—71no—
221C2×C4×C8—71no—
222#391—141no—
223#391—141no—
224#390C2×D711yescharacteristic
225#390C2×D711yescharacteristic
226#389—71no—
227#389—71no—
228#389C2×D711yescharacteristic
229#389C2×D711yescharacteristic
230#388—21no—
231#388—21no—
232#387—141no—
233#387—141no—
234#382—71no—
235#381—21no—
236#380—21no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4C2×D711yescharacteristic
240#376C2×D711yescharacteristic
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#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—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—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—84no—
289C2×C4×C7C4×D412yes—
290C2×C4×C7C4×D412yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7H(5,𝔽2)11yescharacteristic
294C2×C4×C72−1+411yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—22no—
298C2×Dic7—22no—
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—
309Dic14—84no—
310C7⋊C8—22no—
311C7⋊C8—22no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C22×Q8—141no—
315C22×D4C22×D711yescharacteristic
316C23×C4—21no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8—71no—
321C4⋊Q8—74no—
322C4⋊Q8—142no—
323C4×D4—22no—
324C4×D4C4×D714yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C42—21no—
343C2×C42—141no—
344C2×C42—141no—
345C2×C42C22×D711yescharacteristic
346C2×C42C22×D711yescharacteristic
347C2×C42C22×D711yescharacteristic
348#107—281no—
349#106—74no—
350#106—74no—
351#106—281no—
352C4⋊C8—74no—
353C4⋊C8—142no—
354#102—74no—
355#102—74no—
356#102—281no—
357#102—281no—
358#97—281no—
359C4×C8—281no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95C22⋊D711yescharacteristic
370#95C22⋊D711yescharacteristic
371#95C22⋊D711yescharacteristic
372#95C22⋊D711yescharacteristic
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7C2×C4×D411yescharacteristic
381C22×C7#53611yescharacteristic
382C22×C7#53711yescharacteristic
383C22×C7#57611yescharacteristic
384C22×C7#57711yescharacteristic
385C22×C7#57711yescharacteristic
386C22×C7#57711yescharacteristic
387C4×C7—22no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7#51911yescharacteristic
397C4×C7#52011yescharacteristic
398C4×C7#54111yescharacteristic
399C4×C7#54211yescharacteristic
400Dic7—42no—
401Dic7—42no—
402Dic7—81no—
403Dic7—81no—
404C24—21no—
405C2×Q8—144no—
406C2×Q8—561no—
407C2×Q8—561no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×C4×D712yes—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431C2×C8—72no—
432C2×C8—72no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—72no—
449C4⋊C4—72no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—144no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—561no—
460C4⋊C4—561no—
461C4⋊C4C2×C4×D712yes—
462C22⋊C4—24no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—81no—
472C42—561no—
473C42—561no—
474C42C2×C4×D712yes—
475C42#115712yes—
476C2×C7—82no—
477C2×C7#232311yescharacteristic
478C2×C7#294611yescharacteristic
479C2×C7#294911yescharacteristic
480C2×C7#324111yescharacteristic
481C2×C7#324411yescharacteristic
482C2×C7#325711yescharacteristic
483C2×C7#326311yescharacteristic
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C22×C22⋊D711yescharacteristic
492Q8—564no—
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
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—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4—142no—
524C2×C4—142no—
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×C4C4×C22⋊D712yes—
536C2×C4C4×C22⋊D712yes—
537C2×C4C22×C4×D711yescharacteristic
538C2×C4C2×#115711yescharacteristic
539C2×C4#655311yescharacteristic
540C2×C4#655411yescharacteristic
541C2×C4#655411yescharacteristic
542C2×C4#655411yescharacteristic
543C8—142no—
544C8—142no—
545C8—281no—
546C8—281no—
547C7#2212311yescharacteristic, complemented, Sylow
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×C4×C22⋊D711yescharacteristic
556C22#9051611yescharacteristic
557C22#9052411yescharacteristic
558C22#9052611yescharacteristic
559C22#9052711yescharacteristic
560C22#9052711yescharacteristic
561C22#9052811yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—282no—
572C4—282no—
573C4—561no—
574C4—561no—
575C4#9020711yescharacteristic
576C4#9020911yescharacteristic
577C4#9021411yescharacteristic
578C4#9022511yescharacteristic
579C2—82no—
580C2#1173968011yescharacteristic
581C2#1173998811yescharacteristic
582C2#1174000911yescharacteristic
583C2#1174087511yescharacteristic
584C2#1174088211yescharacteristic
585C2#1174090011yescharacteristic
586C2#1174091411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2319C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#2319D7
hypocenterC7#22123
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
3. #11730836C2semidirect
9. #11727375C2semidirect
10. #11727372C2semidirect
11. #11727356C2semidirect
12. #11727355C2semidirect
13. #11727244C2semidirect
14. #11727222C2semidirect
15. #11727190C2semidirect
32. #89780C4semidirect
547. C7#22123semidirect

Maximal subgroups

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

subgroupindex
C7×#23192
C2×#897802
#117308362
#117293592
#117273992
#117273982
#117273912
#117273902
#117273752
#117273722
#117273562
#117273552
#117272442
#117272222
#117271902
#221237