Finite Group Databasebeta

← #33158592

Subgroups — #33158592

Order 18688.

690 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
1#22613025C211yesmaximal, characteristic, complemented
2#22611319C211yesmaximal, characteristic
3#22611269C211yesmaximal, characteristic, complemented
4#22611266C211yesmaximal, characteristic, complemented
5#22611077C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610671C211yesmaximal, characteristic, complemented
8#22610555C211yesmaximal, characteristic
9#22610548C211yesmaximal, characteristic
10#22610506C211yesmaximal, characteristic, complemented
11#22610499C211yesmaximal, characteristic, complemented
12#22610497C211yesmaximal, characteristic, complemented
13#22610192C211yesmaximal, characteristic
14#22609756C211yesmaximal, characteristic
15C73×#2454C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17C2×#14556524C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556502C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988485C2211yescharacteristic
29#17988482—21no—
30#17988469—21no—
31#17988468—21no—
32C4×C4⋊Dic73C2211yescharacteristic
33C4×C4⋊Dic73C2211yescharacteristic
34C4×C4⋊Dic73C2211yescharacteristic
35#17988438—21no—
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#125×C73C2211yescharacteristic
49C2×C4×D4×C73C2211yescharacteristic
50#399×C73—21no—
51#396×C73—21no—
52#390×C73C2211yescharacteristic
53#390×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#385×C73—21no—
57#380×C73—21no—
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556583—22no—
64#14556583—22no—
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×#11924007—21no—
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×C4×Dic73—21no—
74C2×C4×Dic73—21no—
75C2×C4×Dic73—21no—
76#14556524—22no—
77#14556524—41no—
78#14556522—22no—
79#14556522—22no—
80C4⋊#10596831—41no—
81#14556512—22no—
82#14556512—22no—
83#14556512—41no—
84#14556512Q814yes—
85#14556502—41no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×D4×C73C2311yescharacteristic
92C23×C4×C73—21no—
93#125×C73—22no—
94#125×C73—41no—
95C4×D4×C73—22no—
96C4×D4×C73—41no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73D411yescharacteristic
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C42×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114#95×C73—21no—
115#95×C73—21no—
116#95×C73D411yescharacteristic
117#95×C73D411yescharacteristic
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C2×#10596831—21no—
123C2×#10596831—21no—
124C2×#10596831—21no—
125C2×#10596831—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831C2×Q812yes—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73C2×Q812yes—
140C4⋊Dic73#4112yes—
141C4⋊Dic73#4112yes—
142#11924007—24no—
143#11924007—24no—
144#11924007—42no—
145#11924007—81no—
146C4×Dic73—42no—
147C4×Dic73—81no—
148C4×Dic73—81no—
149C24×C73—21no—
150C2×D4×C73—21no—
151C2×D4×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73C2×Q812yes—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—42no—
160C22×C4×C73—42no—
161C22×C4×C73—81no—
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2411yescharacteristic
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—41no—
175C4⋊C4×C73—41no—
176C4⋊C4×C73—42no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73—81no—
179C4⋊C4×C73—81no—
180C4⋊C4×C73#4112yes—
181C22⋊C4×C73—42no—
182C22⋊C4×C73—42no—
183C22⋊C4×C73—81no—
184C22⋊C4×C73—81no—
185C22⋊C4×C73—81no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C23×C73—22no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73C22×D411yescharacteristic
200D4×C73—22no—
201D4×C73—22no—
202D4×C73—41no—
203D4×C73—41no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—22no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—84no—
232C2×C4×C73C22×Q811yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73H(5,𝔽2)11yescharacteristic
237C2×C4×C732−1+411yescharacteristic
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—22no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250#10596831—22no—
251#10596831—41no—
252#10596831—41no—
253#10596831—41no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73#52511yescharacteristic
262C22×C73#54611yescharacteristic
263C22×C73D4×Q811yescharacteristic
264C22×C73#57511yescharacteristic
265C22×C73#57511yescharacteristic
266C22×C73#57611yescharacteristic
267C22×C73#57711yescharacteristic
268C4×C73—41no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—42no—
275C4×C73—81no—
276C4×C73—81no—
277C4×C73#53311yescharacteristic
278C4×C73#54211yescharacteristic
279C4×C73#55411yescharacteristic
280C4×C73#55611yescharacteristic
281Dic73—42no—
282Dic73—42no—
283Dic73—42no—
284Dic73—81no—
285#27027—731nomaximal, Sylow
286C2×C73—82no—
287C2×C73#246511yescharacteristic
288C2×C73#311111yescharacteristic
289C2×C73#311911yescharacteristic
290C2×C73#330911yescharacteristic
291C2×C73#332311yescharacteristic
292C2×C73#338811yescharacteristic
293C2×C73#338911yescharacteristic
294#2454D7311yescharacteristic
295#2370—731no—
296#2067—731no—
297#2059—731no—
298#2059—731no—
299#2025—731no—
300#1989—731no—
301#1944—731no—
302#1926—731no—
303#1925—731no—
304#1919—731no—
305#1918—731no—
306#1917—731no—
307#1864—731no—
308#1762—731no—
309C73#2702711yescharacteristic, complemented, Sylow
310C2×#125—731no—
311C2×#125C2×D7311yescharacteristic
312C2×C4×D4C2×D7311yescharacteristic
313C2×#107—731no—
314C2×#107—731no—
315C2×#106—731no—
316C2×#106—731no—
317C2×C4⋊C8—731no—
318C2×#102—731no—
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#97—731no—
323#399—21no—
324#396—21no—
325#390C2×D7311yescharacteristic
326#390C2×D7311yescharacteristic
327#389—731no—
328#389—731no—
329#389—1461no—
330#389C2×D7311yescharacteristic
331#389C2×D7311yescharacteristic
332#387—1461no—
333#387—1461no—
334#385—21no—
335#380—21no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4—1461no—
340C4×C4⋊C4C2×D7311yescharacteristic
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#336—731no—
350#336—731no—
351#336—731no—
352#336—731no—
353C22×D4C22×D7311yescharacteristic
354C23×C4—21no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8—731no—
358C22×C8—731no—
359#125—22no—
360#125—41no—
361#125—1462no—
362#125—1462no—
363C4×D4—22no—
364C4×D4—41no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22⋊D7311yescharacteristic
378C2×C4⋊C4C22⋊D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C42—1461no—
388C2×C42—1461no—
389C2×C42—1461no—
390C2×C42C22×D7311yescharacteristic
391C2×C42C22×D7311yescharacteristic
392C2×C42C22×D7311yescharacteristic
393#107—2921no—
394#107—1462no—
395#106—1462no—
396#106—1462no—
397C4⋊C8—2921no—
398#102—2921no—
399#102—734no—
400#102—1462no—
401#102—1462no—
402#97—2921no—
403#95—21no—
404#95—21no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95C22⋊D7311yescharacteristic
411#95C22⋊D7311yescharacteristic
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4Q8×D7312yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4C2×C22⋊D7311yescharacteristic
430C22×C4C2×C22⋊D7311yescharacteristic
431C22×C4C2×C22⋊D7311yescharacteristic
432C22×C4C2×C22⋊D7311yescharacteristic
433C22×C4C2×C22⋊D7311yescharacteristic
434C22×C4C2×C22⋊D7311yescharacteristic
435C22×C4C23×D7311yescharacteristic
436C2×C8—732no—
437C2×C8—1461no—
438C2×C8—1461no—
439C2×C8—1461no—
440C2×C8—1461no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—2922no—
460C4⋊C4—5841no—
461C4⋊C4—732no—
462C4⋊C4—732no—
463C4⋊C4—732no—
464C4⋊C4—1464no—
465C4⋊C4—1464no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4#1192402712yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—2922no—
484C42—5841no—
485C42—5841no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C22×C22⋊D7311yescharacteristic
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
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—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4C2×Q8×D7311yescharacteristic
539C2×C4#41×D7311yescharacteristic
540C2×C4#41×D7311yescharacteristic
541C2×C4C2×#1192402711yescharacteristic
542C2×C4#1455663611yescharacteristic
543C2×C4#1455663911yescharacteristic
544C8—2921no—
545C8—2921no—
546C8—2921no—
547C8—1462no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#1798909411yescharacteristic
556C22#1798923111yescharacteristic
557C22Q8×C22⋊D7311yescharacteristic
558C22#1798938511yescharacteristic
559C22#1798938511yescharacteristic
560C22#1798939011yescharacteristic
561C22#1798939811yescharacteristic
562C4—41no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—2922no—
572C4—2922no—
573C4—2922no—
574C4—5841no—
575C4#1798913911yescharacteristic
576C4#1798920711yescharacteristic
577C4#1798929111yescharacteristic
578C4#1798930811yescharacteristic
579C2—82no—
580C2#2261394811yescharacteristic
581C2#2262003411yescharacteristic
582C2#2262012411yescharacteristic
583C2#2262193711yescharacteristic
584C2#2262214711yescharacteristic
585C2#2262287611yescharacteristic
586C2#2262288511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2454C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2454D73
hypocenterC73#27027
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. #22613025C2semidirect
3. #22611269C2semidirect
4. #22611266C2semidirect
6. #22610889C2semidirect
7. #22610671C2semidirect
10. #22610506C2semidirect
11. #22610499C2semidirect
12. #22610497C2semidirect
309. C73#27027semidirect

Maximal subgroups

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

subgroupindex
#226130252
#226113192
#226112692
#226112662
#226110772
#226108892
#226106712
#226105552
#226105482
#226105062
#226104992
#226104972
#226101922
#226097562
C73×#24542
#2702773