Finite Group Databasebeta

← #33098246

Subgroups — #33098246

Order 18688.

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
1C2×#17988867C211yesmaximal, characteristic
2#22612679C211yesmaximal, characteristic, complemented
3#22610879C211yesmaximal, characteristic, complemented
4#22610699C211yesmaximal, characteristic, complemented
5#22610666C211yesmaximal, characteristic, complemented
6#22610661C211yesmaximal, characteristic, complemented
7#22610631C211yesmaximal, characteristic
8#22610617C211yesmaximal, characteristic
9#22610555C211yesmaximal, characteristic
10#22610548C211yesmaximal, characteristic
11#22610520C211yesmaximal, characteristic, complemented
12#22610499C211yesmaximal, characteristic, complemented
13#22609941C211yesmaximal, characteristic
14#22609788C211yesmaximal, characteristic, complemented
15C73×#2319C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17#17988867C418yescomplemented
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988491—21no—
31#17988491—21no—
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988470—21no—
35#17988470—21no—
36#17988453C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×C4×D4×C73C2211yescharacteristic
50#390×C73C2211yescharacteristic
51#390×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#388×C73—21no—
56#381×C73—21no—
57#380×C73—21no—
58C4×C4⋊C4×C73C2211yescharacteristic
59#376×C73C2211yescharacteristic
60C22×Dic146—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556591—22no—
66#14556591C2×C414yes—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C4×Dic73—21no—
75C2×C4×Dic73—21no—
76#14556524—41no—
77#14556524C2×C414yes—
78#14556524C2×C414yes—
79#14556522—41no—
80C4⋊#10596831—22no—
81C4⋊#10596831C2×C414yes—
82#14556512—41no—
83#14556512—41no—
84#14556512C2×C414yes—
85#14556512C2×C414yes—
86#14556502—41no—
87C4×#10596831—41no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94C22×D4×C73C2311yescharacteristic
95C23×C4×C73—21no—
96C4×D4×C73—22no—
97C4×D4×C73C2×C414yes—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C42×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73D411yescharacteristic
116#95×C73D411yescharacteristic
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C2×Dic146—24no—
120C2×Dic146—81no—
121C2×Dic146—81no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831C22×C412yes—
137C2×#10596831C22×C412yes—
138C4⋊Dic73—24no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73C22×C412yes—
146C4⋊Dic73C22×C412yes—
147#11924007—81no—
148#11924007—81no—
149C4×Dic73—81no—
150C4×Dic73—81no—
151C24×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73C22×C412yes—
158C22×C4×C73—21no—
159C22×C4×C73—24no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—81no—
163C22×C4×C73—81no—
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2411yescharacteristic
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—81no—
177C4⋊C4×C73—81no—
178C4⋊C4×C73C22×C412yes—
179C22⋊C4×C73—24no—
180C22⋊C4×C73—81no—
181C22⋊C4×C73—81no—
182C22⋊C4×C73—81no—
183C22⋊C4×C73—81no—
184C42×C73—21no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—81no—
189C42×C73C22×C412yes—
190C42×C73#4112yes—
191C23×C73—22no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73C22×D411yescharacteristic
199D4×C73—22no—
200D4×C73—22no—
201D4×C73—41no—
202D4×C73—41no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—22no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
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—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—84no—
229C2×C4×C73C4×D412yes—
230C2×C4×C73C4×D412yes—
231C2×C4×C73C23×C411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73H(5,𝔽2)11yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×C4×C732−1+411yescharacteristic
236C2×C4×C732−1+411yescharacteristic
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249Dic146—84no—
250#10596831—22no—
251#10596831—22no—
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×C73C2×C4×D411yescharacteristic
262C22×C73#53611yescharacteristic
263C22×C73#53711yescharacteristic
264C22×C73#57611yescharacteristic
265C22×C73#57611yescharacteristic
266C22×C73#57711yescharacteristic
267C22×C73#57811yescharacteristic
268C4×C73—22no—
269C4×C73—22no—
270C4×C73—42no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73—81no—
276C4×C73—81no—
277C4×C73#51911yescharacteristic
278C4×C73#52011yescharacteristic
279C4×C73#54111yescharacteristic
280C4×C73#54211yescharacteristic
281Dic73—42no—
282Dic73—42no—
283Dic73—81no—
284Dic73—81no—
285#22127—731nomaximal, Sylow
286C2×C73—82no—
287C2×C73#232311yescharacteristic
288C2×C73#294711yescharacteristic
289C2×C73#294811yescharacteristic
290C2×C73#324111yescharacteristic
291C2×C73#324311yescharacteristic
292C2×C73#325711yescharacteristic
293C2×C73#326211yescharacteristic
294C2×#474—731no—
295#2331—731no—
296#2319D7311yescharacteristic
297#1987—731no—
298#1949—731no—
299#1943—731no—
300#1942—731no—
301#1936—731no—
302#1934—731no—
303#1926—731no—
304#1925—731no—
305#1921—731no—
306#1918—731no—
307#1804—731no—
308#1768—731no—
309C73#2212711yescharacteristic, complemented, Sylow
310C2×C4⋊Q8—731no—
311C2×C4×D4C2×D7311yescharacteristic
312#474—738no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#107—731no—
316C2×#106—731no—
317C2×C4⋊C8—731no—
318C2×C4⋊C8—731no—
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#102—731no—
323C2×#97—731no—
324C2×C4×C8—731no—
325#391—1461no—
326#391—1461no—
327#390C2×D7311yescharacteristic
328#390C2×D7311yescharacteristic
329#389—731no—
330#389—731no—
331#389C2×D7311yescharacteristic
332#389C2×D7311yescharacteristic
333#388—21no—
334#388—21no—
335#387—1461no—
336#387—1461no—
337#382—731no—
338#381—21no—
339#380—21no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4C2×D7311yescharacteristic
343#376C2×D7311yescharacteristic
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#336—731no—
353#336—731no—
354C22×Q8—1461no—
355C22×D4C22×D7311yescharacteristic
356C23×C4—21no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8—731no—
361C4⋊Q8—734no—
362C4⋊Q8—1462no—
363C4×D4—22no—
364C4×D4C4×D7314yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C42—21no—
383C2×C42—1461no—
384C2×C42—1461no—
385C2×C42C22×D7311yescharacteristic
386C2×C42C22×D7311yescharacteristic
387C2×C42C22×D7311yescharacteristic
388#107—2921no—
389#107—734no—
390#107—734no—
391#106—2921no—
392C4⋊C8—734no—
393C4⋊C8—1462no—
394#102—2921no—
395#102—2921no—
396#102—734no—
397#102—734no—
398#97—2921no—
399C4×C8—2921no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1461no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95C22⋊D7311yescharacteristic
410#95C22⋊D7311yescharacteristic
411#95C22⋊D7311yescharacteristic
412#95C22⋊D7311yescharacteristic
413C24—21no—
414C2×Q8—5841no—
415C2×Q8—5841no—
416C2×Q8—1464no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D7312yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4C2×C22⋊D7311yescharacteristic
434C22×C4C2×C22⋊D7311yescharacteristic
435C22×C4C2×C22⋊D7311yescharacteristic
436C22×C4C2×C22⋊D7311yescharacteristic
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—732no—
441C2×C8—732no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—5841no—
460C4⋊C4—5841no—
461C4⋊C4—732no—
462C4⋊C4—732no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1464no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4C2×C4×D7312yes—
471C22⋊C4—24no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—5841no—
482C42—5841no—
483C42C2×C4×D7312yes—
484C42#1192402812yes—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C22×C22⋊D7311yescharacteristic
493Q8—5844no—
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—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—84no—
524C2×C4—2921no—
525C2×C4—2921no—
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—1462no—
535C2×C4—1462no—
536C2×C4C4×C22⋊D7312yes—
537C2×C4C4×C22⋊D7312yes—
538C2×C4C22×C4×D7311yescharacteristic
539C2×C4C2×#1192402811yescharacteristic
540C2×C4#1455663711yescharacteristic
541C2×C4#1455663811yescharacteristic
542C2×C4#1455663811yescharacteristic
543C2×C4#1455663811yescharacteristic
544C8—2921no—
545C8—2921no—
546C8—1462no—
547C8—1462no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×C4×C22⋊D7311yescharacteristic
556C22#1798915811yescharacteristic
557C22#1798916511yescharacteristic
558C22#1798939011yescharacteristic
559C22#1798939011yescharacteristic
560C22#1798939811yescharacteristic
561C22#1798940211yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—2922no—
572C4—2922no—
573C4—5841no—
574C4—5841no—
575C4#1798906411yescharacteristic
576C4#1798906711yescharacteristic
577C4#1798919911yescharacteristic
578C4#1798920811yescharacteristic
579C2—82no—
580C2#2261261311yescharacteristic
581C2#2261877011yescharacteristic
582C2#2261878111yescharacteristic
583C2#2262117811yescharacteristic
584C2#2262120811yescharacteristic
585C2#2262133611yescharacteristic
586C2#2262138711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2319C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#2319D73
hypocenterC73#22127
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #22612679C2semidirect
3. #22610879C2semidirect
4. #22610699C2semidirect
5. #22610666C2semidirect
6. #22610661C2semidirect
11. #22610520C2semidirect
12. #22610499C2semidirect
14. #22609788C2semidirect
17. #17988867C4semidirect
309. C73#22127semidirect

Maximal subgroups

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

subgroupindex
C2×#179888672
#226126792
#226108792
#226106992
#226106662
#226106612
#226106312
#226106172
#226105552
#226105482
#226105202
#226104992
#226099412
#226097882
C73×#23192
#2212773