Finite Group Databasebeta

← #33088531

Subgroups — #33088531

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618448C211yesmaximal, characteristic
2C2×#17988581C211yesmaximal, characteristic
3C2×#17988569C211yesmaximal, characteristic
4C2×#17988543C211yesmaximal, characteristic, complemented
5#22610592C214yesmaximal, complemented
6#22610585C214yesmaximal, complemented
7#22609670C211yesmaximal, characteristic, complemented
8C2×#17988351C211yesmaximal, characteristic, complemented
9C73×#2911C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12C22×#11924012C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14#17988595C2212yes—
15#17988591C2212yes—
16#17988586—22no—
17#17988581—21no—
18#17988581—21no—
19#17988581—22no—
20#17988581—22no—
21C2×#14556514C2212yes—
22C2×#14556511C2212yes—
23#17988569C418yescomplemented
24C2×#14556509C2212yes—
25C2×#14556507C2212yes—
26#17988543—22no—
27#17988543C2214yes—
28C2×#14556502—21no—
29C2×C4×#10596831—21no—
30#17988356C2214yes—
31#17988351C2214yes—
32C22×M16×C73C2211yescharacteristic
33C2×#117×C73C2211yescharacteristic
34C2×#117×C73C2211yescharacteristic
35#428×C73—22no—
36#428×C73C2214yes—
37#427×C73—22no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028—22no—
43C2×#11924028—22no—
44C2×#11924028—22no—
45C2×#11924028C2×C414yes—
46C22×Dic146C2311yescharacteristic
47C22×D292C2311yescharacteristic
48C22×C22⋊D73—21no—
49C22×C4×D73—21no—
50C2×#11924012—21no—
51C2×#11924012—21no—
52C2×#11924012—22no—
53C2×#11924012—22no—
54C2×#11924012—22no—
55C2×#11924012C2×C414yes—
56C2×#11924012C2312yes—
57C2×#11924012C2312yes—
58C22×#10596831—21no—
59C22×#10596831—21no—
60C22×#10596831—22no—
61C22×#10596831—22no—
62C22×#10596831D411yescharacteristic
63C22×#10596831D411yescharacteristic
64C22×#10596831D412yes—
65C22×#10596831C2312yes—
66#14556514—22no—
67#14556514—22no—
68#14556514—24no—
69#14556511—22no—
70#14556511—22no—
71#14556511—24no—
72#14556509C2×C418yescomplemented
73#14556507C2×C418yescomplemented
74#14556502—24no—
75C4×#10596831—24no—
76C23×C4×C73C2311yescharacteristic
77C2×M16×C73—21no—
78C2×M16×C73—22no—
79C2×M16×C73C2×C414yes—
80C2×M16×C73C2312yes—
81C22×C8×C73—21no—
82#117×C73—21no—
83#117×C73—21no—
84#117×C73—21no—
85#117×C73—21no—
86#117×C73—21no—
87#117×C73—21no—
88#117×C73D412yes—
89#117×C73C2312yes—
90C2×C4⋊C4×C73D412yes—
91C2×C4⋊C4×C73C2312yes—
92C2×C22⋊C4×C73—21no—
93C2×C22⋊C4×C73—21no—
94C2×C42×C73—21no—
95C2×C42×C73—21no—
96#107×C73—24no—
97#106×C73—24no—
98C23×D73—21no—
99#11924028—24no—
100#11924028—24no—
101#11924028—24no—
102#11924028—24no—
103#11924028—42no—
104#11924028—42no—
105#11924028—42no—
106#11924028—42no—
107C2×Dic146—21no—
108C2×Dic146—21no—
109C2×Dic146—21no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146C22×C412yes—
113C2×D292—21no—
114C2×D292—21no—
115C2×D292—21no—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292C22×C412yes—
119C2×C22⋊D73—24no—
120C2×C22⋊D73—41no—
121C2×C22⋊D73—41no—
122C2×C22⋊D73—42no—
123C22×Dic73—21no—
124C2×C4×D73—24no—
125C2×C4×D73—41no—
126C2×C4×D73—41no—
127C2×C4×D73—42no—
128#11924012—24no—
129#11924012—22no—
130#11924012—22no—
131#11924012—22no—
132#11924012—42no—
133#11924012C22×C414yes—
134C2×#10596831—24no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831—24no—
138C2×#10596831—22no—
139C2×#10596831—22no—
140C2×#10596831—22no—
141C2×#10596831—22no—
142C2×#10596831—42no—
143C2×#10596831C2×D412yes—
144C2×#10596831C2×D412yes—
145C24×C73#4111yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73C22×C412yes—
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73#4111yescharacteristic
154C22×C4×C73#4112yes—
155C22×C4×C73#4112yes—
156C22×C4×C73#4112yes—
157C22×C4×C73C2411yescharacteristic
158M16×C73—24no—
159M16×C73—22no—
160M16×C73C22×C414yes—
161C2×C8×C73—24no—
162C2×C8×C73—22no—
163C4⋊C4×C73—22no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—22no—
166C4⋊C4×C73C2×D414yes—
167C22⋊C4×C73—22no—
168C22⋊C4×C73—22no—
169C22⋊C4×C73—41no—
170C22⋊C4×C73—41no—
171C42×C73—22no—
172C42×C73—22no—
173C42×C73—41no—
174C42×C73—41no—
175C23×C73—42no—
176C23×C73#11712yes—
177C23×C73#12111yescharacteristic
178C23×C73#12111yescharacteristic
179C23×C73#12612yes—
180C23×C73C2×#4111yescharacteristic
181C22×D73—22no—
182C22×D73—41no—
183C22×D73—41no—
184C22×D73—82no—
185C2×C4×C73—24no—
186C2×C4×C73—44no—
187C2×C4×C73—42no—
188C2×C4×C73—82no—
189C2×C4×C73#11712yes—
190C2×C4×C73C4×D412yes—
191C2×C4×C73C4×D412yes—
192C2×C4×C73C4×D412yes—
193C2×C4×C73C4×D412yes—
194C2×C4×C73#12111yescharacteristic
195C2×C4×C73#12111yescharacteristic
196C2×C4×C73#12312yes—
197C2×C4×C73#12312yes—
198C2×C4×C73#12412yes—
199C2×C4×C73#12412yes—
200C2×C4×C73#12612yes—
201C2×C4×C73C23×C411yescharacteristic
202C2×C4×C73C22×D411yescharacteristic
203C2×C4×C73C22×D411yescharacteristic
204C2×C4×C73C2×#4111yescharacteristic
205C2×C4×C73C2×#4111yescharacteristic
206C2×C4×C73C2×#4111yescharacteristic
207C22⋊D73—44no—
208C22⋊D73—44no—
209C2×Dic73—22no—
210C2×Dic73—41no—
211C2×Dic73—41no—
212D292—22no—
213D292—22no—
214D292—41no—
215D292—41no—
216C4×D73—44no—
217C4×D73—44no—
218Dic146—22no—
219Dic146—22no—
220Dic146—41no—
221Dic146—41no—
222C8×C73—24no—
223#10596831—24no—
224#10596831—22no—
225#10596831—22no—
226#10596831—42no—
227C22×C73—22no—
228C22×C73—22no—
229C22×C73—22no—
230C22×C73—42no—
231C22×C73—42no—
232C22×C73#38812yes—
233C22×C73#38812yes—
234C22×C73C2×#11711yescharacteristic
235C22×C73C2×#12111yescharacteristic
236C22×C73C2×#12611yescharacteristic
237C2×D73—42no—
238C2×D73—82no—
239C2×D73—82no—
240C2×D73—82no—
241C4×C73—44no—
242C4×C73—82no—
243C4×C73#38812yes—
244C4×C73#38812yes—
245C4×C73C2×C4×D411yescharacteristic
246C4×C73C2×C4×D411yescharacteristic
247C4×C73C2×#12311yescharacteristic
248C4×C73C2×#12411yescharacteristic
249Dic73—42no—
250#21075—731nomaximal, Sylow
251C2×C73—21no—
252C2×C73—21no—
253C2×C73—22no—
254C2×C73—22no—
255C2×C73C2×#38811yescharacteristic
256C2×C73#306712yes—
257D73—82no—
258#2911D7311yescharacteristic, complemented
259#2896—731no—
260C2×#416—731no—
261C2×#413—731no—
262C2×#405—731no—
263#1931—734no—
264#1930—734no—
265#1742—731no—
266C2×#341—731no—
267C73#2107511yescharacteristic, complemented, Sylow
268C22×#41—731no—
269C22×M16—731no—
270C22×M16—731no—
271C22×M16C2×D7311yescharacteristic
272C23×C8—731no—
273C2×#117C2×D7311yescharacteristic
274C2×#117C2×D7311yescharacteristic
275#428—22no—
276#428C2×D7314yes—
277#427—22no—
278#419—732no—
279#418—732no—
280#417—1462no—
281#416—1461no—
282#416—1461no—
283#416—1462no—
284#416—1462no—
285C2×#103—732no—
286C2×#102—732no—
287#413—738no—
288C2×#101—732no—
289C2×#100—732no—
290#405—734no—
291#405—1462no—
292C2×#97—1461no—
293C2×C4×C8—1461no—
294#343—734no—
295#341—734no—
296C2×#41—734no—
297C2×#41—1461no—
298C2×#41—1461no—
299C2×#41—1461no—
300C2×#41—1461no—
301C2×#41—1462no—
302C2×#41—1462no—
303C2×#41—1462no—
304C22×Q8—731no—
305C22×D4—731no—
306C22×D4—1461no—
307C23×C4—1461no—
308C23×C4C22×D7311yescharacteristic
309C2×M16—21no—
310C2×M16—22no—
311C2×M16—734no—
312C2×M16—732no—
313C2×M16—732no—
314C2×M16—1462no—
315C2×M16—1462no—
316C2×M16—1461no—
317C2×M16—1461no—
318C2×M16—1462no—
319C2×M16C4×D7314yes—
320C2×M16C22×D7312yes—
321C22×C8—21no—
322C22×C8—731no—
323C22×C8—731no—
324C22×C8—732no—
325C22×C8—732no—
326C22×C8—1461no—
327C22×C8—1461no—
328C22×C8—1462no—
329C22×C8—1462no—
330#117—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334#117—21no—
335#117—21no—
336#117C22⋊D7312yes—
337#117C22×D7312yes—
338C2×C4⋊C4C22⋊D7312yes—
339C2×C4⋊C4C22×D7312yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C42—21no—
343C2×C42—21no—
344#107—24no—
345#106—24no—
346#103—1462no—
347#103—1462no—
348#103—1464no—
349#102—1462no—
350#102—1462no—
351#102—1464no—
352#101—738no—
353#100—738no—
354#97—1464no—
355C4×C8—1464no—
356C24—1461no—
357C24#1192402711yescharacteristic
358#41—2922no—
359#41—2922no—
360#41—2922no—
361#41—2922no—
362#41—1464no—
363#41—1464no—
364#41—1464no—
365#41—1464no—
366C2×Q8—732no—
367C2×Q8—1461no—
368C2×Q8—1461no—
369C2×Q8—1461no—
370C2×Q8—1461no—
371C2×Q8—1461no—
372C2×D4—2921no—
373C2×D4—2921no—
374C2×D4—2922no—
375C2×D4—732no—
376C2×D4—1464no—
377C2×D4—1461no—
378C2×D4—1461no—
379C2×D4—1461no—
380C2×D4—1461no—
381C2×D4—1461no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—2921no—
385C22×C4—2921no—
386C22×C4—2922no—
387C22×C4—1461no—
388C22×C4—1464no—
389C22×C4C2×C4×D7312yes—
390C22×C4D4×D7311yescharacteristic
391C22×C4D4×D7311yescharacteristic
392C22×C4C2×C22⋊D7311yescharacteristic
393C22×C4C2×C22⋊D7311yescharacteristic
394C22×C4#1192402711yescharacteristic
395C22×C4#1192402812yes—
396C22×C4#1192402912yes—
397C22×C4#1192402912yes—
398C22×C4C23×D7311yescharacteristic
399M16—24no—
400M16—22no—
401M16—2922no—
402M16—734no—
403M16—1462no—
404M16—1462no—
405M16—1464no—
406M16—1462no—
407M16C2×C4×D7314yes—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—2922no—
411C2×C8—732no—
412C2×C8—732no—
413C2×C8—1462no—
414C2×C8—1464no—
415C2×C8—1462no—
416C2×C8—1462no—
417C2×C8—1464no—
418C2×C8—1462no—
419C2×C8—1462no—
420C2×C8—1462no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4C2×C22⋊D7314yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C42—22no—
430C42—22no—
431C42—41no—
432C42—41no—
433C23—42no—
434C23—2921no—
435C23—2921no—
436C23—5842no—
437C23—1462no—
438C23#1455654912yes—
439C23#1455656311yescharacteristic
440C23#1455656311yescharacteristic
441C23#1455658612yes—
442C23C2×#1192402711yescharacteristic
443Q8—2921no—
444Q8—2921no—
445Q8—1462no—
446Q8—1462no—
447D4—2924no—
448D4—2924no—
449D4—2921no—
450D4—2921no—
451D4—1462no—
452D4—1462no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—2921no—
458C2×C4—2921no—
459C2×C4—2924no—
460C2×C4—2924no—
461C2×C4—1462no—
462C2×C4#1455654912yes—
463C2×C4C4×C22⋊D7312yes—
464C2×C4C4×C22⋊D7312yes—
465C2×C4#1455655312yes—
466C2×C4#1455655312yes—
467C2×C4#1455656311yescharacteristic
468C2×C4#1455656311yescharacteristic
469C2×C4#1455657612yes—
470C2×C4#1455657612yes—
471C2×C4#1455658012yes—
472C2×C4#1455658012yes—
473C2×C4#1455658612yes—
474C2×C4C22×C4×D7311yescharacteristic
475C2×C4C2×D4×D7311yescharacteristic
476C2×C4C22×C22⋊D7311yescharacteristic
477C2×C4C2×#1192402811yescharacteristic
478C2×C4C2×#1192402911yescharacteristic
479C2×C4C2×#1192402911yescharacteristic
480C8—24no—
481C8—2922no—
482C8—1464no—
483C8—1462no—
484C8—1462no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—2922no—
491C22—5842no—
492C22—5842no—
493C22—5842no—
494C22#1798847712yes—
495C22#1798847712yes—
496C22C2×#1455654911yescharacteristic
497C22C2×#1455656311yescharacteristic
498C22C2×#1455658611yescharacteristic
499C4—44no—
500C4—82no—
501C4—2922no—
502C4#1798847712yes—
503C4#1798847712yes—
504C4C2×C4×C22⋊D7311yescharacteristic
505C4C2×#1455655311yescharacteristic
506C4C2×#1455657611yescharacteristic
507C4C2×#1455658011yescharacteristic
508C2—21no—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—5842no—
513C2C2×#1798847711yescharacteristic
514C2#2261964812yes—
subgroupG/H
centerC2×C4#14556563
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2911C2
socleC22×C73C2×#121
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenter#2911D73
hypocenterC73#21075
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
4. C2×#17988543C2semidirect
5. #22610592C2semidirect
6. #22610585C2semidirect
7. #22609670C2semidirect
8. C2×#17988351C2semidirect
9. C73×#2911C2semidirect
23. #17988569C4semidirect
72. #14556509C2×C4semidirect
73. #14556507C2×C4semidirect
258. #2911D73semidirect
267. C73#21075semidirect

Maximal subgroups

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

subgroupindex
#226184482
C2×#179885812
C2×#179885692
C2×#179885432
#226105922
#226105852
#226096702
C2×#179883512
C73×#29112
#2107573