Finite Group Databasebeta

← #33088540

Subgroups — #33088540

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618448C211yesmaximal, characteristic
2C2×#17988586C211yesmaximal, characteristic
3C2×#17988569C211yesmaximal, characteristic
4#22618350C211yesmaximal, characteristic, complemented
5#22610592C214yesmaximal, complemented
6#22610585C214yesmaximal, complemented
7C2×#17988356C211yesmaximal, characteristic, complemented
8#22609657C211yesmaximal, characteristic, complemented
9C2×#428×C73C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12C22×#11924012C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14#17988595C2212yes—
15#17988591C2212yes—
16#17988586—22no—
17#17988586—22no—
18#17988586—21no—
19#17988586—21no—
20#17988581—22no—
21C2×#14556514C2212yes—
22C2×#14556511C2212yes—
23#17988569C418yescomplemented
24C2×#14556509C2212yes—
25C2×#14556507C2212yes—
26#17988543—22no—
27#17988543C2214yes—
28C4×#11924012—22no—
29#17988356C2214yes—
30#17988351C2214yes—
31C22×M16×C73C2211yescharacteristic
32C2×#117×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34#428×C73—22no—
35#428×C73C2214yes—
36C2×#107×C73—21no—
37C2×#106×C73—21no—
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×#11924012—22no—
56C2×#11924012—22no—
57C2×#11924012C2×C414yes—
58C2×#11924012D412yes—
59C2×#11924012D412yes—
60C2×#11924012C2312yes—
61C2×#11924012C2312yes—
62C22×#10596831—21no—
63C22×#10596831—21no—
64C22×#10596831C2312yes—
65#14556514—24no—
66#14556514—22no—
67#14556514—22no—
68#14556511—24no—
69#14556511—22no—
70#14556511—22no—
71#14556509C2×C418yescomplemented
72#14556507C2×C418yescomplemented
73#14556502—24no—
74C4×#10596831—24no—
75C23×C4×C73—21no—
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×C73D412yes—
85#117×C73C2312yes—
86C2×C4⋊C4×C73—21no—
87C2×C4⋊C4×C73—21no—
88C2×C4⋊C4×C73—21no—
89C2×C4⋊C4×C73—21no—
90C2×C4⋊C4×C73—21no—
91C2×C4⋊C4×C73D411yescharacteristic
92C2×C4⋊C4×C73D411yescharacteristic
93C2×C4⋊C4×C73C2312yes—
94C2×C22⋊C4×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—24no—
130#11924012—24no—
131#11924012—22no—
132#11924012—22no—
133#11924012—22no—
134#11924012—42no—
135#11924012C22×C414yes—
136C2×#10596831—24no—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C2×#10596831—22no—
140C2×#10596831—22no—
141C2×#10596831—42no—
142C2×#10596831C2×D414yes—
143C24×C73#4111yescharacteristic
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—22no—
147C22×C4×C73—41no—
148C22×C4×C73—41no—
149C22×C4×C73—82no—
150C22×C4×C73C22×C412yes—
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73#4111yescharacteristic
156C22×C4×C73#4112yes—
157C22×C4×C73#4112yes—
158C22×C4×C73#4112yes—
159C22×C4×C73C2411yescharacteristic
160M16×C73—24no—
161M16×C73—22no—
162M16×C73C22×C414yes—
163C2×C8×C73—24no—
164C2×C8×C73—22no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—22no—
169C4⋊C4×C73—41no—
170C4⋊C4×C73—41no—
171C4⋊C4×C73C2×D412yes—
172C4⋊C4×C73C2×D412yes—
173C22⋊C4×C73—22no—
174C22⋊C4×C73—22no—
175C42×C73—22no—
176C42×C73—22no—
177C23×C73—41no—
178C23×C73—41no—
179C23×C73#11712yes—
180C23×C73#12111yescharacteristic
181C23×C73#12111yescharacteristic
182C23×C73#12612yes—
183C23×C73C2×#4111yescharacteristic
184C22×D73—22no—
185C22×D73—41no—
186C22×D73—41no—
187C22×D73—81no—
188C22×D73—81no—
189C2×C4×C73—22no—
190C2×C4×C73—22no—
191C2×C4×C73—42no—
192C2×C4×C73—81no—
193C2×C4×C73—81no—
194C2×C4×C73—82no—
195C2×C4×C73—82no—
196C2×C4×C73—82no—
197C2×C4×C73#11712yes—
198C2×C4×C73C4×D412yes—
199C2×C4×C73C4×D412yes—
200C2×C4×C73C4×D412yes—
201C2×C4×C73C4×D412yes—
202C2×C4×C73#12111yescharacteristic
203C2×C4×C73#12111yescharacteristic
204C2×C4×C73#12312yes—
205C2×C4×C73#12312yes—
206C2×C4×C73#12412yes—
207C2×C4×C73#12412yes—
208C2×C4×C73#12612yes—
209C2×C4×C73C23×C411yescharacteristic
210C2×C4×C73C22×D411yescharacteristic
211C2×C4×C73C22×D411yescharacteristic
212C2×C4×C73C2×#4111yescharacteristic
213C2×C4×C73C2×#4111yescharacteristic
214C2×C4×C73C2×#4111yescharacteristic
215C22⋊D73—44no—
216C22⋊D73—44no—
217C2×Dic73—22no—
218C2×Dic73—41no—
219C2×Dic73—41no—
220D292—22no—
221D292—22no—
222D292—41no—
223D292—41no—
224C4×D73—44no—
225C4×D73—44no—
226Dic146—22no—
227Dic146—22no—
228Dic146—41no—
229Dic146—41no—
230C8×C73—24no—
231#10596831—24no—
232#10596831—24no—
233#10596831—42no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73—21no—
239C22×C73—21no—
240C22×C73—42no—
241C22×C73—42no—
242C22×C73#38812yes—
243C22×C73#38812yes—
244C22×C73C2×#11711yescharacteristic
245C22×C73C2×#12111yescharacteristic
246C22×C73C2×#12611yescharacteristic
247C2×D73—42no—
248C2×D73—82no—
249C2×D73—81no—
250C2×D73—81no—
251C2×D73—82no—
252C4×C73—42no—
253C4×C73—42no—
254C4×C73—82no—
255C4×C73#38812yes—
256C4×C73#38812yes—
257C4×C73C2×C4×D411yescharacteristic
258C4×C73C2×C4×D411yescharacteristic
259C4×C73C2×#12311yescharacteristic
260C4×C73C2×#12411yescharacteristic
261Dic73—42no—
262#21076—731nomaximal, Sylow
263C2×C73—21no—
264C2×C73—21no—
265C2×C73—22no—
266C2×C73—22no—
267C2×C73C2×#38811yescharacteristic
268C2×C73#306811yescharacteristic
269C2×C73#306911yescharacteristic
270D73—82no—
271C2×#428D7311yescharacteristic, complemented
272#2896—731no—
273C2×#417—731no—
274C2×#413—731no—
275#2873—731no—
276#1931—734no—
277#1930—734no—
278C2×#343—731no—
279#1738—731no—
280C73#2107611yescharacteristic, complemented, Sylow
281C22×#41—731no—
282C22×M16—731no—
283C22×M16—731no—
284C22×M16—731no—
285C22×M16C2×D7311yescharacteristic
286C2×#117C2×D7311yescharacteristic
287C22×C4⋊C4C2×D7311yescharacteristic
288#428—22no—
289#428C2×D7314yes—
290C2×#107—21no—
291C2×#106—21no—
292#419—732no—
293#418—732no—
294#417—1462no—
295#417—1462no—
296#417—1461no—
297#417—1461no—
298#416—1462no—
299C2×#103—732no—
300C2×#102—732no—
301#413—738no—
302C2×#101—732no—
303C2×#100—732no—
304#405—734no—
305#405—1462no—
306C4×M16—1462no—
307#343—734no—
308#341—734no—
309C2×#41—734no—
310C2×#41—1461no—
311C2×#41—1461no—
312C2×#41—1461no—
313C2×#41—1461no—
314C2×#41—1462no—
315C2×#41—1462no—
316C2×#41—1462no—
317C22×Q8—731no—
318C22×D4—731no—
319C22×D4—1461no—
320C23×C4—21no—
321C23×C4—1461no—
322C23×C4C22×D7311yescharacteristic
323C2×M16—21no—
324C2×M16—22no—
325C2×M16—732no—
326C2×M16—732no—
327C2×M16—734no—
328C2×M16—732no—
329C2×M16—732no—
330C2×M16—1462no—
331C2×M16—1462no—
332C2×M16—1461no—
333C2×M16—1461no—
334C2×M16—1462no—
335C2×M16—1462no—
336C2×M16—1462no—
337C2×M16C4×D7314yes—
338C2×M16C22×D7312yes—
339C22×C8—21no—
340C22×C8—732no—
341C22×C8—1461no—
342C22×C8—1461no—
343#117—21no—
344#117—21no—
345#117C22⋊D7312yes—
346#117C22×D7312yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C22⋊D7311yescharacteristic
353C2×C4⋊C4C22⋊D7311yescharacteristic
354C2×C4⋊C4C22×D7312yes—
355C2×C22⋊C4—21no—
356C2×C42—21no—
357#107—24no—
358#106—24no—
359#103—1464no—
360#103—1462no—
361#103—1462no—
362#102—1464no—
363#102—1462no—
364#102—1462no—
365#101—738no—
366#100—738no—
367#97—1464no—
368C4×C8—1464no—
369C24—1461no—
370C24#1192402711yescharacteristic
371#41—2922no—
372#41—2922no—
373#41—2922no—
374#41—2922no—
375#41—1464no—
376#41—1464no—
377#41—1464no—
378#41—1464no—
379C2×Q8—732no—
380C2×Q8—1461no—
381C2×Q8—1461no—
382C2×Q8—1461no—
383C2×Q8—1461no—
384C2×Q8—1461no—
385C2×D4—2921no—
386C2×D4—2921no—
387C2×D4—2922no—
388C2×D4—732no—
389C2×D4—1464no—
390C2×D4—1461no—
391C2×D4—1461no—
392C2×D4—1461no—
393C2×D4—1461no—
394C2×D4—1461no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—82no—
401C22×C4—2921no—
402C22×C4—2921no—
403C22×C4—2922no—
404C22×C4—1461no—
405C22×C4—1464no—
406C22×C4C2×C4×D7312yes—
407C22×C4D4×D7311yescharacteristic
408C22×C4D4×D7311yescharacteristic
409C22×C4C2×C22⋊D7311yescharacteristic
410C22×C4C2×C22⋊D7311yescharacteristic
411C22×C4#1192402711yescharacteristic
412C22×C4#1192402812yes—
413C22×C4#1192402912yes—
414C22×C4#1192402912yes—
415C22×C4C23×D7311yescharacteristic
416M16—24no—
417M16—22no—
418M16—2922no—
419M16—734no—
420M16—1462no—
421M16—1462no—
422M16—1464no—
423M16—1464no—
424M16—1464no—
425M16—1462no—
426M16C2×C4×D7314yes—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—2922no—
430C2×C8—734no—
431C2×C8—1462no—
432C2×C8—1464no—
433C2×C8—1462no—
434C2×C8—1462no—
435C2×C8—1462no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×C22⋊D7312yes—
443C4⋊C4C2×C22⋊D7312yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42—22no—
447C42—22no—
448C23—41no—
449C23—41no—
450C23—2921no—
451C23—2921no—
452C23—5841no—
453C23—5841no—
454C23—1462no—
455C23#1455654912yes—
456C23#1455656311yescharacteristic
457C23#1455656311yescharacteristic
458C23#1455658612yes—
459C23C2×#1192402711yescharacteristic
460Q8—2921no—
461Q8—2921no—
462Q8—1462no—
463Q8—1462no—
464D4—2924no—
465D4—2924no—
466D4—2921no—
467D4—2921no—
468D4—1462no—
469D4—1462no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—2921no—
479C2×C4—2921no—
480C2×C4—2924no—
481C2×C4—2924no—
482C2×C4—1462no—
483C2×C4#1455654912yes—
484C2×C4C4×C22⋊D7312yes—
485C2×C4C4×C22⋊D7312yes—
486C2×C4#1455655312yes—
487C2×C4#1455655312yes—
488C2×C4#1455656311yescharacteristic
489C2×C4#1455656311yescharacteristic
490C2×C4#1455657612yes—
491C2×C4#1455657612yes—
492C2×C4#1455658012yes—
493C2×C4#1455658012yes—
494C2×C4#1455658612yes—
495C2×C4C22×C4×D7311yescharacteristic
496C2×C4C2×D4×D7311yescharacteristic
497C2×C4C22×C22⋊D7311yescharacteristic
498C2×C4C2×#1192402811yescharacteristic
499C2×C4C2×#1192402911yescharacteristic
500C2×C4C2×#1192402911yescharacteristic
501C8—24no—
502C8—2922no—
503C8—1464no—
504C8—1464no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—2922no—
514C22—5842no—
515C22—5842no—
516C22—5841no—
517C22—5841no—
518C22#1798847712yes—
519C22#1798847712yes—
520C22C2×#1455654911yescharacteristic
521C22C2×#1455656311yescharacteristic
522C22C2×#1455658611yescharacteristic
523C4—42no—
524C4—42no—
525C4—82no—
526C4—2922no—
527C4#1798847712yes—
528C4#1798847712yes—
529C4C2×C4×C22⋊D7311yescharacteristic
530C4C2×#1455655311yescharacteristic
531C4C2×#1455657611yescharacteristic
532C4C2×#1455658011yescharacteristic
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—5842no—
538C2C2×#1798847711yescharacteristic
539C2#2261965611yescharacteristic
540C2#2261966511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#428×C73C2
socleC22×C73C2×#121
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#428D73
hypocenterC73#21076
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #22618350C2semidirect
5. #22610592C2semidirect
6. #22610585C2semidirect
7. C2×#17988356C2semidirect
8. #22609657C2semidirect
9. C2×#428×C73C2semidirect
23. #17988569C4semidirect
71. #14556509C2×C4semidirect
72. #14556507C2×C4semidirect
271. C2×#428D73semidirect
280. C73#21076semidirect

Maximal subgroups

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

subgroupindex
#226184482
C2×#179885862
C2×#179885692
#226183502
#226105922
#226105852
C2×#179883562
#226096572
C2×#428×C732
#2107673