Finite Group Databasebeta

← #33088532

Subgroups — #33088532

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#22618520C211yesmaximal, characteristic, complemented
2#22618448C211yesmaximal, characteristic
3C2×#17988584C211yesmaximal, characteristic
4C2×#17988569C211yesmaximal, characteristic
5#22610593C214yesmaximal, complemented
6#22610586C214yesmaximal, complemented
7#22609669C211yesmaximal, characteristic, complemented
8C2×#17988352C211yesmaximal, characteristic, complemented
9C2×#405×C73C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12C22×#11924012C2211yescharacteristic
13C2×#14556550C2211yescharacteristic
14#17988630C2214yes—
15#17988627—22no—
16#17988623—22no—
17#17988595C2212yes—
18#17988591C2212yes—
19#17988586—22no—
20#17988584—21no—
21#17988584—21no—
22#17988584—22no—
23#17988581—22no—
24C2×#14556516C2212yes—
25C2×#14556513C2212yes—
26#17988569C418yescomplemented
27C2×#14556509C2212yes—
28C2×#14556507C2212yes—
29#17988355C2214yes—
30#17988352C2214yes—
31C22×M16×C73C2211yescharacteristic
32C23×C8×C73C2211yescharacteristic
33C2×#117×C73C2211yescharacteristic
34#405×C73—22no—
35#405×C73C2214yes—
36C2×#97×C73—21no—
37C2×C4×C8×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×#11924012C2×C414yes—
56C2×#11924012C2312yes—
57C2×#11924012C2312yes—
58C22×#10596831—21no—
59C22×#10596831—21no—
60#14556550—21no—
61#14556550—21no—
62#14556550—21no—
63#14556550—21no—
64C2×C4⋊Dic73C2312yes—
65C2×C22⋊Dic73—21no—
66C2×C4×Dic73—21no—
67#14556524—24no—
68#14556522—24no—
69#14556516—22no—
70#14556516—22no—
71#14556514—24no—
72#14556513—22no—
73#14556513—22no—
74#14556511—24no—
75#14556509C2×C418yescomplemented
76#14556507C2×C418yescomplemented
77C23×C4×C73C2311yescharacteristic
78C2×M16×C73—21no—
79C2×M16×C73—22no—
80C2×M16×C73C2×C414yes—
81C2×M16×C73C2312yes—
82C22×C8×C73—21no—
83C22×C8×C73—22no—
84C22×C8×C73—22no—
85C22×C8×C73D411yescharacteristic
86C22×C8×C73D411yescharacteristic
87C22×C8×C73D412yes—
88C22×C8×C73C2312yes—
89#117×C73—21no—
90#117×C73—21no—
91#117×C73D412yes—
92#117×C73C2312yes—
93C2×C4⋊C4×C73D412yes—
94C2×C22⋊C4×C73—21no—
95C2×C42×C73—21no—
96#97×C73—24no—
97C4×C8×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—
124C22×Dic73—22no—
125C2×C4×D73—24no—
126C2×C4×D73—41no—
127C2×C4×D73—41no—
128C2×C4×D73—42no—
129#11924012—24no—
130#11924012—22no—
131#11924012—22no—
132#11924012—22no—
133#11924012—42no—
134#11924012C22×C414yes—
135C2×#10596831—24no—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—42no—
139C4⋊Dic73—22no—
140C4⋊Dic73—22no—
141C22⋊Dic73—41no—
142C22⋊Dic73—41no—
143C4×Dic73—41no—
144C4×Dic73—41no—
145C24×C73#4111yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73C22×C412yes—
148C22×C4×C73C2×D411yescharacteristic
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73#4111yescharacteristic
153C22×C4×C73#4112yes—
154C22×C4×C73#4112yes—
155C22×C4×C73#4112yes—
156C22×C4×C73C2411yescharacteristic
157M16×C73—24no—
158M16×C73—22no—
159M16×C73C22×C414yes—
160C2×C8×C73—24no—
161C2×C8×C73—22no—
162C2×C8×C73—22no—
163C2×C8×C73—24no—
164C2×C8×C73—22no—
165C2×C8×C73—22no—
166C2×C8×C73—22no—
167C2×C8×C73C2×D412yes—
168C2×C8×C73C2×D412yes—
169C4⋊C4×C73—22no—
170C4⋊C4×C73C2×D414yes—
171C22⋊C4×C73—22no—
172C22⋊C4×C73—22no—
173C42×C73—22no—
174C42×C73—22no—
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#11712yes—
188C2×C4×C73C4×D412yes—
189C2×C4×C73C4×D412yes—
190C2×C4×C73C4×D412yes—
191C2×C4×C73C4×D412yes—
192C2×C4×C73#12111yescharacteristic
193C2×C4×C73#12111yescharacteristic
194C2×C4×C73#12312yes—
195C2×C4×C73#12312yes—
196C2×C4×C73#12412yes—
197C2×C4×C73#12412yes—
198C2×C4×C73#12612yes—
199C2×C4×C73C23×C411yescharacteristic
200C2×C4×C73C22×D411yescharacteristic
201C2×C4×C73C22×D411yescharacteristic
202C2×C4×C73C2×#4111yescharacteristic
203C2×C4×C73C2×#4111yescharacteristic
204C2×C4×C73C2×#4111yescharacteristic
205C22⋊D73—44no—
206C22⋊D73—44no—
207C2×Dic73—22no—
208C2×Dic73—41no—
209C2×Dic73—41no—
210C2×Dic73—42no—
211C2×Dic73—82no—
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—
223C8×C73—22no—
224C8×C73—22no—
225#10596831—24no—
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#38812yes—
243C4×C73#38812yes—
244C4×C73C2×C4×D411yescharacteristic
245C4×C73C2×C4×D411yescharacteristic
246C4×C73C2×#12311yescharacteristic
247C4×C73C2×#12411yescharacteristic
248Dic73—42no—
249Dic73—82no—
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#2911—731no—
259#2896—731no—
260C2×#416—731no—
261C2×#413—731no—
262C2×#405D7311yescharacteristic, complemented
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×C8C2×D7311yescharacteristic
273C2×#117—731no—
274C2×#117C2×D7311yescharacteristic
275#428—734no—
276#428—1462no—
277#427—1462no—
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—22no—
291#405C2×D7314yes—
292C2×#97—21no—
293C2×C4×C8—21no—
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—22no—
323C22×C8—22no—
324C22×C8—1461no—
325C22×C8—1461no—
326C22×C8C22⋊D7311yescharacteristic
327C22×C8C22⋊D7311yescharacteristic
328C22×C8C22⋊D7312yes—
329C22×C8C22×D7312yes—
330#117—21no—
331#117—21no—
332#117—1461no—
333#117—1461no—
334#117—1461no—
335#117—1461no—
336#117D29212yes—
337#117C22×D7312yes—
338C2×C4⋊C4—732no—
339C2×C4⋊C4D29212yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1461no—
342C2×C42—21no—
343C2×C42—1461no—
344#107—1464no—
345#106—1464no—
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—24no—
355C4×C8—24no—
356C24—1461no—
357C24#1192402811yescharacteristic
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—2921no—
384C22×C4—2921no—
385C22×C4—2922no—
386C22×C4—1461no—
387C22×C4—1464no—
388C22×C4—1462no—
389C22×C4C2×C4×D7312yes—
390C22×C4C2×C22⋊D7311yescharacteristic
391C22×C4C2×C22⋊D7311yescharacteristic
392C22×C4C2×D29211yescharacteristic
393C22×C4C2×D29211yescharacteristic
394C22×C4#1192402811yescharacteristic
395C22×C4#1192402812yes—
396C22×C4#1192402812yes—
397C22×C4#1192402812yes—
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—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—2922no—
416C2×C8—1462no—
417C2×C8—1464no—
418C2×C8—1462no—
419C2×C8C2×C22⋊D7312yes—
420C2×C8C2×C22⋊D7312yes—
421C4⋊C4—22no—
422C4⋊C4—1462no—
423C4⋊C4—1462no—
424C4⋊C4C2×D29214yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—2921no—
428C22⋊C4—2921no—
429C42—22no—
430C42—22no—
431C42—2921no—
432C42—2921no—
433C23—42no—
434C23—2921no—
435C23—2921no—
436C23—5842no—
437C23—1462no—
438C23#1455654812yes—
439C23#1455656511yescharacteristic
440C23#1455656511yescharacteristic
441C23#1455658512yes—
442C23C2×#1192402811yescharacteristic
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—2921no—
456C2×C4—2921no—
457C2×C4—2924no—
458C2×C4—2924no—
459C2×C4—2922no—
460C2×C4—5842no—
461C2×C4—1462no—
462C2×C4#1455654812yes—
463C2×C4C4×C22⋊D7312yes—
464C2×C4C4×C22⋊D7312yes—
465C2×C4C4×D29212yes—
466C2×C4C4×D29212yes—
467C2×C4#1455656511yescharacteristic
468C2×C4#1455656511yescharacteristic
469C2×C4#1455657412yes—
470C2×C4#1455657412yes—
471C2×C4#1455657912yes—
472C2×C4#1455657912yes—
473C2×C4#1455658512yes—
474C2×C4C22×C4×D7311yescharacteristic
475C2×C4C22×C22⋊D7311yescharacteristic
476C2×C4C22×D29211yescharacteristic
477C2×C4C2×#1192402811yescharacteristic
478C2×C4C2×#1192402811yescharacteristic
479C2×C4C2×#1192402811yescharacteristic
480C8—24no—
481C8—22no—
482C8—22no—
483C8—2922no—
484C8—1464no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—2922no—
491C22—5842no—
492C22—5842no—
493C22—5842no—
494C22#1798847812yes—
495C22#1798847812yes—
496C22C2×#1455654811yescharacteristic
497C22C2×#1455656511yescharacteristic
498C22C2×#1455658511yescharacteristic
499C4—44no—
500C4—2922no—
501C4—5842no—
502C4#1798847812yes—
503C4#1798847812yes—
504C4C2×C4×C22⋊D7311yescharacteristic
505C4C2×C4×D29211yescharacteristic
506C4C2×#1455657411yescharacteristic
507C4C2×#1455657911yescharacteristic
508C2—21no—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—5842no—
513C2C2×#1798847811yescharacteristic
514C2#2261965112yes—
subgroupG/H
centerC2×C4#14556565
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#405×C73C2
socleC22×C73C2×#121
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#405D73
hypocenterC73#21075
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. #22618520C2semidirect
5. #22610593C2semidirect
6. #22610586C2semidirect
7. #22609669C2semidirect
8. C2×#17988352C2semidirect
9. C2×#405×C73C2semidirect
26. #17988569C4semidirect
75. #14556509C2×C4semidirect
76. #14556507C2×C4semidirect
262. C2×#405D73semidirect
267. C73#21075semidirect

Maximal subgroups

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

subgroupindex
#226185202
#226184482
C2×#179885842
C2×#179885692
#226105932
#226105862
#226096692
C2×#179883522
C2×#405×C732
#2107573