Finite Group Databasebeta

← #33016896

Subgroups — #33016896

Order 18688.

1226 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22608193C212yesmaximal, complemented
2#22608184C212yesmaximal, complemented
3#22608184C212yesmaximal, complemented
4#22608180C212yesmaximal
5#22608180C211yesmaximal, characteristic
6#22608176C212yesmaximal
7#22608168C211yesmaximal, characteristic
8#22608163C212yesmaximal, complemented
9C2×#399×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2212yes—
11C22×C22⋊Dic73C2212yes—
12C22×C22⋊Dic73C2211yescharacteristic
13C22×C22⋊Dic73C2211yescharacteristic
14C22×C4×Dic73C2211yescharacteristic
15C2×#14556498C2212yes—
16C2×#14556498C2212yes—
17C2×#14556498C2212yes—
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2211yescharacteristic
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C22×C4⋊C4×C73C2212yes—
27C22×C4⋊C4×C73C2211yescharacteristic
28C22×C22⋊C4×C73C2211yescharacteristic
29C22×C22⋊C4×C73C2212yes—
30#399×C73C418yescomplemented
31C2×#95×C73C2211yescharacteristic
32C23×Dic73C2312yes—
33C23×Dic73C2312yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C2×C4⋊Dic73—42no—
37C2×C4⋊Dic73—42no—
38C2×C4⋊Dic73—42no—
39C2×C22⋊Dic73—24no—
40C2×C22⋊Dic73—42no—
41C2×C22⋊Dic73—41no—
42C2×C22⋊Dic73—42no—
43C2×C22⋊Dic73—41no—
44C2×C22⋊Dic73—42no—
45C2×C22⋊Dic73—42no—
46C2×C4×Dic73—41no—
47C2×C4×Dic73—42no—
48#14556498—24no—
49#14556498—24no—
50#14556498—42no—
51#14556498—41no—
52#14556498—42no—
53#14556498—42no—
54#14556498—42no—
55#14556498—42no—
56#14556498—42no—
57#14556498—42no—
58#14556498—42no—
59C25×C73C2311yescharacteristic
60C23×C4×C73C2312yes—
61C23×C4×C73C2311yescharacteristic
62C23×C4×C73C2311yescharacteristic
63C23×C4×C73C2312yes—
64C2×C4⋊C4×C73—42no—
65C2×C4⋊C4×C73—41no—
66C2×C4⋊C4×C73—42no—
67C2×C4⋊C4×C73—41no—
68C2×C4⋊C4×C73C2×C418yes—
69C2×C4⋊C4×C73C2×C414yes—
70C2×C22⋊C4×C73—41no—
71C2×C22⋊C4×C73—42no—
72C2×C22⋊C4×C73—41no—
73C2×C22⋊C4×C73—42no—
74C2×C22⋊C4×C73C2×C414yes—
75C2×C22⋊C4×C73C2×C418yes—
76#95×C73C2×C414yes—
77C22×Dic73—22no—
78C22×Dic73—22no—
79C22×Dic73—22no—
80C22×Dic73—22no—
81C22×Dic73—22no—
82C22×Dic73—22no—
83C22×Dic73—22no—
84C22×Dic73—22no—
85C22×Dic73—22no—
86C22×Dic73—22no—
87C22×Dic73—22no—
88C22×Dic73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73#4114yes—
105C4⋊Dic73—48no—
106C22⋊Dic73—48no—
107C22⋊Dic73—44no—
108C22⋊Dic73—44no—
109C4×Dic73—44no—
110C24×C73—22no—
111C24×C73—22no—
112C24×C73—21no—
113C24×C73—22no—
114C24×C73—21no—
115C24×C73—22no—
116C24×C73—22no—
117C24×C73—22no—
118C24×C73C22×C412yes—
119C24×C73C2411yescharacteristic
120C22×C4×C73—22no—
121C22×C4×C73—22no—
122C22×C4×C73—22no—
123C22×C4×C73—22no—
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—22no—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73—22no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—22no—
134C22×C4×C73—22no—
135C22×C4×C73—22no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—22no—
140C22×C4×C73C22×C414yes—
141C22×C4×C73C22×C412yes—
142C22×C4×C73C22×C412yes—
143C22×C4×C73C22×C414yes—
144C4⋊C4×C73—48no—
145C4⋊C4×C73—44no—
146C22⋊C4×C73—48no—
147C22⋊C4×C73—44no—
148C23×C73—22no—
149C23×C73—22no—
150C23×C73—24no—
151C23×C73—22no—
152C23×C73—24no—
153C23×C73—42no—
154C23×C73—42no—
155C23×C73—42no—
156C23×C73—42no—
157C23×C73—42no—
158C23×C73—42no—
159C23×C73—41no—
160C23×C73—42no—
161C23×C73—41no—
162C23×C73—41no—
163C23×C73—42no—
164C23×C73—42no—
165C23×C73—42no—
166C23×C73—42no—
167C23×C73—42no—
168C23×C73—41no—
169C23×C73C23×C411yescharacteristic
170C23×C73C2×#4112yes—
171C23×C73H(5,𝔽2)12yes—
172C23×C73H(5,𝔽2)11yescharacteristic
173C23×C73H(5,𝔽2)12yes—
174C23×C73H(5,𝔽2)11yescharacteristic
175C23×C73H(5,𝔽2)12yes—
176C23×C732−1+412yes—
177C23×C732−1+412yes—
178C2×C4×C73—24no—
179C2×C4×C73—24no—
180C2×C4×C73—22no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—24no—
184C2×C4×C73—24no—
185C2×C4×C73—24no—
186C2×C4×C73—24no—
187C2×C4×C73—24no—
188C2×C4×C73—42no—
189C2×C4×C73—42no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73—42no—
193C2×C4×C73—42no—
194C2×C4×C73—42no—
195C2×C4×C73—41no—
196C2×C4×C73—42no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—42no—
200C2×C4×C73—42no—
201C2×C4×C73—41no—
202C2×Dic73—24no—
203C2×Dic73—24no—
204C2×Dic73—24no—
205C2×Dic73—42no—
206C2×Dic73—42no—
207C2×Dic73—42no—
208C2×Dic73—42no—
209C2×Dic73—42no—
210C2×Dic73—42no—
211C2×Dic73—42no—
212C2×Dic73—42no—
213C2×Dic73—42no—
214C2×Dic73—42no—
215C2×Dic73—42no—
216C2×Dic73—42no—
217C2×Dic73—42no—
218C2×Dic73—42no—
219C2×Dic73—42no—
220C2×Dic73—42no—
221C2×Dic73—42no—
222C2×Dic73—42no—
223C2×Dic73—42no—
224C2×Dic73—42no—
225C2×Dic73—42no—
226C2×Dic73—42no—
227C2×Dic73—42no—
228C2×Dic73—42no—
229C2×Dic73—42no—
230C22×C73—42no—
231C22×C73—44no—
232C22×C73—42no—
233C22×C73—44no—
234C22×C73—42no—
235C22×C73—82no—
236C22×C73—82no—
237C22×C73—82no—
238C22×C73—82no—
239C22×C73C4×#4111yescharacteristic
240C22×C73#51811yescharacteristic
241C22×C73#51812yes—
242C22×C73#51911yescharacteristic
243C22×C73#52012yes—
244C22×C73#53812yes—
245C22×C73#53912yes—
246C22×C73#53912yes—
247C22×C73#54012yes—
248C22×C73#54212yes—
249C22×C73#54212yes—
250C22×C73#56011yescharacteristic
251C22×C73#56012yes—
252C22×C73#56011yescharacteristic
253C22×C73#56011yescharacteristic
254C22×C73#56212yes—
255C22×C73#56212yes—
256C22×C73#56212yes—
257C22×C73#56312yes—
258C22×C73#56312yes—
259C22×C73#56311yescharacteristic
260C4×C73—44no—
261C4×C73—42no—
262C4×C73—42no—
263C4×C73—44no—
264Dic73—44no—
265Dic73—82no—
266Dic73—82no—
267Dic73—82no—
268#14646—731nomaximal, Sylow
269C2×C73—82no—
270C2×C73#233211yescharacteristic
271C2×C73#233512yes—
272C2×C73#237611yescharacteristic
273C2×C73#237812yes—
274C2×C73#238011yescharacteristic
275C2×C73#282712yes—
276C2×C73#282912yes—
277C2×C73#283712yes—
278C2×C73#283812yes—
279C2×#399D7311yescharacteristic
280#1443—732no—
281#1441—732no—
282#1441—732no—
283#1440—731no—
284#1440—732no—
285#1439—732no—
286#1437—731no—
287#1436—732no—
288C73#1464611yescharacteristic, complemented, Sylow
289C22×C4⋊C4—732no—
290C22×C4⋊C4C2×D7312yes—
291C22×C4⋊C4C2×D7311yescharacteristic
292C22×C22⋊C4—731no—
293C22×C22⋊C4—731no—
294C22×C22⋊C4—732no—
295C22×C22⋊C4C2×D7311yescharacteristic
296C22×C22⋊C4C2×D7312yes—
297C22×C42—731no—
298#399Dic7318yescomplemented
299C2×#95—732no—
300C2×#95—732no—
301C2×#95—732no—
302C2×#95—731no—
303C2×#95—732no—
304C2×#95—732no—
305C2×#95—732no—
306C2×#95—732no—
307C2×#95—732no—
308C2×#95—732no—
309C2×#95—732no—
310C2×#95C2×D7311yescharacteristic
311C25C22×D7311yescharacteristic
312C23×C4—732no—
313C23×C4—732no—
314C23×C4—732no—
315C23×C4—732no—
316C23×C4C22×D7312yes—
317C23×C4C22×D7311yescharacteristic
318C23×C4C22×D7312yes—
319C23×C4C22×D7311yescharacteristic
320C2×C4⋊C4—42no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4—2922no—
325C2×C4⋊C4—2922no—
326C2×C4⋊C4—2922no—
327C2×C4⋊C4C2×Dic7318yes—
328C2×C4⋊C4C2×Dic7314yes—
329C2×C22⋊C4—41no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—41no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—2921no—
334C2×C22⋊C4—2922no—
335C2×C22⋊C4—2921no—
336C2×C22⋊C4—2922no—
337C2×C22⋊C4—2922no—
338C2×C22⋊C4—2922no—
339C2×C22⋊C4—1464no—
340C2×C22⋊C4C2×Dic7314yes—
341C2×C22⋊C4C2×Dic7318yes—
342C2×C42—2921no—
343C2×C42—2922no—
344#95—2921no—
345#95—2922no—
346#95—2922no—
347#95—2922no—
348#95—2922no—
349#95—2922no—
350#95—2922no—
351#95—2922no—
352#95—2922no—
353#95—1464no—
354#95—1464no—
355#95C2×Dic7314yes—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24C22×Dic7312yes—
365C24C23×D7311yescharacteristic
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—734no—
387C22×C4—1462no—
388C22×C4—1462no—
389C22×C4—1462no—
390C22×C4—1462no—
391C22×C4—1462no—
392C22×C4—1462no—
393C22×C4—1462no—
394C22×C4—1462no—
395C22×C4—1462no—
396C22×C4—1462no—
397C22×C4—1462no—
398C22×C4—1462no—
399C22×C4—1462no—
400C22×C4—1462no—
401C22×C4—1462no—
402C22×C4—1462no—
403C22×C4—1462no—
404C22×C4—1462no—
405C22×C4—1462no—
406C22×C4—1462no—
407C22×C4—1462no—
408C22×C4—1462no—
409C22×C4—1462no—
410C22×C4—1462no—
411C22×C4—1462no—
412C22×C4—1462no—
413C22×C4—1462no—
414C22×C4C22×Dic7314yes—
415C22×C4C22×Dic7312yes—
416C22×C4C22×Dic7314yes—
417C22×C4C22×Dic7312yes—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—2928no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—2924no—
424C22⋊C4—2924no—
425C22⋊C4—2928no—
426C42—2924no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—24no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23C23×Dic7311yescharacteristic
449C23#41×D7312yes—
450C23#1455663612yes—
451C23#1455663611yescharacteristic
452C23#1455663612yes—
453C23#1455663611yescharacteristic
454C23#1455663712yes—
455C23#1455663812yes—
456C23#1455663912yes—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—2922no—
482C2×C4—2922no—
483C2×C4—2922no—
484C2×C4—2922no—
485C2×C4—2922no—
486C2×C4—2922no—
487C2×C4—2922no—
488C2×C4—2922no—
489C2×C4—2922no—
490C2×C4—2922no—
491C2×C4—2922no—
492C2×C4—2922no—
493C2×C4—2922no—
494C2×C4—2922no—
495C2×C4—2922no—
496C2×C4—2922no—
497C2×C4—2922no—
498C2×C4—2922no—
499C2×C4—2922no—
500C2×C4—2922no—
501C2×C4—2922no—
502C2×C4—2922no—
503C2×C4—2922no—
504C2×C4—2922no—
505C2×C4—2922no—
506C2×C4—1464no—
507C2×C4—1464no—
508C2×C4—1464no—
509C22—42no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22#41×Dic7311yescharacteristic
519C22#1798906211yescharacteristic
520C22#1798906212yes—
521C22#1798906511yescharacteristic
522C22#1798907012yes—
523C22#1798917312yes—
524C22#1798918012yes—
525C22#1798918312yes—
526C22#1798918912yes—
527C22#1798920512yes—
528C22#1798920712yes—
529C22#1798932411yescharacteristic
530C22#1798932412yes—
531C22#1798932411yescharacteristic
532C22#1798932711yescharacteristic
533C22#1798933312yes—
534C22#1798933312yes—
535C22#1798933412yes—
536C22#1798933812yes—
537C22#1798933812yes—
538C22#1798934011yescharacteristic
539C4—42no—
540C4—42no—
541C4—44no—
542C4—44no—
543C4—2924no—
544C4—5842no—
545C4—5842no—
546C4—5842no—
547C2—82no—
548C2#2261268911yescharacteristic
549C2#2261272512yes—
550C2#2261307911yescharacteristic
551C2#2261309712yes—
552C2#2261310811yescharacteristic
553C2#2261803212yes—
554C2#2261807012yes—
555C2#2261814012yes—
556C2#2261815312yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#399×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#399D73
hypocenterC73#14646
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. #22608193C2semidirect
2. #22608184C2semidirect
3. #22608184C2semidirect
8. #22608163C2semidirect
30. #399×C73C4semidirect
288. C73#14646semidirect
298. #399Dic73semidirect

Maximal subgroups

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

subgroupindex
#226081932
#226081842
#226081802
#226081762
#226081682
#226081632
C2×#399×C732
#1464673