Finite Group Databasebeta

← #33008596

Subgroups — #33008596

Order 18688.

1486 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22608193C212yesmaximal, complemented
2#22608180C212yesmaximal
3#22608168C212yesmaximal
4#22608158C211yesmaximal, characteristic, complemented
5#22608158C211yesmaximal, characteristic, complemented
6#22608158C211yesmaximal, characteristic, complemented
7#22608156C212yesmaximal, complemented
8#22608154C211yesmaximal, characteristic
9#22608154C211yesmaximal, characteristic
10C4×#14556498C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C73C211yesmaximal, characteristic
12C22×C22⋊Dic73C2212yes—
13C22×C22⋊Dic73C2212yes—
14C22×C4×Dic73C2212yes—
15C2×#14556498C2211yescharacteristic
16C2×#14556498C2211yescharacteristic
17C2×#14556498C2212yes—
18C2×#14556498C2211yescharacteristic
19C2×#14556498C2211yescharacteristic
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C24×C4×C73C2211yescharacteristic
29C22×C22⋊C4×C73C2211yescharacteristic
30C22×C22⋊C4×C73C2211yescharacteristic
31C22×C42×C73C2211yescharacteristic
32C22×C42×C73C2211yescharacteristic
33C4×C22⋊C4×C73C418yescomplemented
34C2×#95×C73C2211yescharacteristic
35C2×#95×C73C2211yescharacteristic
36C23×Dic73C2312yes—
37C23×Dic73C2312yes—
38C23×Dic73C2312yes—
39C23×Dic73C2312yes—
40C2×C22⋊Dic73—24no—
41C2×C22⋊Dic73—24no—
42C2×C22⋊Dic73—24no—
43C2×C22⋊Dic73—24no—
44C2×C22⋊Dic73—42no—
45C2×C22⋊Dic73—42no—
46C2×C4×Dic73—24no—
47C2×C4×Dic73—24no—
48C2×C4×Dic73—42no—
49#14556498—24no—
50#14556498—24no—
51#14556498—24no—
52#14556498—24no—
53#14556498—22no—
54#14556498—22no—
55#14556498—22no—
56#14556498—22no—
57#14556498—42no—
58#14556498—42no—
59#14556498—42no—
60#14556498—42no—
61#14556498—42no—
62C25×C73C2311yescharacteristic
63C23×C4×C73—22no—
64C23×C4×C73—22no—
65C23×C4×C73—42no—
66C23×C4×C73—42no—
67C23×C4×C73C2×C414yes—
68C23×C4×C73C2311yescharacteristic
69C23×C4×C73C2311yescharacteristic
70C23×C4×C73C2311yescharacteristic
71C23×C4×C73C2311yescharacteristic
72C23×C4×C73C2311yescharacteristic
73C23×C4×C73C2311yescharacteristic
74C2×C22⋊C4×C73—22no—
75C2×C22⋊C4×C73—22no—
76C2×C22⋊C4×C73—22no—
77C2×C22⋊C4×C73—22no—
78C2×C22⋊C4×C73C2×C414yes—
79C2×C22⋊C4×C73C2×C414yes—
80C2×C42×C73—42no—
81C2×C42×C73—42no—
82C2×C42×C73C2×C414yes—
83C2×C42×C73C2×C414yes—
84#95×C73C2×C414yes—
85#95×C73C2×C414yes—
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—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73#4114yes—
113C22×Dic73#4114yes—
114C22⋊Dic73—48no—
115C22⋊Dic73—48no—
116C4×Dic73—48no—
117C24×C73—21no—
118C24×C73—21no—
119C24×C73—21no—
120C24×C73—21no—
121C24×C73—22no—
122C24×C73—22no—
123C24×C73—22no—
124C24×C73—22no—
125C24×C73C22×C412yes—
126C24×C73#4112yes—
127C24×C73#4112yes—
128C24×C73C2411yescharacteristic
129C22×C4×C73—22no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—21no—
138C22×C4×C73—21no—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—28no—
150C22×C4×C73—44no—
151C22×C4×C73—44no—
152C22×C4×C73—48no—
153C22×C4×C73—48no—
154C22×C4×C73C22×C412yes—
155C22×C4×C73C22×C412yes—
156C22×C4×C73C22×C412yes—
157C22×C4×C73C22×C412yes—
158C22×C4×C73C22×C412yes—
159C22×C4×C73C22×C412yes—
160C22×C4×C73#4112yes—
161C22×C4×C73#4112yes—
162C22×C4×C73#4112yes—
163C22×C4×C73#4112yes—
164C22⋊C4×C73—28no—
165C22⋊C4×C73—28no—
166C42×C73—44no—
167C42×C73—44no—
168C23×C73—22no—
169C23×C73—22no—
170C23×C73—22no—
171C23×C73—22no—
172C23×C73—22no—
173C23×C73—22no—
174C23×C73—22no—
175C23×C73—22no—
176C23×C73—22no—
177C23×C73—22no—
178C23×C73—24no—
179C23×C73—24no—
180C23×C73—24no—
181C23×C73—24no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73#11714yes—
191C23×C73C23×C411yescharacteristic
192C23×C73C2×#4112yes—
193C23×C73C2×#4112yes—
194C23×C73C2×#4111yescharacteristic
195C23×C73C2×#4111yescharacteristic
196C23×C73C2×#4111yescharacteristic
197C23×C73C2×#4111yescharacteristic
198C23×C73C2×#4111yescharacteristic
199C23×C73C2×#4111yescharacteristic
200C23×C73H(5,𝔽2)12yes—
201C23×C73H(5,𝔽2)12yes—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—24no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—416no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73#11714yes—
232C2×C4×C73#11714yes—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C22×C73—24no—
262C22×C73—24no—
263C22×C73—24no—
264C22×C73—42no—
265C22×C73—44no—
266C22×C73—44no—
267C22×C73—44no—
268C22×C73—44no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73C2×#11711yescharacteristic
273C22×C73C2×#11711yescharacteristic
274C22×C73C2×#11711yescharacteristic
275C22×C73C4×#4111yescharacteristic
276C22×C73C4×#4111yescharacteristic
277C22×C73#51811yescharacteristic
278C22×C73#51811yescharacteristic
279C22×C73#52911yescharacteristic
280C22×C73#52911yescharacteristic
281C22×C73#52911yescharacteristic
282C22×C73#52911yescharacteristic
283C22×C73#52912yes—
284C22×C73#52912yes—
285C22×C73#53812yes—
286C22×C73#53812yes—
287C22×C73#55112yes—
288C22×C73#55112yes—
289C22×C73#55112yes—
290C22×C73#55312yes—
291C22×C73#55312yes—
292C22×C73#55312yes—
293C22×C73#55312yes—
294C22×C73#55912yes—
295C4×C73—24no—
296C4×C73—24no—
297C4×C73—42no—
298C4×C73—42no—
299C4×C73—42no—
300C4×C73—42no—
301Dic73—44no—
302Dic73—44no—
303Dic73—82no—
304Dic73—82no—
305#13947—731nomaximal, Sylow
306C2×C73—44no—
307C2×C73#228311yescharacteristic
308C2×C73#229811yescharacteristic
309C2×C73#231211yescharacteristic
310C2×C73#235311yescharacteristic
311C2×C73#235311yescharacteristic
312C2×C73#235311yescharacteristic
313C2×C73#235711yescharacteristic
314C2×C73#257612yes—
315C2×C73#257812yes—
316C2×C73#259512yes—
317C2×C73#259712yes—
318C2×C4×C22⋊C4D7311yescharacteristic
319#1443—732no—
320#1440—732no—
321#1437—732no—
322#1434—731no—
323#1434—731no—
324#1434—731no—
325#1434—732no—
326C22⋊#95—731no—
327C22⋊#95—731no—
328C4×#95—731no—
329C73#1394711yescharacteristic, complemented, Sylow
330C24×C4C2×D7311yescharacteristic
331C22×C22⋊C4—732no—
332C22×C22⋊C4—732no—
333C22×C22⋊C4C2×D7311yescharacteristic
334C22×C22⋊C4C2×D7311yescharacteristic
335C22×C42—732no—
336C22×C42C2×D7311yescharacteristic
337C22×C42C2×D7311yescharacteristic
338C4×C22⋊C4Dic7318yescomplemented
339C2×#95—732no—
340C2×#95—732no—
341C2×#95—732no—
342C2×#95—732no—
343C2×#95—732no—
344C2×#95—731no—
345C2×#95—731no—
346C2×#95—732no—
347C2×#95—731no—
348C2×#95—731no—
349C2×#95—732no—
350C2×#95—732no—
351C2×#95—732no—
352C2×#95C2×D7311yescharacteristic
353C2×#95C2×D7311yescharacteristic
354C25C22×D7311yescharacteristic
355C23×C4—22no—
356C23×C4—22no—
357C23×C4—42no—
358C23×C4—42no—
359C23×C4—732no—
360C23×C4—732no—
361C23×C4—732no—
362C23×C4—732no—
363C23×C4C2×Dic7314yes—
364C23×C4C22×D7311yescharacteristic
365C23×C4C22×D7311yescharacteristic
366C23×C4C22×D7311yescharacteristic
367C23×C4C22×D7311yescharacteristic
368C23×C4C22×D7311yescharacteristic
369C23×C4C22×D7311yescharacteristic
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—2922no—
375C2×C22⋊C4—2922no—
376C2×C22⋊C4—1464no—
377C2×C22⋊C4—1464no—
378C2×C22⋊C4—1464no—
379C2×C22⋊C4—1464no—
380C2×C22⋊C4C2×Dic7314yes—
381C2×C22⋊C4C2×Dic7314yes—
382C2×C42—42no—
383C2×C42—42no—
384C2×C42—2922no—
385C2×C42—1464no—
386C2×C42—1464no—
387C2×C42C2×Dic7314yes—
388C2×C42C2×Dic7314yes—
389#95—2922no—
390#95—2922no—
391#95—2922no—
392#95—2922no—
393#95—2922no—
394#95—1464no—
395#95—1464no—
396#95—1464no—
397#95—1464no—
398#95—1462no—
399#95—1462no—
400#95—1462no—
401#95—1462no—
402#95C2×Dic7314yes—
403#95C2×Dic7314yes—
404C24—22no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24C22×Dic7312yes—
413C24#1192402812yes—
414C24#1192402812yes—
415C24C23×D7311yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—734no—
442C22×C4—734no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4C22×Dic7312yes—
470C22×C4C22×Dic7312yes—
471C22×C4C22×Dic7312yes—
472C22×C4C22×Dic7312yes—
473C22×C4C22×Dic7312yes—
474C22×C4C22×Dic7312yes—
475C22×C4#1192402812yes—
476C22×C4#1192402812yes—
477C22×C4#1192402812yes—
478C22×C4#1192402812yes—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—2928no—
482C22⋊C4—2928no—
483C42—44no—
484C42—44no—
485C42—2928no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#1455655014yes—
509C23C23×Dic7311yescharacteristic
510C23#41×D7312yes—
511C23#41×D7312yes—
512C23C2×#1192402811yescharacteristic
513C23C2×#1192402811yescharacteristic
514C23C2×#1192402811yescharacteristic
515C23C2×#1192402811yescharacteristic
516C23C2×#1192402811yescharacteristic
517C23C2×#1192402811yescharacteristic
518C23#1455663612yes—
519C23#1455663612yes—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—2922no—
550C2×C4—2922no—
551C2×C4—2922no—
552C2×C4—2922no—
553C2×C4—2922no—
554C2×C4—2922no—
555C2×C4—2922no—
556C2×C4—2922no—
557C2×C4—2922no—
558C2×C4—2922no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—1464no—
572C2×C4—1464no—
573C2×C4—1464no—
574C2×C4—1464no—
575C2×C4—1464no—
576C2×C4—1464no—
577C2×C4#1455655014yes—
578C2×C4#1455655014yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22C2×#1455655011yescharacteristic
591C22C2×#1455655011yescharacteristic
592C22C2×#1455655011yescharacteristic
593C22#41×Dic7311yescharacteristic
594C22#41×Dic7311yescharacteristic
595C22#1798906211yescharacteristic
596C22#1798906211yescharacteristic
597C22#1798911512yes—
598C22#1798911512yes—
599C22#1798912111yescharacteristic
600C22#1798912111yescharacteristic
601C22#1798912111yescharacteristic
602C22#1798912111yescharacteristic
603C22#1798917512yes—
604C22#1798917512yes—
605C22#1798926612yes—
606C22#1798926612yes—
607C22#1798926612yes—
608C22#1798928612yes—
609C22#1798928612yes—
610C22#1798928612yes—
611C22#1798928712yes—
612C22#1798932012yes—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—2924no—
620C4—2924no—
621C4—5842no—
622C4—5842no—
623C2—44no—
624C2#2261232911yescharacteristic
625C2#2261242911yescharacteristic
626C2#2261252911yescharacteristic
627C2#2261287111yescharacteristic
628C2#2261287111yescharacteristic
629C2#2261287111yescharacteristic
630C2#2261290411yescharacteristic
631C2#2261522912yes—
632C2#2261525912yes—
633C2#2261544612yes—
634C2#2261547612yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×C4×C22⋊C4×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×C4×C22⋊C4D73
hypocenterC73#13947
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22608193C2semidirect
4. #22608158C2semidirect
5. #22608158C2semidirect
6. #22608158C2semidirect
7. #22608156C2semidirect
10. C4×#14556498C2semidirect
33. C4×C22⋊C4×C73C4semidirect
329. C73#13947semidirect
338. C4×C22⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
#226081932
#226081802
#226081682
#226081582
#226081562
#226081542
C4×#145564982
C2×C4×C22⋊C4×C732
#1394773