Finite Group Databasebeta

← #33017118

Subgroups — #33017118

Order 18688.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988448C211yesmaximal, characteristic
2C2×#17988445C212yesmaximal
3#22608195C212yesmaximal, complemented
4#22608194C212yesmaximal, complemented
5#22608189C212yesmaximal, complemented
6#22608186C212yesmaximal, complemented
7#22608176C211yesmaximal, characteristic
8#22608174C212yesmaximal
9C73×#1439C211yesmaximal, characteristic
10C24×Dic73C2211yescharacteristic
11C22×C4⋊Dic73C2211yescharacteristic
12C22×#11924007C2212yes—
13C22×#11924007C2212yes—
14C22×#11924007C2212yes—
15C22×C22⋊Dic73C2212yes—
16C22×C22⋊Dic73C2211yescharacteristic
17#17988448C418yescomplemented
18#17988445—28no—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C22×C4⋊C4×C73C2211yescharacteristic
29C22×C22⋊C4×C73C2212yes—
30C22×C22⋊C4×C73C2211yescharacteristic
31C2×#95×C73C2212yes—
32C2×#95×C73C2211yescharacteristic
33C23×Dic73—41no—
34C23×Dic73—41no—
35C23×Dic73—41no—
36C23×Dic73—41no—
37C23×Dic73C2×C414yes—
38C23×Dic73D414yes—
39C23×Dic73Q814yes—
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2312yes—
43C23×Dic73C2312yes—
44C2×C4⋊Dic73—42no—
45C2×C4⋊Dic73—41no—
46C2×#11924007—24no—
47C2×#11924007—24no—
48C2×#11924007—42no—
49C2×#11924007—42no—
50C2×#11924007—42no—
51C2×#11924007—42no—
52C2×#11924007—42no—
53C2×#11924007—42no—
54C2×#11924007—42no—
55C2×C22⋊Dic73—24no—
56C2×C22⋊Dic73—42no—
57C2×C22⋊Dic73—42no—
58C2×C22⋊Dic73—41no—
59C2×C22⋊Dic73—41no—
60C2×C22⋊Dic73C2×C414yes—
61#14556498—24no—
62#14556498—24no—
63#14556498—41no—
64#14556498—42no—
65#14556498—42no—
66#14556498—42no—
67#14556498—42no—
68#14556498C2×C418yes—
69#14556498C2×C418yes—
70C25×C73C2311yescharacteristic
71C23×C4×C73C2312yes—
72C23×C4×C73C2312yes—
73C23×C4×C73C2311yescharacteristic
74C23×C4×C73C2311yescharacteristic
75C2×C4⋊C4×C73—41no—
76C2×C4⋊C4×C73—42no—
77C2×C22⋊C4×C73—24no—
78C2×C22⋊C4×C73—41no—
79C2×C22⋊C4×C73—41no—
80C2×C22⋊C4×C73—42no—
81C2×C22⋊C4×C73—42no—
82C2×C22⋊C4×C73C2×C414yes—
83#95×C73—41no—
84#95×C73—42no—
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—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—44no—
110C22×Dic73—44no—
111C22×Dic73—44no—
112C22×Dic73—44no—
113C22×Dic73—44no—
114C22×Dic73—44no—
115C22×Dic73C4⋊C4116yescomplemented
116C22×Dic73C22×C414yes—
117C22×Dic73C22×C414yes—
118C22×Dic73C2×D414yes—
119C22×Dic73C2×Q814yes—
120C4⋊Dic73—44no—
121#11924007—48no—
122#11924007—48no—
123#11924007—48no—
124C22⋊Dic73—44no—
125C22⋊Dic73—48no—
126C24×C73—22no—
127C24×C73—22no—
128C24×C73—21no—
129C24×C73—21no—
130C24×C73—21no—
131C24×C73—21no—
132C24×C73—22no—
133C24×C73—22no—
134C24×C73C22×C412yes—
135C24×C73C2×D412yes—
136C24×C73C2×Q812yes—
137C24×C73C2411yescharacteristic
138C22×C4×C73—21no—
139C22×C4×C73—22no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
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—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73C22×C412yes—
161C22×C4×C73C22×C412yes—
162C4⋊C4×C73—44no—
163C22⋊C4×C73—44no—
164C22⋊C4×C73—48no—
165C23×C73—22no—
166C23×C73—22no—
167C23×C73—24no—
168C23×C73—24no—
169C23×C73—22no—
170C23×C73—22no—
171C23×C73—24no—
172C23×C73—24no—
173C23×C73—22no—
174C23×C73—22no—
175C23×C73—24no—
176C23×C73—24no—
177C23×C73—41no—
178C23×C73—42no—
179C23×C73—41no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—41no—
185C23×C73—42no—
186C23×C73—41no—
187C23×C73C2×C4⋊C414yes—
188C23×C73C23×C411yescharacteristic
189C23×C73C22×D411yescharacteristic
190C23×C73C22×Q811yescharacteristic
191C23×C73H(5,𝔽2)11yescharacteristic
192C23×C73H(5,𝔽2)11yescharacteristic
193C23×C73H(5,𝔽2)12yes—
194C23×C73H(5,𝔽2)12yes—
195C23×C73H(5,𝔽2)11yescharacteristic
196C23×C73H(5,𝔽2)11yescharacteristic
197C23×C732−1+412yes—
198C23×C732−1+412yes—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×Dic73—24no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—24no—
228C2×Dic73—24no—
229C2×Dic73—24no—
230C2×Dic73—24no—
231C2×Dic73—24no—
232C2×Dic73—416no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—42no—
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×Dic73C2×C4⋊C418yes—
252C22×C73—24no—
253C22×C73—28no—
254C22×C73—44no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—44no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—44no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73C22×C4⋊C411yescharacteristic
264C22×C73#51811yescharacteristic
265C22×C73#51811yescharacteristic
266C22×C73#52012yes—
267C22×C73#52012yes—
268C22×C73#53411yescharacteristic
269C22×C73#53411yescharacteristic
270C22×C73#53512yes—
271C22×C73#53612yes—
272C22×C73#54311yescharacteristic
273C22×C73#54311yescharacteristic
274C22×C73#54412yes—
275C22×C73#54412yes—
276C22×C73#56011yescharacteristic
277C22×C73#56012yes—
278C22×C73#56011yescharacteristic
279C22×C73#56212yes—
280C22×C73#56212yes—
281C22×C73#56212yes—
282C22×C73#56212yes—
283C22×C73#56311yescharacteristic
284C22×C73#56312yes—
285C22×C73#56311yescharacteristic
286C4×C73—42no—
287C4×C73—42no—
288C4×C73—82no—
289C4×C73—82no—
290Dic73—28no—
291Dic73—44no—
292Dic73—82no—
293Dic73—82no—
294#14674—731nomaximal, Sylow
295C2×C73—44no—
296C2×C73#231511yescharacteristic
297C2×C73#231712yes—
298C2×C73#237611yescharacteristic
299C2×C73#237612yes—
300C2×C73#237611yescharacteristic
301C2×C73#281511yescharacteristic
302C2×C73#281612yes—
303C2×C73#281711yescharacteristic
304C2×C73#284111yescharacteristic
305C2×C73#284112yes—
306C2×C73#284111yescharacteristic
307C2×#381—731no—
308C2×#380—732no—
309#1444—732no—
310#1444—732no—
311#1442—732no—
312#1442—732no—
313#1439—732no—
314#1439—731no—
315#1439D7311yescharacteristic
316C73#1467411yescharacteristic, complemented, Sylow
317C24×C4—731no—
318C22×C4⋊C4—732no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—732no—
321C22×C4⋊C4—732no—
322C22×C4⋊C4C2×D7311yescharacteristic
323C22×C22⋊C4—731no—
324C22×C22⋊C4—732no—
325C22×C22⋊C4C2×D7312yes—
326C22×C22⋊C4C2×D7311yescharacteristic
327#381—738no—
328#380—1468no—
329C2×#95—732no—
330C2×#95—731no—
331C2×#95—732no—
332C2×#95—732no—
333C2×#95—732no—
334C2×#95—732no—
335C2×#95—732no—
336C2×#95—732no—
337C2×#95—732no—
338C2×#95C2×D7311yescharacteristic
339C2×#95C2×D7312yes—
340C25C22×D7311yescharacteristic
341C23×C4—2921no—
342C23×C4—2921no—
343C23×C4—2921no—
344C23×C4—2921no—
345C23×C4—734no—
346C23×C4—734no—
347C23×C4—734no—
348C23×C4—732no—
349C23×C4—732no—
350C23×C4—732no—
351C23×C4—732no—
352C23×C4C22×D7312yes—
353C23×C4C22×D7312yes—
354C23×C4C22×D7311yescharacteristic
355C23×C4C22×D7311yescharacteristic
356C2×C4⋊C4—42no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—2922no—
359C2×C4⋊C4—2922no—
360C2×C4⋊C4—2922no—
361C2×C4⋊C4—2922no—
362C2×C4⋊C4—2922no—
363C2×C4⋊C4—2922no—
364C2×C4⋊C4—2922no—
365C2×C4⋊C4—2922no—
366C2×C4⋊C4—2921no—
367C2×C4⋊C4—1464no—
368C2×C4⋊C4—1464no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—41no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—2922no—
375C2×C22⋊C4—2922no—
376C2×C22⋊C4—2921no—
377C2×C22⋊C4—2921no—
378C2×C22⋊C4—734no—
379C2×C22⋊C4—1464no—
380C2×C22⋊C4C4×D7314yes—
381#95—41no—
382#95—42no—
383#95—2921no—
384#95—2922no—
385#95—2922no—
386#95—2922no—
387#95—2922no—
388#95—738no—
389#95—738no—
390#95—1464no—
391#95—1464no—
392C24—22no—
393C24—22no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24C2×C4×D7312yes—
401C24D4×D7312yes—
402C24Q8×D7312yes—
403C24C23×D7311yescharacteristic
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—2924no—
427C22×C4—2924no—
428C22×C4—2924no—
429C22×C4—2924no—
430C22×C4—2924no—
431C22×C4—2924no—
432C22×C4—7316no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—734no—
436C22×C4—734no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
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×C4C2×C4×D7312yes—
462C22×C4C2×C4×D7312yes—
463C4⋊C4—44no—
464C4⋊C4—2928no—
465C4⋊C4—2924no—
466C4⋊C4—2928no—
467C4⋊C4—2928no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—2928no—
471C22⋊C4—2924no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23C4⋊C4×D7314yes—
495C23C22×C4×D7311yescharacteristic
496C23C2×D4×D7311yescharacteristic
497C23C2×Q8×D7311yescharacteristic
498C23#1455663611yescharacteristic
499C23#1455663611yescharacteristic
500C23#1455663612yes—
501C23#1455663612yes—
502C23#1455663711yescharacteristic
503C23#1455663711yescharacteristic
504C23#1455663812yes—
505C23#1455663812yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2922no—
544C2×C4—2922no—
545C2×C4—2922no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—29216no—
549C2×C4—738no—
550C2×C4—1464no—
551C2×C4—1464no—
552C2×C4—1464no—
553C2×C4—1464no—
554C2×C4—1464no—
555C2×C4—1464no—
556C2×C4—1464no—
557C2×C4—1464no—
558C2×C4—1464no—
559C22—24no—
560C22—28no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22C2×C4⋊C4×D7311yescharacteristic
571C22#1798906011yescharacteristic
572C22#1798906111yescharacteristic
573C22#1798906612yes—
574C22#1798906612yes—
575C22#1798914511yescharacteristic
576C22#1798914511yescharacteristic
577C22#1798915212yes—
578C22#1798915912yes—
579C22#1798921211yescharacteristic
580C22#1798921211yescharacteristic
581C22#1798921612yes—
582C22#1798921612yes—
583C22#1798932411yescharacteristic
584C22#1798932512yes—
585C22#1798932611yescharacteristic
586C22#1798933112yes—
587C22#1798933212yes—
588C22#1798933212yes—
589C22#1798933412yes—
590C22#1798933711yescharacteristic
591C22#1798933912yes—
592C22#1798934011yescharacteristic
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—2924no—
598C4—5842no—
599C4—5842no—
600C4—1468no—
601C2—44no—
602C2#2261254411yescharacteristic
603C2#2261255012yes—
604C2#2261307411yescharacteristic
605C2#2261307712yes—
606C2#2261308111yescharacteristic
607C2#2261795611yescharacteristic
608C2#2261796312yes—
609C2#2261796911yescharacteristic
610C2#2261819111yescharacteristic
611C2#2261819212yes—
612C2#2261819311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#14674
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #22608195C2semidirect
4. #22608194C2semidirect
5. #22608189C2semidirect
6. #22608186C2semidirect
17. #17988448C4semidirect
115. C22×Dic73C4⋊C4semidirect
316. C73#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#179884482
C2×#179884452
#226081952
#226081942
#226081892
#226081862
#226081762
#226081742
C73×#14392
#1467473