Finite Group Databasebeta

← #33104091

Subgroups — #33104091

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988652C211yesmaximal, characteristic
2C2×#17988624C211yesmaximal, characteristic, complemented
3C2×#106×D73C211yesmaximal, characteristic, complemented
4#22612863C211yesmaximal, characteristic
5#22612857C211yesmaximal, characteristic
6#22610701C211yesmaximal, characteristic, complemented
7#22610131C211yesmaximal, characteristic, complemented
8#22610129C211yesmaximal, characteristic, complemented
9#22610120C211yesmaximal, characteristic, complemented
10#22610117C211yesmaximal, characteristic, complemented
11#22610093C211yesmaximal, characteristic, complemented
12#22609955C211yesmaximal, characteristic, complemented
13#22609955C211yesmaximal, characteristic, complemented
14C4×#14556522C211yesmaximal, characteristic, complemented
15C73×#1949C211yesmaximal, characteristic, complemented
16C22×C8×D73C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×#14556549C2211yescharacteristic
20C2×#14556549C2211yescharacteristic
21C2×C4⋊C4×D73C2211yescharacteristic
22C2×C4⋊C4×D73C2211yescharacteristic
23#17988652C418yescomplemented
24#17988624—24no—
25C2×#14556525C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556523C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556522C2211yescharacteristic
31C2×#14556522C2211yescharacteristic
32C2×#14556522C2211yescharacteristic
33#106×D73—24no—
34C2×#14556503C2211yescharacteristic
35C2×#14556503C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37#17988478—21no—
38#17988476—21no—
39#17988472—21no—
40#17988470—21no—
41#17988465—21no—
42#17988465—21no—
43#17988453C2211yescharacteristic
44#17988444C2211yescharacteristic
45#17988444C2211yescharacteristic
46#17988444C2211yescharacteristic
47#17988444C2211yescharacteristic
48C4×C4⋊Dic73C2211yescharacteristic
49C4×#11924005—21no—
50#17988434—21no—
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×#107×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×C4⋊C8×C73C2211yescharacteristic
58#382×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×D292D411yescharacteristic
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C22×C4×D73C2311yescharacteristic
66C22×C4×D73C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C2×C8×D73—22no—
70C2×C8×D73—22no—
71C2×C8×D73—22no—
72C2×C8×D73—22no—
73C2×C8×D73C2×C414yes—
74C2×C8×D73C2×C414yescomplemented
75C4×D292C2×C414yes—
76C4×D292Q814yes—
77#14556549—22no—
78#14556549—22no—
79#14556549—41no—
80#14556549—41no—
81C2×C4⋊Dic73D411yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×#11924007—21no—
87C2×#11924007—21no—
88C4⋊C4×D73—22no—
89C4⋊C4×D73—22no—
90C4⋊C4×D73—41no—
91C4⋊C4×D73—41no—
92C2×#11924005—21no—
93C2×#11924005—21no—
94C2×#11924005—21no—
95C2×#11924005—21no—
96C2×#11924005—21no—
97C2×#11924005—21no—
98C2×#11924005D411yescharacteristic
99C2×#11924005D411yescharacteristic
100C2×C4×Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103#14556525—22no—
104#14556524—41no—
105#14556524—41no—
106#14556523—22no—
107#14556522—22no—
108#14556522—22no—
109#14556522—22no—
110#14556522D414yescomplemented
111#14556503C2×C414yes—
112#14556503C2×C414yescomplemented
113C4×#10596831C2×C414yescomplemented
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120#14556498—21no—
121C22×C8×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C42×C73—21no—
128C2×C42×C73C2311yescharacteristic
129#107×C73—22no—
130#106×C73—22no—
131C4⋊C8×C73C2×C414yescomplemented
132#95×C73—21no—
133#95×C73—21no—
134#95×C73—21no—
135C23×D73C2×D411yescharacteristic
136C23×D73C2×D411yescharacteristic
137C2×D292—22no—
138C2×D292—22no—
139C2×D292C4⋊C414yes—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73C2×D411yescharacteristic
145C22×Dic73C2×D411yescharacteristic
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—42no—
155C2×C4×D73—42no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73—81no—
159C2×C4×D73—81no—
160C2×C4×D73C22×C412yes—
161C2×C4×D73C22×C412yes—
162C2×C4×D73C2×Q812yes—
163C2×C4×D73C2×Q812yes—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831C22×C412yes—
167C2×#10596831C22×C412yes—
168C2×#10596831C2×D412yes—
169C2×#10596831#4112yes—
170C8×D73—28no—
171C8×D73—28no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73C4⋊C414yes—
180C4⋊Dic73C2×D412yes—
181#11924007—42no—
182#11924007—42no—
183#11924005—42no—
184#11924005—42no—
185#11924005—42no—
186#11924005—42no—
187#11924005—81no—
188#11924005—81no—
189#11924005C4⋊C414yes—
190#11924005C4⋊C414yes—
191C4×Dic73—42no—
192C4×Dic73—42no—
193C4×Dic73—81no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2411yescharacteristic
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C22×C412yes—
206C2×C8×C73C22×C412yes—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73C2×D412yes—
215C42×C73—81no—
216C42×C73C22×C412yes—
217C42×C73C2×Q812yes—
218C23×C73C22×D411yescharacteristic
219C22×D73—21no—
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C22×D73—21no—
229C22×D73C2×C4⋊C412yes—
230C22×D73C2×C4⋊C412yes—
231C22×D73#13512yes—
232C22×D73#13512yes—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—22no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73C2×C4⋊C412yes—
250C2×C4×C73C2×C4⋊C412yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×Q811yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×C4×C732−1+411yescharacteristic
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73C2×C4⋊C412yes—
274C2×Dic73C2×C4⋊C412yes—
275D292—28no—
276C4×D73—22no—
277C4×D73—22no—
278C4×D73—22no—
279C4×D73—22no—
280C4×D73—22no—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—22no—
284C4×D73—84no—
285C4×D73—84no—
286C8×C73—22no—
287C8×C73—22no—
288#10596831—22no—
289#10596831C4×D414yes—
290C22×C73C22×C4⋊C411yescharacteristic
291C22×C73D4211yescharacteristic
292C22×C73#54711yescharacteristic
293C22×C73C2×#13511yescharacteristic
294C22×C73C2×#13511yescharacteristic
295C22×C73#57711yescharacteristic
296C22×C73#57711yescharacteristic
297C2×D73—22no—
298C2×D73—22no—
299C2×D73—22no—
300C2×D73—22no—
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—41no—
307C2×D73—41no—
308C2×D73—41no—
309C2×D73—41no—
310C2×D73#42714yescomplemented
311C4×C73—22no—
312C4×C73—22no—
313C4×C73—42no—
314C4×C73—81no—
315C4×C73—81no—
316C4×C73—81no—
317C4×C73C2×C4×D411yescharacteristic
318C4×C73#52011yescharacteristic
319C4×C73D4×Q811yescharacteristic
320C4×C73#55411yescharacteristic
321Dic73—22no—
322Dic73—22no—
323Dic73—42no—
324Dic73—81no—
325Dic73—81no—
326Dic73—81no—
327#22654—731nomaximal, Sylow
328C2×C73D4×C4⋊C411yescharacteristic
329C2×C73C2×#42711yescharacteristic
330C2×C73#291211yescharacteristic
331C2×C73#328211yescharacteristic
332C2×C73#328211yescharacteristic
333C2×C73#333711yescharacteristic
334C2×C73#333711yescharacteristic
335D73—24no—
336D73—42no—
337C2×C8×D4—731no—
338C2×#427—731no—
339C22×#106—731no—
340#2352—731no—
341#2352—731no—
342#1949—731no—
343#1949D7311yescharacteristic, complemented
344#1854—731no—
345#1854—731no—
346#1852—731no—
347#1852—731no—
348#1846—731no—
349#1807—731no—
350#1807—731no—
351C4×#106—731no—
352C73#2265411yescharacteristic, complemented, Sylow
353C23×C8—731no—
354C23×C8—731no—
355C2×C4×D4—731no—
356C2×#117—731no—
357C2×#117—731no—
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C8×D4—738no—
361#427—1464no—
362C2×#107—731no—
363C2×#107—731no—
364C2×#107—731no—
365C2×#107C2×D7311yescharacteristic
366C2×#106—731no—
367C2×#106—731no—
368C2×#106—731no—
369C2×#106—731no—
370C2×#106—731no—
371C2×#106—1464no—
372C2×#106C2×D7311yescharacteristic
373C2×C4⋊C8C2×D7311yescharacteristic
374C2×C22⋊C8—731no—
375C2×C22⋊C8—731no—
376C2×C4×C8—731no—
377#388—1461no—
378#388—1461no—
379#387—1461no—
380#387—1461no—
381#386—1461no—
382#386—1461no—
383#382—731no—
384#382C2×D7311yescharacteristic
385#380—731no—
386#380—731no—
387#380—731no—
388#380—731no—
389C4×C4⋊C4—731no—
390C4×C4⋊C4C2×D7311yescharacteristic
391C4×C22⋊C4—1461no—
392C4×C22⋊C4—1461no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340C2×D7311yescharacteristic
398#340C2×D7311yescharacteristic
399C22×D4—731no—
400C23×C4—731no—
401C23×C4—731no—
402C23×C4—1461no—
403C23×C4—1461no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—734no—
407C22×C8—734no—
408C22×C8—1462no—
409C22×C8—1462no—
410C22×C8—1462no—
411C22×C8—1462no—
412C22×C8C22×D7311yescharacteristic
413C22×C8C22×D7311yescharacteristic
414C4×D4—734no—
415C4×D4—734no—
416#117—2921no—
417#117—2921no—
418#117—1462no—
419#117—1462no—
420C2×C4⋊C4—2921no—
421C2×C4⋊C4—2921no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—1462no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1462no—
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C22⋊C4—731no—
436C2×C22⋊C4—731no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4—1461no—
442C2×C22⋊C4—1461no—
443C2×C42—21no—
444C2×C42—1461no—
445C2×C42—1461no—
446C2×C42—1461no—
447C2×C42C22×D7311yescharacteristic
448#107—22no—
449#107—2921no—
450#107—2921no—
451#107—1462no—
452#106—22no—
453#106—734no—
454#106—1462no—
455#106—1462no—
456#106—1462no—
457#106—1462no—
458C4⋊C8C4×D7314yescomplemented
459C22⋊C8—734no—
460C22⋊C8—734no—
461C4×C8—734no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472C24—731no—
473C24—731no—
474C2×D4—734no—
475C2×D4—1462no—
476C2×D4—1462no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—2922no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—5841no—
486C22×C4—5841no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—732no—
490C22×C4—732no—
491C22×C4—732no—
492C22×C4—732no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4D4×D7311yescharacteristic
506C22×C4D4×D7311yescharacteristic
507C22×C4C23×D7311yescharacteristic
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—732no—
513C2×C8—732no—
514C2×C8—732no—
515C2×C8—732no—
516C2×C8—1468no—
517C2×C8—1468no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8C2×C4×D7312yes—
521C2×C8C2×C4×D7312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—2922no—
530C4⋊C4—2922no—
531C4⋊C4—734no—
532C4⋊C4—732no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4D4×D7312yes—
541C22⋊C4—2922no—
542C22⋊C4—2922no—
543C22⋊C4—2922no—
544C22⋊C4—2922no—
545C22⋊C4—5841no—
546C22⋊C4—5841no—
547C22⋊C4—734no—
548C22⋊C4—734no—
549C42—81no—
550C42—2922no—
551C42—2922no—
552C42—5841no—
553C42C2×C4×D7312yes—
554C42Q8×D7312yes—
555C23—732no—
556C23—732no—
557C23—732no—
558C23—732no—
559C23—1461no—
560C23—1461no—
561C23—1461no—
562C23—1461no—
563C23—1461no—
564C23—1461no—
565C23—1461no—
566C23—1461no—
567C23—1461no—
568C23—1461no—
569C23C2×D4×D7311yescharacteristic
570D4—1468no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—5844no—
599C2×C4—5844no—
600C2×C4—732no—
601C2×C4—732no—
602C2×C4—1462no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4C4⋊C4×D7312yes—
616C2×C4C4⋊C4×D7312yes—
617C2×C4C22×C4×D7311yescharacteristic
618C2×C4C2×D4×D7311yescharacteristic
619C2×C4C2×Q8×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#1455663711yescharacteristic
622C2×C4#1455663911yescharacteristic
623C8—22no—
624C8—22no—
625C8—734no—
626C8—1462no—
627C22—2921no—
628C22—2921no—
629C22—2921no—
630C22—2921no—
631C22—734no—
632C22—1462no—
633C22—1462no—
634C22—1462no—
635C22—1462no—
636C22—1462no—
637C22—1462no—
638C22—1462no—
639C22—1462no—
640C22—1462no—
641C22C2×C4⋊C4×D7311yescharacteristic
642C22#1798922111yescharacteristic
643C22#1798923911yescharacteristic
644C22#135×D7311yescharacteristic
645C22#135×D7311yescharacteristic
646C22#1798939611yescharacteristic
647C22#1798939711yescharacteristic
648C4—22no—
649C4—22no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—2922no—
655C4—5841no—
656C4—5841no—
657C4—5841no—
658C4—1462no—
659C4—1462no—
660C4C4×D4×D7311yescharacteristic
661C4#1798906911yescharacteristic
662C4#1798925211yescharacteristic
663C4#1798929311yescharacteristic
664C2—2922no—
665C2—1464no—
666C2#2261283211yescharacteristic
667C2#427×D7311yescharacteristic
668C2#2261852511yescharacteristic
669C2#2262163311yescharacteristic
670C2#2262164411yescharacteristic
671C2#2262230711yescharacteristic
672C2#2262231711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1949C2
socleC23×C73C22×D4
prime radicalC22×D292D4
cosocleC22×C4×C73C24
hypercenter#1949D73
hypocenterC73#22654
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. C2×#17988624C2semidirect
3. C2×#106×D73C2semidirect
6. #22610701C2semidirect
7. #22610131C2semidirect
8. #22610129C2semidirect
9. #22610120C2semidirect
10. #22610117C2semidirect
11. #22610093C2semidirect
12. #22609955C2semidirect
13. #22609955C2semidirect
14. C4×#14556522C2semidirect
15. C73×#1949C2semidirect
23. #17988652C4semidirect
74. C2×C8×D73C2×C4semidirect
110. #14556522D4semidirect
112. #14556503C2×C4semidirect
113. C4×#10596831C2×C4semidirect
131. C4⋊C8×C73C2×C4semidirect
310. C2×D73#427semidirect
343. #1949D73semidirect
352. C73#22654semidirect
458. C4⋊C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886522
C2×#179886242
C2×#106×D732
#226128632
#226128572
#226107012
#226101312
#226101292
#226101202
#226101172
#226100932
#226099552
C4×#145565222
C73×#19492
#2265473