Finite Group Databasebeta

← #33096018

Subgroups — #33096018

Order 18688.

1094 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Dic73C211yesmaximal, characteristic
2#22610547C212yesmaximal, complemented
3#22610541C212yesmaximal, complemented
4#22610540C212yesmaximal, complemented
5#22610534C212yesmaximal, complemented
6#22609909C211yesmaximal, characteristic, complemented
7#22609909C211yesmaximal, characteristic, complemented
8#22609841C211yesmaximal, characteristic
9#22609695C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic73C211yesmaximal, characteristic, complemented
11C73×#1779C211yesmaximal, characteristic
12C22×C22⋊Dic73C411yescharacteristic
13C22×C22⋊Dic73C411yescharacteristic
14C22×C4×Dic73C2211yescharacteristic
15C22×C4×Dic73C2211yescharacteristic
16C2×#14556520C2212yescomplemented
17C2×#14556520C2212yescomplemented
18C2×C22⋊#10596831C2211yescharacteristic
19C2×C22⋊#10596831C2211yescharacteristic
20C2×C22⋊#10596831C2212yes—
21C2×C22⋊#10596831C2212yes—
22C2×#14556501C2212yescomplemented
23C2×C8×Dic73C2212yescomplemented
24C4×C22⋊Dic73C411yescharacteristic
25C4×C22⋊Dic73C411yescharacteristic
26C4×C22⋊Dic73C411yescharacteristic
27C4×C22⋊Dic73C411yescharacteristic
28C4×C22⋊Dic73C2211yescharacteristic
29C4×C22⋊Dic73C2211yescharacteristic
30C4×C22⋊Dic73C2211yescharacteristic
31C4×C22⋊Dic73C2211yescharacteristic
32C2×#14556498C411yescharacteristic
33C2×#14556498C411yescharacteristic
34#17988341C2212yescomplemented
35#17988341C2212yescomplemented
36#17988341C2212yescomplemented
37#17988341C2212yescomplemented
38C24×C4×C73C2211yescharacteristic
39C2×C22⋊C8×C73C2211yescharacteristic
40C2×C22⋊C8×C73C2211yescharacteristic
41C2×C22⋊C8×C73C2212yes—
42C2×C22⋊C8×C73C2212yes—
43C23×Dic73C2×C411yescharacteristic
44C23×Dic73C2×C411yescharacteristic
45C23×Dic73C2×C411yescharacteristic
46C23×Dic73C2×C411yescharacteristic
47C22×#10596831C2312yes—
48C22×#10596831C2312yes—
49C2×C22⋊Dic73—24no—
50C2×C22⋊Dic73—24no—
51C2×C22⋊Dic73C2×C411yescharacteristic
52C2×C22⋊Dic73C2×C411yescharacteristic
53C2×C22⋊Dic73C2×C411yescharacteristic
54C2×C22⋊Dic73C2×C411yescharacteristic
55C2×C22⋊Dic73C2×C411yescharacteristic
56C2×C22⋊Dic73C2×C411yescharacteristic
57C2×C22⋊Dic73C2×C411yescharacteristic
58C2×C22⋊Dic73C2×C411yescharacteristic
59C2×C4×Dic73—42no—
60C2×C4×Dic73—42no—
61C2×C4×Dic73C2×C411yescharacteristic
62C2×C4×Dic73C2×C411yescharacteristic
63C2×C4×Dic73C2×C411yescharacteristic
64C2×C4×Dic73C2×C411yescharacteristic
65C2×C4×Dic73C2311yescharacteristic
66C2×C4×Dic73C2311yescharacteristic
67C2×C4×Dic73C2311yescharacteristic
68C2×C4×Dic73C2311yescharacteristic
69#14556520—42no—
70#14556520—42no—
71C22⋊#10596831—22no—
72C22⋊#10596831—22no—
73C22⋊#10596831—42no—
74C22⋊#10596831—42no—
75#14556501—42no—
76C8×Dic73—42no—
77#14556498C2×C411yescharacteristic
78#14556498C2×C411yescharacteristic
79#14556498C2×C411yescharacteristic
80#14556498C2×C411yescharacteristic
81#14556498C2×C411yescharacteristic
82#14556498C2×C411yescharacteristic
83#14556498C2×C411yescharacteristic
84#14556498C2×C411yescharacteristic
85C25×C73C2×C411yescharacteristic
86C23×C4×C73—22no—
87C23×C4×C73—22no—
88C23×C4×C73—42no—
89C23×C4×C73—42no—
90C23×C4×C73C2×C411yescharacteristic
91C23×C4×C73C2×C411yescharacteristic
92C23×C4×C73C2×C411yescharacteristic
93C23×C4×C73C2311yescharacteristic
94C23×C4×C73C2311yescharacteristic
95C23×C4×C73C2311yescharacteristic
96C22×C8×C73C2312yes—
97C22×C8×C73C2312yes—
98C22⋊C8×C73—22no—
99C22⋊C8×C73—22no—
100C22⋊C8×C73—42no—
101C22⋊C8×C73—42no—
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—41no—
111C22×Dic73—41no—
112C22×Dic73—41no—
113C22×Dic73—41no—
114C22×Dic73C22×C411yescharacteristic
115C22×Dic73C22×C411yescharacteristic
116C22×Dic73C22×C411yescharacteristic
117C22×Dic73C22×C411yescharacteristic
118C22×Dic73C22×C411yescharacteristic
119C22×Dic73C22×C411yescharacteristic
120C22×Dic73C22×C411yescharacteristic
121C22×Dic73C22×C411yescharacteristic
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C22⋊Dic73—22no—
129C22⋊Dic73—22no—
130C22⋊Dic73—22no—
131C22⋊Dic73—22no—
132C22⋊Dic73—22no—
133C22⋊Dic73—22no—
134C22⋊Dic73M1614yes—
135C22⋊Dic73M1614yes—
136C4×Dic73—22no—
137C4×Dic73—22no—
138C4×Dic73—22no—
139C4×Dic73—22no—
140C4×Dic73—41no—
141C4×Dic73—41no—
142C4×Dic73—41no—
143C4×Dic73—41no—
144C24×C73—22no—
145C24×C73—22no—
146C24×C73—22no—
147C24×C73—22no—
148C24×C73—22no—
149C24×C73—41no—
150C24×C73—41no—
151C24×C73C22×C411yescharacteristic
152C24×C73C22×C411yescharacteristic
153C24×C73C22×C411yescharacteristic
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—41no—
168C22×C4×C73—41no—
169C22×C4×C73—41no—
170C22×C4×C73—41no—
171C22×C4×C73—41no—
172C22×C4×C73—41no—
173C22×C4×C73—48no—
174C22×C4×C73—48no—
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C411yescharacteristic
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73#4112yes—
179C22×C4×C73#4112yes—
180C22×C4×C73C2411yescharacteristic
181C2×C8×C73—22no—
182C2×C8×C73—22no—
183C2×C8×C73—22no—
184C2×C8×C73—22no—
185C2×C8×C73—22no—
186C2×C8×C73—22no—
187C23×C73—21no—
188C23×C73—21no—
189C23×C73—21no—
190C23×C73—21no—
191C23×C73—28no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—44no—
205C23×C73—44no—
206C23×C73—44no—
207C23×C73—44no—
208C23×C73—44no—
209C23×C73—44no—
210C23×C73#11712yes—
211C23×C73C2×M1612yes—
212C23×C73C2×M1612yes—
213C23×C73#13112yes—
214C23×C73#13112yes—
215C23×C73C23×C411yescharacteristic
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—44no—
227C2×C4×C73—44no—
228C2×C4×C73—44no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73#11712yes—
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73H(5,𝔽2)12yes—
237C2×C4×C73H(5,𝔽2)12yes—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—44no—
259C2×Dic73—44no—
260C2×Dic73—44no—
261C2×Dic73—44no—
262C2×Dic73C2×M1612yes—
263C2×Dic73C2×M1612yes—
264C2×Dic73C2×M1612yes—
265C2×Dic73C2×M1612yes—
266C8×C73—42no—
267C8×C73—42no—
268#10596831—42no—
269#10596831—42no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—416no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73#43314yes—
292C22×C73C2×#11711yescharacteristic
293C22×C73#51811yescharacteristic
294C22×C73#51811yescharacteristic
295C22×C73C22×M1611yescharacteristic
296C22×C73C22×M1611yescharacteristic
297C22×C73C2×#13111yescharacteristic
298C22×C73C2×#13111yescharacteristic
299C4×C73—22no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73#53812yes—
303C4×C73#53812yes—
304Dic73—22no—
305Dic73—22no—
306Dic73—22no—
307Dic73—22no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—41no—
312#21904—731nomaximal, Sylow
313C2×C73—24no—
314C2×C73—42no—
315C2×C73—42no—
316C2×C73#231211yescharacteristic
317C2×C73C2×#43311yescharacteristic
318C2×C73#292011yescharacteristic
319C2×C73#297411yescharacteristic
320C2×C73#297411yescharacteristic
321C2×C73#297411yescharacteristic
322C2×C73#297411yescharacteristic
323C2×C4×C22⋊C4—731no—
324#1924—732no—
325#1924—732no—
326#1923—732no—
327#1923—732no—
328#1797—731no—
329#1797—731no—
330#1779—731no—
331#1779D7311yescharacteristic
332#1749—731no—
333C4×C22⋊C8—731no—
334C73#2190411yescharacteristic, complemented, Sylow
335C24×C4C2×D7311yescharacteristic
336C22×C22⋊C4—731no—
337C22×C22⋊C4—731no—
338C22×C42—731no—
339C22×C42—731no—
340C2×C4⋊C8—732no—
341C2×C4⋊C8—732no—
342C2×C22⋊C8—732no—
343C2×C22⋊C8—732no—
344C2×C22⋊C8—731no—
345C2×C22⋊C8—731no—
346C2×C22⋊C8C2×D7311yescharacteristic
347C2×C22⋊C8C2×D7311yescharacteristic
348C2×C22⋊C8C2×D7312yes—
349C2×C22⋊C8C2×D7312yes—
350C2×#97—732no—
351C2×C4×C8—732no—
352C4×C22⋊C4—731no—
353C4×C22⋊C4—731no—
354C4×C22⋊C4—731no—
355C4×C22⋊C4—731no—
356C4×C22⋊C4—731no—
357C4×C22⋊C4—731no—
358C4×C22⋊C4—731no—
359C4×C22⋊C4—731no—
360C2×#95—731no—
361C2×#95—731no—
362#336—732no—
363#336—732no—
364#336—732no—
365#336—732no—
366C25C4×D7311yescharacteristic
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—42no—
371C23×C4—731no—
372C23×C4—731no—
373C23×C4—731no—
374C23×C4—731no—
375C23×C4C4×D7311yescharacteristic
376C23×C4C4×D7311yescharacteristic
377C23×C4C4×D7311yescharacteristic
378C23×C4C22×D7311yescharacteristic
379C23×C4C22×D7311yescharacteristic
380C23×C4C22×D7311yescharacteristic
381C22×C8—732no—
382C22×C8—732no—
383C22×C8C22×D7312yes—
384C22×C8C22×D7312yes—
385C2×C22⋊C4—731no—
386C2×C22⋊C4—731no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4—731no—
391C2×C22⋊C4—731no—
392C2×C22⋊C4—731no—
393C2×C22⋊C4—1464no—
394C2×C22⋊C4—1464no—
395C2×C42—2922no—
396C2×C42—2922no—
397C2×C42—731no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42—731no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C4⋊C8—2922no—
406C4⋊C8—2922no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—42no—
410C22⋊C8—42no—
411C22⋊C8—2922no—
412C22⋊C8—2922no—
413C22⋊C8—1462no—
414C22⋊C8—1462no—
415#97—2922no—
416C4×C8—2922no—
417#95—731no—
418#95—731no—
419#95—731no—
420#95—731no—
421#95—731no—
422#95—731no—
423#95—731no—
424#95—731no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—41no—
431C24—41no—
432C24C2×C4×D7311yescharacteristic
433C24C2×C4×D7311yescharacteristic
434C24C2×C4×D7311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—2921no—
457C22×C4—2921no—
458C22×C4—2921no—
459C22×C4—2921no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4C2×C4×D7311yescharacteristic
477C22×C4C2×C4×D7311yescharacteristic
478C22×C4C2×C4×D7311yescharacteristic
479C22×C4#1192402712yes—
480C22×C4#1192402712yes—
481C22×C4C23×D7311yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—1462no—
489C2×C8—1462no—
490C2×C8—1462no—
491C2×C8—1462no—
492C2×C8—1462no—
493C2×C8—1462no—
494C22⋊C4—734no—
495C22⋊C4—734no—
496C22⋊C4—1462no—
497C22⋊C4—1462no—
498C22⋊C4—1462no—
499C22⋊C4—1462no—
500C22⋊C4—1462no—
501C22⋊C4—1462no—
502C42—2921no—
503C42—2921no—
504C42—2921no—
505C42—2921no—
506C42—1462no—
507C42—1462no—
508C42—1462no—
509C42—1462no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—28no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23#1455654712yes—
534C23M16×D7312yes—
535C23M16×D7312yes—
536C23#1455659712yes—
537C23#1455659912yes—
538C23C22×C4×D7311yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—2924no—
557C2×C4—2924no—
558C2×C4—2924no—
559C2×C4—2924no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4#1455654712yes—
585C2×C4C2×#1192402711yescharacteristic
586C2×C4C2×#1192402711yescharacteristic
587C2×C4#1455663612yes—
588C2×C4#1455663612yes—
589C8—42no—
590C8—42no—
591C8—2922no—
592C8—2922no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—416no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22#1798864514yes—
615C22C2×#1455654711yescharacteristic
616C22#1798906011yescharacteristic
617C22#1798906011yescharacteristic
618C22C2×M16×D7311yescharacteristic
619C22C2×M16×D7311yescharacteristic
620C22C2×#1455659711yescharacteristic
621C22C2×#1455659911yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—2921no—
626C4—2921no—
627C4—2921no—
628C4—2921no—
629C4—1462no—
630C4—1462no—
631C4—1462no—
632C4—1462no—
633C4#1798917212yes—
634C4#1798917212yes—
635C2—24no—
636C2—42no—
637C2—42no—
638C2#2261252711yescharacteristic
639C2C2×#1798864511yescharacteristic
640C2#2261856011yescharacteristic
641C2#2261896211yescharacteristic
642C2#2261896211yescharacteristic
643C2#2261896711yescharacteristic
644C2#2261896711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1779C2
prime radicalC25×C73C2×C4
cosocleC22×C4×C73C24
hypercenter#1779D73
hypocenterC73#21904
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #22610547C2semidirect
3. #22610541C2semidirect
4. #22610540C2semidirect
5. #22610534C2semidirect
6. #22609909C2semidirect
7. #22609909C2semidirect
9. #22609695C2semidirect
10. C22⋊C8×Dic73C2semidirect
16. C2×#14556520C22semidirect
17. C2×#14556520C22semidirect
22. C2×#14556501C22semidirect
23. C2×C8×Dic73C22semidirect
34. #17988341C22semidirect
35. #17988341C22semidirect
36. #17988341C22semidirect
37. #17988341C22semidirect
334. C73#21904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic732
#226105472
#226105412
#226105402
#226105342
#226099092
#226098412
#226096952
C22⋊C8×Dic732
C73×#17792
#2190473