Finite Group Databasebeta

← #33084062

Subgroups — #33084062

Order 18688.

1126 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988628C211yesmaximal, characteristic
2C2×#17988626C211yesmaximal, characteristic
3C2×#17988544C211yesmaximal, characteristic
4#22612285C211yesmaximal, characteristic
5#22612285C211yesmaximal, characteristic
6#22609828C211yesmaximal, characteristic, complemented
7#22609827C211yesmaximal, characteristic, complemented
8#22609827C211yesmaximal, characteristic, complemented
9#107×Dic73C211yesmaximal, characteristic, complemented
10C4×#14556524C211yesmaximal, characteristic, complemented
11#106×Dic73C211yesmaximal, characteristic, complemented
12C4×#14556522C211yesmaximal, characteristic, complemented
13#22609646C211yesmaximal, characteristic
14#340×D73C211yesmaximal, characteristic
15C73×#1776C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19C2×#14556549C2211yescharacteristic
20C2×#14556548C2211yescharacteristic
21C2×C4⋊C4×D73C2211yescharacteristic
22C2×C4⋊C4×D73C2211yescharacteristic
23#17988628C418yescomplemented
24#17988626C418yescomplemented
25C2×#14556525C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556523C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556522C2211yescharacteristic
31#17988544C418yescomplemented
32C2×#14556501C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35C4⋊C4×Dic73C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×#11924007—21no—
40C4×#11924007—21no—
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988438—21no—
44#17988438—21no—
45C4×#11924005—21no—
46C4×#11924005—21no—
47#17988434—21no—
48#17988434—21no—
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×#107×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×#97×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C22×C4×D73C2311yescharacteristic
65C2×#11924011C2×C414yes—
66C2×#11924011C2×C414yes—
67C2×#11924011C2×C414yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C2×C8×D73—22no—
71C2×C8×D73—22no—
72C2×C8×D73C2×C414yes—
73#14556549—22no—
74#14556549—22no—
75#14556549C2×C414yes—
76#14556549C2×C414yes—
77#14556548C2×C414yes—
78#14556548Q814yes—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007D411yescharacteristic
86C2×#11924007D411yescharacteristic
87C4⋊C4×D73—22no—
88C4⋊C4×D73—22no—
89C4⋊C4×D73C2×C414yes—
90C4⋊C4×D73C2×C414yes—
91C2×#11924005—21no—
92C2×#11924005—21no—
93C2×#11924005D411yescharacteristic
94C2×#11924005D411yescharacteristic
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73—21no—
100C2×C4×Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103C2×C4×Dic73C2311yescharacteristic
104#14556525C2×C414yescomplemented
105#14556524C2×C414yescomplemented
106#14556524C2×C414yescomplemented
107#14556523C2×C414yescomplemented
108#14556522C2×C414yescomplemented
109#14556522C2×C414yescomplemented
110#14556501C2×C414yescomplemented
111C4×#10596831C2×C414yescomplemented
112C8×Dic73C2×C414yescomplemented
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C42×C73—21no—
126C2×C42×C73—21no—
127C2×C42×C73C2311yescharacteristic
128#107×C73C2×C414yescomplemented
129#106×C73C2×C414yescomplemented
130#97×C73C2×C414yescomplemented
131#95×C73—21no—
132#95×C73—21no—
133C23×D73C2×D411yescharacteristic
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C2×C4×D73—24no—
142C2×C4×D73—24no—
143C2×C4×D73—81no—
144C2×C4×D73—81no—
145C2×C4×D73—81no—
146C2×C4×D73—81no—
147C2×C4×D73C22×C412yes—
148C2×C4×D73C22×C412yes—
149C2×C4×D73C22×C412yes—
150C2×C4×D73C2×Q812yes—
151C2×C4×D73#4112yes—
152C2×C4×D73#4112yes—
153#11924011C42116yescomplemented
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831C22×C412yes—
157C2×#10596831C22×C412yes—
158C2×#10596831C22×C412yes—
159C2×#10596831C22×C412yes—
160C8×D73—28no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73C22×C412yes—
166C4⋊Dic73C22×C412yes—
167C4⋊Dic73C22×C412yes—
168C4⋊Dic73C22×C412yes—
169#11924007—24no—
170#11924007—24no—
171#11924007C4⋊C414yes—
172#11924007C4⋊C414yes—
173#11924005—24no—
174#11924005—24no—
175#11924005C4⋊C414yes—
176#11924005C4⋊C414yes—
177C4×Dic73—24no—
178C4×Dic73—24no—
179C4×Dic73—81no—
180C4×Dic73—81no—
181C4×Dic73—81no—
182C4×Dic73—81no—
183C4×Dic73—81no—
184C4×Dic73—81no—
185C4×Dic73C22×C412yes—
186C4×Dic73C2×Q812yes—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2411yescharacteristic
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73C22×C412yes—
197C2×C8×C73C22×C412yes—
198C2×C8×C73C22×C412yes—
199C2×C8×C73C22×C412yes—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C412yes—
205C4⋊C4×C73C22×C412yes—
206C4⋊C4×C73C22×C412yes—
207C4⋊C4×C73C22×C412yes—
208C42×C73—81no—
209C42×C73—81no—
210C42×C73C22×C412yes—
211C42×C73C2×Q812yes—
212C23×C73C22×D411yescharacteristic
213C22×D73—21no—
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—21no—
217C22×D73C2×C4⋊C412yes—
218C22×D73C4×D412yes—
219C22×D73C4×D412yes—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73C2×C4⋊C412yes—
237C2×C4×C73C2×C4⋊C412yes—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C23×C411yescharacteristic
240C2×C4×C73C23×C411yescharacteristic
241C2×C4×C73C22×Q811yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73C2×C4⋊C412yes—
261C2×Dic73C2×C4⋊C412yes—
262C2×Dic73C2×C4⋊C412yes—
263C2×Dic73C4×D412yes—
264C2×Dic73C4×D412yes—
265C4×D73—84no—
266C4×D73—84no—
267C4×D73C2×C4214yes—
268C4×D73#11714yes—
269C4×D73C4×Q814yes—
270C4×D73C4×Q814yes—
271C8×C73—22no—
272C8×C73C2×C4214yes—
273#10596831—22no—
274#10596831C2×C4214yes—
275C22×C73C22×C4⋊C411yescharacteristic
276C22×C73C2×C4×D411yescharacteristic
277C22×C73C2×C4×D411yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57711yescharacteristic
281C22×C73#57811yescharacteristic
282C2×D73—22no—
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—22no—
286C2×D73—22no—
287C2×D73—22no—
288C2×D73C4×C4⋊C414yes—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—42no—
292C4×C73—42no—
293C4×C73—42no—
294C4×C73—42no—
295C4×C73C22×C4211yescharacteristic
296C4×C73C2×#11711yescharacteristic
297C4×C73C2×C4×Q811yescharacteristic
298C4×C73C2×C4×Q811yescharacteristic
299Dic73—22no—
300Dic73—22no—
301Dic73—42no—
302Dic73—42no—
303Dic73—42no—
304Dic73—42no—
305Dic73C4×C4⋊C414yes—
306#20453—731nomaximal, Sylow
307C2×C73C2×C4×C4⋊C411yescharacteristic
308C2×C73#291211yescharacteristic
309C2×C73#291311yescharacteristic
310C2×C73#294611yescharacteristic
311C2×C73#294611yescharacteristic
312C2×C73#294711yescharacteristic
313C2×C73#294711yescharacteristic
314D73—24no—
315C2×#428—731no—
316C2×#428—731no—
317C2×#405—731no—
318#2274—731no—
319#2274—731no—
320#1776—731no—
321#1776—731no—
322#1776—731no—
323#1776D7311yescharacteristic, complemented
324C4×#107—731no—
325C4×#107—731no—
326C4×#106—731no—
327C4×#106—731no—
328#1735—731no—
329C2×#340—731no—
330C73#2045311yescharacteristic, complemented, Sylow
331C22×M16—731no—
332C23×C8—731no—
333C2×#117—731no—
334C2×#117—731no—
335C2×#117—731no—
336C22×C4⋊C4—731no—
337C22×C4⋊C4—731no—
338#428—738no—
339#428—738no—
340C2×#107—731no—
341C2×#107—731no—
342C2×#107—731no—
343C2×#107C2×D7311yescharacteristic
344C2×#106—731no—
345C2×#106—731no—
346C2×#106—731no—
347C2×#106C2×D7311yescharacteristic
348#405—738no—
349C2×#97—731no—
350C2×#97C2×D7311yescharacteristic
351C2×C4×C8—731no—
352C2×C4×C8—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—1461no—
360C4×C4⋊C4—1461no—
361C4×C4⋊C4—1461no—
362C4×C4⋊C4—1461no—
363C4×C4⋊C4C2×D7311yescharacteristic
364C4×C4⋊C4C2×D7311yescharacteristic
365C4×C22⋊C4—1461no—
366C4×C22⋊C4—1461no—
367C4×C22⋊C4—1461no—
368C4×C22⋊C4—1461no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340C2×D7311yescharacteristic
376#340C2×D7311yescharacteristic
377C23×C4—731no—
378C23×C4—1461no—
379C23×C4—1461no—
380C2×M16—734no—
381C2×M16—734no—
382C2×M16—734no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—734no—
386C22×C8—1462no—
387C22×C8—1462no—
388C22×C8C22×D7311yescharacteristic
389C22×C8C22×D7311yescharacteristic
390#117—734no—
391#117—734no—
392#117—734no—
393#117—734no—
394#117—1462no—
395#117—1462no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—734no—
403C2×C4⋊C4—734no—
404C2×C4⋊C4—1462no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C4⋊C4C22×D7311yescharacteristic
411C2×C4⋊C4C22×D7311yescharacteristic
412C2×C22⋊C4—731no—
413C2×C22⋊C4—731no—
414C2×C22⋊C4—1461no—
415C2×C22⋊C4—1461no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—731no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422C2×C42—1461no—
423C2×C42—1461no—
424C2×C42—1461no—
425C2×C42—1461no—
426C2×C42—1461no—
427C2×C42C22×D7311yescharacteristic
428#107—734no—
429#107—734no—
430#107—734no—
431#107C4×D7314yescomplemented
432#106—734no—
433#106—734no—
434#106—734no—
435#106C4×D7314yescomplemented
436#97—734no—
437#97C4×D7314yescomplemented
438C4×C8—734no—
439C4×C8—734no—
440#95—21no—
441#95—21no—
442#95—1461no—
443#95—1461no—
444#95—1461no—
445#95—1461no—
446#95—1461no—
447#95—1461no—
448C24—731no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—5841no—
454C22×C4—5841no—
455C22×C4—5841no—
456C22×C4—5841no—
457C22×C4—731no—
458C22×C4—732no—
459C22×C4—732no—
460C22×C4—732no—
461C22×C4—732no—
462C22×C4—732no—
463C22×C4—732no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—1464no—
467C22×C4—1464no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4D4×D7311yescharacteristic
473C22×C4D4×D7311yescharacteristic
474C22×C4C23×D7311yescharacteristic
475M16—7316no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—732no—
479C2×C8—732no—
480C2×C8—732no—
481C2×C8—732no—
482C2×C8—1468no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8C2×C4×D7312yes—
486C2×C8C2×C4×D7312yes—
487C2×C8C2×C4×D7312yes—
488C2×C8C2×C4×D7312yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—734no—
494C4⋊C4—734no—
495C4⋊C4—732no—
496C4⋊C4—732no—
497C4⋊C4—732no—
498C4⋊C4—732no—
499C4⋊C4—1464no—
500C4⋊C4—1464no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4C2×C4×D7312yes—
506C4⋊C4C2×C4×D7312yes—
507C4⋊C4C2×C4×D7312yes—
508C4⋊C4C2×C4×D7312yes—
509C22⋊C4—734no—
510C22⋊C4—734no—
511C22⋊C4—1464no—
512C22⋊C4—1464no—
513C42—81no—
514C42—81no—
515C42—5841no—
516C42—5841no—
517C42—5841no—
518C42—5841no—
519C42—5841no—
520C42—5841no—
521C42—732no—
522C42—732no—
523C42—1464no—
524C42—1464no—
525C42C2×C4×D7312yes—
526C42Q8×D7312yes—
527C23—732no—
528C23—732no—
529C23—732no—
530C23—1461no—
531C23—1461no—
532C23—1461no—
533C23—1461no—
534C23C2×D4×D7311yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—5844no—
560C2×C4—5844no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—734no—
565C2×C4—734no—
566C2×C4—734no—
567C2×C4—734no—
568C2×C4—732no—
569C2×C4—732no—
570C2×C4—1462no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4C4⋊C4×D7312yes—
579C2×C4C4⋊C4×D7312yes—
580C2×C4C22×C4×D7311yescharacteristic
581C2×C4C22×C4×D7311yescharacteristic
582C2×C4C22×C4×D7311yescharacteristic
583C2×C4C2×Q8×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4#41×D7311yescharacteristic
586C8—22no—
587C8—734no—
588C8—1462no—
589C8C42×D7314yes—
590C22—734no—
591C22—1462no—
592C22—1462no—
593C22—1462no—
594C22—1462no—
595C22—1462no—
596C22—1462no—
597C22C2×C4⋊C4×D7311yescharacteristic
598C22C4×D4×D7311yescharacteristic
599C22C4×D4×D7311yescharacteristic
600C22#1798939111yescharacteristic
601C22#1798939611yescharacteristic
602C22#1798939711yescharacteristic
603C22#1798940311yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—2922no—
611C4—2922no—
612C4—2922no—
613C4—2922no—
614C4—734no—
615C4—1462no—
616C4—1462no—
617C4C2×C42×D7311yescharacteristic
618C4#117×D7311yescharacteristic
619C4C4×Q8×D7311yescharacteristic
620C4C4×Q8×D7311yescharacteristic
621C2—1464no—
622C2C4×C4⋊C4×D7311yescharacteristic
623C2#2261852511yescharacteristic
624C2#2261853011yescharacteristic
625C2#2261876011yescharacteristic
626C2#2261876211yescharacteristic
627C2#2261877111yescharacteristic
628C2#2261877311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC4×C73C22×C42
FrattiniC22×C4C23×D73
FittingC73×#1776C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1776D73
hypocenterC73#20453
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
6. #22609828C2semidirect
7. #22609827C2semidirect
8. #22609827C2semidirect
9. #107×Dic73C2semidirect
10. C4×#14556524C2semidirect
11. #106×Dic73C2semidirect
12. C4×#14556522C2semidirect
15. C73×#1776C2semidirect
23. #17988628C4semidirect
24. #17988626C4semidirect
31. #17988544C4semidirect
104. #14556525C2×C4semidirect
105. #14556524C2×C4semidirect
106. #14556524C2×C4semidirect
107. #14556523C2×C4semidirect
108. #14556522C2×C4semidirect
109. #14556522C2×C4semidirect
110. #14556501C2×C4semidirect
111. C4×#10596831C2×C4semidirect
112. C8×Dic73C2×C4semidirect
128. #107×C73C2×C4semidirect
129. #106×C73C2×C4semidirect
130. #97×C73C2×C4semidirect
153. #11924011C42semidirect
323. #1776D73semidirect
330. C73#20453semidirect
431. #107C4×D73semidirect
435. #106C4×D73semidirect
437. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886282
C2×#179886262
C2×#179885442
#226122852
#226098282
#226098272
#107×Dic732
C4×#145565242
#106×Dic732
C4×#145565222
#226096462
#340×D732
C73×#17762
#2045373