Finite Group Databasebeta

← #33095776

Subgroups — #33095776

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989197C211yesmaximal, characteristic
2C2×#17988645C211yesmaximal, characteristic
3#22612551C211yesmaximal, characteristic
4#22610898C211yesmaximal, characteristic, complemented
5#22610894C211yesmaximal, characteristic, complemented
6#22610749C211yesmaximal, characteristic, complemented
7#22610731C211yesmaximal, characteristic, complemented
8#22610243C211yesmaximal, characteristic
9#22610239C211yesmaximal, characteristic
10#22610236C211yesmaximal, characteristic
11#22609945C211yesmaximal, characteristic, complemented
12#22609944C211yesmaximal, characteristic, complemented
13#22609935C211yesmaximal, characteristic, complemented
14#22609934C211yesmaximal, characteristic, complemented
15C73×#1873C211yesmaximal, characteristic
16#17989197—24no—
17#17989197C418yescomplemented
18C2×#14556589C2211yescharacteristic
19C2×#14556584—21no—
20C2×#14556584—21no—
21C2×#14556582—21no—
22C2×#14556582C2211yescharacteristic
23C2×#14556581—21no—
24C2×#14556570C2211yescharacteristic
25C2×Q8×Dic73C2211yescharacteristic
26C2×#14556556C2211yescharacteristic
27C2×#14556550C2211yescharacteristic
28C22×C4⋊Dic73C2211yescharacteristic
29#17988645—24no—
30C2×#14556520C2211yescharacteristic
31C2×#14556520C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556516C2211yescharacteristic
34C2×#14556515C2211yescharacteristic
35C2×#14556515C2211yescharacteristic
36C2×#14556514C2211yescharacteristic
37C2×#14556514C2211yescharacteristic
38C2×C22⋊#10596831C2211yescharacteristic
39C2×#14556501C2211yescharacteristic
40C2×C8×Dic73C2211yescharacteristic
41#17988485—21no—
42#17988485—21no—
43#17988459—21no—
44#17988459—21no—
45#17988456C2211yescharacteristic
46#17988456C2211yescharacteristic
47#17988452C2211yescharacteristic
48#17988452C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×C22⋊Q8×C73C2211yescharacteristic
56C2×#117×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59C2×C22⋊C8×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556589—22no—
67#14556589C2×C414yescomplemented
68#14556584—28no—
69#14556584—28no—
70#14556582—28no—
71#14556582—22no—
72#14556582C2×C414yescomplemented
73#14556581—28no—
74#14556570—22no—
75#14556570C2×C414yes—
76Q8×Dic73—22no—
77Q8×Dic73C2×C414yescomplemented
78#14556556—22no—
79#14556556C2×C414yescomplemented
80#14556550C2×C414yes—
81#14556550D414yes—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73—41no—
86C2×C4⋊Dic73—41no—
87C2×C4⋊Dic73D411yescharacteristic
88C2×C4⋊Dic73D411yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2311yescharacteristic
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×#11924007—21no—
97C2×#11924007—21no—
98C2×#11924007—21no—
99C2×#11924007—21no—
100C2×#11924007—21no—
101C2×C22⋊Dic73—21no—
102C2×C22⋊Dic73D411yescharacteristic
103C2×C22⋊Dic73D411yescharacteristic
104C2×C4×Dic73—21no—
105C2×C4×Dic73—21no—
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108#14556520—22no—
109#14556520—22no—
110#14556516—41no—
111#14556516—41no—
112#14556515—41no—
113#14556515—41no—
114#14556514—41no—
115#14556514—41no—
116C22⋊#10596831—22no—
117#14556501—22no—
118C8×Dic73—22no—
119#14556498—21no—
120#14556498—21no—
121#14556498—21no—
122#14556498—21no—
123C22×Q8×C73C2311yescharacteristic
124C23×C4×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126C22×C8×C73C2311yescharacteristic
127C22⋊Q8×C73—22no—
128C22⋊Q8×C73C2×C414yes—
129#117×C73—41no—
130#117×C73—41no—
131C2×C4⋊C4×C73—21no—
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C4⋊C4×C73C2311yescharacteristic
134C2×C4⋊C4×C73C2311yescharacteristic
135C2×C22⋊C4×C73—21no—
136C2×C22⋊C4×C73—21no—
137C2×C42×C73—21no—
138#103×C73—41no—
139#103×C73—41no—
140C22⋊C8×C73—22no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146C22×C412yes—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—42no—
152C22×Dic73—81no—
153C22×Dic73—81no—
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156C22×Dic73C2×D411yescharacteristic
157C22×Dic73C2×D411yescharacteristic
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—24no—
166C4⋊Dic73—24no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—42no—
172C4⋊Dic73C22⋊C414yes—
173C4⋊Dic73C22⋊C414yes—
174C4⋊Dic73C22×C412yes—
175#11924007—24no—
176#11924007—24no—
177#11924007—24no—
178#11924007—24no—
179#11924007—24no—
180#11924007—81no—
181#11924007—81no—
182#11924007—81no—
183#11924007—81no—
184C22⋊Dic73—24no—
185C22⋊Dic73C22⋊C414yes—
186C22⋊Dic73C22⋊C414yes—
187C4×Dic73—24no—
188C4×Dic73—24no—
189C4×Dic73C22×C412yes—
190C4×Dic73C22×C412yes—
191C4×Dic73C2×D412yes—
192C4×Dic73C2×D412yes—
193C24×C73C2×D411yescharacteristic
194C2×Q8×C73—21no—
195C2×Q8×C73—21no—
196C2×Q8×C73—21no—
197C2×Q8×C73—21no—
198C2×Q8×C73—21no—
199C2×Q8×C73C22×C412yes—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73—21no—
208C22×C4×C73C22×C412yes—
209C22×C4×C73C2×D411yescharacteristic
210C22×C4×C73C2×D412yes—
211C22×C4×C73C2411yescharacteristic
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C4⋊C4×C73—24no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C4⋊C4×C73—21no—
223C4⋊C4×C73—21no—
224C4⋊C4×C73C22×C412yes—
225C22⋊C4×C73—24no—
226C22⋊C4×C73—42no—
227C42×C73—42no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73—21no—
233C23×C73—21no—
234C23×C73C2×C22⋊C412yes—
235C23×C73C22×D411yescharacteristic
236Q8×C73—22no—
237Q8×C73—22no—
238Q8×C73—41no—
239Q8×C73—41no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73C2×C22⋊C412yes—
259C2×C4×C73C23×C411yescharacteristic
260C2×C4×C73C22×D411yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
262C2×C4×C73H(5,𝔽2)11yescharacteristic
263C2×C4×C732−1+411yescharacteristic
264C2×C4×C732−1+411yescharacteristic
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—84no—
286C2×Dic73C2×C22⋊C412yes—
287C2×Dic73C2×C22⋊C412yes—
288C2×Dic73C2×C22⋊C412yes—
289C2×Dic73C2×C22⋊C412yes—
290Dic146—22no—
291Dic146—22no—
292Dic146—41no—
293Dic146—41no—
294C8×C73—41no—
295C8×C73—41no—
296#10596831—41no—
297#10596831—41no—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73C22×C22⋊C411yescharacteristic
306C22×C73#53511yescharacteristic
307C22×C73#53611yescharacteristic
308C22×C73#57711yescharacteristic
309C22×C73#57711yescharacteristic
310C22×C73#57811yescharacteristic
311C22×C73#57811yescharacteristic
312C4×C73—22no—
313C4×C73—42no—
314C4×C73—42no—
315C4×C73—81no—
316C4×C73—81no—
317C4×C73#52011yescharacteristic
318C4×C73#52011yescharacteristic
319C4×C73#53711yescharacteristic
320C4×C73#53711yescharacteristic
321Dic73—22no—
322Dic73—22no—
323Dic73—22no—
324Dic73—22no—
325Dic73—42no—
326Dic73—42no—
327Dic73—81no—
328Dic73—81no—
329#21873—731nomaximal, Sylow
330C2×C73—42no—
331C2×C73#231111yescharacteristic
332C2×C73#289811yescharacteristic
333C2×C73#289811yescharacteristic
334C2×C73#322211yescharacteristic
335C2×C73#322311yescharacteristic
336C2×C73#324211yescharacteristic
337C2×C73#324211yescharacteristic
338C2×#541—731no—
339C2×#433—731no—
340#2317—731no—
341#1992—731no—
342#1991—731no—
343#1960—731no—
344#1956—731no—
345#1873—731no—
346#1873D7311yescharacteristic
347#1872—731no—
348#1872—731no—
349#1805—731no—
350#1805—731no—
351#1803—731no—
352#1803—731no—
353C73#2187311yescharacteristic, complemented, Sylow
354#541—738no—
355#541—1464no—
356C2×C4⋊Q8—731no—
357C2×#126—1461no—
358C2×#126—1461no—
359C2×#125—731no—
360C2×#125—1461no—
361C2×#125—1461no—
362C2×C22⋊Q8—731no—
363C2×C22⋊Q8C2×D7311yescharacteristic
364C2×C4×Q8—731no—
365C2×C4×Q8—731no—
366C2×#117—731no—
367C2×#117C2×D7311yescharacteristic
368C22×C4⋊C4—731no—
369#433—1464no—
370C2×C4⋊C8—731no—
371C2×C4⋊C8—731no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×#103—731no—
377C2×#103—731no—
378C2×#103C2×D7311yescharacteristic
379C2×#103C2×D7311yescharacteristic
380C2×C22⋊C8—731no—
381C2×C22⋊C8C2×D7311yescharacteristic
382C2×#97—731no—
383C2×C4×C8—731no—
384#389—1461no—
385#389—1461no—
386#385—1461no—
387#385—1461no—
388#384—731no—
389#384—731no—
390#382—731no—
391#382—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340C2×D7311yescharacteristic
399#340C2×D7311yescharacteristic
400C22×Q8—731no—
401C22×Q8C22×D7311yescharacteristic
402C23×C4—1461no—
403C23×C4C22×D7311yescharacteristic
404C22×C8—731no—
405C22×C8—731no—
406C22×C8C22×D7311yescharacteristic
407C22×C8C22×D7311yescharacteristic
408C4⋊Q8—734no—
409C4⋊Q8—1462no—
410#126—1468no—
411#126—1468no—
412#125—734no—
413#125—1462no—
414#125—1468no—
415#125—1468no—
416C22⋊Q8—22no—
417C22⋊Q8—734no—
418C22⋊Q8—1462no—
419C22⋊Q8C4×D7314yes—
420C4×Q8—734no—
421C4×Q8—734no—
422C4×Q8—1462no—
423C4×Q8—1462no—
424#117—41no—
425#117—41no—
426#117—734no—
427#117—734no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—2921no—
430C2×C4⋊C4—2921no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4—1461no—
446C2×C4⋊C4—1461no—
447C2×C4⋊C4—1461no—
448C2×C4⋊C4C22×D7311yescharacteristic
449C2×C4⋊C4C22×D7311yescharacteristic
450C2×C4⋊C4C22×D7311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—731no—
454C2×C22⋊C4—731no—
455C2×C22⋊C4—1461no—
456C2×C42—21no—
457C2×C42—731no—
458C2×C42—731no—
459C2×C42—1461no—
460C2×C42—1461no—
461C4⋊C8—1462no—
462C4⋊C8—1462no—
463#103—41no—
464#103—41no—
465#103—2921no—
466#103—2921no—
467#103—2921no—
468#103—2921no—
469#103—2921no—
470#103—2921no—
471C22⋊C8—22no—
472C22⋊C8—1462no—
473#97—1462no—
474C4×C8—1462no—
475#95—1461no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479C24D4×D7311yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—732no—
486C2×Q8—1461no—
487C2×Q8—1461no—
488C2×Q8—1461no—
489C2×Q8—1461no—
490C2×Q8—1461no—
491C2×Q8C2×C4×D7312yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—2922no—
501C22×C4—5841no—
502C22×C4—5841no—
503C22×C4—731no—
504C22×C4—731no—
505C22×C4—731no—
506C22×C4—731no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4—1461no—
511C22×C4C2×C4×D7312yes—
512C22×C4D4×D7311yescharacteristic
513C22×C4D4×D7312yes—
514C22×C4C23×D7311yescharacteristic
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—2922no—
534C4⋊C4—5841no—
535C4⋊C4—5841no—
536C4⋊C4—5841no—
537C4⋊C4—5841no—
538C4⋊C4—734no—
539C4⋊C4—734no—
540C4⋊C4—732no—
541C4⋊C4—1464no—
542C4⋊C4—1464no—
543C4⋊C4—1464no—
544C4⋊C4—1464no—
545C4⋊C4—1464no—
546C4⋊C4—1464no—
547C4⋊C4—1461no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1464no—
552C4⋊C4—1461no—
553C4⋊C4C2×C4×D7312yes—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—734no—
557C22⋊C4—734no—
558C22⋊C4—1464no—
559C42—42no—
560C42—732no—
561C42—732no—
562C42—732no—
563C42—732no—
564C42—1464no—
565C42—1464no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23C22⋊C4×D7312yes—
573C23C2×D4×D7311yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—2921no—
579Q8—2921no—
580Q8—1462no—
581Q8—1462no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—5844no—
611C2×C4—732no—
612C2×C4—732no—
613C2×C4—732no—
614C2×C4—732no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4—1461no—
618C2×C4—1461no—
619C2×C4—1461no—
620C2×C4—1461no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4C22⋊C4×D7312yes—
626C2×C4C22×C4×D7311yescharacteristic
627C2×C4C2×D4×D7311yescharacteristic
628C2×C4#1455663611yescharacteristic
629C2×C4#1455663611yescharacteristic
630C2×C4#1455663811yescharacteristic
631C2×C4#1455663811yescharacteristic
632C8—41no—
633C8—41no—
634C8—2921no—
635C8—2921no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22C2×C22⋊C4×D7311yescharacteristic
644C22#1798915211yescharacteristic
645C22#1798915911yescharacteristic
646C22#1798939411yescharacteristic
647C22#1798939611yescharacteristic
648C22#1798940111yescharacteristic
649C22#1798940311yescharacteristic
650C4—22no—
651C4—42no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—2922no—
656C4—2922no—
657C4—5841no—
658C4—5841no—
659C4—1462no—
660C4—1462no—
661C4—1462no—
662C4—1462no—
663C4#1798906611yescharacteristic
664C4#1798906611yescharacteristic
665C4#1798916411yescharacteristic
666C4#1798916411yescharacteristic
667C2—42no—
668C2#2261251811yescharacteristic
669C2#2261846011yescharacteristic
670C2#2261846311yescharacteristic
671C2#2262100111yescharacteristic
672C2#2262101211yescharacteristic
673C2#2262119711yescharacteristic
674C2#2262120511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1873C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1873D73
hypocenterC73#21873
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #22610898C2semidirect
5. #22610894C2semidirect
6. #22610749C2semidirect
7. #22610731C2semidirect
11. #22609945C2semidirect
12. #22609944C2semidirect
13. #22609935C2semidirect
14. #22609934C2semidirect
17. #17989197C4semidirect
67. #14556589C2×C4semidirect
72. #14556582C2×C4semidirect
77. Q8×Dic73C2×C4semidirect
79. #14556556C2×C4semidirect
353. C73#21873semidirect

Maximal subgroups

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

subgroupindex
C2×#179891972
C2×#179886452
#226125512
#226108982
#226108942
#226107492
#226107312
#226102432
#226102392
#226102362
#226099452
#226099442
#226099352
#226099342
C73×#18732
#2187373