Finite Group Databasebeta

← #34021134

Subgroups — #34021134

Order 18688.

1350 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989608C211yesmaximal, characteristic, complemented
2C2×#17989603C211yesmaximal, characteristic, complemented
3#22625824C212yesmaximal, complemented
4#22625786C212yesmaximal, complemented
5#22625785C212yesmaximal, complemented
6#22625755C212yesmaximal
7#22625661C212yesmaximal, complemented
8#22625660C212yesmaximal, complemented
9C2×C4×#11924027C211yesmaximal, characteristic
10#17989608—24no—
11#17989603—24no—
12C2×C4⋊#10596833C2212yes—
13C2×C4⋊#10596833C2212yes—
14C2×#14556648C2214yes—
15C2×C22⋊#10596833C2212yes—
16C2×C22⋊#10596833C2212yes—
17C2×C22⋊#10596833C2214yes—
18C2×#14556646C2212yes—
19C2×C4×#10596833C2212yes—
20#17989453C2214yes—
21#17989453C2214yes—
22C22×#11924027C2211yescharacteristic
23C4×#11924027—24no—
24C4×#11924027—24no—
25C2×C4×Dic146C411yescharacteristic
26C2×#14556556C411yescharacteristic
27C2×D4×Dic73C411yescharacteristic
28C2×C4×C22⋊D73C411yescharacteristic
29C2×C4×C22⋊D73C411yescharacteristic
30C2×#14556551C411yescharacteristic
31C2×#14556551C411yescharacteristic
32C2×#14556549C2211yescharacteristic
33C2×#14556547C2211yescharacteristic
34C2×#14556547C2211yescharacteristic
35C22×C4×Dic73C2211yescharacteristic
36C22×C4×Dic73C2211yescharacteristic
37C2×C42×D73C2211yescharacteristic
38C2×C4×D4×C73C411yescharacteristic
39C22×#10596833C2314yes—
40C22×#10596833C2314yes—
41C4⋊#10596833—24no—
42C4⋊#10596833—24no—
43#14556648—28no—
44C22⋊#10596833—24no—
45C22⋊#10596833—24no—
46C22⋊#10596833—28no—
47#14556646—24no—
48C4×#10596833—24no—
49C2×#11924027—22no—
50C2×#11924027—22no—
51C2×#11924027—24no—
52C2×#11924027—24no—
53C22×Dic146C2×C411yescharacteristic
54C22×C22⋊D73C2×C411yescharacteristic
55C22×C22⋊D73C2×C411yescharacteristic
56C23×Dic73C2×C411yescharacteristic
57C23×Dic73C2×C411yescharacteristic
58C23×Dic73C2311yescharacteristic
59C23×Dic73C2311yescharacteristic
60C22×C4×D73C2×C411yescharacteristic
61C22×C4×D73C2×C411yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C4×Dic146—24no—
64#14556556—24no—
65D4×Dic73—24no—
66C4×C22⋊D73—24no—
67C4×C22⋊D73—24no—
68#14556551—24no—
69#14556551—24no—
70#14556549—22no—
71#14556549—22no—
72#14556547—22no—
73#14556547—22no—
74#14556547—22no—
75#14556547—22no—
76C2×C4⋊Dic73C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×C22⋊Dic73C2×C411yescharacteristic
82C2×C22⋊Dic73C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×C4×Dic73—22no—
86C2×C4×Dic73—22no—
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2×C411yescharacteristic
91C2×C4×Dic73C2×C411yescharacteristic
92C2×C4×Dic73D414yes—
93C2×C4×Dic73D414yes—
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C42×D73—22no—
99C42×D73—22no—
100C22×D4×C73C2×C411yescharacteristic
101C23×C4×C73C2×C411yescharacteristic
102C23×C4×C73C2×C411yescharacteristic
103C4×D4×C73—24no—
104C2×C4⋊C4×C73C2×C411yescharacteristic
105C2×C22⋊C4×C73C2×C411yescharacteristic
106C2×C22⋊C4×C73C2×C411yescharacteristic
107C2×C42×C73C2×C411yescharacteristic
108C2×#10596833—24no—
109C2×#10596833—24no—
110C2×#10596833—24no—
111C2×#10596833—24no—
112C2×#10596833—24no—
113C2×#10596833—24no—
114C23×D73C22×C411yescharacteristic
115#11924027—216no—
116#11924027—216no—
117C2×Dic146—22no—
118C2×Dic146—22no—
119C2×Dic146—22no—
120C2×C22⋊D73—22no—
121C2×C22⋊D73—22no—
122C2×C22⋊D73—22no—
123C2×C22⋊D73—22no—
124C2×C22⋊D73—22no—
125C2×C22⋊D73—22no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—42no—
133C22×Dic73—42no—
134C22×Dic73C22⋊C414yes—
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C2×D412yes—
142C22×Dic73C2×D412yes—
143C22×Dic73#4114yes—
144C22×Dic73C2411yescharacteristic
145C2×C4×D73—21no—
146C2×C4×D73—21no—
147C2×C4×D73—22no—
148C2×C4×D73—22no—
149C2×C4×D73—24no—
150C2×C4×D73—22no—
151C2×C4×D73—42no—
152C2×C4×D73#4114yes—
153C4⋊Dic73—22no—
154#11924007—22no—
155#11924007—22no—
156#11924007—22no—
157#11924007—22no—
158C22⋊Dic73—22no—
159C22⋊Dic73—22no—
160#11924005—22no—
161#11924005—22no—
162C4×Dic73—22no—
163C4×Dic73—22no—
164C4×Dic73—22no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73C22⋊C414yes—
170C4×Dic73C22⋊C414yes—
171C4×Dic73C2×D412yes—
172C4×Dic73C2×D412yes—
173C4×Dic73C2×D412yes—
174C4×Dic73C2×D412yes—
175C24×C73C22×C411yescharacteristic
176C24×C73C22×C411yescharacteristic
177C2×D4×C73—22no—
178C2×D4×C73—22no—
179C2×D4×C73—22no—
180C22×C4×C73—24no—
181C22×C4×C73—22no—
182C22×C4×C73—24no—
183C22×C4×C73C22⋊C414yes—
184C22×C4×C73C22×C411yescharacteristic
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C4⋊C4×C73—22no—
190C22⋊C4×C73—22no—
191C22⋊C4×C73—22no—
192C42×C73—22no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73—22no—
200C23×C73—22no—
201C23×C73C2×C22⋊C412yes—
202C23×C73#11712yes—
203C23×C73C23×C411yescharacteristic
204C22×D73—21no—
205C22×D73—21no—
206C22×D73—22no—
207C22×D73—22no—
208C22×D73#11712yes—
209D4×C73—28no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—22no—
213C2×C4×C73—28no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—44no—
219C2×C4×C73C2×C22⋊C412yes—
220C2×C4×C73C2×C22⋊C412yes—
221C2×C4×C73#11712yes—
222C2×C4×C73#13114yes—
223C22⋊D73—28no—
224C22⋊D73—28no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—44no—
240C2×Dic73—44no—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×C22⋊C412yes—
243C2×Dic73C2×C22⋊C412yes—
244C2×Dic73#11712yes—
245C2×Dic73C22×D411yescharacteristic
246C2×Dic73C22×D411yescharacteristic
247C2×Dic73C2×#4112yes—
248C2×Dic73C2×#4112yes—
249C4×D73—24no—
250C4×D73—24no—
251C4×D73—44no—
252C4×D73—44no—
253Dic146—28no—
254#10596833—44no—
255#10596833—44no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—24no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—42no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73C22×C22⋊C411yescharacteristic
267C22×C73C2×#11711yescharacteristic
268C22×C73C2×#11711yescharacteristic
269C22×C73C2×#13112yes—
270C22×C73#56812yes—
271C2×D73—24no—
272C2×D73—22no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—42no—
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73#40814yes—
281Dic73—22no—
282Dic73—22no—
283Dic73—22no—
284Dic73—22no—
285Dic73—41no—
286Dic73—41no—
287Dic73#52512yes—
288Dic73#52512yes—
289#20721—731nomaximal, Sylow
290C2×C73—42no—
291C2×C73—42no—
292C2×C73#228011yescharacteristic
293C2×C73C2×#40811yescharacteristic
294C2×C73#287911yescharacteristic
295C2×C73#292512yes—
296C2×C73#292512yes—
297D73—42no—
298C2×C4×#41—731no—
299C2×#432—731no—
300C2×#431—731no—
301#1818—732no—
302#1797—732no—
303#1797—732no—
304#1780—732no—
305#1750—732no—
306#1750—732no—
307C73#2072111yescharacteristic, complemented, Sylow
308C22×#41—731no—
309C4×#41—1464no—
310C4×#41—1464no—
311C2×C4×Q8—731no—
312C2×C4×Q8—731no—
313C2×C4×D4—731no—
314C2×C4×D4—731no—
315C2×C4×D4—731no—
316C2×C4×D4—731no—
317C2×C4×D4—731no—
318C2×C4×D4#6440811yescharacteristic
319C2×#117—731no—
320C2×#117—731no—
321C2×#117—731no—
322C22×C42—731no—
323C22×C42—731no—
324C22×C42—731no—
325#432—1464no—
326#431—1464no—
327C2×C4⋊C8—732no—
328C2×C4⋊C8—732no—
329C2×C22⋊C8—734no—
330C2×C22⋊C8—734no—
331C2×C22⋊C8—732no—
332C2×C22⋊C8—732no—
333C2×#97—732no—
334C2×C4×C8—732no—
335#336—734no—
336#336—734no—
337C2×#41—1462no—
338C2×#41—1462no—
339C2×#41—1464no—
340C2×#41—1464no—
341C22×Q8—731no—
342C22×D4—731no—
343C22×D4—731no—
344C22×D4C2×#6440811yescharacteristic
345C23×C4—731no—
346C23×C4—731no—
347C23×C4—731no—
348C23×C4—731no—
349C23×C4—731no—
350C23×C4—731no—
351C23×C4—731no—
352C23×C4C2×#6440811yescharacteristic
353C23×C4C2×#6440811yescharacteristic
354C22×C8—734no—
355C22×C8—734no—
356C4×Q8—1464no—
357C4×Q8—1464no—
358C4×D4—24no—
359C4×D4—1464no—
360C4×D4—1464no—
361C4×D4—1464no—
362C4×D4—1464no—
363C4×D4—1464no—
364#117—1462no—
365#117—1462no—
366#117—1462no—
367#117—1462no—
368#117—1462no—
369#117—1462no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4C2×#6440811yescharacteristic
376C2×C22⋊C4—731no—
377C2×C22⋊C4—731no—
378C2×C22⋊C4—731no—
379C2×C22⋊C4—731no—
380C2×C22⋊C4C2×#6440811yescharacteristic
381C2×C22⋊C4C2×#6440811yescharacteristic
382C2×C42—734no—
383C2×C42—734no—
384C2×C42—731no—
385C2×C42—731no—
386C2×C42—731no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—731no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—1462no—
394C2×C42—1462no—
395C2×C42—1462no—
396C2×C42—1462no—
397C2×C42C2×#6440811yescharacteristic
398C4⋊C8—1464no—
399C4⋊C8—1464no—
400C22⋊C8—1464no—
401C22⋊C8—1464no—
402C22⋊C8—1468no—
403C22⋊C8—1468no—
404#97—1464no—
405C4×C8—1464no—
406C24—731no—
407C24C22×#6440811yescharacteristic
408C24C22×#6440811yescharacteristic
409#41—14616no—
410#41—14616no—
411C2×Q8—1462no—
412C2×Q8—1462no—
413C2×Q8—1462no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—1462no—
418C2×D4—1462no—
419C2×D4—1462no—
420C2×D4—1462no—
421C2×D4—1462no—
422C2×D4—1462no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—2922no—
427C22×C4—2922no—
428C22×C4—2922no—
429C22×C4—734no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—732no—
436C22×C4—732no—
437C22×C4—734no—
438C22×C4—734no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—1462no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1464no—
447C22×C4—1462no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4C22⋊#6440814yes—
454C22×C4C22×#6440811yescharacteristic
455C22×C4C22×#6440811yescharacteristic
456C22×C4C22×#6440811yescharacteristic
457C22×C4C22×#6440811yescharacteristic
458C22×C4C22×#6440811yescharacteristic
459C2×C8—1464no—
460C2×C8—1464no—
461C2×C8—1464no—
462C2×C8—1464no—
463C2×C8—1464no—
464C2×C8—1464no—
465C4⋊C4—22no—
466C4⋊C4—1462no—
467C4⋊C4—1462no—
468C4⋊C4—1462no—
469C4⋊C4—1462no—
470C4⋊C4—1462no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—1462no—
474C22⋊C4—1462no—
475C22⋊C4—1462no—
476C22⋊C4—1462no—
477C42—22no—
478C42—734no—
479C42—734no—
480C42—732no—
481C42—732no—
482C42—732no—
483C42—732no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42—1462no—
489C42—1462no—
490C42—1462no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—732no—
500C23—1461no—
501C23—1461no—
502C23—1462no—
503C23—1462no—
504C23C2×C22⋊#6440812yes—
505C23#1455667012yes—
506C23C23×#6440811yescharacteristic
507Q8—1468no—
508D4—28no—
509D4—1468no—
510D4—1468no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—2924no—
521C2×C4—2924no—
522C2×C4—2924no—
523C2×C4—2924no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—732no—
532C2×C4—1464no—
533C2×C4—1464no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4—1464no—
539C2×C4—1464no—
540C2×C4—1464no—
541C2×C4—1464no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4C2×C22⋊#6440812yes—
549C2×C4C2×C22⋊#6440812yes—
550C2×C4#1455667012yes—
551C2×C4#1455667814yes—
552C8—2924no—
553C8—2924no—
554C22—24no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—2922no—
565C22—2921no—
566C22—2921no—
567C22—1464no—
568C22—1462no—
569C22C22×C22⋊#6440811yescharacteristic
570C22C2×#1455667011yescharacteristic
571C22C2×#1455667011yescharacteristic
572C22C2×#1455667812yes—
573C22#1798966612yes—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—2921no—
579C4—2921no—
580C4—732no—
581C4—732no—
582C4—1462no—
583C4—1462no—
584C4—1462no—
585C4—1462no—
586C4#1798956014yes—
587C2—42no—
588C2—42no—
589C2—2922no—
590C2#2262637511yescharacteristic
591C2C2×#1798956011yescharacteristic
592C2#2262651611yescharacteristic
593C2#2262662612yes—
594C2#2262662712yes—
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×C4×D4×C73C4
prime radicalC22×#11924027C22
cosocleC22×Dic73C24
hypercenterC2×C4×D4#64408
hypocenterC73#20721
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#17989608C2semidirect
2. C2×#17989603C2semidirect
3. #22625824C2semidirect
4. #22625786C2semidirect
5. #22625785C2semidirect
7. #22625661C2semidirect
8. #22625660C2semidirect
307. C73#20721semidirect

Maximal subgroups

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

subgroupindex
C2×#179896082
C2×#179896032
#226258242
#226257862
#226257852
#226257552
#226256612
#226256602
C2×C4×#119240272
#2072173