Finite Group Databasebeta

← #32978708

Subgroups — #32978708

Order 18688.

742 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556565C211yesmaximal, characteristic
2C2×#17988322C212yesmaximal
3#22608161C211yesmaximal, characteristic, complemented
4#22607811C212yesmaximal, complemented
5C73×#1797C211yesmaximal, characteristic, complemented
6C2×#14556565C2211yescharacteristic
7C2×C4×D292C411yescharacteristic, complemented
8C2×C4×C22⋊D73—21no—
9C22×C4⋊Dic73—21no—
10#17988489C411yescharacteristic, complemented
11#17988485C411yescharacteristic, complemented
12#17988478—21no—
13#17988466—21no—
14#17988447C2211yescharacteristic
15C4×C4⋊Dic73C411yescharacteristic, complemented
16C4×#11924005—21no—
17C2×#14556498—21no—
18C2×#14556498C2212yes—
19#17988322—24no—
20C22×C42×C73C2211yescharacteristic
21C2×C4⋊C8×C73—21no—
22C2×C22⋊C8×C73C2212yes—
23#336×C73—21no—
24C22×D292C2×C411yescharacteristic
25C22×C22⋊D73—21no—
26C23×Dic73—22no—
27C22×C4×D73—21no—
28#14556565—21no—
29#14556565—21no—
30#14556565D412yes—
31#14556565Q812yes—
32C4×D292—42no—
33C4×C22⋊D73—42no—
34C4×C22⋊D73—42no—
35C2×C4⋊Dic73—24no—
36C2×C4⋊Dic73—42no—
37C2×C4⋊Dic73—42no—
38C2×C4⋊Dic73C2×C411yescharacteristic
39C2×C4⋊Dic73C2×C411yescharacteristic
40C2×C4⋊Dic73C2×C411yescharacteristic
41C2×#11924007—21no—
42C2×C22⋊Dic73—21no—
43C2×#11924005—21no—
44C2×#11924005—21no—
45C2×#11924005—21no—
46C2×#11924005—21no—
47C2×C4×Dic73—21no—
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51#14556498—22no—
52#14556498—22no—
53#14556498—42no—
54C23×C4×C73D411yescharacteristic
55C23×C4×C73D411yescharacteristic
56C23×C4×C73C2311yescharacteristic
57C22×C8×C73—22no—
58C2×C42×C73—42no—
59C2×C42×C73C2×C411yescharacteristic
60C2×C42×C73C2×C411yescharacteristic
61C2×C42×C73D411yescharacteristic
62C2×C42×C73D411yescharacteristic
63C4⋊C8×C73—42no—
64C22⋊C8×C73—22no—
65C22⋊C8×C73—22no—
66C23×D73—21no—
67C2×D292—21no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292C4⋊C412yescomplemented
71C2×C22⋊D73—22no—
72C2×C22⋊D73—22no—
73C2×C22⋊D73—41no—
74C2×C22⋊D73—41no—
75C2×C22⋊D73—41no—
76C2×C22⋊D73—41no—
77C22×Dic73—21no—
78C22×Dic73—24no—
79C22×Dic73—21no—
80C22×Dic73—21no—
81C22×Dic73—42no—
82C22×Dic73—42no—
83C22×Dic73—42no—
84C22×Dic73—42no—
85C22×Dic73—42no—
86C22×Dic73—42no—
87C2×C4×D73—42no—
88C2×C4×D73—42no—
89C2×C4×D73—42no—
90C4⋊Dic73—21no—
91C4⋊Dic73—48no—
92C4⋊Dic73—41no—
93C4⋊Dic73—41no—
94C4⋊Dic73C4⋊C412yescomplemented
95#11924007—42no—
96C22⋊Dic73—41no—
97C22⋊Dic73—41no—
98#11924005—22no—
99#11924005—22no—
100#11924005—42no—
101#11924005—42no—
102#11924005—42no—
103C4×Dic73—42no—
104C24×C73C2×D411yescharacteristic
105C22×C4×C73—22no—
106C22×C4×C73—22no—
107C22×C4×C73—21no—
108C22×C4×C73—21no—
109C22×C4×C73—41no—
110C22×C4×C73—41no—
111C22×C4×C73—41no—
112C22×C4×C73—41no—
113C22×C4×C73C22⋊C411yescharacteristic
114C22×C4×C73C22⋊C411yescharacteristic
115C22×C4×C73C22⋊C411yescharacteristic
116C22×C4×C73C22⋊C411yescharacteristic
117C22×C4×C73D812yes—
118C22×C4×C73SD1612yes—
119C22×C4×C73C22×C411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122C22×C4×C73C2×Q811yescharacteristic
123C22×C4×C73#4111yescharacteristic
124C22×C4×C73#4111yescharacteristic
125C2×C8×C73—42no—
126C2×C8×C73—42no—
127C2×C8×C73—42no—
128C42×C73—22no—
129C42×C73—22no—
130C42×C73—41no—
131C42×C73—41no—
132C23×C73—21no—
133C23×C73—21no—
134C23×C73—21no—
135C23×C73—21no—
136C23×C73C2×C22⋊C411yescharacteristic
137C23×C73#12111yescharacteristic
138C23×C73C22⋊Q811yescharacteristic
139C23×C73C2×D811yescharacteristic
140C23×C73C2×SD1611yescharacteristic
141C23×C73AGL(1,ℤ/8)11yescharacteristic
142C23×C73#13711yescharacteristic
143C22×D73—22no—
144C22×D73—41no—
145C22×D73—41no—
146C22×D73—41no—
147C22×D73—41no—
148C22×D73—41no—
149C22×D73—41no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—24no—
157C2×C4×C73—24no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73—42no—
169C2×C4×C73—42no—
170C2×C4×C73#9912yes—
171C2×C4×C73#10212yes—
172C2×C4×C73#10212yes—
173C2×C4×C73#10412yes—
174C2×C4×C73C2×C4⋊C411yescharacteristic
175C2×C4×C73#11711yescharacteristic
176C2×C4×C73#12111yescharacteristic
177C2×C4×C73C22⋊Q811yescharacteristic
178C22⋊D73—42no—
179C22⋊D73—42no—
180C22⋊D73—42no—
181C22⋊D73—42no—
182C2×Dic73—22no—
183C2×Dic73—41no—
184C2×Dic73—41no—
185C2×Dic73—44no—
186C2×Dic73—44no—
187C2×Dic73—44no—
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73—41no—
191C2×Dic73—41no—
192C2×Dic73—41no—
193C2×Dic73—41no—
194C2×Dic73—82no—
195C2×Dic73—82no—
196C2×Dic73—82no—
197C2×Dic73—82no—
198D292—42no—
199D292—42no—
200C4×D73—84no—
201C8×C73—82no—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—21no—
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—21no—
213C22×C73—21no—
214C22×C73#38011yescharacteristic
215C22×C73C2×#9911yescharacteristic
216C22×C73#41011yescharacteristic
217C22×C73C2×#10211yescharacteristic
218C22×C73#41711yescharacteristic
219C22×C73C2×#10411yescharacteristic
220C22×C73#42111yescharacteristic
221C22×C73#45911yescharacteristic
222C22×C73#46111yescharacteristic
223C22×C73#47411yescharacteristic
224C22×C73#47611yescharacteristic
225C2×D73—42no—
226C2×D73—42no—
227C2×D73—42no—
228C2×D73—81no—
229C2×D73—81no—
230C2×D73—81no—
231C2×D73—81no—
232C4×C73—21no—
233C4×C73—21no—
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—22no—
237C4×C73—41no—
238C4×C73—41no—
239C4×C73#32714yescomplemented
240Dic73—42no—
241Dic73—84no—
242Dic73—81no—
243Dic73—81no—
244#8571—731nomaximal, Sylow
245C2×C73—21no—
246C2×C73—21no—
247C2×C73—21no—
248C2×C73—21no—
249C2×C73C2×#32711yescharacteristic
250C2×C73#150911yescharacteristic
251C2×C73#151311yescharacteristic
252C2×C73#151711yescharacteristic
253C2×C73#179911yescharacteristic
254C2×C73#180211yescharacteristic
255C2×C73#180611yescharacteristic
256D73—82no—
257C4×#121—731no—
258#1797D7311yescharacteristic, complemented
259C2×#327—732no—
260C4⋊#95—731no—
261#1294—732no—
262C73#857111yescharacteristic, complemented, Sylow
263C2×#121—731no—
264C2×C4×D4—731no—
265C2×C4×D4—1461no—
266C22×C4⋊C4—1461no—
267C22×C42C2×D7311yescharacteristic
268C2×C4⋊C8—21no—
269C2×C22⋊C8C2×D7312yes—
270#390—731no—
271#389—731no—
272#388—1461no—
273#386—1461no—
274#380—731no—
275C4×C4⋊C4—731no—
276C4×C22⋊C4—1461no—
277C2×#95—732no—
278C2×#95—1461no—
279#336—21no—
280#327—1464no—
281C22×D4—731no—
282C22×D4—1461no—
283C23×C4—1462no—
284C23×C4—1461no—
285C23×C4D29211yescharacteristic
286C23×C4D29211yescharacteristic
287C23×C4C22×D7311yescharacteristic
288C22×C8—22no—
289#121—732no—
290#121—732no—
291#121—1461no—
292#121—1461no—
293C4×D4—2922no—
294C4×D4—2922no—
295C4×D4—2922no—
296C2×C4⋊C4—2922no—
297C2×C4⋊C4—2922no—
298C2×C4⋊C4—731no—
299C2×C4⋊C4—731no—
300C2×C4⋊C4—731no—
301C2×C4⋊C4—1464no—
302C2×C4⋊C4—1461no—
303C2×C22⋊C4—1461no—
304C2×C22⋊C4—1461no—
305C2×C22⋊C4—1461no—
306C2×C22⋊C4—1461no—
307C2×C22⋊C4—1461no—
308C2×C42—42no—
309C2×C42—1461no—
310C2×C42C4×D7311yescharacteristic
311C2×C42C4×D7311yescharacteristic
312C2×C42C22⋊D7311yescharacteristic
313C2×C42C22⋊D7311yescharacteristic
314C4⋊C8—42no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317#95—2922no—
318#95—1462no—
319#95—1462no—
320#95—1461no—
321#95—1461no—
322#95—1461no—
323C24—1461no—
324C24C2×D29211yescharacteristic
325C2×D4—2921no—
326C2×D4—2921no—
327C2×D4—2921no—
328C2×D4—2921no—
329C2×D4—2921no—
330C2×D4—2921no—
331C2×D4—732no—
332C2×D4—1462no—
333C2×D4—1462no—
334C2×D4—1461no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—2922no—
344C22×C4—2922no—
345C22×C4—2922no—
346C22×C4—2922no—
347C22×C4—2922no—
348C22×C4—2922no—
349C22×C4—2922no—
350C22×C4—2922no—
351C22×C4—2922no—
352C22×C4—1464no—
353C22×C4—1461no—
354C22×C4—1461no—
355C22×C4—1461no—
356C22×C4#1192400511yescharacteristic
357C22×C4#1192400511yescharacteristic
358C22×C4#1192400511yescharacteristic
359C22×C4#1192400511yescharacteristic
360C22×C4D58412yes—
361C22×C4#1192401712yes—
362C22×C4C2×C4×D7311yescharacteristic
363C22×C4D4×D7311yescharacteristic
364C22×C4C2×C22⋊D7311yescharacteristic
365C22×C4Q8×D7311yescharacteristic
366C22×C4#1192402711yescharacteristic
367C22×C4#1192402911yescharacteristic
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—2921no—
372C4⋊C4—2921no—
373C4⋊C4—2928no—
374C4⋊C4—2922no—
375C4⋊C4—732no—
376C4⋊C4—1461no—
377C22⋊C4—2922no—
378C22⋊C4—2922no—
379C22⋊C4—2921no—
380C22⋊C4—2921no—
381C22⋊C4—2922no—
382C22⋊C4—1462no—
383C22⋊C4—1462no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—2922no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—2921no—
394C23—2921no—
395C23—2921no—
396C23—2921no—
397C23—2921no—
398C23—2921no—
399C23—1462no—
400C23C2×#1192400511yescharacteristic
401C23#1455656611yescharacteristic
402C23#1455657111yescharacteristic
403C23C2×D58411yescharacteristic
404C23C2×#1192401711yescharacteristic
405C23#1455661911yescharacteristic
406C23#1455662411yescharacteristic
407D4—2922no—
408D4—2922no—
409D4—2922no—
410D4—2922no—
411D4—2922no—
412D4—2922no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—2921no—
434C2×C4—2921no—
435C2×C4—2924no—
436C2×C4—2924no—
437C2×C4—2924no—
438C2×C4—2921no—
439C2×C4—2921no—
440C2×C4—2921no—
441C2×C4—2921no—
442C2×C4—2921no—
443C2×C4—2921no—
444C2×C4—5842no—
445C2×C4—5842no—
446C2×C4—5842no—
447C2×C4—5842no—
448C2×C4—5844no—
449C2×C4—1462no—
450C2×C4#1455650512yes—
451C2×C4#1455651312yes—
452C2×C4#1455651312yes—
453C2×C4#1455651712yes—
454C2×C4C4⋊C4×D7311yescharacteristic
455C2×C4#1455654911yescharacteristic
456C2×C4#1455656211yescharacteristic
457C2×C4#1455656711yescharacteristic
458C8—82no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—2922no—
472C22—2922no—
473C22—2922no—
474C22—5841no—
475C22—5841no—
476C22—5841no—
477C22—5841no—
478C22#1798844411yescharacteristic
479C22C2×#1455650511yescharacteristic
480C22#1798856011yescharacteristic
481C22C2×#1455651311yescharacteristic
482C22#1798858911yescharacteristic
483C22C2×#1455651711yescharacteristic
484C22#1798860311yescharacteristic
485C22#1798878311yescharacteristic
486C22#1798879311yescharacteristic
487C22#1798886811yescharacteristic
488C22#1798887811yescharacteristic
489C4—22no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—41no—
495C4—41no—
496C4—2922no—
497C4—5844no—
498C4—5841no—
499C4—5841no—
500C4#1798832214yescomplemented
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—5842no—
506C2C2×#1798832211yescharacteristic
507C2#2260845711yescharacteristic
508C2#2260847511yescharacteristic
509C2#2260849211yescharacteristic
510C2#2260991611yescharacteristic
511C2#2260993211yescharacteristic
512C2#2260994811yescharacteristic
subgroupG/H
centerC23C2×#11924005
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1797C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#14556565C22
cosocleC23×C4×C73C23
hypercenter#1797D73
hypocenterC73#8571
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #22608161C2semidirect
4. #22607811C2semidirect
5. C73×#1797C2semidirect
7. C2×C4×D292C4semidirect
10. #17988489C4semidirect
11. #17988485C4semidirect
15. C4×C4⋊Dic73C4semidirect
70. C2×D292C4⋊C4semidirect
94. C4⋊Dic73C4⋊C4semidirect
239. C4×C73#327semidirect
258. #1797D73semidirect
262. C73#8571semidirect
500. C4#17988322semidirect

Maximal subgroups

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

subgroupindex
C4×#145565652
C2×#179883222
#226081612
#226078112
C73×#17972
#857173