Finite Group Databasebeta

← #33055445

Subgroups — #33055445

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619012C211yesmaximal, characteristic, complemented
2#22618954C211yesmaximal, characteristic
3C2×C4×#11924015C211yesmaximal, characteristic
4#22608746C214yesmaximal, complemented
5#22608728C214yesmaximal, complemented
6#22608677C211yesmaximal, characteristic, complemented
7#22608323C211yesmaximal, characteristic
8#22608215C211yesmaximal, characteristic, complemented
9C73×#1479C211yesmaximal, characteristic, complemented
10C22×#11924015C411yescharacteristic
11C2×C4×D292C2211yescharacteristic
12Q8×#10596831—22no—
13Q8×#10596831C2214yescomplemented
14C4×#11924015C412yes—
15C4×#11924015C412yes—
16C4×#11924015C2212yes—
17C4×#11924015C2212yes—
18C8×D292—22no—
19#17988652C2214yes—
20#17988640—22no—
21#17988640C2211yescharacteristic
22#17988640C2211yescharacteristic
23C2×#14556522C411yescharacteristic
24C2×#14556515C411yescharacteristic
25C2×#14556511C411yescharacteristic
26C22⋊C8×D73—22no—
27C2×C4×#10596831C2211yescharacteristic
28#17988339C2212yescomplemented
29#17988337C2212yescomplemented
30#17988316C2214yes—
31#17988309C2212yescomplemented
32C8×#10596831C2212yescomplemented
33C2×C4×Q8×C73C2211yescharacteristic
34#431×C73C2211yescharacteristic
35#431×C73C2211yescharacteristic
36Q8⋊C8×C73C2214yescomplemented
37C22×D292C2×C411yescharacteristic
38C22×C4×D73—21no—
39C2×#11924015C818yescomplemented
40C2×#11924015C2×C412yes—
41C2×#11924015C2×C412yes—
42C22×#10596831C2×C411yescharacteristic
43C22×#10596831C2×C411yescharacteristic
44C2×C8×D73—24no—
45C4×D292—21no—
46C4×D292—21no—
47C4×D292C2×C412yes—
48C4×D292C2312yes—
49C2×C4⋊Dic73C2×C411yescharacteristic
50C2×#11924005—21no—
51#14556522C2×C412yes—
52#14556522C2×C412yes—
53C4⋊#10596831—24no—
54C4⋊#10596831D412yes—
55C4⋊#10596831C2312yes—
56#14556515C2×C412yes—
57#14556515C2×C412yes—
58#14556511C2×C412yes—
59#14556511C2×C412yes—
60C22⋊#10596831—21no—
61C22⋊#10596831D412yes—
62#14556503—24no—
63C4×#10596831—24no—
64C4×#10596831—21no—
65C4×#10596831C2×C411yescharacteristic
66C4×#10596831C2×C411yescharacteristic
67C4×#10596831C2311yescharacteristic
68C4×#10596831C2311yescharacteristic
69C4×#10596831C2312yes—
70C22×Q8×C73C2×C411yescharacteristic
71C4×Q8×C73—21no—
72C4×Q8×C73—21no—
73C4×Q8×C73C2×C412yes—
74C4×Q8×C73C2312yes—
75C2×C4⋊C4×C73—21no—
76C2×C4⋊C4×C73C2×C411yescharacteristic
77C2×C42×C73—21no—
78C2×C42×C73C2311yescharacteristic
79C4⋊C8×C73—21no—
80C4⋊C8×C73C2312yes—
81C22⋊C8×C73—21no—
82C22⋊C8×C73—21no—
83C4×C8×C73—21no—
84C4×C8×C73C2312yes—
85C23×D73—21no—
86C2×D292—21no—
87C2×D292—22no—
88C2×D292C2×C814yes—
89C2×D292C22×C412yes—
90C22×Dic73—21no—
91C2×C4×D73—22no—
92C2×C4×D73—22no—
93C2×C4×D73—82no—
94#11924015C2×C818yescomplemented
95#11924015C2×C818yescomplemented
96C2×#10596831—22no—
97C2×#10596831—22no—
98C2×#10596831C2×C814yescomplemented
99C2×#10596831C22×C411yescharacteristic
100C2×#10596831C22×C411yescharacteristic
101C2×#10596831C22×C411yescharacteristic
102C2×#10596831C22×C411yescharacteristic
103C2×#10596831C2×D414yes—
104C8×D73—84no—
105C4⋊Dic73—21no—
106C4⋊Dic73C22×C412yes—
107#11924005—22no—
108#11924005—22no—
109C2×Q8×C73—21no—
110C2×Q8×C73—22no—
111C2×Q8×C73C2×C814yescomplemented
112C2×Q8×C73C22×C412yes—
113C22×C4×C73—21no—
114C22×C4×C73—21no—
115C22×C4×C73C22×C411yescharacteristic
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73#4111yescharacteristic
118C2×C8×C73—22no—
119C2×C8×C73—22no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—21no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73C22×C412yes—
124C42×C73—22no—
125C42×C73—22no—
126C42×C73C22×C411yescharacteristic
127C42×C73C2×D411yescharacteristic
128C42×C73#4111yescharacteristic
129C42×C73C2411yescharacteristic
130C23×C73C4×D411yescharacteristic
131C22×D73—22no—
132C22×D73—24no—
133C22×D73—81no—
134C22×D73—81no—
135Q8×C73—22no—
136Q8×C73—24no—
137Q8×C73C22×C814yes—
138C2×C4×C73—22no—
139C2×C4×C73—24no—
140C2×C4×C73—22no—
141C2×C4×C73—22no—
142C2×C4×C73—41no—
143C2×C4×C73—81no—
144C2×C4×C73C4×D411yescharacteristic
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73C22×C812yes—
148C2×C4×C73#13112yes—
149C2×C4×C73C23×C411yescharacteristic
150C2×C4×C73C22×D411yescharacteristic
151C2×C4×C73C2×#4111yescharacteristic
152C2×Dic73—22no—
153C2×Dic73—81no—
154D292—22no—
155D292—24no—
156D292C22×C814yes—
157C4×D73—44no—
158C4×D73—82no—
159C8×C73—42no—
160C8×C73—42no—
161#10596831—21no—
162#10596831—21no—
163#10596831—24no—
164#10596831—42no—
165#10596831C22×C812yes—
166#10596831C22×C812yes—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73C8×D412yes—
170C22×C73C2×C4×D411yescharacteristic
171C2×D73—24no—
172C2×D73—42no—
173C2×D73—42no—
174C2×D73—84no—
175C4×C73—24no—
176C4×C73—21no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—42no—
180C4×C73C8×D412yes—
181C4×C73C23×C811yescharacteristic
182C4×C73C2×#13111yescharacteristic
183C4×C73#57712yes—
184Dic73—42no—
185#17855—731nomaximal, Sylow
186C2×C73—22no—
187C2×C73C2×C8×D411yescharacteristic
188C2×C73#294911yescharacteristic
189C2×C73#295711yescharacteristic
190D73—44no—
191#2982—731no—
192#2972—731no—
193C2×C4×SD16—731no—
194SD16⋊C8—734no—
195C8×SD16—734no—
196#1564—731no—
197#1479D7311yescharacteristic, complemented
198#1478—731no—
199#1450—731no—
200C73#1785511yescharacteristic, complemented, Sylow
201C22×SD16—731no—
202C2×C4×Q8C2×D7311yescharacteristic
203C2×C4×D4—731no—
204C8×Q8—734no—
205C8×Q8—1462no—
206C4×SD16—732no—
207C4×SD16—732no—
208C4×SD16—732no—
209C4×SD16—732no—
210C8×D4—734no—
211C8×D4—1462no—
212#431—731no—
213#431—731no—
214#431—1462no—
215#431C2×D7311yescharacteristic
216#431C2×D7311yescharacteristic
217C2×#106—731no—
218C2×#103—731no—
219C2×#102—731no—
220C2×C22⋊C8—1462no—
221C2×C4×C8—731no—
222#335—732no—
223#334—732no—
224Q8⋊C8C2×D7314yescomplemented
225D4⋊C8—734no—
226#322—732no—
227C82—732no—
228C22×Q8C4×D7311yescharacteristic
229C22×D4—731no—
230C23×C4—1461no—
231C2×SD16—738no—
232C2×SD16—732no—
233C2×SD16—732no—
234C22×C8—731no—
235C22×C8—731no—
236C22×C8—1464no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C4×D7312yes—
240C4×Q8C22×D7312yes—
241C4×D4—732no—
242C4×D4—732no—
243C4×D4—1461no—
244C4×D4—1461no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—731no—
247C2×C4⋊C4C4×D7311yescharacteristic
248C2×C22⋊C4—1461no—
249C2×C42—21no—
250C2×C42C22×D7311yescharacteristic
251#106—732no—
252#106—732no—
253C4⋊C8—21no—
254C4⋊C8—732no—
255C4⋊C8—732no—
256C4⋊C8—1464no—
257C4⋊C8C22×D7312yes—
258#103—732no—
259#103—732no—
260#102—732no—
261#102—732no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—732no—
265C22⋊C8—1464no—
266C22⋊C8—1461no—
267C4×C8—21no—
268C4×C8—731no—
269C4×C8—731no—
270C4×C8—731no—
271C4×C8—732no—
272C4×C8—731no—
273C4×C8—1464no—
274C4×C8—1461no—
275C4×C8C22×D7312yes—
276C24—1461no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8C8×D7314yescomplemented
280C2×Q8C2×C4×D7312yes—
281C2×D4—734no—
282C2×D4—732no—
283C2×D4—1461no—
284C2×D4—1462no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—5842no—
288C22×C4—1462no—
289C22×C4—1461no—
290C22×C4—1462no—
291C22×C4C2×C4×D7311yescharacteristic
292C22×C4D4×D7311yescharacteristic
293C22×C4#1192402711yescharacteristic
294SD16—738no—
295SD16—738no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—5844no—
299C2×C8—734no—
300C2×C8—731no—
301C2×C8—731no—
302C2×C8—731no—
303C2×C8—731no—
304C2×C8—734no—
305C2×C8—1462no—
306C2×C8—1462no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—732no—
311C4⋊C4—1461no—
312C4⋊C4C2×C4×D7312yes—
313C22⋊C4—1462no—
314C22⋊C4—1462no—
315C42—22no—
316C42—22no—
317C42C2×C4×D7311yescharacteristic
318C42D4×D7311yescharacteristic
319C42#1192402711yescharacteristic
320C42C23×D7311yescharacteristic
321C23—5841no—
322C23—5841no—
323C23—1462no—
324C23—1464no—
325C23#1455655111yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8C2×C8×D7314yes—
329D4—734no—
330D4—1464no—
331D4—1462no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—2924no—
339C2×C4—5842no—
340C2×C4—5841no—
341C2×C4—1462no—
342C2×C4#1455655111yescharacteristic
343C2×C4#1455655111yescharacteristic
344C2×C4#1455655111yescharacteristic
345C2×C4C2×C8×D7312yes—
346C2×C4#1455659912yes—
347C2×C4C22×C4×D7311yescharacteristic
348C2×C4C2×D4×D7311yescharacteristic
349C2×C4C2×#1192402711yescharacteristic
350C8—42no—
351C8—42no—
352C8—2922no—
353C8—732no—
354C8—732no—
355C8—1464no—
356C8—1461no—
357C8—1461no—
358C22—21no—
359C22—21no—
360C22—2922no—
361C22—2922no—
362C22—5844no—
363C22—1464no—
364C22#1798865012yes—
365C22C2×#1455655111yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—24no—
370C4—42no—
371C4—2922no—
372C4#1798865012yes—
373C4C22×C8×D7311yescharacteristic
374C4C2×#1455659911yescharacteristic
375C4#1798939612yes—
376C2—22no—
377C2—2924no—
378C2C2×#1798865011yescharacteristic
379C2#2261878611yescharacteristic
380C2#2261884411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC4×C73C23×C8
FrattiniC42C23×D73
FittingC73×#1479C2
socleC22×C73C2×C4×D4
prime radicalC22×D292C2×C4
cosocleC42×C73C24
hypercenter#1479D73
hypocenterC73#17855
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
1. #22619012C2semidirect
4. #22608746C2semidirect
5. #22608728C2semidirect
6. #22608677C2semidirect
8. #22608215C2semidirect
9. C73×#1479C2semidirect
13. Q8×#10596831C22semidirect
28. #17988339C22semidirect
29. #17988337C22semidirect
31. #17988309C22semidirect
32. C8×#10596831C22semidirect
36. Q8⋊C8×C73C22semidirect
39. C2×#11924015C8semidirect
94. #11924015C2×C8semidirect
95. #11924015C2×C8semidirect
98. C2×#10596831C2×C8semidirect
111. C2×Q8×C73C2×C8semidirect
197. #1479D73semidirect
200. C73#17855semidirect
224. Q8⋊C8C2×D73semidirect
279. C2×Q8C8×D73semidirect

Maximal subgroups

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

subgroupindex
#226190122
#226189542
C2×C4×#119240152
#226087462
#226087282
#226086772
#226083232
#226082152
C73×#14792
#1785573