Finite Group Databasebeta

← #33071174

Subgroups — #33071174

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620876C211yesmaximal, characteristic, complemented
2#22619724C211yesmaximal, characteristic
3C2×#17988868C211yesmaximal, characteristic
4#22618954C211yesmaximal, characteristic
5#22609385C214yesmaximal, complemented
6#22609295C214yesmaximal, complemented
7#22608705C211yesmaximal, characteristic, complemented
8#22608615C211yesmaximal, characteristic, complemented
9C73×#1542C211yesmaximal, characteristic, complemented
10C2×#14556591C2211yescharacteristic
11C2×C4×D292C2211yescharacteristic
12#17988966C2212yescomplemented
13#17988939C2212yescomplemented
14#17988884—22no—
15#17988874C2212yes—
16#17988868—22no—
17#17988868C2214yes—
18#17988864C2212yes—
19#17988797—22no—
20#17988742—22no—
21C8×D292—22no—
22#17988652C2214yes—
23#17988640C2211yescharacteristic
24#17988630—21no—
25C2×#14556525—21no—
26#17988611C2211yescharacteristic
27#17988591—21no—
28C2×#14556513—21no—
29C22⋊C8×D73—22no—
30#17988339C2214yescomplemented
31#17988334C2214yescomplemented
32#533×C73C2211yescharacteristic
33#431×C73C2211yescharacteristic
34C2×C4⋊C8×C73C2211yescharacteristic
35#333×C73C2212yescomplemented
36#332×C73C2212yescomplemented
37C22×Dic146—21no—
38C22×D292D411yescharacteristic
39C22×C4×D73—21no—
40C2×#11924018—24no—
41C2×#11924012—21no—
42C2×C8×D73—24no—
43#14556591—21no—
44#14556591—21no—
45#14556591—21no—
46#14556591C2312yes—
47C4×D292—21no—
48C4×D292—21no—
49C4×D292D412yes—
50C4×D292C2312yes—
51C2×C4⋊Dic73—21no—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73D411yescharacteristic
54C2×#11924005—21no—
55#14556525—24no—
56#14556524—22no—
57#14556522—22no—
58C4⋊#10596831—21no—
59C4⋊#10596831D412yes—
60C4⋊#10596831C2312yes—
61#14556514—24no—
62#14556513—24no—
63#14556511—22no—
64#14556511—22no—
65C22⋊#10596831D412yes—
66#14556503—24no—
67C4×#10596831C2312yes—
68C22×C8×C73—21no—
69C4⋊Q8×C73C2311yescharacteristic
70C4⋊Q8×C73C2311yescharacteristic
71#125×C73—21no—
72C22⋊Q8×C73—21no—
73C22⋊Q8×C73—21no—
74C4×Q8×C73—21no—
75C4×Q8×C73—21no—
76#117×C73—21no—
77C2×C42×C73C2311yescharacteristic
78C4⋊C8×C73—21no—
79C4⋊C8×C73C2312yes—
80C4⋊C8×C73C2312yes—
81C22⋊C8×C73—21no—
82C4×C8×C73—21no—
83C23×D73—21no—
84C2×Dic146—22no—
85C2×Dic146—41no—
86C2×Dic146—82no—
87C2×D292—21no—
88C2×D292—22no—
89C2×D292Q1614yescomplemented
90C2×D292C2×D412yes—
91C22×Dic73—21no—
92C22×Dic73—41no—
93C2×C4×D73—22no—
94C2×C4×D73—22no—
95C2×C4×D73—82no—
96#11924018—84no—
97#11924012—42no—
98C2×#10596831—22no—
99C2×#10596831C2×D414yes—
100C8×D73—84no—
101C4⋊Dic73—21no—
102C4⋊Dic73—22no—
103C4⋊Dic73—22no—
104C4⋊Dic73—41no—
105C4⋊Dic73—41no—
106C4⋊Dic73C2×D412yes—
107#11924005—22no—
108#11924005—22no—
109C2×Q8×C73—21no—
110C2×Q8×C73—21no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4111yescharacteristic
114C2×C8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73—42no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C42×C73—41no—
125C42×C73C2×D411yescharacteristic
126C42×C73C2×D411yescharacteristic
127C42×C73#4111yescharacteristic
128C42×C73C2411yescharacteristic
129C23×C73#12111yescharacteristic
130C22×D73—22no—
131C22×D73—24no—
132C22×D73—81no—
133C22×D73—81no—
134Q8×C73—42no—
135Q8×C73—81no—
136C2×C4×C73—22no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73#12111yescharacteristic
141C2×C4×C73#12111yescharacteristic
142C2×C4×C73#12111yescharacteristic
143C2×C4×C73C2×Q1612yes—
144C2×C4×C73#13512yes—
145C2×C4×C73C22×D411yescharacteristic
146C2×C4×C73C22×D411yescharacteristic
147C2×C4×C73C2×#4111yescharacteristic
148C2×Dic73—22no—
149C2×Dic73—42no—
150C2×Dic73—81no—
151C2×Dic73—82no—
152D292—22no—
153D292—24no—
154D292C2×Q1614yes—
155C4×D73—44no—
156C4×D73—82no—
157Dic146—44no—
158Dic146—82no—
159C8×C73—42no—
160C8×C73—42no—
161#10596831—24no—
162#10596831—42no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73#46712yes—
166C22×C73C2×#12111yescharacteristic
167C2×D73—24no—
168C2×D73—42no—
169C2×D73—42no—
170C2×D73—84no—
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—81no—
175C4×C73—81no—
176C4×C73#46712yes—
177C4×C73C22×Q1611yescharacteristic
178C4×C73C2×#13511yescharacteristic
179C4×C73#57612yes—
180Dic73—42no—
181Dic73—82no—
182#19387—731nomaximal, Sylow
183C2×C73—22no—
184C2×C73#304111yescharacteristic
185C2×C73C2×#46711yescharacteristic
186C2×C73#306911yescharacteristic
187D73—44no—
188#3209—731no—
189#3077—731no—
190C2×#474—731no—
191#2972—731no—
192#1683—734no—
193#1669—734no—
194#1571—731no—
195#1549—731no—
196#1542D7311yescharacteristic, complemented
197C73#1938711yescharacteristic, complemented, Sylow
198#533C2×D7311yescharacteristic
199C2×C4⋊Q8—731no—
200C2×C4×D4—731no—
201#494—732no—
202#487—732no—
203#478—1462no—
204#476—732no—
205#474—732no—
206#474—734no—
207#474—1462no—
208#462—1462no—
209#451—1462no—
210C8×D4—734no—
211C8×D4—1462no—
212#431—731no—
213#431C2×D7311yescharacteristic
214#428—1461no—
215C2×#107—1461no—
216#423—731no—
217C2×C4⋊C8C2×D7311yescharacteristic
218#418—1461no—
219C2×#102—1461no—
220C2×C22⋊C8—1462no—
221#335—734no—
222#333C2×D7312yescomplemented
223#332—734no—
224#332C2×D7312yescomplemented
225C22×Q8—1461no—
226C22×D4—731no—
227C23×C4—1461no—
228C2×Q16—1464no—
229C2×M16—1461no—
230C22×C8—21no—
231C22×C8—1464no—
232C4⋊Q8—732no—
233C4⋊Q8—1461no—
234C4⋊Q8—1461no—
235C4⋊Q8—1461no—
236C4⋊Q8C22×D7311yescharacteristic
237C4⋊Q8C22×D7311yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—732no—
244C4×D4—732no—
245C4×D4—1461no—
246C4×D4—1461no—
247#117—21no—
248C2×C4⋊C4—731no—
249C2×C4⋊C4—1461no—
250C2×C4⋊C4—1461no—
251C2×C22⋊C4—1461no—
252C2×C42C22×D7311yescharacteristic
253#107—1462no—
254#107—1464no—
255#106—1462no—
256C4⋊C8—21no—
257C4⋊C8—732no—
258C4⋊C8—732no—
259C4⋊C8—1461no—
260C4⋊C8C22×D7312yes—
261C4⋊C8C22×D7312yes—
262#103—1464no—
263#102—1462no—
264#102—1462no—
265#102—1464no—
266C22⋊C8—21no—
267C22⋊C8—732no—
268C22⋊C8—1464no—
269C4×C8—21no—
270C4×C8—732no—
271C24—1461no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—2921no—
275C2×Q8—5842no—
276C2×Q8—1462no—
277C2×D4—734no—
278C2×D4—732no—
279C2×D4—1461no—
280C2×D4—1462no—
281C22×C4—2921no—
282C22×C4—5842no—
283C22×C4—1462no—
284C22×C4—1461no—
285C22×C4—1462no—
286C22×C4D4×D7311yescharacteristic
287C22×C4D4×D7311yescharacteristic
288C22×C4#1192402811yescharacteristic
289Q16—5844no—
290M16—2922no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—5844no—
295C2×C8—734no—
296C2×C8—1462no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—2921no—
304C4⋊C4—2921no—
305C4⋊C4—732no—
306C4⋊C4—1462no—
307C4⋊C4—1462no—
308C4⋊C4—1461no—
309C22⋊C4—41no—
310C22⋊C4—1462no—
311C22⋊C4—1462no—
312C42—41no—
313C42D4×D7311yescharacteristic
314C42D4×D7311yescharacteristic
315C42#1192402811yescharacteristic
316C42C23×D7311yescharacteristic
317C23—5841no—
318C23—5841no—
319C23—1462no—
320C23—1464no—
321C23#1455656411yescharacteristic
322Q8—42no—
323Q8—81no—
324Q8—2924no—
325Q8—5842no—
326D4—734no—
327D4—1464no—
328D4—1462no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—2924no—
334C2×C4—2922no—
335C2×C4—5842no—
336C2×C4—5841no—
337C2×C4—5842no—
338C2×C4—1462no—
339C2×C4#1455656411yescharacteristic
340C2×C4#1455656411yescharacteristic
341C2×C4#1455656411yescharacteristic
342C2×C4Q16×D7312yes—
343C2×C4#1455661212yes—
344C2×C4C2×D4×D7311yescharacteristic
345C2×C4C2×D4×D7311yescharacteristic
346C2×C4C2×#1192402811yescharacteristic
347C8—42no—
348C8—42no—
349C8—2922no—
350C8—1464no—
351C22—21no—
352C22—21no—
353C22—2922no—
354C22—2922no—
355C22—5844no—
356C22—1464no—
357C22#1798882712yes—
358C22C2×#1455656411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—2922no—
365C4—5842no—
366C4#1798882712yes—
367C4C2×Q16×D7311yescharacteristic
368C4C2×#1455661211yescharacteristic
369C4#1798938912yes—
370C2—22no—
371C2—2924no—
372C2#2261943711yescharacteristic
373C2C2×#1798882711yescharacteristic
374C2#2261967111yescharacteristic
subgroupG/H
centerC22C2×#14556564
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#1542C2
socleC22×C73C2×#121
prime radicalC22×D292D4
hypercenter#1542D73
hypocenterC73#19387
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #22620876C2semidirect
5. #22609385C2semidirect
6. #22609295C2semidirect
7. #22608705C2semidirect
8. #22608615C2semidirect
9. C73×#1542C2semidirect
12. #17988966C22semidirect
13. #17988939C22semidirect
30. #17988339C22semidirect
31. #17988334C22semidirect
35. #333×C73C22semidirect
36. #332×C73C22semidirect
89. C2×D292Q16semidirect
196. #1542D73semidirect
197. C73#19387semidirect
222. #333C2×D73semidirect
224. #332C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226208762
#226197242
C2×#179888682
#226189542
#226093852
#226092952
#226087052
#226086152
C73×#15422
#1938773