Finite Group Databasebeta

← #33085186

Subgroups — #33085186

Order 18688.

5982 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×D292C211yesmaximal, characteristic
2C2×#17988591C214yesmaximal, complemented
3C2×#17988551C211yesmaximal, characteristic, complemented
4#22609861C218yesmaximal, complemented
5C73×#2280C211yesmaximal, characteristic, complemented
6C25×D73—21no—
7C23×D292—216no—
8C23×D292—24no—
9C23×D292—22no—
10C23×D292—21no—
11C23×D292C418yescomplemented
12C23×D292C2214yes—
13C23×D292C2212yes—
14C22×#11924012C2212yescomplemented
15#17988591—216no—
16C2×#14556511C22116yescomplemented
17#17988551—24no—
18C2×C22⋊#10596831C2214yescomplemented
19C24×C4×C73C2211yescharacteristic
20C2×#117×C73C2212yescomplemented
21#381×C73—21no—
22#380×C73C2214yescomplemented
23#379×C73—21no—
24C4×C22⋊C4×C73—22no—
25C24×D73—216no—
26C24×D73—28no—
27C24×D73—22no—
28C24×D73—24no—
29C24×D73—416no—
30C22×D292—216no—
31C22×D292—232no—
32C22×D292—264no—
33C22×D292—264no—
34C22×D292—232no—
35C22×D292—28no—
36C22×D292—216no—
37C22×D292—28no—
38C22×D292—24no—
39C22×D292—22no—
40C22×D292—28no—
41C22×D292—24no—
42C22×D292C2×C4116yes—
43C22×D292C2×C418yes—
44C22×D292D4116yes—
45C22×D292C2314yes—
46C2×#11924012—24no—
47C2×#11924012—28no—
48C22×#10596831C2314yescomplemented
49#14556511—232no—
50C22⋊#10596831—28no—
51C25×C73D411yescharacteristic
52C23×C4×C73—28no—
53C23×C4×C73C2×C414yes—
54C23×C4×C73D412yes—
55C23×C4×C73D411yescharacteristic
56C23×C4×C73D414yes—
57C23×C4×C73C2312yes—
58C23×C4×C73C2311yescharacteristic
59#117×C73—28no—
60C2×C4⋊C4×C73C2314yescomplemented
61C2×C22⋊C4×C73—24no—
62C2×C22⋊C4×C73—22no—
63C2×C42×C73—22no—
64#95×C73—24no—
65C23×D73—232no—
66C23×D73—264no—
67C23×D73—216no—
68C23×D73—28no—
69C23×D73—216no—
70C23×D73—24no—
71C23×D73—4128no—
72C23×D73—416no—
73C23×D73—416no—
74C23×D73—464no—
75C23×D73—416no—
76C2×D292—232no—
77C2×D292—264no—
78C2×D292—264no—
79C2×D292—2128no—
80C2×D292—216no—
81C2×D292—216no—
82C2×D292—28no—
83C2×D292—216no—
84C2×D292—232no—
85C2×D292—28no—
86C2×D292—24no—
87C2×D292—264no—
88C2×D292C22⋊C4132yescomplemented
89C2×D292C22×C418yes—
90C2×D292C2×D4116yes—
91#11924012—48no—
92C2×#10596831—24no—
93C2×#10596831—28no—
94C24×C73—24no—
95C24×C73—28no—
96C24×C73C22⋊C414yes—
97C24×C73C2×D411yescharacteristic
98C24×C73C2×D412yes—
99C22×C4×C73—24no—
100C22×C4×C73—216no—
101C22×C4×C73—28no—
102C22×C4×C73—216no—
103C22×C4×C73—28no—
104C22×C4×C73—24no—
105C22×C4×C73C22⋊C414yes—
106C22×C4×C73C22⋊C418yes—
107C22×C4×C73C22×C414yes—
108C22×C4×C73C22×C412yes—
109C22×C4×C73C2×D418yes—
110C22×C4×C73C2×D412yes—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D414yes—
113C22×C4×C73C2×D412yes—
114C22×C4×C73C2411yescharacteristic
115C4⋊C4×C73—28no—
116C22⋊C4×C73—44no—
117C22⋊C4×C73—48no—
118C42×C73—44no—
119C23×C73—232no—
120C23×C73—24no—
121C23×C73—24no—
122C23×C73—24no—
123C23×C73—28no—
124C23×C73—216no—
125C23×C73C2×C22⋊C412yes—
126C23×C73C2×C22⋊C414yes—
127C23×C73#12014yes—
128C23×C73AGL(1,ℤ/8)18yes—
129C23×C73C22×D411yescharacteristic
130C22×D73—264no—
131C22×D73—28no—
132C22×D73—216no—
133C22×D73—232no—
134C22×D73—4256no—
135C22×D73—432no—
136C22×D73—432no—
137C22×D73—464no—
138C22×D73—4128no—
139C22×D73—416no—
140C22×D73—432no—
141C2×C4×C73—24no—
142C2×C4×C73—28no—
143C2×C4×C73—28no—
144C2×C4×C73—28no—
145C2×C4×C73—216no—
146C2×C4×C73—44no—
147C2×C4×C73—48no—
148C2×C4×C73C2×C22⋊C418yes—
149C2×C4×C73C2×C22⋊C414yes—
150C2×C4×C73C2×C22⋊C412yes—
151C2×C4×C73#12014yes—
152C2×C4×C73#12018yes—
153C2×C4×C73C23×C411yescharacteristic
154C2×C4×C73C22×D414yes—
155C2×C4×C73C22×D411yescharacteristic
156D292—264no—
157D292—28no—
158D292—216no—
159D292—232no—
160D292C2×C22⋊C4116yes—
161#10596831—44no—
162C22×C73—232no—
163C22×C73—24no—
164C22×C73—24no—
165C22×C73—28no—
166C22×C73—216no—
167C22×C73—24no—
168C22×C73#37914yes—
169C22×C73#41818yescomplemented
170C22×C73C22×C22⋊C411yescharacteristic
171C22×C73C2×#12012yes—
172C22×C73C2×AGL(1,ℤ/8)14yes—
173C2×D73—216no—
174C2×D73—464no—
175C2×D73—4128no—
176C2×D73—416no—
177C2×D73—432no—
178C4×C73—24no—
179C4×C73—84no—
180C4×C73#37914yes—
181C4×C73C22×C22⋊C412yes—
182C4×C73C2×#12012yes—
183#20598—731nomaximal, Sylow
184C2×C73—24no—
185C2×C73—28no—
186C2×C73C2×#37911yescharacteristic
187C2×C73C2×#41812yes—
188C2×C73#300314yes—
189D73—416no—
190C24×D4—731no—
191C2×#418—734no—
192C2×#407—731no—
193#2280D7311yescharacteristic, complemented
194#1786—738no—
195C73#2059811yescharacteristic, complemented, Sylow
196C26—1461no—
197C23×D4—738no—
198C23×D4—732no—
199C23×D4—734no—
200C23×D4—14616no—
201C23×D4—1464no—
202C23×D4—1462no—
203C23×D4—1461no—
204C24×C4C2×D7311yescharacteristic
205C22×M16—732no—
206C2×#117C2×D7312yescomplemented
207#418—14616no—
208C2×#102—7316no—
209#407—1464no—
210C2×C22⋊C8—734no—
211#381—21no—
212#380C2×D7314yescomplemented
213#379—21no—
214C4×C22⋊C4—22no—
215C25—29216no—
216C25—14616no—
217C25—1468no—
218C25—1462no—
219C25—1464no—
220C25C22⋊D7311yescharacteristic
221C22×D4—738no—
222C22×D4—7316no—
223C22×D4—7316no—
224C22×D4—734no—
225C22×D4—14664no—
226C22×D4—14632no—
227C22×D4—14616no—
228C22×D4—14632no—
229C22×D4—14664no—
230C22×D4—1468no—
231C22×D4—14616no—
232C22×D4—1468no—
233C22×D4—1464no—
234C22×D4—1462no—
235C22×D4—1468no—
236C22×D4—1464no—
237C23×C4—28no—
238C23×C4C4×D7314yes—
239C23×C4D29214yes—
240C23×C4C22⋊D7312yes—
241C23×C4C22⋊D7311yescharacteristic
242C23×C4C22×D7312yes—
243C23×C4C22×D7311yescharacteristic
244C2×M16—1468no—
245C2×M16—1464no—
246C22×C8—734no—
247#117—28no—
248C2×C4⋊C4C22×D7314yescomplemented
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C42—22no—
252#102—14632no—
253C22⋊C8—1468no—
254#95—24no—
255C24—24no—
256C24—28no—
257C24—292128no—
258C24—29264no—
259C24—29216no—
260C24—29216no—
261C24—29216no—
262C24—14664no—
263C24—14632no—
264C24—1468no—
265C24—14616no—
266C24—14616no—
267C24—1464no—
268C24#1192400514yes—
269C24C2×C22⋊D7311yescharacteristic
270C24C2×C22⋊D7312yes—
271C2×D4—7332no—
272C2×D4—738no—
273C2×D4—7316no—
274C2×D4—14664no—
275C2×D4—14664no—
276C2×D4—14632no—
277C2×D4—146128no—
278C2×D4—14664no—
279C2×D4—14616no—
280C2×D4—14616no—
281C2×D4—1468no—
282C2×D4—14616no—
283C2×D4—14632no—
284C2×D4—1468no—
285C2×D4—1464no—
286C22×C4—24no—
287C22×C4—216no—
288C22×C4—28no—
289C22×C4—216no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4#1192400514yes—
293C22×C4#1192400518yes—
294C22×C4C2×C4×D7314yes—
295C22×C4C2×C4×D7312yes—
296C22×C4D4×D7318yes—
297C22×C4C2×C22⋊D7312yes—
298C22×C4C2×C22⋊D7311yescharacteristic
299C22×C4C2×D29214yes—
300C22×C4C2×D29212yes—
301C22×C4C23×D7311yescharacteristic
302M16—2928no—
303C2×C8—1468no—
304C2×C8—1464no—
305C4⋊C4—28no—
306C22⋊C4—44no—
307C22⋊C4—48no—
308C42—44no—
309C23—232no—
310C23—28no—
311C23—24no—
312C23—216no—
313C23—24no—
314C23—24no—
315C23—292256no—
316C23—29264no—
317C23—292128no—
318C23—29232no—
319C23—29232no—
320C23—29232no—
321C23—29216no—
322C23—14664no—
323C23—14632no—
324C23—1468no—
325C23—14616no—
326C23C2×#1192400512yes—
327C23C2×#1192400514yes—
328C23#1455655914yes—
329C23#1455661618yes—
330C23C22×C22⋊D7311yescharacteristic
331D4—7316no—
332D4—14664no—
333D4—14632no—
334D4—1468no—
335D4—14616no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—216no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4C22⋊C4×D7318yes—
344C2×C4C2×#1192400514yes—
345C2×C4C2×#1192400512yes—
346C2×C4#1455655914yes—
347C2×C4#1455656118yes—
348C2×C4C22×C4×D7311yescharacteristic
349C2×C4C2×D4×D7314yes—
350C2×C4C22×D29211yescharacteristic
351C8—2924no—
352C22—232no—
353C22—28no—
354C22—216no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—29264no—
359C22—292128no—
360C22—29232no—
361C22—29216no—
362C22—14616no—
363C22#1798844214yes—
364C22#1798859118yescomplemented
365C22C22×#1192400511yescharacteristic
366C22C2×#1455655912yes—
367C22C2×#1455661614yes—
368C4—24no—
369C4—84no—
370C4#1798844214yes—
371C4C2×C22⋊C4×D7312yes—
372C4C2×#1455656112yes—
373C2—24no—
374C2—28no—
375C2—29216no—
376C2C2×#1798844211yescharacteristic
377C2C2×#1798859112yes—
378C2#2261915714yes—
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2280C2
socleC23×C73C22×D4
prime radicalC24×D292C2
cosocleC22×C4×C73C24
hypercenter#2280D73
hypocenterC73#20598
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#17988591C2semidirect
3. C2×#17988551C2semidirect
4. #22609861C2semidirect
5. C73×#2280C2semidirect
11. C23×D292C4semidirect
14. C22×#11924012C22semidirect
16. C2×#14556511C22semidirect
18. C2×C22⋊#10596831C22semidirect
20. C2×#117×C73C22semidirect
22. #380×C73C22semidirect
48. C22×#10596831C23semidirect
60. C2×C4⋊C4×C73C23semidirect
88. C2×D292C22⋊C4semidirect
169. C22×C73#418semidirect
193. #2280D73semidirect
195. C73#20598semidirect
206. C2×#117C2×D73semidirect
212. #380C2×D73semidirect
248. C2×C4⋊C4C22×D73semidirect
364. C22#17988591semidirect

Maximal subgroups

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

subgroupindex
C24×D2922
C2×#179885912
C2×#179885512
#226098612
C73×#22802
#2059873