Finite Group Databasebeta

← #32976192

Subgroups — #32976192

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618564C211yesmaximal, characteristic
2C2×#17988644C211yesmaximal, characteristic
3#22607824C214yesmaximal, complemented
4C4×#117×C73C211yesmaximal, characteristic
5#17988644—21no—
6#17988644—21no—
7#17988644C412yescomplemented
8#17988644C412yescomplemented
9#17988644C412yescomplemented
10#17988644C2212yes—
11#17988640—21no—
12#17988640—21no—
13#17988611—21no—
14C2×C4⋊#10596831C412yescomplemented
15#17988551—21no—
16C2×C22⋊#10596831—21no—
17C4×#11924012C2212yes—
18C2×#14556502—21no—
19C2×#117×C73—21no—
20C22×C42×C73C2211yescharacteristic
21C4×C4⋊C4×C73—21no—
22C4×C4⋊C4×C73C412yes—
23C4×C22⋊C4×C73—21no—
24C4×C22⋊C4×C73C412yes—
25#376×C73C2212yes—
26C43×C73—21no—
27C2×#11924012—22no—
28C22×#10596831—22no—
29C4⋊#10596831—22no—
30C4⋊#10596831—22no—
31C4⋊#10596831—22no—
32C4⋊#10596831C2×C414yes—
33C22⋊#10596831—22no—
34C22⋊#10596831—22no—
35C22⋊#10596831—22no—
36C22⋊#10596831—22no—
37#14556502—22no—
38#14556502—22no—
39#14556502C2×C414yes—
40C4×#10596831—22no—
41C23×C4×C73C2×C411yescharacteristic
42C23×C4×C73D411yescharacteristic
43C23×C4×C73D411yescharacteristic
44#117×C73—28no—
45C2×C4⋊C4×C73—22no—
46C2×C22⋊C4×C73—22no—
47C2×C42×C73—22no—
48C2×C42×C73—22no—
49C2×C42×C73—22no—
50C2×C42×C73C2×C411yescharacteristic
51C2×C42×C73C2×C411yescharacteristic
52C2×C42×C73C2×C411yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54C2×C42×C73C2×C411yescharacteristic
55C2×C42×C73D411yescharacteristic
56C2×C42×C73D411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C42×C73D411yescharacteristic
59C2×C42×C73Q811yescharacteristic
60C2×C42×C73Q811yescharacteristic
61C2×C42×C73C2311yescharacteristic
62#95×C73—22no—
63#95×C73C2×C414yes—
64#11924012—44no—
65C2×#10596831—24no—
66C2×#10596831—24no—
67C2×#10596831—44no—
68C24×C73C22⋊C411yescharacteristic
69C22×C4×C73—24no—
70C22×C4×C73—24no—
71C22×C4×C73—42no—
72C22×C4×C73—42no—
73C22×C4×C73C4211yescharacteristic
74C22×C4×C73C4211yescharacteristic
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C4⋊C411yescharacteristic
83C22×C4×C73C4⋊C411yescharacteristic
84C22×C4×C73C4⋊C411yescharacteristic
85C22×C4×C73C4⋊C411yescharacteristic
86C22×C4×C73C4⋊C411yescharacteristic
87C22×C4×C73C4⋊C411yescharacteristic
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C4⋊C4×C73—28no—
92C22⋊C4×C73—28no—
93C42×C73—28no—
94C42×C73—44no—
95C42×C73—44no—
96C42×C73C4211yescharacteristic
97C42×C73C4211yescharacteristic
98C42×C73C22⋊C411yescharacteristic
99C42×C73C22⋊C411yescharacteristic
100C42×C73C22⋊C411yescharacteristic
101C42×C73C22⋊C411yescharacteristic
102C42×C73C4⋊C411yescharacteristic
103C42×C73C4⋊C411yescharacteristic
104C42×C73C4⋊C411yescharacteristic
105C42×C73C4⋊C411yescharacteristic
106C42×C73C4⋊C411yescharacteristic
107C42×C73C4⋊C411yescharacteristic
108C42×C73C22×C411yescharacteristic
109C42×C73C22×C411yescharacteristic
110C42×C73C2×D411yescharacteristic
111C42×C73C2×Q811yescharacteristic
112C23×C73—22no—
113C23×C73—22no—
114C23×C73—41no—
115C23×C73#9511yescharacteristic
116C23×C73#9511yescharacteristic
117C23×C73C2×C22⋊C411yescharacteristic
118C2×C4×C73—22no—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—22no—
122C2×C4×C73—22no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—22no—
128C2×C4×C73—28no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—44no—
137C2×C4×C73—44no—
138C2×C4×C73#9511yescharacteristic
139C2×C4×C73#9511yescharacteristic
140C2×C4×C73#9511yescharacteristic
141C2×C4×C73#9511yescharacteristic
142C2×C4×C73#9511yescharacteristic
143C2×C4×C73#9511yescharacteristic
144C2×C4×C73#10414yes—
145C2×C4×C73C2×C4211yescharacteristic
146C2×C4×C73C2×C22⋊C411yescharacteristic
147C2×C4×C73C2×C22⋊C411yescharacteristic
148C2×C4×C73C2×C4⋊C411yescharacteristic
149C2×C4×C73C2×C4⋊C411yescharacteristic
150C2×C4×C73C2×C4⋊C411yescharacteristic
151#10596831—44no—
152#10596831—44no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—44no—
161C22×C73#33912yes—
162C22×C73C2×#9511yescharacteristic
163C22×C73#42112yes—
164C4×C73—21no—
165C4×C73—21no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—48no—
174C4×C73#33912yes—
175C4×C73#34412yes—
176C4×C73C2×#10412yes—
177#8071—731nomaximal, Sylow
178C2×C73—22no—
179C2×C73—22no—
180C2×C73—41no—
181C2×C73C2×#33911yescharacteristic
182C2×C73#173311yescharacteristic
183C2×C73C2×#34411yescharacteristic
184#2920—731no—
185C2×#432—731no—
186C4×#117D7311yescharacteristic
187#1300—734no—
188C73#807111yescharacteristic, complemented, Sylow
189C2×#117—21no—
190C22×C42C2×D7311yescharacteristic
191#432—732no—
192#432—732no—
193#432—732no—
194#432—732no—
195#432—1461no—
196#432—1461no—
197#431—1461no—
198#431—1461no—
199#423—1461no—
200C2×C4⋊C8—732no—
201#407—1461no—
202C2×C22⋊C8—1461no—
203C4×M16—732no—
204C2×#97—1461no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4Dic7312yes—
207C4×C22⋊C4—21no—
208C4×C22⋊C4Dic7312yes—
209#376C2×D7312yes—
210C43—21no—
211C23×C4C2×Dic7311yescharacteristic
212C23×C4C22⋊D7311yescharacteristic
213C23×C4C22⋊D7311yescharacteristic
214C2×M16—1462no—
215C22×C8—1462no—
216#117—28no—
217C2×C4⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42Dic14611yescharacteristic
223C2×C42Dic14611yescharacteristic
224C2×C42C4×D7311yescharacteristic
225C2×C42C4×D7311yescharacteristic
226C2×C42C4×D7311yescharacteristic
227C2×C42C4×D7311yescharacteristic
228C2×C42D29211yescharacteristic
229C2×C42D29211yescharacteristic
230C2×C42C2×Dic7311yescharacteristic
231C2×C42C22⋊D7311yescharacteristic
232C2×C42C22⋊D7311yescharacteristic
233C2×C42C22×D7311yescharacteristic
234C4⋊C8—734no—
235C4⋊C8—1462no—
236C4⋊C8—1462no—
237C4⋊C8—1462no—
238C22⋊C8—1462no—
239C22⋊C8—1462no—
240C22⋊C8—1462no—
241C22⋊C8—1462no—
242#97—734no—
243#97—1462no—
244#97—1462no—
245C4×C8—1462no—
246#95—22no—
247#95C2×Dic7314yes—
248C24C22⋊Dic7311yescharacteristic
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C4×Dic7311yescharacteristic
254C22×C4C4×Dic7311yescharacteristic
255C22×C4#1192400511yescharacteristic
256C22×C4#1192400511yescharacteristic
257C22×C4#1192400511yescharacteristic
258C22×C4#1192400511yescharacteristic
259C22×C4C22⋊Dic7311yescharacteristic
260C22×C4C22⋊Dic7311yescharacteristic
261C22×C4C22⋊Dic7311yescharacteristic
262C22×C4#1192400711yescharacteristic
263C22×C4#1192400711yescharacteristic
264C22×C4#1192400711yescharacteristic
265C22×C4#1192400711yescharacteristic
266C22×C4C4⋊Dic7311yescharacteristic
267C22×C4C4⋊Dic7311yescharacteristic
268C22×C4C22×Dic7311yescharacteristic
269C22×C4C2×C22⋊D7311yescharacteristic
270C22×C4C2×C22⋊D7311yescharacteristic
271M16—2924no—
272C2×C8—2924no—
273C2×C8—1464no—
274C2×C8—1464no—
275C4⋊C4—28no—
276C22⋊C4—28no—
277C42—28no—
278C42—44no—
279C42—44no—
280C42C4×Dic7311yescharacteristic
281C42C4×Dic7311yescharacteristic
282C42#1192400511yescharacteristic
283C42#1192400511yescharacteristic
284C42#1192400511yescharacteristic
285C42#1192400511yescharacteristic
286C42#1192400711yescharacteristic
287C42#1192400711yescharacteristic
288C42#1192400711yescharacteristic
289C42#1192400711yescharacteristic
290C42C4⋊Dic7311yescharacteristic
291C42C4⋊Dic7311yescharacteristic
292C42C2×C4×D7311yescharacteristic
293C42C2×C4×D7311yescharacteristic
294C42C2×D29211yescharacteristic
295C42C2×Dic14611yescharacteristic
296C23—22no—
297C23—22no—
298C23—41no—
299C23#1455649811yescharacteristic
300C23#1455649811yescharacteristic
301C23C2×C22⋊Dic7311yescharacteristic
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—28no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4#1455649811yescharacteristic
323C2×C4#1455649811yescharacteristic
324C2×C4#1455649811yescharacteristic
325C2×C4#1455649811yescharacteristic
326C2×C4#1455649811yescharacteristic
327C2×C4#1455649811yescharacteristic
328C2×C4#1455651914yes—
329C2×C4C2×C4×Dic7311yescharacteristic
330C2×C4C2×#1192400511yescharacteristic
331C2×C4C2×#1192400511yescharacteristic
332C2×C4C2×#1192400711yescharacteristic
333C2×C4C2×#1192400711yescharacteristic
334C2×C4C2×C4⋊Dic7311yescharacteristic
335C8—2924no—
336C8—2924no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22#1798834712yes—
346C22C2×#1455649811yescharacteristic
347C22#1798860412yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—48no—
358C4#1798834712yes—
359C4#1798835712yes—
360C4C2×#1455651912yes—
361C2—22no—
362C2—22no—
363C2—41no—
364C2C2×#1798834711yescharacteristic
365C2#2260964011yescharacteristic
366C2C2×#1798835711yescharacteristic
subgroupG/H
centerC2×C4C2×#11924005
commutatorC2×C4×C73C2×C42
FrattiniC2×C42C22×D73
FittingC4×#117×C73C2
socleC22×C73C2×#95
prime radicalC24×C73C22⋊C4
cosocleC2×C42×C73C23
hypercenterC4×#117D73
hypocenterC73#8071
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
3. #22607824C2semidirect
7. #17988644C4semidirect
8. #17988644C4semidirect
9. #17988644C4semidirect
14. C2×C4⋊#10596831C4semidirect
188. C73#8071semidirect

Maximal subgroups

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

subgroupindex
#226185642
C2×#179886442
#226078242
C4×#117×C732
#807173