Finite Group Databasebeta

← #36475880

Subgroups — #36475880

Order 22784.

1486 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151464C211yesmaximal, characteristic
2C2×#19151444C211yesmaximal, characteristic
3#26018486C214yesmaximal
4#26018479C212yesmaximal
5#26018477C212yesmaximal
6C4×#15667919C212yesmaximal
7#95×Dic89C212yesmaximal
8C2×#376×C89C211yesmaximal, characteristic
9C22×C4⋊Dic89C2212yes—
10C22×C4×Dic89C2214yes—
11C22×C4×Dic89C2212yes—
12#19151464C418yescomplemented
13#19151444C418yescomplemented
14C2×#15667919C2214yes—
15C2×#15667919C2214yes—
16C2×#15667919C2214yes—
17C2×#15667919C2212yes—
18C2×#15667919C2211yescharacteristic
19C2×#15667919C2212yes—
20C2×#15667919C2211yescharacteristic
21C2×#15667919C2212yes—
22C22×C42×C89C2212yes—
23C22×C42×C89C2211yescharacteristic
24#376×C89C418yescomplemented
25C2×#95×C89C2211yescharacteristic
26C2×#95×C89C2211yescharacteristic
27C2×#95×C89C2212yes—
28C23×Dic89C2312yes—
29C23×Dic89C2314yes—
30C23×Dic89C2312yes—
31C2×C4⋊Dic89—42no—
32C2×C4⋊Dic89—42no—
33C2×C4⋊Dic89C2×C418yes—
34C2×C4×Dic89—28no—
35C2×C4×Dic89—28no—
36C2×C4×Dic89—28no—
37C2×C4×Dic89—44no—
38C2×C4×Dic89C2×C418yes—
39#15667919—28no—
40#15667919—28no—
41#15667919—44no—
42#15667919C2×C414yes—
43#15667919C2×C418yes—
44#15667919C2×C414yes—
45#15667919C2×C418yes—
46#15667919C2×C418yes—
47C23×C4×C89C2312yes—
48C23×C4×C89C2311yescharacteristic
49C23×C4×C89C2311yescharacteristic
50C23×C4×C89C2311yescharacteristic
51C23×C4×C89C2312yes—
52C2×C42×C89—28no—
53C2×C42×C89C2×C414yes—
54C2×C42×C89C2×C414yes—
55C2×C42×C89C2×C418yes—
56C2×C42×C89C2×C414yes—
57#95×C89C2×C414yes—
58#95×C89C2×C418yes—
59#95×C89C2×C414yes—
60C22×Dic89—24no—
61C22×Dic89—22no—
62C22×Dic89—22no—
63C22×Dic89—24no—
64C22×Dic89—24no—
65C22×Dic89—24no—
66C22×Dic89—24no—
67C22×Dic89—24no—
68C22×Dic89—24no—
69C22×Dic89—22no—
70C22×Dic89—24no—
71C22×Dic89—24no—
72C22×Dic89—22no—
73C22×Dic89C22×C414yes—
74C22×Dic89C22×C418yes—
75C22×Dic89C22×C414yes—
76C22×Dic89#4118yes—
77C4⋊Dic89—48no—
78C4×Dic89—216no—
79C4×Dic89—416no—
80C24×C89C2411yescharacteristic
81C22×C4×C89—24no—
82C22×C4×C89—24no—
83C22×C4×C89—24no—
84C22×C4×C89—22no—
85C22×C4×C89—21no—
86C22×C4×C89—21no—
87C22×C4×C89—21no—
88C22×C4×C89—21no—
89C22×C4×C89—22no—
90C22×C4×C89—24no—
91C22×C4×C89—24no—
92C22×C4×C89—22no—
93C22×C4×C89—22no—
94C22×C4×C89C22×C414yes—
95C22×C4×C89C22×C412yes—
96C22×C4×C89C22×C412yes—
97C22×C4×C89C22×C412yes—
98C22×C4×C89C22×C412yes—
99C22×C4×C89C22×C412yes—
100C22×C4×C89C22×C412yes—
101C22×C4×C89C22×C412yes—
102C22×C4×C89C22×C412yes—
103C22×C4×C89C22×C412yes—
104C22×C4×C89C22×C414yes—
105C22×C4×C89#4118yes—
106C42×C89—216no—
107C42×C89C42116yescomplemented
108C23×C89C23×C411yescharacteristic
109C23×C89C23×C411yescharacteristic
110C23×C89C23×C411yescharacteristic
111C23×C89C2×#4114yes—
112C23×C89C2×#4114yes—
113C23×C89H(5,𝔽2)11yescharacteristic
114C23×C89H(5,𝔽2)11yescharacteristic
115C23×C892−1+411yescharacteristic
116C23×C892−1+411yescharacteristic
117C2×C4×C89—24no—
118C2×C4×C89—24no—
119C2×C4×C89—24no—
120C2×C4×C89—24no—
121C2×C4×C89—24no—
122C2×C4×C89—24no—
123C2×C4×C89—24no—
124C2×C4×C89—22no—
125C2×C4×C89—22no—
126C2×C4×C89—22no—
127C2×C4×C89—22no—
128C2×C4×C89—24no—
129C2×C4×C89—22no—
130C2×C4×C89—22no—
131C2×C4×C89—28no—
132C2×C4×C89—28no—
133C2×C4×C89—24no—
134C2×C4×C89—24no—
135C2×C4×C89—24no—
136C2×C4×C89—44no—
137C2×C4×C89—44no—
138C2×C4×C89C2×C4214yes—
139C2×C4×C89C2×C4214yes—
140C2×C4×C89C2×C4214yes—
141C2×C4×C89#11718yes—
142C2×Dic89—24no—
143C2×Dic89—24no—
144C2×Dic89—28no—
145C2×Dic89—28no—
146C2×Dic89—24no—
147C2×Dic89—28no—
148C2×Dic89—24no—
149C2×Dic89—28no—
150C2×Dic89—28no—
151C2×Dic89—24no—
152C2×Dic89—44no—
153C2×Dic89—44no—
154C2×Dic89—44no—
155C2×Dic89—44no—
156C2×Dic89—44no—
157C2×Dic89—44no—
158C2×Dic89#11718yes—
159C22×C89C22×C4211yescharacteristic
160C22×C89C2×#11712yes—
161C22×C89C2×#11712yes—
162C22×C89C4×#4112yes—
163C22×C89C4×#4112yes—
164C22×C89C4×#4112yes—
165C22×C89C4×#4112yes—
166C22×C89#51811yescharacteristic
167C22×C89#51911yescharacteristic
168C22×C89#52011yescharacteristic
169C22×C89#52011yescharacteristic
170C22×C89#52011yescharacteristic
171C22×C89#52011yescharacteristic
172C22×C89#55114yes—
173C22×C89#55214yes—
174C22×C89#55214yes—
175C22×C89#55514yes—
176C4×C89—28no—
177C4×C89—24no—
178C4×C89—24no—
179C4×C89—24no—
180C4×C89—44no—
181Dic89—28no—
182Dic89—44no—
183Dic89—48no—
184#13805—891nomaximal, Sylow
185C2×C89C4×#11711yescharacteristic
186C2×C89C4×#11711yescharacteristic
187C2×C89#227411yescharacteristic
188C2×C89#234312yes—
189C2×C89#234312yes—
190C2×C89#234612yes—
191C2×C89#234612yes—
192C2×C89#237112yes—
193C2×C89#237212yes—
194C2×#382—891no—
195C2×#376—891no—
196C2×#376D8911yescharacteristic
197#1436—894no—
198#1434—892no—
199#1434—892no—
200C4×#95—892no—
201C4×#95—892no—
202C89#1380511yescharacteristic, complemented, Sylow
203C22×C4⋊C4—892no—
204C22×C42—892no—
205C22×C42—894no—
206C22×C42C2×D8912yes—
207C22×C42C2×D8911yescharacteristic
208#382—898no—
209#376—898no—
210#376Dic8918yescomplemented
211C2×#95—891no—
212C2×#95—892no—
213C2×#95—892no—
214C2×#95—891no—
215C2×#95—892no—
216C2×#95—894no—
217C2×#95—894no—
218C2×#95—894no—
219C2×#95C2×D8911yescharacteristic
220C2×#95C2×D8911yescharacteristic
221C2×#95C2×D8912yes—
222C23×C4—892no—
223C23×C4—894no—
224C23×C4—892no—
225C23×C4C22×D8912yes—
226C23×C4C22×D8911yescharacteristic
227C23×C4C22×D8911yescharacteristic
228C23×C4C22×D8911yescharacteristic
229C23×C4C22×D8912yes—
230C2×C4⋊C4—898no—
231C2×C4⋊C4—3562no—
232C2×C4⋊C4—3562no—
233C2×C42—28no—
234C2×C42—898no—
235C2×C42—3564no—
236C2×C42—1788no—
237C2×C42—1788no—
238C2×C42—1788no—
239C2×C42C4×D8914yes—
240C2×C42C4×D8914yes—
241C2×C42C2×Dic8918yes—
242C2×C42C2×Dic8914yes—
243#95—894no—
244#95—898no—
245#95—894no—
246#95—898no—
247#95—898no—
248#95—3564no—
249#95—1788no—
250#95—1788no—
251#95C2×Dic8914yes—
252#95C2×Dic8918yes—
253#95C2×Dic8914yes—
254C24C23×D8911yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—894no—
269C22×C4—898no—
270C22×C4—894no—
271C22×C4—898no—
272C22×C4—1784no—
273C22×C4—1784no—
274C22×C4—1784no—
275C22×C4—1784no—
276C22×C4—1784no—
277C22×C4—1784no—
278C22×C4—1782no—
279C22×C4—1784no—
280C22×C4—1784no—
281C22×C4—1782no—
282C22×C4—1784no—
283C22×C4—1782no—
284C22×C4—1782no—
285C22×C4C2×C4×D8912yes—
286C22×C4C2×C4×D8912yes—
287C22×C4C2×C4×D8912yes—
288C22×C4C2×C4×D8912yes—
289C22×C4C2×C4×D8912yes—
290C22×C4C2×C4×D8912yes—
291C22×C4C22×Dic8914yes—
292C22×C4C22×Dic8914yes—
293C22×C4C22×Dic8912yes—
294C22×C4C22×Dic8912yes—
295C22×C4C22×Dic8912yes—
296C22×C4#1308357418yes—
297C4⋊C4—3568no—
298C42—216no—
299C42—35616no—
300C42—17816no—
301C42C4×Dic89116yescomplemented
302C23C22×C4×D8911yescharacteristic
303C23C22×C4×D8911yescharacteristic
304C23C23×Dic8911yescharacteristic
305C23#41×D8914yes—
306C23C2×#1308357414yes—
307C23#1566805811yescharacteristic
308C23#1566805811yescharacteristic
309C23#1566805911yescharacteristic
310C23#1566805911yescharacteristic
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—898no—
333C2×C4—3564no—
334C2×C4—3564no—
335C2×C4—3564no—
336C2×C4—3564no—
337C2×C4—3564no—
338C2×C4—3564no—
339C2×C4—1784no—
340C2×C4—1788no—
341C2×C4—1788no—
342C2×C4—1784no—
343C2×C4—1788no—
344C2×C4—1784no—
345C2×C4—1788no—
346C2×C4—1788no—
347C2×C4—1784no—
348C2×C4—1784no—
349C2×C4C2×C4×Dic8914yes—
350C2×C4C2×C4×Dic8914yes—
351C2×C4C2×C4×Dic8914yes—
352C2×C4#1566797118yes—
353C22C22×C4×Dic8911yescharacteristic
354C22#117×D8912yes—
355C22C2×#1566797112yes—
356C22#1915206612yes—
357C22C4×#1308357412yes—
358C22C4×#1308357412yes—
359C22#41×Dic8912yes—
360C22#1915207211yescharacteristic
361C22#1915207511yescharacteristic
362C22#1915207811yescharacteristic
363C22#1915207811yescharacteristic
364C22#1915208111yescharacteristic
365C22#1915208111yescharacteristic
366C22#1915227214yes—
367C22#1915228114yes—
368C22#1915228714yes—
369C22#1915231114yes—
370C4—24no—
371C4—24no—
372C4—28no—
373C4—24no—
374C4—44no—
375C4—3568no—
376C4—3564no—
377C4—1788no—
378C2C4×#1566797111yescharacteristic
379C2#117×Dic8911yescharacteristic
380C2#2602260811yescharacteristic
381C2#2602311212yes—
382C2#2602311312yes—
383C2#2602313712yes—
384C2#2602313812yes—
385C2#2602335712yes—
386C2#2602336212yes—
subgroupG/H
center, FrattiniC24C23×D89
commutatorC22×C89C22×C42
FittingC2×#376×C89C2
socle, prime radical, cosocleC24×C89C24
hypercenterC2×#376D89
hypocenterC89#13805
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
12. #19151464C4semidirect
13. #19151444C4semidirect
24. #376×C89C4semidirect
107. C42×C89C42semidirect
202. C89#13805semidirect
210. #376Dic89semidirect
301. C42C4×Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191514642
C2×#191514442
#260184862
#260184792
#260184772
C4×#156679192
#95×Dic892
C2×#376×C892
#1380589