Finite Group Databasebeta

← #36714552

Subgroups — #36714552

Order 22784.

2858 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15667974C211yesmaximal, characteristic
2#26023221C218yesmaximal, complemented
3#26023197C218yesmaximal, complemented
4#26022847C214yesmaximal, complemented
5#26022832C212yesmaximal
6#26022655C212yesmaximal, complemented
7#26022633C214yesmaximal, complemented
8#26022609C211yesmaximal, characteristic, complemented
9C89×#2274C211yesmaximal, characteristic, complemented
10C23×D356C2211yescharacteristic
11C23×C4×D89C2212yes—
12C2×#15667974C4116yescomplemented
13C2×#15667974C2214yes—
14C2×#15667974C2214yes—
15C2×#15667971C2212yes—
16C2×#15667969C2218yescomplemented
17C22×#13083551C2212yes—
18C22×C4×Dic89C2211yescharacteristic
19#19151500C2218yes—
20#19151499C2218yes—
21#19151489C22116yescomplemented
22#19151487C22116yescomplemented
23#19151467C2214yescomplemented
24#19151463C2214yescomplemented
25#19151460C2218yescomplemented
26#19151457C2218yescomplemented
27#19151456C2214yes—
28#19151456C2214yes—
29#19151453C2218yes—
30C4⋊C4×Dic89C2218yescomplemented
31C4×#13083553C2214yescomplemented
32C4×#13083553C2214yescomplemented
33C4×#13083551C2218yescomplemented
34C4×C22⋊Dic89C2214yes—
35C2×#117×C89C2212yes—
36C22×C4⋊C4×C89C2211yescharacteristic
37C4×C4⋊C4×C89C2218yescomplemented
38C4×C22⋊C4×C89C2214yes—
39C24×D89C2312yes—
40C22×D356—28no—
41C22×D356C2×C418yes—
42C22×D356C2314yes—
43C23×Dic89C2311yescharacteristic
44C23×Dic89C2311yescharacteristic
45C22×C4×D89—48no—
46C22×C4×D89C2×C4116yes—
47C22×C4×D89C2314yes—
48C22×C4×D89C2314yes—
49#15667974C2×C4132yes—
50#15667974C2×C4132yes—
51#15667971—28no—
52#15667969—416no—
53C2×C4⋊Dic89C2314yes—
54C2×#13083553C2318yes—
55C2×#13083553C2318yes—
56C2×C22⋊Dic89C2314yes—
57C2×#13083551C2×C4116yes—
58C2×#13083551C23116yes—
59C2×#13083551C23116yes—
60C2×#13083551C2318yes—
61C2×C4×Dic89C2×C418yes—
62C2×C4×Dic89C2314yes—
63C2×C4×Dic89C2318yes—
64C2×C4×Dic89C2312yes—
65C2×C4×Dic89C2312yes—
66#15667919C2318yes—
67#15667919C2318yes—
68#15667919C2318yes—
69C23×C4×C89C2312yes—
70C23×C4×C89C2311yescharacteristic
71#117×C89—28no—
72C2×C4⋊C4×C89C2×C418yes—
73C2×C4⋊C4×C89C2314yes—
74C2×C4⋊C4×C89C2314yes—
75C2×C22⋊C4×C89C2314yes—
76C2×C42×C89C2314yes—
77C2×C42×C89C2318yes—
78#95×C89C2318yes—
79C23×D89—28no—
80C23×D89—44no—
81C23×D89—44no—
82C23×D89C22×C418yes—
83C23×D89C2414yes—
84C2×D356—232no—
85C2×D356—28no—
86C2×D356—28no—
87C2×D356C22×C4116yes—
88C22×Dic89—24no—
89C22×Dic89—42no—
90C22×Dic89C22×C414yes—
91C22×Dic89C22×C414yes—
92C22×Dic89C2418yes—
93C22×Dic89C2411yescharacteristic
94C22×Dic89C2412yes—
95C22×Dic89C2411yescharacteristic
96C2×C4×D89—216no—
97C2×C4×D89—432no—
98C2×C4×D89—48no—
99C2×C4×D89C22×C4116yes—
100C2×C4×D89C22×C4116yes—
101C4⋊Dic89—28no—
102#13083553—216no—
103#13083553—48no—
104C22⋊Dic89—28no—
105#13083551—416no—
106#13083551—416no—
107#13083551C22×C4132yes—
108C4×Dic89—28no—
109C4×Dic89—48no—
110C4×Dic89C22×C418yes—
111C4×Dic89C22×C418yes—
112C24×C89C2411yescharacteristic
113C22×C4×C89—44no—
114C22×C4×C89C22×C418yes—
115C22×C4×C89C22×C414yes—
116C22×C4×C89#4118yes—
117C22×C4×C89C2418yes—
118C22×C4×C89C2414yes—
119C22×C4×C89C2412yes—
120C4⋊C4×C89—28no—
121C4⋊C4×C89C22×C4116yes—
122C22⋊C4×C89—28no—
123C42×C89—28no—
124C42×C89—48no—
125C23×C89—22no—
126C23×C89—22no—
127C23×C89C23×C412yes—
128C23×C89C2×#4114yes—
129C23×C89C2511yescharacteristic
130C22×D89—216no—
131C22×D89—28no—
132C22×D89—28no—
133C22×D89—28no—
134C22×D89—48no—
135C22×D89—48no—
136C22×D89—432no—
137C22×D89C23×C418yes—
138C2×C4×C89—28no—
139C2×C4×C89—28no—
140C2×C4×C89—28no—
141C2×C4×C89—28no—
142C2×C4×C89—48no—
143C2×C4×C89#117116yescomplemented
144C2×C4×C89C23×C418yes—
145C2×C4×C89C23×C414yes—
146C2×C4×C89C2×#4114yes—
147C2×C4×C89C2×#4114yes—
148C2×Dic89—22no—
149C2×Dic89—24no—
150C2×Dic89—28no—
151C2×Dic89—28no—
152C2×Dic89—28no—
153C2×Dic89—28no—
154C2×Dic89—44no—
155C2×Dic89C23×C414yes—
156C2×Dic89C23×C412yes—
157C2×Dic89C23×C412yes—
158C2×Dic89H(5,𝔽2)14yes—
159D356—232no—
160C4×D89—232no—
161C4×D89—416no—
162C22×C89—22no—
163C22×C89—22no—
164C22×C89—28no—
165C22×C89C2×#11714yes—
166C22×C89C24×C411yescharacteristic
167C22×C89C22×#4112yes—
168C22×C89C2×H(5,𝔽2)12yes—
169C22×C89#58512yes—
170C2×D89—28no—
171C2×D89—28no—
172C2×D89—28no—
173C2×D89—432no—
174C2×D89—416no—
175C4×C89—28no—
176C4×C89—48no—
177C4×C89C2×#11718yes—
178Dic89—22no—
179Dic89—24no—
180Dic89—48no—
181Dic89#51814yes—
182#34186—891nomaximal, Sylow
183C2×C89—24no—
184C2×C89C22×#11711yescharacteristic
185C2×C89C2×#51811yescharacteristic
186C2×C89#342811yescharacteristic
187C2×C89#350614yes—
188D89—48no—
189C22×C4×D4—891no—
190#2357—898no—
191#2354—898no—
192#2312—894no—
193#2309—892no—
194#2285—892no—
195#2280—894no—
196#2274—891no—
197#2274D8911yescharacteristic, complemented
198C89#3418611yescharacteristic, complemented, Sylow
199C23×D4—891no—
200C24×C4—892no—
201C2×C4×D4—8916no—
202C2×C4×D4—894no—
203C2×C4×D4—894no—
204C2×#117—892no—
205C2×#117—898no—
206C2×#117C2×D8912yes—
207C22×C4⋊C4C2×D8911yescharacteristic
208C22×C22⋊C4—892no—
209C22×C42—891no—
210#390—898no—
211#390—898no—
212#388—8916no—
213#388—8916no—
214#384—894no—
215#382—894no—
216#381—898no—
217#380—894no—
218#380—894no—
219#380—898no—
220#379—898no—
221C4×C4⋊C4—898no—
222C4×C4⋊C4—894no—
223C4×C4⋊C4—894no—
224C4×C4⋊C4C2×D8918yescomplemented
225C4×C22⋊C4—894no—
226C4×C22⋊C4—898no—
227C4×C22⋊C4C2×D8914yes—
228C25—892no—
229C22×D4—898no—
230C22×D4—894no—
231C22×D4—1788no—
232C23×C4—891no—
233C23×C4—891no—
234C23×C4—8916no—
235C23×C4—894no—
236C23×C4—894no—
237C23×C4—3568no—
238C23×C4C22×D8912yes—
239C23×C4C22×D8911yescharacteristic
240C4×D4—8932no—
241C4×D4—8932no—
242#117—28no—
243#117—35616no—
244#117—1788no—
245C2×C4⋊C4—898no—
246C2×C4⋊C4—898no—
247C2×C4⋊C4—894no—
248C2×C4⋊C4C4×D8918yes—
249C2×C4⋊C4C22×D8914yes—
250C2×C4⋊C4C22×D8914yes—
251C2×C22⋊C4—8916no—
252C2×C22⋊C4—898no—
253C2×C22⋊C4—8916no—
254C2×C22⋊C4—8916no—
255C2×C22⋊C4—894no—
256C2×C22⋊C4C22×D8914yes—
257C2×C42—898no—
258C2×C42—892no—
259C2×C42—892no—
260C2×C42—898no—
261C2×C42—894no—
262C2×C42C22×D8914yes—
263C2×C42C22×D8918yes—
264#95—898no—
265#95—898no—
266#95—898no—
267#95C22×D8918yes—
268C24—898no—
269C24—894no—
270C24—3564no—
271C24—3564no—
272C24—1788no—
273C24C23×D8911yescharacteristic
274C2×D4—8916no—
275C2×D4—17832no—
276C2×D4—1788no—
277C2×D4—1788no—
278C22×C4—44no—
279C22×C4—891no—
280C22×C4—894no—
281C22×C4—894no—
282C22×C4—891no—
283C22×C4—892no—
284C22×C4—8916no—
285C22×C4—8916no—
286C22×C4—898no—
287C22×C4—3568no—
288C22×C4—35632no—
289C22×C4—3562no—
290C22×C4—17816no—
291C22×C4—1784no—
292C22×C4C2×C4×D8918yes—
293C22×C4C2×C4×D8914yes—
294C22×C4#1308357418yes—
295C22×C4C23×D8918yes—
296C22×C4C23×D8914yes—
297C22×C4C23×D8912yes—
298C4⋊C4—28no—
299C4⋊C4—3568no—
300C4⋊C4—17816no—
301C4⋊C4—1788no—
302C4⋊C4C2×C4×D89116yes—
303C22⋊C4—28no—
304C22⋊C4—8932no—
305C22⋊C4—35616no—
306C22⋊C4—35616no—
307C22⋊C4—1788no—
308C42—28no—
309C42—48no—
310C42—898no—
311C42—898no—
312C42—3568no—
313C42—1788no—
314C23—22no—
315C23—22no—
316C23—898no—
317C23—35632no—
318C23—3568no—
319C23—3568no—
320C23—1788no—
321C23—1788no—
322C23—1788no—
323C23—17816no—
324C23C22×C4×D8912yes—
325C23C2×#1308357414yes—
326C23C24×D8911yescharacteristic
327D4—17832no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—48no—
333C2×C4—892no—
334C2×C4—894no—
335C2×C4—894no—
336C2×C4—892no—
337C2×C4—35616no—
338C2×C4—3564no—
339C2×C4—17832no—
340C2×C4—1788no—
341C2×C4—1788no—
342C2×C4—1788no—
343C2×C4—1788no—
344C2×C4—1782no—
345C2×C4—1784no—
346C2×C4#15667969116yescomplemented
347C2×C4C22×C4×D8918yes—
348C2×C4C22×C4×D8914yes—
349C2×C4C2×#1308357414yes—
350C2×C4C2×#1308357414yes—
351C22—22no—
352C22—22no—
353C22—28no—
354C22—35632no—
355C22—35616no—
356C22—1788no—
357C22—1788no—
358C22—1788no—
359C22C2×#1566796914yes—
360C22C23×C4×D8911yescharacteristic
361C22C22×#1308357412yes—
362C22H(5,𝔽2)×D8912yes—
363C22#1915243512yes—
364C4—28no—
365C4—48no—
366C4—894no—
367C4—3568no—
368C4—1782no—
369C4—1784no—
370C4C2×#1566796918yes—
371C2—24no—
372C2—3568no—
373C2C22×#1566796911yescharacteristic
374C2#518×D8911yescharacteristic
375C2#2603345411yescharacteristic
376C2#2603409714yes—
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2274C2
socle, cosocleC23×C89C25
prime radicalC23×D356C22
hypercenter#2274D89
hypocenterC89#34186
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2274

Complements

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

NH ≅ G/Ntype
2. #26023221C2semidirect
3. #26023197C2semidirect
4. #26022847C2semidirect
6. #26022655C2semidirect
7. #26022633C2semidirect
8. #26022609C2semidirect
9. C89×#2274C2semidirect
12. C2×#15667974C4semidirect
16. C2×#15667969C22semidirect
21. #19151489C22semidirect
22. #19151487C22semidirect
23. #19151467C22semidirect
24. #19151463C22semidirect
25. #19151460C22semidirect
26. #19151457C22semidirect
30. C4⋊C4×Dic89C22semidirect
31. C4×#13083553C22semidirect
32. C4×#13083553C22semidirect
33. C4×#13083551C22semidirect
37. C4×C4⋊C4×C89C22semidirect
143. C2×C4×C89#117semidirect
197. #2274D89semidirect
198. C89#34186semidirect
224. C4×C4⋊C4C2×D89semidirect
346. C2×C4#15667969semidirect

Maximal subgroups

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

subgroupindex
C22×#156679742
#260232212
#260231972
#260228472
#260228322
#260226552
#260226332
#260226092
C89×#22742
#3418689