Finite Group Databasebeta

← #36556728

Subgroups — #36556728

Order 22784.

1122 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151624C211yesmaximal, characteristic, complemented
2C2×#19151564C211yesmaximal, characteristic
3C2×#19151564C211yesmaximal, characteristic
4#26022742C211yesmaximal, characteristic
5#26020865C218yesmaximal, complemented
6#26019948C211yesmaximal, characteristic, complemented
7#26019946C211yesmaximal, characteristic, complemented
8C22×#97×C89C211yesmaximal, characteristic, complemented
9C22×#13083574C2211yescharacteristic
10C22×#13083558C2211yescharacteristic
11C22×#13083558C2211yescharacteristic
12C2×#15667971C2211yescharacteristic
13#19151624—24no—
14C2×#15667941C2214yes—
15#19151564—24no—
16#19151564—24no—
17C2×#15667924C2214yes—
18C2×#15667924C2214yes—
19#19151504C412yes—
20#19151500C412yes—
21#19151477C414yes—
22C4×#13083551C2214yes—
23#19151353C2214yes—
24#19151352C2214yes—
25C23×C8×C89C2211yescharacteristic
26C23×C8×C89C2211yescharacteristic
27C22×C42×C89C2211yescharacteristic
28C2×#97×C89—24no—
29C2×#97×C89C2214yes—
30C2×#13083574—22no—
31C2×#13083574—22no—
32C2×#13083574—22no—
33C2×#13083574—22no—
34C2×#13083574—22no—
35C2×#13083574—22no—
36C22×Dic178C2×C411yescharacteristic
37C22×D356C2×C411yescharacteristic
38C22×C22⋊D89C2×C412yes—
39C22×C4×D89C2312yes—
40C2×#13083558—22no—
41C2×#13083558—22no—
42C2×#13083558—22no—
43C2×#13083558—22no—
44C2×#13083558—22no—
45C2×#13083558—22no—
46C22×#10623494C2312yes—
47C22×#10623494C2312yes—
48#15667971—22no—
49#15667971—22no—
50C2×C4⋊Dic89C2×C412yes—
51C2×C22⋊Dic89C2×C412yes—
52C2×#13083551C2×C418yes—
53C2×C4×Dic89C2312yes—
54#15667941—28no—
55#15667924—28no—
56#15667924—28no—
57#15667919C2×C418yes—
58C23×C4×C89C2×C412yes—
59C23×C4×C89C2311yescharacteristic
60C22×C8×C89—22no—
61C22×C8×C89—22no—
62C22×C8×C89—22no—
63C22×C8×C89—22no—
64C22×C8×C89D414yes—
65C22×C8×C89D414yes—
66C22×C8×C89C2312yes—
67C22×C8×C89C2312yes—
68C2×C42×C89—22no—
69C2×C42×C89—22no—
70C2×C42×C89C2×C412yes—
71C2×C42×C89C2312yes—
72#97×C89—28no—
73C23×D89C22×C412yes—
74#13083574—44no—
75#13083574—44no—
76#13083574—44no—
77#13083574—44no—
78C2×Dic178—22no—
79C2×Dic178—22no—
80C2×Dic178—22no—
81C2×D356—22no—
82C2×D356—22no—
83C2×D356—22no—
84C2×C22⋊D89—24no—
85C2×C22⋊D89—24no—
86C2×C22⋊D89—24no—
87C22×Dic89C22×C412yes—
88C22×Dic89C22×C414yes—
89C2×C4×D89—22no—
90C2×C4×D89—22no—
91C2×C4×D89—22no—
92C2×C4×D89—22no—
93C2×C4×D89—22no—
94C2×C4×D89—22no—
95#13083558—44no—
96#13083558—44no—
97C2×#10623494—22no—
98C2×#10623494—22no—
99C2×#10623494—22no—
100C2×#10623494—22no—
101C2×#10623494—22no—
102C2×#10623494—22no—
103C4⋊Dic89—24no—
104C22⋊Dic89—24no—
105#13083551—48no—
106C4×Dic89—22no—
107C4×Dic89—22no—
108C24×C89C22×C411yescharacteristic
109C22×C4×C89—24no—
110C22×C4×C89—44no—
111C22×C4×C89C22×C414yes—
112C22×C4×C89C22×C411yescharacteristic
113C22×C4×C89C2×D412yes—
114C22×C4×C89C2×D412yes—
115C22×C4×C89#4112yes—
116C22×C4×C89#4112yes—
117C22×C4×C89#4112yes—
118C22×C4×C89#4112yes—
119C22×C4×C89C2411yescharacteristic
120C2×C8×C89—28no—
121C2×C8×C89—28no—
122C2×C8×C89—22no—
123C2×C8×C89—22no—
124C2×C8×C89—22no—
125C2×C8×C89—22no—
126C2×C8×C89C2×D414yes—
127C2×C8×C89C2×D414yes—
128C42×C89—24no—
129C42×C89—22no—
130C42×C89—22no—
131C23×C89—24no—
132C23×C89#11712yes—
133C23×C89C4×D412yes—
134C23×C89C4×D412yes—
135C23×C89C23×C411yescharacteristic
136C22×D89—22no—
137C22×D89—22no—
138C22×D89—22no—
139C22×D89—28no—
140C2×C4×C89—24no—
141C2×C4×C89—24no—
142C2×C4×C89—24no—
143C2×C4×C89—48no—
144C2×C4×C89#11712yes—
145C2×C4×C89C4×D412yes—
146C2×C4×C89C4×D412yes—
147C2×C4×C89#12114yes—
148C2×C4×C89#12314yes—
149C2×C4×C89#12414yes—
150C2×C4×C89#12614yes—
151C2×C4×C89C22×D411yescharacteristic
152C2×C4×C89C22×D411yescharacteristic
153C2×C4×C89C2×#4111yescharacteristic
154C2×C4×C89C2×#4111yescharacteristic
155C2×C4×C89C2×#4111yescharacteristic
156C2×C4×C89C2×#4111yescharacteristic
157C22⋊D89—48no—
158C2×Dic89—22no—
159C2×Dic89—24no—
160C2×Dic89—22no—
161C2×Dic89—22no—
162C2×Dic89—28no—
163D356—44no—
164C4×D89—42no—
165C4×D89—42no—
166C4×D89—42no—
167C4×D89—42no—
168Dic178—44no—
169C8×C89—24no—
170C8×C89—24no—
171#10623494—42no—
172#10623494—42no—
173C22×C89—24no—
174C22×C89—24no—
175C22×C89—24no—
176C22×C89#38814yes—
177C22×C89C2×#11711yescharacteristic
178C22×C89C2×C4×D411yescharacteristic
179C22×C89C2×C4×D411yescharacteristic
180C22×C89#56814yes—
181C2×D89—44no—
182C2×D89—44no—
183C2×D89—44no—
184C2×D89—42no—
185C4×C89—44no—
186C4×C89#38814yes—
187C4×C89C2×#12111yescharacteristic
188C4×C89C2×#12311yescharacteristic
189C4×C89C2×#12411yescharacteristic
190C4×C89C2×#12611yescharacteristic
191Dic89—44no—
192Dic89—42no—
193#21047—891nomaximal, Sylow
194C2×C89—24no—
195C2×C89C2×#38811yescharacteristic
196C2×C89#292312yes—
197C2×C89#293012yes—
198C2×C89#293212yes—
199D89—82no—
200C2×#424—891no—
201C2×#408—891no—
202C2×#408—891no—
203C22×#97D8911yescharacteristic, complemented
204#2297—891no—
205#1924—898no—
206#1729—891no—
207#1728—891no—
208C89#2104711yescharacteristic, complemented, Sylow
209C22×#41—891no—
210C22×M16—891no—
211C22×M16—891no—
212C23×C8C2×D8911yescharacteristic
213C23×C8C2×D8911yescharacteristic
214C2×#117—891no—
215C22×C42C2×D8911yescharacteristic
216#424—1784no—
217C2×C4⋊C8—894no—
218#408—1784no—
219#408—1784no—
220C2×C22⋊C8—894no—
221C2×C22⋊C8—894no—
222C2×#97—24no—
223C2×#97C2×D8914yes—
224#391—892no—
225#390—892no—
226#386—894no—
227C4×C22⋊C4—894no—
228#336—894no—
229#336—894no—
230C2×#41—1782no—
231C2×#41—1782no—
232C2×#41—1782no—
233C2×#41—1782no—
234C2×#41—1782no—
235C2×#41—1782no—
236C22×Q8—891no—
237C22×D4—892no—
238C22×D4—891no—
239C23×C4—892no—
240C23×C4C4×D8912yes—
241C23×C4C22×D8911yescharacteristic
242C2×M16—1782no—
243C2×M16—1782no—
244C2×M16—1782no—
245C2×M16—1782no—
246C2×M16—1782no—
247C2×M16—1782no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—892no—
253C22×C8—892no—
254C22×C8D35614yes—
255C22×C8C22⋊D8914yes—
256C22×C8C22×D8912yes—
257C22×C8C22×D8912yes—
258#117—1782no—
259#117—1782no—
260C2×C4⋊C4—892no—
261C2×C22⋊C4—898no—
262C2×C22⋊C4—892no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—892no—
266C2×C42C4×D8912yes—
267C2×C42C22×D8912yes—
268C4⋊C8—1788no—
269C22⋊C8—1788no—
270C22⋊C8—1788no—
271#97—28no—
272#95—898no—
273C24—892no—
274C24C2×C4×D8911yescharacteristic
275#41—3564no—
276#41—3564no—
277#41—3564no—
278#41—3564no—
279C2×Q8—1782no—
280C2×Q8—1782no—
281C2×Q8—1782no—
282C2×D4—1784no—
283C2×D4—1784no—
284C2×D4—1784no—
285C2×D4—1782no—
286C2×D4—1782no—
287C2×D4—1782no—
288C22×C4—24no—
289C22×C4—44no—
290C22×C4—892no—
291C22×C4—894no—
292C22×C4—1782no—
293C22×C4—1782no—
294C22×C4—1782no—
295C22×C4—1782no—
296C22×C4—1782no—
297C22×C4—1782no—
298C22×C4C2×C4×D8914yes—
299C22×C4C2×C4×D8911yescharacteristic
300C22×C4C2×C22⋊D8912yes—
301C22×C4C2×D35612yes—
302C22×C4#1308357412yes—
303C22×C4#1308357412yes—
304C22×C4#1308357412yes—
305C22×C4#1308357412yes—
306C22×C4C23×D8911yescharacteristic
307M16—3564no—
308M16—3564no—
309C2×C8—28no—
310C2×C8—28no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—1782no—
316C2×C8—1782no—
317C2×C8—1782no—
318C2×C8—1782no—
319C2×C8—1782no—
320C2×C8—1782no—
321C2×C8C2×C22⋊D8914yes—
322C2×C8C2×D35614yes—
323C4⋊C4—1784no—
324C22⋊C4—3568no—
325C22⋊C4—1784no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—1782no—
330C42—1782no—
331C23—24no—
332C23—1782no—
333C23—1782no—
334C23—1782no—
335C23—1788no—
336C23#1566796912yes—
337C23C4×C22⋊D8912yes—
338C23C4×D35612yes—
339C23C22×C4×D8911yescharacteristic
340Q8—3564no—
341D4—3568no—
342D4—3564no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—48no—
347C2×C4—3562no—
348C2×C4—3562no—
349C2×C4—3562no—
350C2×C4—3562no—
351C2×C4—1782no—
352C2×C4—1782no—
353C2×C4—1784no—
354C2×C4—1782no—
355C2×C4—1788no—
356C2×C4#1566796912yes—
357C2×C4C4×C22⋊D8912yes—
358C2×C4C4×D35612yes—
359C2×C4#1566798614yes—
360C2×C4#1566799514yes—
361C2×C4#1566800014yes—
362C2×C4#1566800614yes—
363C2×C4C22×C22⋊D8911yescharacteristic
364C2×C4C22×D35611yescharacteristic
365C2×C4C2×#1308357411yescharacteristic
366C2×C4C2×#1308357411yescharacteristic
367C2×C4C2×#1308357411yescharacteristic
368C2×C4C2×#1308357411yescharacteristic
369C8—24no—
370C8—24no—
371C8—3562no—
372C8—3562no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—3562no—
377C22—3564no—
378C22—3564no—
379C22—3564no—
380C22#1915148914yes—
381C22C2×#1566796911yescharacteristic
382C22C2×C4×C22⋊D8911yescharacteristic
383C22C2×C4×D35611yescharacteristic
384C22#1915236414yes—
385C4—44no—
386C4—3562no—
387C4—3564no—
388C4#1915148914yes—
389C4C2×#1566798611yescharacteristic
390C4C2×#1566799511yescharacteristic
391C4C2×#1566800011yescharacteristic
392C4C2×#1566800611yescharacteristic
393C2—24no—
394C2—7122no—
395C2C2×#1915148911yescharacteristic
396C2#2602890112yes—
397C2#2602895512yes—
398C2#2602897112yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC22×#97×C89C2
prime radicalC22×#13083574C22
cosocleC22×C4×C89C24
hypercenterC22×#97D89
hypocenterC89#21047
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
1. C2×#19151624C2semidirect
5. #26020865C2semidirect
6. #26019948C2semidirect
7. #26019946C2semidirect
8. C22×#97×C89C2semidirect
203. C22×#97D89semidirect
208. C89#21047semidirect

Maximal subgroups

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

subgroupindex
C2×#191516242
C2×#191515642
#260227422
#260208652
#260199482
#260199462
C22×#97×C892
#2104789