Finite Group Databasebeta

← C2×C89×#2365 #36411986

Subgroups — C2×C89×#2365 #36411986

Order 22784.

3158 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Q8×C89C211yesmaximal, characteristic
2C22×C4×Q8×C89C212yesmaximal
3C89×#2365C2116yesmaximal, direct product factor
4C2×#391×C89C212yesmaximal
5C2×#389×C89C212yesmaximal
6C2×#387×C89C214yesmaximal
7C2×C4×C4⋊C4×C89C212yesmaximal
8C2×C4×C4⋊C4×C89C211yesmaximal, characteristic
9C2×C4×C4⋊C4×C89C211yesmaximal, characteristic
10C23×Q8×C89C2212yes—
11C2×C4⋊Q8×C89C4116yescomplemented
12C2×C4⋊Q8×C89C2218yes—
13C2×C4×Q8×C89—216no—
14C2×C4×Q8×C89C22116yes—
15C22×C4⋊C4×C89C2211yescharacteristic
16C22×C4⋊C4×C89C2214yes—
17C22×C4⋊C4×C89C2211yescharacteristic
18C22×C4⋊C4×C89C2212yes—
19C22×C4⋊C4×C89C2212yes—
20C22×C4⋊C4×C89C2211yescharacteristic
21C22×C4⋊C4×C89C2211yescharacteristic
22C22×C42×C89C2212yes—
23C22×C42×C89C2214yes—
24C22×C42×C89C2212yes—
25C22×C42×C89C2211yescharacteristic
26#391×C89C22116yes—
27#389×C89C22116yes—
28#387×C89C22132yes—
29C4×C4⋊C4×C89C2218yes—
30C4×C4⋊C4×C89C2218yes—
31C4×C4⋊C4×C89C22116yes—
32C2×#95×C89C2214yes—
33C2×#95×C89C2214yes—
34C2×#95×C89C2212yes—
35C2×#95×C89C2212yes—
36C22×Q8×C89—28no—
37C22×Q8×C89—28no—
38C22×Q8×C89C2×C4116yes—
39C22×Q8×C89C2318yes—
40C23×C4×C89C2312yes—
41C23×C4×C89C2312yes—
42C23×C4×C89C2311yescharacteristic
43C23×C4×C89C2314yes—
44C23×C4×C89C2312yes—
45C23×C4×C89C2312yes—
46C23×C4×C89C2311yescharacteristic
47C23×C4×C89C2311yescharacteristic
48C4⋊Q8×C89C2×C4164yescomplemented
49C4×Q8×C89—264no—
50C2×C4⋊C4×C89—216no—
51C2×C4⋊C4×C89—24no—
52C2×C4⋊C4×C89—24no—
53C2×C4⋊C4×C89C2×C4116yes—
54C2×C4⋊C4×C89C2×C4116yes—
55C2×C4⋊C4×C89D418yes—
56C2×C4⋊C4×C89Q818yescomplemented
57C2×C4⋊C4×C89C2314yes—
58C2×C4⋊C4×C89C2314yes—
59C2×C4⋊C4×C89C23116yes—
60C2×C4⋊C4×C89C2318yes—
61C2×C4⋊C4×C89C2318yes—
62C2×C4⋊C4×C89C2314yes—
63C2×C4⋊C4×C89C2314yes—
64C2×C42×C89—216no—
65C2×C42×C89—28no—
66C2×C42×C89—28no—
67C2×C42×C89C2×C418yes—
68C2×C42×C89C2318yes—
69C2×C42×C89C23116yes—
70C2×C42×C89C2318yes—
71C2×C42×C89C2314yes—
72#95×C89C2318yes—
73#95×C89C2318yes—
74#95×C89C23116yes—
75#95×C89C23116yes—
76C24×C89C2411yescharacteristic
77C2×Q8×C89—264no—
78C2×Q8×C89—216no—
79C2×Q8×C89—216no—
80C2×Q8×C89C22×C4132yes—
81C22×C4×C89—24no—
82C22×C4×C89—28no—
83C22×C4×C89—24no—
84C22×C4×C89—28no—
85C22×C4×C89—22no—
86C22×C4×C89—22no—
87C22×C4×C89—24no—
88C22×C4×C89—24no—
89C22×C4×C89—24no—
90C22×C4×C89—24no—
91C22×C4×C89—24no—
92C22×C4×C89—24no—
93C22×C4×C89—24no—
94C22×C4×C89—24no—
95C22×C4×C89C22×C414yes—
96C22×C4×C89C22×C4116yes—
97C22×C4×C89C22×C414yes—
98C22×C4×C89C22×C414yes—
99C22×C4×C89C2×D418yes—
100C22×C4×C89C2×D414yes—
101C22×C4×C89C2×Q818yes—
102C22×C4×C89C2×Q814yes—
103C22×C4×C89#4114yes—
104C22×C4×C89#4114yes—
105C22×C4×C89C2414yes—
106C22×C4×C89C2414yes—
107C22×C4×C89C2412yes—
108C22×C4×C89C2418yes—
109C22×C4×C89C2414yes—
110C22×C4×C89C2414yes—
111C22×C4×C89C2412yes—
112C22×C4×C89C2412yes—
113C4⋊C4×C89—232no—
114C4⋊C4×C89—28no—
115C4⋊C4×C89—28no—
116C4⋊C4×C89C22×C4132yes—
117C4⋊C4×C89C22×C4132yes—
118C4⋊C4×C89C2×D4116yes—
119C4⋊C4×C89C2×Q8116yescomplemented
120C42×C89—232no—
121C42×C89—216no—
122C42×C89—216no—
123C42×C89C22×C4116yes—
124C23×C89C23×C412yes—
125C23×C89C22×D412yes—
126C23×C89C22×Q812yes—
127C23×C89C2×#4112yes—
128C23×C89C2×#4112yes—
129C23×C892−1+412yes—
130C23×C892−1+412yes—
131C23×C89C2511yescharacteristic
132Q8×C89—264no—
133C2×C4×C89—232no—
134C2×C4×C89—28no—
135C2×C4×C89—216no—
136C2×C4×C89—216no—
137C2×C4×C89—24no—
138C2×C4×C89—28no—
139C2×C4×C89—24no—
140C2×C4×C89—28no—
141C2×C4×C89—22no—
142C2×C4×C89—22no—
143C2×C4×C89—24no—
144C2×C4×C89—24no—
145C2×C4×C89—24no—
146C2×C4×C89—24no—
147C2×C4×C89—24no—
148C2×C4×C89—24no—
149C2×C4×C89—24no—
150C2×C4×C89—24no—
151C2×C4×C89—48no—
152C2×C4×C89—48no—
153C2×C4×C89C4×D4116yes—
154C2×C4×C89C4×Q8116yescomplemented
155C2×C4×C89C23×C414yes—
156C2×C4×C89C23×C4116yes—
157C2×C4×C89C23×C414yes—
158C2×C4×C89C23×C414yes—
159C2×C4×C89C22×D418yes—
160C2×C4×C89C22×D414yes—
161C2×C4×C89C22×Q818yes—
162C2×C4×C89C22×Q814yes—
163C2×C4×C89C2×#4114yes—
164C2×C4×C89C2×#4114yes—
165C22×C89C2×C4×D414yes—
166C22×C89C2×C4×Q814yes—
167C22×C89#51914yes—
168C22×C89#54714yes—
169C22×C89D4×Q814yes—
170C22×C89#55614yes—
171C22×C89#55714yes—
172C22×C89C24×C411yescharacteristic
173C22×C89C23×D411yescharacteristic
174C22×C89C23×Q811yescharacteristic
175C22×C89C22×#4111yescharacteristic
176C22×C89C22×#4111yescharacteristic
177C22×C89C2×2−1+411yescharacteristic
178C22×C89C2×2−1+411yescharacteristic
179C4×C89—216no—
180C4×C89—24no—
181C4×C89—28no—
182C4×C89—28no—
183C4×C89—44no—
184C4×C89—44no—
185C4×C89C2×C4×D418yes—
186C4×C89C2×C4×Q818yescomplemented
187C2×#2365C8911yesmaximal, characteristic, direct product factor, Sylow
188C2×C89#236518yesdirect product factor
189C2×C89C22×C4×D411yescharacteristic
190C2×C89C22×C4×Q811yescharacteristic
191C2×C89C2×#51911yescharacteristic
192C2×C89C2×#54711yescharacteristic
193C2×C89C2×D4×Q811yescharacteristic
194C2×C89C2×#55611yescharacteristic
195C2×C89C2×#55711yescharacteristic
196C22×C4⋊Q8C2×C8911yescharacteristic
197C22×C4×Q8C2×C8912yes—
198#2365C2×C89116yesdirect product factor
199C2×#391C2×C8912yes—
200C2×#389C2×C8912yes—
201C2×#387C2×C8914yes—
202C2×C4×C4⋊C4C2×C8911yescharacteristic
203C2×C4×C4⋊C4C2×C8911yescharacteristic
204C2×C4×C4⋊C4C2×C8912yes—
205C89C2×#236511yescharacteristic, direct product factor, Sylow
206C23×Q8C22×C8912yes—
207C2×C4⋊Q8C4×C89116yescomplemented
208C2×C4⋊Q8C22×C8918yes—
209C2×C4×Q8—216no—
210C2×C4×Q8C22×C89116yes—
211C22×C4⋊C4C22×C8914yes—
212C22×C4⋊C4C22×C8911yescharacteristic
213C22×C4⋊C4C22×C8911yescharacteristic
214C22×C4⋊C4C22×C8912yes—
215C22×C4⋊C4C22×C8912yes—
216C22×C4⋊C4C22×C8911yescharacteristic
217C22×C4⋊C4C22×C8911yescharacteristic
218C22×C42C22×C8914yes—
219C22×C42C22×C8912yes—
220C22×C42C22×C8912yes—
221C22×C42C22×C8911yescharacteristic
222#391C22×C89116yes—
223#389C22×C89116yes—
224#387C22×C89132yes—
225C4×C4⋊C4C22×C8918yes—
226C4×C4⋊C4C22×C8918yes—
227C4×C4⋊C4C22×C89116yes—
228C2×#95C22×C8912yes—
229C2×#95C22×C8912yes—
230C2×#95C22×C8914yes—
231C2×#95C22×C8914yes—
232C22×Q8—28no—
233C22×Q8—28no—
234C22×Q8C2×C4×C89116yes—
235C22×Q8C23×C8918yes—
236C23×C4C23×C8912yes—
237C23×C4C23×C8912yes—
238C23×C4C23×C8914yes—
239C23×C4C23×C8911yescharacteristic
240C23×C4C23×C8912yes—
241C23×C4C23×C8912yes—
242C23×C4C23×C8911yescharacteristic
243C23×C4C23×C8911yescharacteristic
244C4⋊Q8C2×C4×C89164yescomplemented
245C4×Q8—264no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4C2×C4×C89116yes—
250C2×C4⋊C4C2×C4×C89116yes—
251C2×C4⋊C4D4×C8918yes—
252C2×C4⋊C4Q8×C8918yescomplemented
253C2×C4⋊C4C23×C89116yes—
254C2×C4⋊C4C23×C8914yes—
255C2×C4⋊C4C23×C8914yes—
256C2×C4⋊C4C23×C8918yes—
257C2×C4⋊C4C23×C8918yes—
258C2×C4⋊C4C23×C8914yes—
259C2×C4⋊C4C23×C8914yes—
260C2×C42—216no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42C2×C4×C8918yes—
264C2×C42C23×C89116yes—
265C2×C42C23×C8918yes—
266C2×C42C23×C8918yes—
267C2×C42C23×C8914yes—
268#95C23×C8918yes—
269#95C23×C8918yes—
270#95C23×C89116yes—
271#95C23×C89116yes—
272C24C24×C8911yescharacteristic
273C2×Q8—264no—
274C2×Q8—216no—
275C2×Q8—216no—
276C2×Q8C22×C4×C89132yes—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4C22×C4×C8914yes—
292C22×C4C22×C4×C89116yes—
293C22×C4C22×C4×C8914yes—
294C22×C4C22×C4×C8914yes—
295C22×C4C2×D4×C8918yes—
296C22×C4C2×D4×C8914yes—
297C22×C4C2×Q8×C8918yes—
298C22×C4C2×Q8×C8914yes—
299C22×C4#41×C8914yes—
300C22×C4#41×C8914yes—
301C22×C4C24×C8914yes—
302C22×C4C24×C8914yes—
303C22×C4C24×C8912yes—
304C22×C4C24×C8918yes—
305C22×C4C24×C8914yes—
306C22×C4C24×C8914yes—
307C22×C4C24×C8912yes—
308C22×C4C24×C8912yes—
309C4⋊C4—232no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4C22×C4×C89132yes—
313C4⋊C4C22×C4×C89132yes—
314C4⋊C4C2×D4×C89116yes—
315C4⋊C4C2×Q8×C89116yescomplemented
316C42—232no—
317C42—216no—
318C42—216no—
319C42C22×C4×C89116yes—
320C23C23×C4×C8912yes—
321C23C22×D4×C8912yes—
322C23C22×Q8×C8912yes—
323C23C2×#41×C8912yes—
324C23C2×#41×C8912yes—
325C232−1+4×C8912yes—
326C232−1+4×C8912yes—
327C23C25×C8911yescharacteristic
328Q8—264no—
329C2×C4—232no—
330C2×C4—28no—
331C2×C4—216no—
332C2×C4—216no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4C4×D4×C89116yes—
350C2×C4C4×Q8×C89116yescomplemented
351C2×C4C23×C4×C8914yes—
352C2×C4C23×C4×C89116yes—
353C2×C4C23×C4×C8914yes—
354C2×C4C23×C4×C8914yes—
355C2×C4C22×D4×C8918yes—
356C2×C4C22×D4×C8914yes—
357C2×C4C22×Q8×C8918yes—
358C2×C4C22×Q8×C8914yes—
359C2×C4C2×#41×C8914yes—
360C2×C4C2×#41×C8914yes—
361C22C2×C4×D4×C8914yes—
362C22C2×C4×Q8×C8914yes—
363C22#519×C8914yes—
364C22#547×C8914yes—
365C22D4×Q8×C8914yes—
366C22#556×C8914yes—
367C22#557×C8914yes—
368C22C24×C4×C8911yescharacteristic
369C22C23×D4×C8911yescharacteristic
370C22C23×Q8×C8911yescharacteristic
371C22C22×#41×C8911yescharacteristic
372C22C22×#41×C8911yescharacteristic
373C22C2×2−1+4×C8911yescharacteristic
374C22C2×2−1+4×C8911yescharacteristic
375C4—216no—
376C4—24no—
377C4—28no—
378C4—28no—
379C4—44no—
380C4—44no—
381C4C2×C4×D4×C8918yes—
382C4C2×C4×Q8×C8918yescomplemented
383C2C89×#236518yesdirect product factor
384C2C22×C4×D4×C8911yescharacteristic
385C2C22×C4×Q8×C8911yescharacteristic
386C2C2×#519×C8911yescharacteristic
387C2C2×#547×C8911yescharacteristic
388C2C2×D4×Q8×C8911yescharacteristic
389C2C2×#556×C8911yescharacteristic
390C2C2×#557×C8911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C89C24
commutatorC22C24×C4×C89
Frattini, cosocleC23C25×C89
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C89 ◁ G

Complements

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

NH ≅ G/Ntype
3. C89×#2365C2direct (N × H)
11. C2×C4⋊Q8×C89C4semidirect
48. C4⋊Q8×C89C2×C4semidirect
56. C2×C4⋊C4×C89Q8semidirect
119. C4⋊C4×C89C2×Q8semidirect
154. C2×C4×C89C4×Q8semidirect
186. C4×C89C2×C4×Q8semidirect
187. C2×#2365C89direct (N × H)
188. C2×C89#2365direct (N × H)
198. #2365C2×C89direct (N × H)
205. C89C2×#2365direct (N × H)
207. C2×C4⋊Q8C4×C89semidirect
244. C4⋊Q8C2×C4×C89semidirect
252. C2×C4⋊C4Q8×C89semidirect
315. C4⋊C4C2×Q8×C89semidirect
350. C2×C4C4×Q8×C89semidirect
382. C4C2×C4×Q8×C89semidirect
383. C2C89×#2365direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊Q8×C892
C22×C4×Q8×C892
C89×#23652
C2×#391×C892
C2×#389×C892
C2×#387×C892
C2×C4×C4⋊C4×C892
C2×#236589