Finite Group Databasebeta

← C2×#26028563 #36760300

Subgroups — C2×#26028563 #36760300

Order 22784.

3182 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668004C211yesmaximal, characteristic, complemented
2C22×C42×D89C211yesmaximal, characteristic
3#26028563C2116yesmaximal, direct product factor
4C2×#19151497C212yesmaximal, complemented
5C2×#19151483C214yesmaximal, complemented
6C2×#19151472C214yesmaximal
7C2×#19151455C212yesmaximal
8C2×#382×C89C211yesmaximal, characteristic, complemented
9C23×C4×D89C2212yes—
10C23×C4×D89C2211yescharacteristic
11C2×#15668004—28no—
12C2×#15668004C2218yescomplemented
13C22×C4⋊Dic89C2214yes—
14C22×C4⋊Dic89C2212yes—
15C22×#13083553C2214yes—
16C22×#13083551C2214yes—
17C22×C4×Dic89C2212yes—
18C22×C4×Dic89C2211yescharacteristic
19C2×C42×D89—28no—
20C2×C42×D89C2218yes—
21#19151497C22116yescomplemented
22#19151483C22132yescomplemented
23#19151472C22132yes—
24#19151455C22116yes—
25C2×#15667919C2214yes—
26C2×#15667919C2214yes—
27C22×C4⋊C4×C89C2212yes—
28C22×C42×C89C2211yescharacteristic
29#382×C89C2218yescomplemented
30C2×#95×C89C2214yes—
31C24×D89C2311yescharacteristic
32C23×Dic89C2314yes—
33C23×Dic89C2311yescharacteristic
34C23×Dic89C2312yes—
35C22×C4×D89—28no—
36C22×C4×D89—28no—
37C22×C4×D89—28no—
38C22×C4×D89—24no—
39C22×C4×D89—24no—
40C22×C4×D89Q818yes—
41C22×C4×D89C2318yes—
42C22×C4×D89C2314yes—
43#15668004—232no—
44C2×C4⋊Dic89—216no—
45C2×C4⋊Dic89—28no—
46C2×C4⋊Dic89C2318yes—
47C2×C4⋊Dic89C23116yes—
48C2×#13083553—216no—
49C2×#13083553C23116yes—
50C2×#13083551—216no—
51C2×#13083551C23116yes—
52C2×C4×Dic89—28no—
53C2×C4×Dic89—24no—
54C2×C4×Dic89C2318yes—
55C2×C4×Dic89C2314yes—
56C42×D89—232no—
57#15667919C23116yes—
58#15667919C23116yes—
59C23×C4×C89C2314yes—
60C23×C4×C89C2312yes—
61C23×C4×C89C2311yescharacteristic
62C2×C4⋊C4×C89—28no—
63C2×C4⋊C4×C89C2318yes—
64C2×C42×C89D418yes—
65C2×C42×C89C2314yes—
66#95×C89C23116yes—
67C23×D89—22no—
68C23×D89—22no—
69C23×D89—24no—
70C23×D89C2×Q814yes—
71C23×D89#4118yes—
72C23×D89C2412yes—
73C22×Dic89—22no—
74C22×Dic89—24no—
75C22×Dic89—22no—
76C22×Dic89—24no—
77C22×Dic89—28no—
78C22×Dic89—28no—
79C22×Dic89—28no—
80C22×Dic89C2×Q814yes—
81C22×Dic89#4118yes—
82C22×Dic89C2418yes—
83C22×Dic89C2412yes—
84C22×Dic89C2414yes—
85C2×C4×D89—264no—
86C2×C4×D89—216no—
87C2×C4×D89—232no—
88C2×C4×D89—216no—
89C2×C4×D89—28no—
90C2×C4×D89—28no—
91C2×C4×D89—216no—
92C2×C4×D89C2×Q8116yes—
93C4⋊Dic89—216no—
94C4⋊Dic89—232no—
95#13083553—232no—
96#13083551—232no—
97C4×Dic89—216no—
98C4×Dic89—28no—
99C24×C89C2411yescharacteristic
100C22×C4×C89—24no—
101C22×C4×C89—24no—
102C22×C4×C89—28no—
103C22×C4×C89—28no—
104C22×C4×C89—28no—
105C22×C4×C89C2×D418yes—
106C22×C4×C89C2×D414yes—
107C22×C4×C89C2×Q814yes—
108C22×C4×C89#4114yes—
109C22×C4×C89C2418yes—
110C22×C4×C89C2414yes—
111C22×C4×C89C2412yes—
112C4⋊C4×C89—216no—
113C42×C89C2×D4116yes—
114C23×C89C22×D412yes—
115C23×C89C22×Q812yes—
116C23×C89C2×#4114yes—
117C23×C89C2×#4114yes—
118C23×C89C2×#4112yes—
119C23×C89C2511yescharacteristic
120C22×D89—28no—
121C22×D89—216no—
122C22×D89—216no—
123C22×D89—28no—
124C22×D89—22no—
125C22×D89—24no—
126C22×D89—22no—
127C22×D89#125116yescomplemented
128C22×D89C22×Q814yes—
129C22×D89C2×#4118yes—
130C2×C4×C89—216no—
131C2×C4×C89—24no—
132C2×C4×C89—24no—
133C2×C4×C89—28no—
134C2×C4×C89—28no—
135C2×C4×C89—28no—
136C2×C4×C89—416no—
137C2×C4×C89C22⋊Q8116yes—
138C2×C4×C89C22×D418yes—
139C2×C4×C89C22×D414yes—
140C2×C4×C89C22×Q814yes—
141C2×C4×C89C2×#4114yes—
142C2×Dic89—28no—
143C2×Dic89—216no—
144C2×Dic89—22no—
145C2×Dic89—24no—
146C2×Dic89—24no—
147C2×Dic89—22no—
148C2×Dic89—28no—
149C2×Dic89—28no—
150C2×Dic89—28no—
151C2×Dic89—416no—
152C2×Dic89C22×Q814yes—
153C2×Dic89C2×#4118yes—
154C4×D89—264no—
155C4×D89—232no—
156C22×C89C2×C22⋊Q814yes—
157C22×C89#52518yes—
158C22×C89C2×#12514yes—
159C22×C89#52918yes—
160C22×C89#53314yes—
161C22×C89C23×D411yescharacteristic
162C22×C89C23×Q811yescharacteristic
163C22×C89C22×#4111yescharacteristic
164C22×C89C22×#4112yes—
165C22×C89C22×#4112yes—
166C2×D89—232no—
167C2×D89—24no—
168C2×D89—28no—
169C2×D89—28no—
170C2×D89—24no—
171C2×D89C2×#12518yescomplemented
172C4×C89—28no—
173C4×C89—48no—
174C4×C89C2×C22⋊Q818yes—
175Dic89—24no—
176Dic89—28no—
177Dic89—48no—
178C2×#2851—891nomaximal, Sylow
179C2×C89#285118yescomplemented
180C2×C89C22×C22⋊Q811yescharacteristic
181C2×C89C2×#52512yes—
182C2×C89C22×#12511yescharacteristic
183C2×C89C2×#52912yes—
184C2×C89C2×#53311yescharacteristic
185D89—28no—
186C22×#125—891no—
187C23×C42—891no—
188#2851—8916no—
189C2×#389—892no—
190C2×#387—894no—
191C2×#385—894no—
192C2×#382D8911yescharacteristic, complemented
193C2×#380—892no—
194C89C2×#285111yescharacteristic, complemented, Sylow
195C24×C4—891no—
196C24×C4—892no—
197C2×#125—898no—
198C2×#125—1788no—
199C22×C4⋊C4—894no—
200C22×C4⋊C4—892no—
201C22×C4⋊C4—894no—
202C22×C4⋊C4C2×D8912yes—
203C22×C22⋊C4—894no—
204C22×C42—892no—
205C22×C42—891no—
206C22×C42—898no—
207C22×C42—1788no—
208C22×C42C2×D8911yescharacteristic
209#389—8916no—
210#387—8932no—
211#385—8932no—
212#382C2×D8918yescomplemented
213#380—8916no—
214C2×#95—894no—
215C2×#95—894no—
216C2×#95C2×D8914yes—
217C25—891no—
218C23×C4—892no—
219C23×C4—891no—
220C23×C4—898no—
221C23×C4—894no—
222C23×C4—894no—
223C23×C4—898no—
224C23×C4—1788no—
225C23×C4—1788no—
226C23×C4—1788no—
227C23×C4—1784no—
228C23×C4—1784no—
229C23×C4C22×D8914yes—
230C23×C4C22×D8912yes—
231C23×C4C22×D8911yescharacteristic
232#125—17832no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—898no—
235C2×C4⋊C4—8916no—
236C2×C4⋊C4—8916no—
237C2×C4⋊C4—17816no—
238C2×C4⋊C4—17816no—
239C2×C4⋊C4—1788no—
240C2×C4⋊C4C22×D8918yes—
241C2×C22⋊C4—8916no—
242C2×C22⋊C4—17816no—
243C2×C42—898no—
244C2×C42—894no—
245C2×C42—1788no—
246C2×C42—1784no—
247C2×C42—17832no—
248C2×C42D35618yes—
249C2×C42C22×D8914yes—
250#95—8916no—
251#95—8916no—
252#95C22×D89116yes—
253C24—894no—
254C24—898no—
255C24—892no—
256C24—1782no—
257C24—1784no—
258C24—1782no—
259C24C23×D8911yescharacteristic
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—894no—
266C22×C4—898no—
267C22×C4—892no—
268C22×C4—894no—
269C22×C4—8916no—
270C22×C4—898no—
271C22×C4—17816no—
272C22×C4—17816no—
273C22×C4—17864no—
274C22×C4—1788no—
275C22×C4—1788no—
276C22×C4—1788no—
277C22×C4—1782no—
278C22×C4—1784no—
279C22×C4—1784no—
280C22×C4—1782no—
281C22×C4—17832no—
282C22×C4—17816no—
283C22×C4—1788no—
284C22×C4—1788no—
285C22×C4C2×D35618yes—
286C22×C4C2×D35614yes—
287C22×C4Q8×D8914yes—
288C22×C4#1308357314yes—
289C22×C4C23×D8918yes—
290C22×C4C23×D8914yes—
291C22×C4C23×D8912yes—
292C4⋊C4—216no—
293C4⋊C4—17832no—
294C4⋊C4—17832no—
295C4⋊C4—17816no—
296C22⋊C4—17832no—
297C42—17816no—
298C42—1788no—
299C42C2×D356116yes—
300C23—894no—
301C23—898no—
302C23—8916no—
303C23—1788no—
304C23—1782no—
305C23—1784no—
306C23—1782no—
307C23—1788no—
308C23—17816no—
309C23—17816no—
310C23C22×D35612yes—
311C23C2×Q8×D8912yes—
312C23#41×D8914yes—
313C23#41×D8914yes—
314C23C2×#1308357312yes—
315C23C24×D8911yescharacteristic
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—216no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—416no—
323C2×C4—894no—
324C2×C4—898no—
325C2×C4—35616no—
326C2×C4—17864no—
327C2×C4—1788no—
328C2×C4—1788no—
329C2×C4—1788no—
330C2×C4—1782no—
331C2×C4—1784no—
332C2×C4—1784no—
333C2×C4—1782no—
334C2×C4—1788no—
335C2×C4—17816no—
336C2×C4—17832no—
337C2×C4#15667992116yes—
338C2×C4C22×D35618yes—
339C2×C4C22×D35614yes—
340C2×C4C2×Q8×D8914yes—
341C2×C4C2×#1308357314yes—
342C22—898no—
343C22—17832no—
344C22—1784no—
345C22—1788no—
346C22—1788no—
347C22—1784no—
348C22C2×#1566799214yes—
349C22#1915211018yes—
350C22#125×D8914yes—
351C22#1915212718yes—
352C22#1915215414yes—
353C22C23×D35611yescharacteristic
354C22C22×Q8×D8911yescharacteristic
355C22C2×#41×D8912yes—
356C22C2×#41×D8912yes—
357C22C22×#1308357311yescharacteristic
358C4—28no—
359C4—48no—
360C4—3568no—
361C4—1784no—
362C4—1788no—
363C4C2×#1566799218yes—
364C2—1788no—
365C2#2602856318yesdirect product factor
366C2C22×#1566799211yescharacteristic
367C2C2×#1915211012yes—
368C2C2×#125×D8911yescharacteristic
369C2C2×#1915212712yes—
370C2C2×#1915215411yescharacteristic
subgroupG/H
centerC24C23×D89
commutator, cosocleC23×C89C25
FrattiniC23C24×D89
FittingC2×#382×C89C2
socleC24×C89C24
prime radicalC24×D89C23
hypercenterC2×#382D89
hypocenterC89C2×#2851
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
1. C22×#15668004C2semidirect
3. #26028563C2direct (N × H)
4. C2×#19151497C2semidirect
5. C2×#19151483C2semidirect
8. C2×#382×C89C2semidirect
12. C2×#15668004C22semidirect
21. #19151497C22semidirect
22. #19151483C22semidirect
29. #382×C89C22semidirect
127. C22×D89#125semidirect
171. C2×D89C2×#125semidirect
179. C2×C89#2851semidirect
192. C2×#382D89semidirect
194. C89C2×#2851semidirect
212. #382C2×D89semidirect
365. C2#26028563direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156680042
C22×C42×D892
#260285632
C2×#191514972
C2×#191514832
C2×#191514722
C2×#191514552
C2×#382×C892
C2×#285189