Finite Group Databasebeta

← C2×#26019926 #36542223

Subgroups — C2×#26019926 #36542223

Order 22784.

798 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152003C211yesmaximal, characteristic
2C2×#19152001C211yesmaximal, characteristic
3C2×#19151977C211yesmaximal, characteristic
4C2×#19151950C211yesmaximal, characteristic
5C2×#19151948C211yesmaximal, characteristic
6#26019926C218yesmaximal, direct product factor
7C2×#19151319C211yesmaximal, characteristic
8C2×#498×C89C211yesmaximal, characteristic
9C22×#13083564—22no—
10C2×#15668012C2211yescharacteristic
11C2×#15668012C2211yescharacteristic
12#19152003—22no—
13#19152003C2214yes—
14#19152001—22no—
15#19152001C2214yes—
16#19151977—22no—
17#19151977C2214yes—
18#19151950—22no—
19#19151950C2214yes—
20#19151948—22no—
21#19151948C2214yes—
22C2×#15667946—22no—
23C2×#15667945—22no—
24C2×#15667936—22no—
25C2×#15667935—22no—
26C2×C4×#10623494C2211yescharacteristic
27C2×C4×#10623494C2211yescharacteristic
28#19151319C2214yes—
29C2×C4⋊Q8×C89C2211yescharacteristic
30C2×C4⋊Q8×C89C2211yescharacteristic
31#498×C89—22no—
32#498×C89C2214yes—
33C2×#106×C89—22no—
34C2×C4×C8×C89C2211yescharacteristic
35C22×Dic178—21no—
36C22×Dic178—21no—
37C2×#13083564—28no—
38C2×#13083564—48no—
39C22×#10623494D412yes—
40C22×#10623494D412yes—
41#15668012—21no—
42#15668012—21no—
43#15668012—21no—
44#15668012—21no—
45#15668012—21no—
46#15668012—21no—
47#15668012C2312yes—
48#15668012C2312yes—
49C2×C4⋊Dic89—21no—
50C2×C4⋊Dic89—21no—
51C2×C4⋊Dic89—21no—
52C2×C4⋊Dic89—21no—
53#15667946—28no—
54#15667945—28no—
55#15667936—28no—
56#15667935—28no—
57C4×#10623494D412yes—
58C4×#10623494D412yes—
59C4×#10623494C2312yes—
60C4×#10623494C2312yes—
61C22×Q8×C89—21no—
62C22×Q8×C89—21no—
63C22×C8×C89D412yes—
64C4⋊Q8×C89—21no—
65C4⋊Q8×C89—21no—
66C4⋊Q8×C89—21no—
67C4⋊Q8×C89—21no—
68C4⋊Q8×C89—21no—
69C4⋊Q8×C89—21no—
70C4⋊Q8×C89C2312yes—
71C4⋊Q8×C89C2312yes—
72C2×C4⋊C4×C89—21no—
73C2×C4⋊C4×C89—21no—
74C2×C4⋊C4×C89—21no—
75C2×C4⋊C4×C89—21no—
76C2×C42×C89C2311yescharacteristic
77#106×C89—28no—
78C4×C8×C89D412yes—
79C4×C8×C89C2312yes—
80C2×Dic178—22no—
81C2×Dic178—22no—
82C2×Dic178—44no—
83C2×Dic178—44no—
84C2×Dic178—41no—
85C2×Dic178—41no—
86C22×Dic89—41no—
87C22×Dic89—41no—
88#13083564—416no—
89C2×#10623494—24no—
90C2×#10623494—24no—
91C2×#10623494C2×D414yes—
92C2×#10623494C2×D414yes—
93C4⋊Dic89—22no—
94C4⋊Dic89—22no—
95C4⋊Dic89—22no—
96C4⋊Dic89—22no—
97C4⋊Dic89—41no—
98C4⋊Dic89—41no—
99C4⋊Dic89—41no—
100C4⋊Dic89—41no—
101C2×Q8×C89—22no—
102C2×Q8×C89—22no—
103C2×Q8×C89—41no—
104C2×Q8×C89—41no—
105C2×Q8×C89—44no—
106C2×Q8×C89—44no—
107C22×C4×C89—41no—
108C22×C4×C89—41no—
109C22×C4×C89C2×D411yescharacteristic
110C22×C4×C89C2×D411yescharacteristic
111C22×C4×C89C2×D411yescharacteristic
112C2×C8×C89Q1618yescomplemented
113C2×C8×C89C2×D414yes—
114C4⋊C4×C89—22no—
115C4⋊C4×C89—22no—
116C4⋊C4×C89—22no—
117C4⋊C4×C89—22no—
118C4⋊C4×C89—41no—
119C4⋊C4×C89—41no—
120C4⋊C4×C89—41no—
121C4⋊C4×C89—41no—
122C42×C89C2×D411yescharacteristic
123C42×C89C2×D411yescharacteristic
124C42×C89C2×D411yescharacteristic
125C42×C89C2411yescharacteristic
126C23×C89#12711yescharacteristic
127Q8×C89—44no—
128Q8×C89—44no—
129Q8×C89—82no—
130Q8×C89—82no—
131C2×C4×C89—42no—
132C2×C4×C89—42no—
133C2×C4×C89—82no—
134C2×C4×C89—82no—
135C2×C4×C89#12711yescharacteristic
136C2×C4×C89#12711yescharacteristic
137C2×C4×C89#12711yescharacteristic
138C2×C4×C89C2×Q1614yes—
139C2×C4×C89AGL(1,ℤ/8)14yes—
140C2×C4×C89#13714yes—
141C2×C4×C89C22×D411yescharacteristic
142C2×C4×C89C22×D411yescharacteristic
143C2×C4×C89C22×D411yescharacteristic
144C2×Dic89—42no—
145C2×Dic89—42no—
146C2×Dic89—82no—
147C2×Dic89—82no—
148Dic178—44no—
149Dic178—44no—
150Dic178—82no—
151Dic178—82no—
152C8×C89C2×Q1618yescomplemented
153#10623494—24no—
154#10623494—24no—
155C22×C89#49412yes—
156C22×C89#49612yes—
157C22×C89#49712yes—
158C22×C89C2×#12711yescharacteristic
159C4×C89—82no—
160C4×C89—82no—
161C4×C89#49412yes—
162C4×C89#49612yes—
163C4×C89#49712yes—
164C4×C89C22×Q1612yes—
165C4×C89C2×AGL(1,ℤ/8)12yes—
166C4×C89C2×#13712yes—
167Dic89—82no—
168Dic89—82no—
169C2×#1723—891nomaximal, Sylow
170C2×C89#172314yescomplemented
171C2×C89C2×#49411yescharacteristic
172C2×C89C2×#49611yescharacteristic
173C2×C89C2×#49711yescharacteristic
174C2×#500—891no—
175C2×#500—891no—
176C2×#498D8911yescharacteristic
177C2×#494—891no—
178C2×#487—891no—
179C2×#487—891no—
180#1723—898no—
181C2×#322—891no—
182C89C2×#172311yescharacteristic, complemented, Sylow
183C22×Q16—1782no—
184C2×C4⋊Q8—891no—
185C2×C4⋊Q8—891no—
186C2×C4⋊Q8C2×D8911yescharacteristic
187C2×C4⋊Q8C2×D8911yescharacteristic
188#500—894no—
189#500—894no—
190#500—1782no—
191#500—1782no—
192#498—22no—
193#498C2×D8914yes—
194#494—894no—
195#494—1782no—
196#487—894no—
197#487—894no—
198#487—1782no—
199#487—1782no—
200C2×#107—1782no—
201C2×#107—1782no—
202C2×#106—22no—
203C2×#103—1782no—
204C2×#103—1782no—
205C2×C4×C8—891no—
206C2×C4×C8—891no—
207C2×C4×C8C2×D8911yescharacteristic
208#322—894no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—1781no—
212C22×Q8—1781no—
213C2×Q16—3568no—
214C2×Q16—1788no—
215C22×C8—892no—
216C22×C8—892no—
217C22×C8C22⋊D8912yes—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—892no—
225C4⋊Q8—892no—
226C4⋊Q8—1781no—
227C4⋊Q8—1781no—
228C4⋊Q8—1781no—
229C4⋊Q8—1781no—
230C4⋊Q8—1781no—
231C4⋊Q8—1781no—
232C4⋊Q8C22×D8912yes—
233C4⋊Q8C22×D8912yes—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1781no—
239C2×C4⋊C4—1781no—
240C2×C4⋊C4—1781no—
241C2×C4⋊C4—1781no—
242C2×C42C22×D8911yescharacteristic
243#107—1788no—
244#107—1788no—
245#106—28no—
246#103—1788no—
247#103—1788no—
248C4×C8—892no—
249C4×C8—892no—
250C4×C8—892no—
251C4×C8—892no—
252C4×C8C22⋊D8912yes—
253C4×C8C22×D8912yes—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—3561no—
261C2×Q8—3561no—
262C2×Q8—3564no—
263C2×Q8—3564no—
264C2×Q8—1782no—
265C2×Q8—1782no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—3561no—
269C22×C4—3561no—
270C22×C4D4×D8911yescharacteristic
271C22×C4D4×D8911yescharacteristic
272C22×C4C2×C22⋊D8911yescharacteristic
273Q16—35616no—
274C2×C8—894no—
275C2×C8—894no—
276C2×C8—1784no—
277C2×C8—1784no—
278C2×C8#1308356418yescomplemented
279C2×C8C2×C22⋊D8914yes—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—3561no—
289C4⋊C4—3561no—
290C4⋊C4—3561no—
291C4⋊C4—3561no—
292C4⋊C4—1782no—
293C4⋊C4—1782no—
294C4⋊C4—1782no—
295C4⋊C4—1782no—
296C42D4×D8911yescharacteristic
297C42D4×D8911yescharacteristic
298C42C2×C22⋊D8911yescharacteristic
299C42C23×D8911yescharacteristic
300C23#1566800811yescharacteristic
301Q8—44no—
302Q8—44no—
303Q8—82no—
304Q8—82no—
305Q8—3564no—
306Q8—3564no—
307Q8—7122no—
308Q8—7122no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—3562no—
314C2×C4—3562no—
315C2×C4—7122no—
316C2×C4—7122no—
317C2×C4#1566800811yescharacteristic
318C2×C4#1566800811yescharacteristic
319C2×C4#1566800811yescharacteristic
320C2×C4C2×#1308356414yes—
321C2×C4#1566803914yes—
322C2×C4#1566804114yes—
323C2×C4C2×D4×D8911yescharacteristic
324C2×C4C2×D4×D8911yescharacteristic
325C2×C4C22×C22⋊D8911yescharacteristic
326C8—1784no—
327C8—1784no—
328C8C2×#1308356418yescomplemented
329C22#1915197712yes—
330C22#1915198412yes—
331C22#1915198912yes—
332C22C2×#1566800811yescharacteristic
333C4—82no—
334C4—82no—
335C4—7122no—
336C4—7122no—
337C4#1915197712yes—
338C4#1915198412yes—
339C4#1915198912yes—
340C4C22×#1308356412yes—
341C4C2×#1566803912yes—
342C4C2×#1566804112yes—
343C2#2601992614yesdirect product factor
344C2C2×#1915197711yescharacteristic
345C2C2×#1915198411yescharacteristic
346C2C2×#1915198911yescharacteristic
subgroupG/H
centerC23#15668008
commutator, cosocleC42×C89C24
FrattiniC42C23×D89
FittingC2×#498×C89C2
socle, prime radicalC23×C89#127
hypercenterC2×#498D89
hypocenterC89C2×#1723
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
6. #26019926C2direct (N × H)
112. C2×C8×C89Q16semidirect
152. C8×C89C2×Q16semidirect
170. C2×C89#1723semidirect
182. C89C2×#1723semidirect
278. C2×C8#13083564semidirect
328. C8C2×#13083564semidirect
343. C2#26019926direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191520032
C2×#191520012
C2×#191519772
C2×#191519502
C2×#191519482
#260199262
C2×#191513192
C2×#498×C892
C2×#172389