Finite Group Databasebeta

← #36362959

Subgroups — #36362959

Order 22144.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149724C211yesmaximal, characteristic, complemented
2#26012050C212yesmaximal, complemented
3#26011680C214yesmaximal, complemented
4#26011677C214yesmaximal, complemented
5C2×#19149594C211yesmaximal, characteristic, complemented
6C2×#19149593C212yesmaximal, complemented
7#534×C173C211yesmaximal, characteristic, complemented
8#19149724—22no—
9#19149724—24no—
10#19149724C2214yes—
11C2×#14583826—21no—
12C2×#14583825—22no—
13C22×C22⋊D173—21no—
14C22×C22⋊D173C2212yes—
15C2×D4×D173—22no—
16#19149662C2214yescomplemented
17#19149660—22no—
18#19149651—24no—
19#19149648C2214yescomplemented
20#19149647—22no—
21C2×C22⋊Dic173C2212yescomplemented
22#19149594C2214yescomplemented
23#19149593C2218yescomplemented
24C22×D4×C173C2211yescharacteristic
25C22×D4×C173C2211yescharacteristic, complemented
26#123×C173—21no—
27#123×C173C2212yescomplemented
28#121×C173—22no—
29#120×C173—21no—
30#120×C173C2212yescomplemented
31C2×C22⋊C4×C173C2211yescharacteristic, complemented
32C23×D173—21no—
33C23×D173D412yescomplemented
34#14583826—24no—
35#14583826—24no—
36#14583825—28no—
37#14583825—28no—
38C2×Dic346—21no—
39C2×D692—21no—
40C2×C22⋊D173—22no—
41C2×C22⋊D173—24no—
42C2×C22⋊D173—24no—
43C2×C22⋊D173—22no—
44C2×C22⋊D173—22no—
45C2×C22⋊D173—24no—
46C2×C22⋊D173—22no—
47C2×C22⋊D173—22no—
48C2×C22⋊D173D414yescomplemented
49C2×C22⋊D173C2314yes—
50D4×D173—28no—
51D4×D173—28no—
52C22×Dic173—21no—
53C22×Dic173—22no—
54C22×Dic173D412yescomplemented
55C2×C4×D173—22no—
56#14583805—24no—
57C22⋊Dic173—24no—
58C22⋊Dic173—22no—
59C22⋊Dic173C2314yescomplemented
60#14583803—24no—
61C24×C173D411yescharacteristic, complemented
62C24×C173C2311yescharacteristic
63C24×C173C2311yescharacteristic, complemented
64C2×D4×C173—22no—
65C2×D4×C173—22no—
66C2×D4×C173—22no—
67C2×D4×C173—22no—
68C2×D4×C173—21no—
69C2×D4×C173—21no—
70C2×D4×C173—22no—
71C2×D4×C173D412yescomplemented
72C2×D4×C173C2312yes—
73C22×C4×C173—21no—
74C22×C4×C173D411yescharacteristic, complemented
75C22×C4×C173C2311yescharacteristic, complemented
76C4⋊C4×C173—22no—
77C22⋊C4×C173—21no—
78C22⋊C4×C173—22no—
79C22⋊C4×C173—22no—
80C22⋊C4×C173C2312yescomplemented
81C22×D173—22no—
82C22×D173—24no—
83C22×D173—22no—
84C22×D173—82no—
85C22×D173—82no—
86C22×D173C2×D414yes—
87Dic346—24no—
88D692—24no—
89C22⋊D173—28no—
90C22⋊D173—24no—
91C22⋊D173—24no—
92C22⋊D173—24no—
93C22⋊D173—24no—
94C22⋊D173—84no—
95C2×Dic173—24no—
96C2×Dic173—22no—
97C2×Dic173—22no—
98C2×Dic173—24no—
99C2×Dic173—82no—
100C2×Dic173C2×D414yes—
101C4×D173—28no—
102C23×C173—21no—
103C23×C173—21no—
104C23×C173—22no—
105C23×C173—21no—
106C23×C173—81no—
107C23×C173—81no—
108C23×C173—81no—
109C23×C173C2×D412yes—
110C23×C173C2×D412yes—
111C23×C173C2×D412yes—
112C23×C173C2×D411yescharacteristic
113C23×C173C2×D411yescharacteristic
114C23×C173C2411yescharacteristic
115D4×C173—24no—
116D4×C173—24no—
117D4×C173—41no—
118D4×C173—41no—
119D4×C173—42no—
120D4×C173—82no—
121C2×C4×C173—21no—
122C2×C4×C173—21no—
123C2×C4×C173—22no—
124C2×C4×C173—21no—
125C2×C4×C173—22no—
126C2×C4×C173—81no—
127C2×C4×C173C2×D412yes—
128C22×C173—22no—
129C22×C173—41no—
130C22×C173—41no—
131C22×C173—41no—
132C22×C173—42no—
133C22×C173—42no—
134C22×C173—42no—
135C22×C173—42no—
136C22×C173—42no—
137C22×C173—44no—
138C22×C173—41no—
139C22×C173—81no—
140C22×C173#12011yescharacteristic
141C22×C173#12011yescharacteristic
142C22×C173#12012yes—
143C22×C173C22×D412yes—
144C22×C173C22×D411yescharacteristic
145C2×D173—24no—
146C2×D173—44no—
147C2×D173—44no—
148C2×D173—82no—
149C2×D173—84no—
150C4×C173—22no—
151C4×C173—42no—
152C4×C173—41no—
153Dic173—24no—
154Dic173—44no—
155C2×C173—22no—
156C2×C173—21no—
157C2×C173—21no—
158C2×C173—22no—
159C2×C173—21no—
160C2×C173—42no—
161C2×C173—81no—
162C2×C173C2×#12011yescharacteristic
163D173—44no—
164C173#302611yescharacteristic, complemented, Sylow
165#3026—1731nomaximal, Sylow
166C2×H(5,𝔽2)—1731no—
167#534—1732no—
168#534D17311yescharacteristic, complemented
169#458—1734no—
170B(4,𝔽2)—1734no—
171C2×#99—1731no—
172C2×#99—1732no—
173H(5,𝔽2)—3464no—
174H(5,𝔽2)—3462no—
175H(5,𝔽2)—1734no—
176C2×#41—3462no—
177C2×#41—3461no—
178C22×D4—3462no—
179C22×D4—3461no—
180C22×D4—1732no—
181C22×D4C2×D17311yescharacteristic
182C22×D4C2×D17311yescharacteristic, complemented
183#123—21no—
184#123—3462no—
185#123—1734no—
186#123C2×D17312yescomplemented
187#121—22no—
188#121—3464no—
189#120—21no—
190#120—3462no—
191#120—1734no—
192#120C2×D17312yescomplemented
193C2×C22⋊C4—1732no—
194C2×C22⋊C4C2×D17311yescharacteristic, complemented
195#99—1734no—
196#99—1738no—
197C24—3461no—
198C24—1732no—
199C24C22⋊D17311yescharacteristic, complemented
200C24C22×D17311yescharacteristic
201C24C22×D17311yescharacteristic, complemented
202#41—3464no—
203#41—3468no—
204#41—3468no—
205#41—3464no—
206C2×Q8—3461no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—3461no—
215C2×D4—3468no—
216C2×D4—3468no—
217C2×D4—3462no—
218C2×D4—3464no—
219C2×D4—3462no—
220C2×D4—3462no—
221C2×D4—3462no—
222C2×D4—3464no—
223C2×D4—3462no—
224C2×D4—3464no—
225C2×D4—1734no—
226C2×D4—1734no—
227C2×D4C22⋊D17312yescomplemented
228C2×D4C22×D17312yes—
229C22×C4—21no—
230C22×C4—3462no—
231C22×C4—3461no—
232C22×C4—3462no—
233C22×C4—1732no—
234C22×C4C22⋊D17311yescharacteristic, complemented
235C22×C4C22×D17311yescharacteristic, complemented
236C4⋊C4—22no—
237C4⋊C4—3464no—
238C22⋊C4—21no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—3464no—
242C22⋊C4—3462no—
243C22⋊C4—3464no—
244C22⋊C4—1734no—
245C22⋊C4C22×D17312yescomplemented
246C23—21no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—3464no—
254C23—3462no—
255C23—3462no—
256C23—13842no—
257C23—13842no—
258C23—1734no—
259C23D4×D17312yes—
260C23D4×D17312yes—
261C23C2×C22⋊D17312yes—
262C23C2×C22⋊D17311yescharacteristic
263C23C2×C22⋊D17311yescharacteristic
264C23C23×D17311yescharacteristic
265Q8—3464no—
266D4—24no—
267D4—24no—
268D4—41no—
269D4—41no—
270D4—42no—
271D4—82no—
272D4—3464no—
273D4—3468no—
274D4—3464no—
275D4—3464no—
276D4—3464no—
277D4—3464no—
278D4—13844no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—3468no—
286C2×C4—3464no—
287C2×C4—3462no—
288C2×C4—3462no—
289C2×C4—3464no—
290C2×C4—13842no—
291C2×C4—1734no—
292C2×C4C2×C22⋊D17312yes—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—41no—
304C22—81no—
305C22—3464no—
306C22—13842no—
307C22—13844no—
308C22—6924no—
309C22—6924no—
310C22#1914964711yescharacteristic
311C22#1914964711yescharacteristic
312C22#1914964712yes—
313C22C2×D4×D17312yes—
314C22C22×C22⋊D17311yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—3464no—
319C4—6924no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—6924no—
328C2C2×#1914964711yescharacteristic
subgroupG/H
centerC2C2×#19149647
commutator, cosocleC23×C173C24
FrattiniC23C23×D173
Fitting#534×C173C2
socleC2×C173C2×#120
hypercenter#534D173
hypocenterC173#3026
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#19149724C2semidirect
2. #26012050C2semidirect
3. #26011680C2semidirect
4. #26011677C2semidirect
5. C2×#19149594C2semidirect
6. C2×#19149593C2semidirect
7. #534×C173C2semidirect
16. #19149662C22semidirect
19. #19149648C22semidirect
21. C2×C22⋊Dic173C22semidirect
22. #19149594C22semidirect
23. #19149593C22semidirect
25. C22×D4×C173C22semidirect
27. #123×C173C22semidirect
30. #120×C173C22semidirect
31. C2×C22⋊C4×C173C22semidirect
33. C23×D173D4semidirect
48. C2×C22⋊D173D4semidirect
54. C22×Dic173D4semidirect
59. C22⋊Dic173C23semidirect
61. C24×C173D4semidirect
63. C24×C173C23semidirect
71. C2×D4×C173D4semidirect
74. C22×C4×C173D4semidirect
75. C22×C4×C173C23semidirect
80. C22⋊C4×C173C23semidirect
164. C173#3026semidirect
168. #534D173semidirect
182. C22×D4C2×D173semidirect
186. #123C2×D173semidirect
192. #120C2×D173semidirect
194. C2×C22⋊C4C2×D173semidirect
199. C24C22⋊D173semidirect
201. C24C22×D173semidirect
227. C2×D4C22⋊D173semidirect
234. C22×C4C22⋊D173semidirect
235. C22×C4C22×D173semidirect
245. C22⋊C4C22×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
#260120502
#260116802
#260116772
C2×#191495942
C2×#191495932
#534×C1732
#3026173