Finite Group Databasebeta

← #36362962

Subgroups — #36362962

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#26012053C212yesmaximal, complemented
3#26012050C211yesmaximal, characteristic, complemented
4#26011682C214yesmaximal, complemented
5#26011679C214yesmaximal, complemented
6C2×#19149593C212yesmaximal, complemented
7C2×#99×C173C211yesmaximal, characteristic, complemented
8#19149724—22no—
9#19149724—24no—
10#19149724C2214yes—
11C2×#14583826—21no—
12C2×#14583825—22no—
13C22×C22⋊D173—21no—
14C22×C22⋊D173C2212yes—
15C22×C22⋊D173C2211yescharacteristic, complemented
16C2×D4×D173—22no—
17#19149663—22no—
18#19149663C2214yescomplemented
19#19149662—21no—
20#19149660C2212yescomplemented
21#19149653—24no—
22#19149651—22no—
23#19149649—22no—
24#19149649C2214yescomplemented
25#19149648—21no—
26#19149647C2212yescomplemented
27C2×C22⋊Dic173C2211yescharacteristic, complemented
28#19149593C2218yescomplemented
29C22×D4×C173C2211yescharacteristic
30C2×C22⋊C4×C173C2212yescomplemented
31#99×C173C2214yescomplemented
32C23×D173—21no—
33C23×D173D412yescomplemented
34C23×D173C2311yescharacteristic, complemented
35#14583826—24no—
36#14583826—24no—
37#14583825—28no—
38#14583825—28no—
39C2×Dic346—21no—
40C2×D692—21no—
41C2×D692—24no—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—24no—
44C2×C22⋊D173—22no—
45C2×C22⋊D173—22no—
46C2×C22⋊D173—24no—
47C2×C22⋊D173—22no—
48C2×C22⋊D173—22no—
49C2×C22⋊D173—21no—
50C2×C22⋊D173—22no—
51C2×C22⋊D173—21no—
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54C2×C22⋊D173D414yescomplemented
55C2×C22⋊D173C2314yes—
56D4×D173—28no—
57D4×D173—28no—
58C22×Dic173—21no—
59C22×Dic173—21no—
60C22×Dic173D412yescomplemented
61C22×Dic173C2311yescharacteristic, complemented
62C2×C4×D173—22no—
63C4⋊Dic173—24no—
64#14583805—22no—
65C22⋊Dic173—21no—
66C22⋊Dic173—22no—
67C22⋊Dic173C2312yescomplemented
68#14583803—24no—
69#14583803—24no—
70#14583803—22no—
71C24×C173D411yescharacteristic, complemented
72C24×C173C2311yescharacteristic
73C2×D4×C173—22no—
74C2×D4×C173—22no—
75C2×D4×C173D412yescomplemented
76C2×D4×C173C2312yes—
77C22×C4×C173—22no—
78C22×C4×C173D411yescharacteristic, complemented
79C22⋊C4×C173—22no—
80C22⋊C4×C173C2314yescomplemented
81C22×D173—22no—
82C22×D173—24no—
83C22×D173—22no—
84C22×D173—21no—
85C22×D173—22no—
86C22×D173—81no—
87C22×D173—82no—
88C22×D173—82no—
89C22×D173C2×D414yes—
90Dic346—24no—
91D692—24no—
92D692—84no—
93C22⋊D173—28no—
94C22⋊D173—24no—
95C22⋊D173—24no—
96C22⋊D173—24no—
97C22⋊D173—24no—
98C22⋊D173—41no—
99C22⋊D173—41no—
100C22⋊D173—42no—
101C22⋊D173—82no—
102C2×Dic173—24no—
103C2×Dic173—22no—
104C2×Dic173—22no—
105C2×Dic173—22no—
106C2×Dic173—22no—
107C2×Dic173—21no—
108C2×Dic173—81no—
109C2×Dic173C2×D414yes—
110C4×D173—28no—
111C23×C173—21no—
112C23×C173—21no—
113C23×C173—81no—
114C23×C173—81no—
115C23×C173C2×D412yes—
116C23×C173C2×D412yes—
117C23×C173C2×D412yes—
118C23×C173C2×D411yescharacteristic
119C23×C173C2×D411yescharacteristic
120C23×C173C2411yescharacteristic
121D4×C173—24no—
122D4×C173—24no—
123C2×C4×C173—21no—
124C2×C4×C173—21no—
125C2×C4×C173—24no—
126C2×C4×C173—82no—
127C2×C4×C173C2×D412yes—
128C22×C173—22no—
129C22×C173—42no—
130C22×C173—41no—
131C22×C173—41no—
132C22×C173—42no—
133C22×C173—41no—
134C22×C173—42no—
135C22×C173—42no—
136C22×C173—81no—
137C22×C173#12011yescharacteristic
138C22×C173#12011yescharacteristic
139C22×C173#12012yes—
140C22×C173C22×D412yes—
141C22×C173C22×D411yescharacteristic
142C2×D173—24no—
143C2×D173—44no—
144C2×D173—44no—
145C2×D173—42no—
146C2×D173—44no—
147C2×D173—41no—
148C2×D173—82no—
149C2×D173—84no—
150C4×C173—22no—
151C4×C173—44no—
152Dic173—24no—
153Dic173—42no—
154Dic173—41no—
155C2×C173—21no—
156C2×C173—22no—
157C2×C173—21no—
158C2×C173—22no—
159C2×C173—21no—
160C2×C173—42no—
161C2×C173C2×#12011yescharacteristic
162D173—44no—
163D173—81no—
164C173#302611yescharacteristic, complemented, Sylow
165#3026—1731nomaximal, Sylow
166C2×H(5,𝔽2)—1731no—
167#534—1731no—
168#534—1732no—
169#458—1734no—
170B(4,𝔽2)—1734no—
171C2×#99—1732no—
172C2×#99D17311yescharacteristic, complemented
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—1731no—
181C22×D4—1732no—
182C22×D4C2×D17311yescharacteristic
183#123—3462no—
184#123—3461no—
185#123—1732no—
186#123—1734no—
187#121—3464no—
188#121—3462no—
189#120—3462no—
190#120—3461no—
191#120—1732no—
192#120—1734no—
193C2×C22⋊C4—1731no—
194C2×C22⋊C4C2×D17312yescomplemented
195#99—1738no—
196#99C2×D17314yescomplemented
197C24—3461no—
198C24—1731no—
199C24—1732no—
200C24D69211yescharacteristic, complemented
201C24C22×D17311yescharacteristic
202#41—3464no—
203#41—3468no—
204#41—3468no—
205#41—3464no—
206C2×Q8—3461no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—3461no—
210C2×D4—3464no—
211C2×D4—3468no—
212C2×D4—3468no—
213C2×D4—3462no—
214C2×D4—3464no—
215C2×D4—3462no—
216C2×D4—3462no—
217C2×D4—3462no—
218C2×D4—3464no—
219C2×D4—3462no—
220C2×D4—3462no—
221C2×D4—3462no—
222C2×D4—3462no—
223C2×D4—3461no—
224C2×D4—3461no—
225C2×D4—1734no—
226C2×D4—1734no—
227C2×D4D69212yescomplemented
228C2×D4C22×D17312yes—
229C22×C4—22no—
230C22×C4—3462no—
231C22×C4—3461no—
232C22×C4—3461no—
233C22×C4—1731no—
234C22×C4—1732no—
235C22×C4D69211yescharacteristic, complemented
236C4⋊C4—3464no—
237C4⋊C4—3462no—
238C22⋊C4—22no—
239C22⋊C4—3464no—
240C22⋊C4—3464no—
241C22⋊C4—3461no—
242C22⋊C4—3462no—
243C22⋊C4—3462no—
244C22⋊C4—1732no—
245C22⋊C4C22×D17314yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—3464no—
251C23—3462no—
252C23—3462no—
253C23—3462no—
254C23—3461no—
255C23—13842no—
256C23—13842no—
257C23—13841no—
258C23—1734no—
259C23D4×D17312yes—
260C23D4×D17312yes—
261C23C2×D69212yes—
262C23C2×D69211yescharacteristic
263C23C2×D69211yescharacteristic
264C23C23×D17311yescharacteristic
265Q8—3464no—
266D4—24no—
267D4—24no—
268D4—3464no—
269D4—3468no—
270D4—3464no—
271D4—3464no—
272D4—3464no—
273D4—3464no—
274D4—13844no—
275D4—13842no—
276D4—6921no—
277D4—6921no—
278D4—6922no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—3468no—
284C2×C4—3464no—
285C2×C4—3462no—
286C2×C4—3462no—
287C2×C4—3462no—
288C2×C4—3462no—
289C2×C4—3461no—
290C2×C4—13841no—
291C2×C4—1734no—
292C2×C4C2×D69212yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—3464no—
303C22—13842no—
304C22—13844no—
305C22—6924no—
306C22—6922no—
307C22—6924no—
308C22—6924no—
309C22—6921no—
310C22#1914964911yescharacteristic
311C22#1914964911yescharacteristic
312C22#1914964912yes—
313C22C2×D4×D17312yes—
314C22C22×D69211yescharacteristic
315C4—22no—
316C4—44no—
317C4—3464no—
318C4—6922no—
319C4—6921no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—13841no—
327C2—6924no—
328C2C2×#1914964911yescharacteristic
subgroupG/H
centerC2C2×#19149649
commutator, cosocleC23×C173C24
FrattiniC23C23×D173
FittingC2×#99×C173C2
socleC2×C173C2×#120
hypercenterC2×#99D173
hypocenterC173#3026
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#19149724C2semidirect
2. #26012053C2semidirect
3. #26012050C2semidirect
4. #26011682C2semidirect
5. #26011679C2semidirect
6. C2×#19149593C2semidirect
7. C2×#99×C173C2semidirect
15. C22×C22⋊D173C22semidirect
18. #19149663C22semidirect
20. #19149660C22semidirect
24. #19149649C22semidirect
26. #19149647C22semidirect
27. C2×C22⋊Dic173C22semidirect
28. #19149593C22semidirect
30. C2×C22⋊C4×C173C22semidirect
31. #99×C173C22semidirect
33. C23×D173D4semidirect
34. C23×D173C23semidirect
54. C2×C22⋊D173D4semidirect
60. C22×Dic173D4semidirect
61. C22×Dic173C23semidirect
67. C22⋊Dic173C23semidirect
71. C24×C173D4semidirect
75. C2×D4×C173D4semidirect
78. C22×C4×C173D4semidirect
80. C22⋊C4×C173C23semidirect
164. C173#3026semidirect
172. C2×#99D173semidirect
194. C2×C22⋊C4C2×D173semidirect
196. #99C2×D173semidirect
200. C24D692semidirect
227. C2×D4D692semidirect
235. C22×C4D692semidirect
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
#260120532
#260120502
#260116822
#260116792
C2×#191495932
C2×#99×C1732
#3026173