Finite Group Databasebeta

← #36362966

Subgroups — #36362966

Order 22144.

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

normal subgroups only

This group has 772 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
1#26012327C211yesmaximal, characteristic, complemented
2#26012060C212yesmaximal, complemented
3#26012055C211yesmaximal, characteristic, complemented
4#26011679C218yesmaximal, complemented
5#26011464C212yesmaximal, complemented
6#410×C173C211yesmaximal, characteristic, complemented
7#19149727—21no—
8#19149726—22no—
9#19149725—22no—
10#19149724—21no—
11#19149724C2214yes—
12C2×#14583826—22no—
13C2×#14583826C2212yes—
14#41×D173—24no—
15C22×D692C2211yescharacteristic, complemented
16#19149676—22no—
17#19149675—21no—
18#19149668—21no—
19#19149667—22no—
20#19149654—24no—
21#19149653—22no—
22#19149649C2218yescomplemented
23#19149647C2214yescomplemented
24#19149638C2211yescharacteristic, complemented
25#19149593C2218yescomplemented
26C2×#41×C173C2211yescharacteristic
27#117×C173C2212yescomplemented
28#99×C173C2214yescomplemented
29C23×D173C2312yes—
30#14583827—24no—
31#14583826—24no—
32#14583826—24no—
33#14583826—22no—
34#14583826—22no—
35#14583826—24no—
36#14583826—22no—
37#14583826—22no—
38#14583825—24no—
39#14583825—28no—
40C2×Dic346—22no—
41C2×Dic346D412yescomplemented
42Q8×D173—24no—
43C2×D692—22no—
44C2×D692—28no—
45C2×D692—21no—
46C2×D692—22no—
47C2×D692—22no—
48C2×D692—21no—
49C2×D692D412yescomplemented
50C2×C22⋊D173—24no—
51C2×C22⋊D173—24no—
52C2×C22⋊D173C2314yes—
53D4×D173—24no—
54D4×D173—28no—
55C2×C4×D173—24no—
56C2×C4×D173D414yescomplemented
57C4⋊Dic173—21no—
58C22⋊Dic173C2312yes—
59#14583803—28no—
60#14583803—24no—
61C4×Dic173—21no—
62#41×C173—22no—
63#41×C173—22no—
64C2×Q8×C173D411yescharacteristic, complemented
65C2×D4×C173D411yescharacteristic, complemented
66C2×D4×C173C2312yes—
67C22×C4×C173D412yescomplemented
68C22×C4×C173C2311yescharacteristic
69C4⋊C4×C173—22no—
70C22⋊C4×C173C2314yes—
71C42×C173—22no—
72C22×D173—24no—
73C22×D173—22no—
74C22×D173—24no—
75C22×D173—82no—
76C22×D173C2×D414yes—
77Dic346—24no—
78Dic346—22no—
79Dic346—22no—
80D692—24no—
81D692—22no—
82D692—22no—
83D692—41no—
84D692—41no—
85D692—42no—
86D692—88no—
87C22⋊D173—28no—
88C22⋊D173—24no—
89C22⋊D173—24no—
90C22⋊D173—84no—
91C2×Dic173—24no—
92C2×Dic173—22no—
93C2×Dic173C2×D414yes—
94C4×D173—28no—
95C4×D173—24no—
96C4×D173—24no—
97C23×C173C2×D412yes—
98C23×C173C2411yescharacteristic
99Q8×C173—22no—
100D4×C173—24no—
101D4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173—24no—
105C2×C4×C173C2×D412yes—
106C2×C4×C173C2×D412yes—
107C2×C4×C173C2×D412yes—
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2×D411yescharacteristic
110C22×C173—22no—
111C22×C173—41no—
112C22×C173—42no—
113C22×C173C22×D412yes—
114C22×C173C22×D411yescharacteristic
115C2×D173—24no—
116C2×D173—44no—
117C2×D173—44no—
118C2×D173—48no—
119C2×D173—42no—
120C2×D173—84no—
121C4×C173—22no—
122C4×C173—44no—
123C4×C173#12011yescharacteristic
124C4×C173#12011yescharacteristic
125C4×C173#12012yes—
126Dic173—24no—
127Dic173—42no—
128C2×C173—21no—
129C2×C173—22no—
130C2×C173—42no—
131C2×C173C2×#12011yescharacteristic
132D173—44no—
133D173—82no—
134C173#302711yescharacteristic, complemented, Sylow
135#3027—1731nomaximal, Sylow
136#585—1731no—
137#535—1731no—
138#535—1732no—
139B(4,𝔽2)—1738no—
140#410—1732no—
141#410D17311yescharacteristic, complemented
1422−1+4—3462no—
1432−1+4—3461no—
144H(5,𝔽2)—3462no—
145H(5,𝔽2)—3461no—
146H(5,𝔽2)—1734no—
147C2×#41—3464no—
148C2×#41—3462no—
149C2×#41—1732no—
150C2×#41C2×D17311yescharacteristic
151C22×D4—1731no—
152#127—3462no—
153#127—3461no—
154#124—3462no—
155#124—3461no—
156#121—3464no—
157#121—3462no—
158#120—1734no—
159#120—1738no—
160#117—1731no—
161#117C2×D17312yescomplemented
162#99—1738no—
163#99C2×D17314yescomplemented
164C24—1732no—
165#41—22no—
166#41—22no—
167#41—3464no—
168#41—3464no—
169#41—3464no—
170#41—3464no—
171#41—3468no—
172#41—3462no—
173#41—3462no—
174#41—3462no—
175#41—3464no—
176#41—3462no—
177C2×Q8—3462no—
178C2×Q8—3464no—
179C2×Q8—1732no—
180C2×Q8D69211yescharacteristic, complemented
181C2×D4—3462no—
182C2×D4—3468no—
183C2×D4—3464no—
184C2×D4—3468no—
185C2×D4—3464no—
186C2×D4—3464no—
187C2×D4—3462no—
188C2×D4—3462no—
189C2×D4—3461no—
190C2×D4—3461no—
191C2×D4—1734no—
192C2×D4—1732no—
193C2×D4D69211yescharacteristic, complemented
194C2×D4C22×D17312yes—
195C22×C4—3464no—
196C22×C4—1734no—
197C22×C4D69212yescomplemented
198C22×C4C22×D17311yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—3461no—
201C22⋊C4—3468no—
202C22⋊C4—3464no—
203C22⋊C4—1732no—
204C22⋊C4C22×D17314yes—
205C42—22no—
206C42—3461no—
207C23—3464no—
208C23—3464no—
209C23—3462no—
210C23—13842no—
211C23—1734no—
212C23C2×D69212yes—
213C23C23×D17311yescharacteristic
214Q8—22no—
215Q8—3464no—
216Q8—3462no—
217Q8—3462no—
218D4—24no—
219D4—22no—
220D4—3464no—
221D4—3468no—
222D4—3464no—
223D4—3464no—
224D4—3462no—
225D4—3462no—
226D4—13848no—
227D4—13844no—
228D4—6921no—
229D4—6921no—
230D4—6922no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—3468no—
235C2×C4—3464no—
236C2×C4—3464no—
237C2×C4—3464no—
238C2×C4—3462no—
239C2×C4—1734no—
240C2×C4D4×D17312yes—
241C2×C4D4×D17312yes—
242C2×C4C2×D69212yes—
243C2×C4C2×D69211yescharacteristic
244C2×C4C2×D69211yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—3464no—
249C22—13844no—
250C22—6928no—
251C22—6924no—
252C22—6924no—
253C22—6922no—
254C22C2×D4×D17312yes—
255C22C22×D69211yescharacteristic
256C4—22no—
257C4—44no—
258C4—3464no—
259C4—6922no—
260C4#1914964911yescharacteristic
261C4#1914964911yescharacteristic
262C4#1914964912yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—13842no—
267C2—6924no—
268C2C2×#1914964911yescharacteristic
subgroupG/H
centerC2C2×#19149649
commutator, cosocleC23×C173C24
FrattiniC23C23×D173
Fitting#410×C173C2
socleC2×C173C2×#120
hypercenter#410D173
hypocenterC173#3027
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012060C2semidirect
3. #26012055C2semidirect
4. #26011679C2semidirect
5. #26011464C2semidirect
6. #410×C173C2semidirect
15. C22×D692C22semidirect
22. #19149649C22semidirect
23. #19149647C22semidirect
24. #19149638C22semidirect
25. #19149593C22semidirect
27. #117×C173C22semidirect
28. #99×C173C22semidirect
41. C2×Dic346D4semidirect
49. C2×D692D4semidirect
56. C2×C4×D173D4semidirect
64. C2×Q8×C173D4semidirect
65. C2×D4×C173D4semidirect
67. C22×C4×C173D4semidirect
134. C173#3027semidirect
141. #410D173semidirect
161. #117C2×D173semidirect
163. #99C2×D173semidirect
180. C2×Q8D692semidirect
193. C2×D4D692semidirect
197. C22×C4D692semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260120602
#260120552
#260116792
#260114642
#410×C1732
#3027173