Finite Group Databasebeta

← #36370163

Subgroups — #36370163

Order 22144.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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#26012558C212yesmaximal, complemented
2#26012472C212yesmaximal, complemented
3#26012470C212yesmaximal, complemented
4C2×#19149738C214yesmaximal, complemented
5D173⋊#99C214yesmaximal, complemented
6C2×#19149725C211yesmaximal, characteristic
7D4×#10621924—28no—
8#19149758C2214yescomplemented
9C2×C22⋊#10621924C2218yescomplemented
10#19149738C2218yescomplemented
11#19149738C2214yescomplemented
12#19149737C2214yescomplemented
13#19149725—28no—
14C2×#14583827C411yescharacteristic, complemented
15C2×#14583827C2211yescharacteristic
16C2×#14583826C412yescomplemented
17C2×#14583826C2211yescharacteristic
18C22×D692C411yescharacteristic, complemented
19C22×D692C2212yes—
20C2×D4×D173C412yescomplemented
21C2×D4×D173C411yescharacteristic, complemented
22C2×D4×D173C2212yes—
23C2×D4×D173C2211yescharacteristic
24C2×#41×C173C411yescharacteristic, complemented
25C22×#10621924—28no—
26C4⋊#10621924—24no—
27C22⋊#10621924—28no—
28C22⋊#10621924C2318yes—
29C4×#10621924—24no—
30C23×D173C2×C412yescomplemented
31C23×D173C2×C412yescomplemented
32C23×D173C2312yes—
33#14583827—24no—
34#14583827—24no—
35#14583826—28no—
36#14583826—24no—
37C2×Dic346C2×C411yescharacteristic, complemented
38C2×D692—28no—
39C2×D692—24no—
40C2×D692C2×C411yescharacteristic
41C2×D692C2×C411yescharacteristic, complemented
42C2×D692C2×C412yescomplemented
43C2×D692C2×C412yescomplemented
44C2×D692C2312yes—
45C2×D692C2311yescharacteristic
46C2×C22⋊D173C2×C412yes—
47C2×C22⋊D173C2×C411yescharacteristic, complemented
48C2×C22⋊D173C2×C412yescomplemented
49C2×C22⋊D173C2311yescharacteristic
50D4×D173—28no—
51D4×D173—28no—
52D4×D173—24no—
53D4×D173D418yes—
54C2×C4×D173C2×C412yescomplemented
55C2×C4×D173C2×C411yescharacteristic, complemented
56C2×C4×D173C2×C412yescomplemented
57C2×C4×D173C2311yescharacteristic
58#41×C173—24no—
59C2×Q8×C173C2×C411yescharacteristic, complemented
60C2×D4×C173C2×C411yescharacteristic
61C2×D4×C173C2×C412yescomplemented
62C22×C4×C173C2×C411yescharacteristic, complemented
63C22×C4×C173C2×C412yescomplemented
64C2×#10621924—28no—
65C2×#10621924—88no—
66C22×D173—24no—
67C22×D173—24no—
68C22×D173—82no—
69C22×D173—82no—
70C22×D173—82no—
71C22×D173C22×C412yes—
72C22×D173C22×C411yescharacteristic
73C22×D173C22×C412yes—
74C22×D173C2×D418yes—
75C22×D173C2411yescharacteristic
76Dic346—22no—
77D692—24no—
78D692—24no—
79D692—24no—
80D692—22no—
81D692—22no—
82D692C22⋊C414yes—
83C22⋊D173—24no—
84C22⋊D173—22no—
85C22⋊D173—22no—
86C22⋊D173C22⋊C418yes—
87C2×Dic173C22×C411yescharacteristic
88C2×Dic173C22×C411yescharacteristic
89C4×D173—24no—
90C4×D173—22no—
91C4×D173—24no—
92C4×D173C2×D414yes—
93C23×C173C22×C411yescharacteristic
94C23×C173C22×C412yes—
95Q8×C173—22no—
96D4×C173—24no—
97D4×C173C22⋊C414yes—
98C2×C4×C173—24no—
99C2×C4×C173—22no—
100C2×C4×C173C22×C412yes—
101C2×C4×C173C22×C411yescharacteristic
102C2×C4×C173C22×C411yescharacteristic
103C22×C173—21no—
104C22×C173—42no—
105C22×C173—41no—
106C22×C173C2×C22⋊C414yes—
107C22×C173C23×C411yescharacteristic
108C2×D173—21no—
109C2×D173—22no—
110C2×D173—42no—
111C2×D173—41no—
112C2×D173—48no—
113C2×D173—42no—
114C2×D173—41no—
115C2×D173—84no—
116C2×D173—84no—
117C2×D173C2×C22⋊C414yes—
118C2×D173C22×D412yes—
119#10621924—48no—
120C4×C173—21no—
121C4×C173—22no—
122C4×C173C2×C22⋊C412yes—
123Dic173—21no—
124Dic173C2×C22⋊C412yes—
125C2×C173—21no—
126C2×C173—24no—
127C2×C173—41no—
128C2×C173C22×C22⋊C411yescharacteristic
129D173—22no—
130D173—24no—
131D173—42no—
132D173—41no—
133C173#288311yescharacteristic, complemented, Sylow
134#2883—1731nomaximal, Sylow
135C2×H(5,𝔽2)—1731no—
136#518—1732no—
137#410—1732no—
138#410—1732no—
139C2×#99—1734no—
140C2×#99—1734no—
141H(5,𝔽2)—3468no—
142C2×#41—1731no—
143C2×#41—1731no—
144C2×#41—1731no—
145C2×#41—1732no—
146C2×#41#1062192411yescharacteristic, complemented
147C22×D4—1731no—
148C22×D4—1732no—
149C22×D4—1732no—
150C22×D4—1731no—
151C22×D4—1731no—
152C22×D4—1732no—
153C4×D4—3468no—
154#117—1734no—
155C2×C22⋊C4—1738no—
156#99—1734no—
157#99—1738no—
158#99—1734no—
159C24—1732no—
160C24—1732no—
161C24—1732no—
162#41—24no—
163#41—3468no—
164#41—3464no—
165#41—3464no—
166#41—3464no—
167C2×Q8—1731no—
168C2×Q8C2×#1062192411yescharacteristic, complemented
169C2×D4—3468no—
170C2×D4—3464no—
171C2×D4—3468no—
172C2×D4—3468no—
173C2×D4—3464no—
174C2×D4—1732no—
175C2×D4—1731no—
176C2×D4—1731no—
177C2×D4—1732no—
178C2×D4—1738no—
179C2×D4—1732no—
180C2×D4—1732no—
181C2×D4—1731no—
182C2×D4—1732no—
183C2×D4—1731no—
184C2×D4—1731no—
185C2×D4C2×#1062192411yescharacteristic
186C2×D4C2×#1062192412yescomplemented
187C22×C4—3468no—
188C22×C4—1732no—
189C22×C4—1732no—
190C22×C4—1731no—
191C22×C4—1731no—
192C22×C4C2×#1062192412yescomplemented
193C22×C4C2×#1062192411yescharacteristic, complemented
194C4⋊C4—3464no—
195C22⋊C4—3468no—
196C22⋊C4—1738no—
197C42—3464no—
198C23—3464no—
199C23—3464no—
200C23—13842no—
201C23—13842no—
202C23—13842no—
203C23—1732no—
204C23—1731no—
205C23—1738no—
206C23—1732no—
207C23—1731no—
208C23C22×#1062192411yescharacteristic
209C23C22×#1062192412yes—
210Q8—22no—
211Q8—3462no—
212D4—24no—
213D4—3462no—
214D4—3464no—
215D4—3464no—
216D4—3464no—
217D4—3462no—
218D4—3464no—
219D4—3462no—
220D4—3462no—
221D4—1738no—
222D4—1734no—
223D4C22⋊#1062192414yes—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—3464no—
227C2×C4—3464no—
228C2×C4—3462no—
229C2×C4—3468no—
230C2×C4—13848no—
231C2×C4—1731no—
232C2×C4—1731no—
233C2×C4—1734no—
234C2×C4C22×#1062192412yes—
235C2×C4C22×#1062192411yescharacteristic
236C2×C4C22×#1062192411yescharacteristic
237C22—21no—
238C22—42no—
239C22—41no—
240C22—3462no—
241C22—3461no—
242C22—13844no—
243C22—13844no—
244C22—1732no—
245C22—1734no—
246C22—6928no—
247C22—6921no—
248C22—6922no—
249C22—6921no—
250C22—6922no—
251C22C2×C22⋊#1062192414yes—
252C22C23×#1062192411yescharacteristic
253C4—21no—
254C4—22no—
255C4—3461no—
256C4—1732no—
257C4—6928no—
258C4C2×C22⋊#1062192412yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—3462no—
263C2—3464no—
264C2—6921no—
265C2—6922no—
266C2C22×C22⋊#1062192411yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10621924
commutatorC22×C173C23×C4
FrattiniC22C23×#10621924
FittingC2×#41×C173C4
socleC2×C173C22×C22⋊C4
prime radicalC2×#19149725C2
cosocleC22×D173C24
hypercenterC2×#41#10621924
hypocenterC173#2883
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26012558C2semidirect
2. #26012472C2semidirect
3. #26012470C2semidirect
4. C2×#19149738C2semidirect
5. D173⋊#99C2semidirect
8. #19149758C22semidirect
9. C2×C22⋊#10621924C22semidirect
10. #19149738C22semidirect
11. #19149738C22semidirect
12. #19149737C22semidirect
14. C2×#14583827C4semidirect
16. C2×#14583826C4semidirect
18. C22×D692C4semidirect
20. C2×D4×D173C4semidirect
21. C2×D4×D173C4semidirect
24. C2×#41×C173C4semidirect
30. C23×D173C2×C4semidirect
31. C23×D173C2×C4semidirect
37. C2×Dic346C2×C4semidirect
41. C2×D692C2×C4semidirect
42. C2×D692C2×C4semidirect
43. C2×D692C2×C4semidirect
47. C2×C22⋊D173C2×C4semidirect
48. C2×C22⋊D173C2×C4semidirect
54. C2×C4×D173C2×C4semidirect
55. C2×C4×D173C2×C4semidirect
56. C2×C4×D173C2×C4semidirect
59. C2×Q8×C173C2×C4semidirect
61. C2×D4×C173C2×C4semidirect
62. C22×C4×C173C2×C4semidirect
63. C22×C4×C173C2×C4semidirect
133. C173#2883semidirect
146. C2×#41#10621924semidirect
168. C2×Q8C2×#10621924semidirect
186. C2×D4C2×#10621924semidirect
192. C22×C4C2×#10621924semidirect
193. C22×C4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125582
#260124722
#260124702
C2×#191497382
D173⋊#992
C2×#191497252
#2883173