Finite Group Databasebeta

← #36362934

Subgroups — #36362934

Order 22144.

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

normal subgroups only

This group has 736 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
2C2×#19149705C212yesmaximal, complemented
3#26011663C214yesmaximal, complemented
4#26011658C214yesmaximal, complemented
5#26011509C212yesmaximal, complemented
6#26011475C211yesmaximal, characteristic, complemented
7#535×C173C211yesmaximal, characteristic, complemented
8#19149727C2212yescomplemented
9#19149726—22no—
10#19149725—22no—
11#19149724—22no—
12#19149724C2212yescomplemented
13C2×#14583826—22no—
14C2×#14583826C2212yes—
15#41×D173—24no—
16#19149705—24no—
17#19149705C2218yescomplemented
18C2×#14583814—22no—
19C2×#14583811—22no—
20C2×#14583810C2212yescomplemented
21#19149607C2218yescomplemented
22#19149598C2212yescomplemented
23#19149596C2212yescomplemented
24C2×#41×C173C2211yescharacteristic
25C22×D4×C173C2211yescharacteristic, complemented
26#127×C173C2212yescomplemented
27#124×C173C2212yescomplemented
28#121×C173—22no—
29#120×C173—22no—
30#117×C173C2211yescharacteristic, complemented
31#14583827—24no—
32#14583826—24no—
33#14583826—24no—
34#14583826—22no—
35#14583826—24no—
36#14583826—22no—
37#14583826D414yescomplemented
38#14583826C2314yes—
39#14583825—28no—
40#14583825—24no—
41C2×Dic346—22no—
42C2×Dic346D412yescomplemented
43Q8×D173—24no—
44C2×D692—22no—
45C2×D692D412yescomplemented
46C2×C22⋊D173—24no—
47C2×C22⋊D173—22no—
48D4×D173—28no—
49D4×D173—24no—
50C2×C4×D173—24no—
51C2×C4×D173—22no—
52#14583814—28no—
53#14583811—28no—
54#14583810—22no—
55#14583810C2314yes—
56C2×#13063436—22no—
57C24×C173—21no—
58#41×C173—22no—
59#41×C173—22no—
60C2×Q8×C173C2311yescharacteristic
61C2×D4×C173—24no—
62C2×D4×C173—22no—
63C2×D4×C173—22no—
64C2×D4×C173—21no—
65C2×D4×C173D412yes—
66C2×D4×C173C2312yes—
67C2×D4×C173C2311yescharacteristic
68C22×C4×C173D412yes—
69C22×C4×C173C2311yescharacteristic
70C4⋊C4×C173—21no—
71C22⋊C4×C173—21no—
72C22⋊C4×C173—24no—
73C42×C173C2312yes—
74C22×D173—24no—
75C22×D173—22no—
76Dic346—24no—
77Dic346—22no—
78Dic346C2×D414yes—
79D692—24no—
80D692—22no—
81D692C2×D414yes—
82C22⋊D173—28no—
83C22⋊D173—24no—
84C22⋊D173—24no—
85C2×Dic173—24no—
86C2×Dic173—22no—
87C4×D173—28no—
88C4×D173—24no—
89C4×D173—24no—
90#13063436—24no—
91C23×C173—22no—
92C23×C173—42no—
93C23×C173—82no—
94C23×C173C2×D412yes—
95C23×C173C2×D411yescharacteristic
96Q8×C173—22no—
97D4×C173—24no—
98D4×C173—22no—
99D4×C173—24no—
100D4×C173—41no—
101D4×C173—41no—
102D4×C173—84no—
103C2×C4×C173—22no—
104C2×C4×C173—22no—
105C2×C4×C173—22no—
106C2×C4×C173C2×D412yes—
107C2×C4×C173C2×D412yes—
108C2×C4×C173C2×D412yes—
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2411yescharacteristic
111C22×C173—22no—
112C22×C173—41no—
113C22×C173—42no—
114C22×C173—44no—
115C22×C173—42no—
116C22×C173—84no—
117C22×C173#12012yes—
118C22×C173C22×D411yescharacteristic
119C2×D173—24no—
120C2×D173—82no—
121C2×D173—84no—
122C4×C173—22no—
123C4×C173—42no—
124C4×C173#12011yescharacteristic
125C4×C173#12011yescharacteristic
126C4×C173C22×D412yes—
127Dic173—24no—
128C2×C173—21no—
129C2×C173—22no—
130C2×C173—42no—
131C2×C173—82no—
132C2×C173C2×#12011yescharacteristic
133D173—44no—
134C173#302111yescharacteristic, complemented, Sylow
135#3021—1731nomaximal, Sylow
136#585—1731no—
137C2×AGL(1,ℤ/8)—1732no—
138#535D17311yescharacteristic, complemented
139#454—1734no—
140#453—1734no—
141#421—1732no—
142#413—1731no—
1432−1+4—3462no—
1442−1+4—1732no—
145H(5,𝔽2)—3462no—
146H(5,𝔽2)—3462no—
147H(5,𝔽2)—1732no—
148C2×#41—3464no—
149C2×#41—3462no—
150C2×#41—1732no—
151C2×#41C2×D17311yescharacteristic
152C22×D4C2×D17311yescharacteristic, complemented
153AGL(1,ℤ/8)—3464no—
154AGL(1,ℤ/8)—1738no—
155C2×SD16—3462no—
156C2×D8—3462no—
157C2×M16—1732no—
158#127C2×D17312yescomplemented
159#124C2×D17312yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D17311yescharacteristic, complemented
163#104—1738no—
164#101—1732no—
165#100—1732no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—3464no—
170#41—3464no—
171#41—3468no—
172#41—3464no—
173#41—3464no—
174#41—3462no—
175#41—3462no—
176#41—3464no—
177#41—1734no—
178#41—1734no—
179C2×Q8—3462no—
180C2×Q8—3464no—
181C2×Q8—1732no—
182C2×Q8C22×D17311yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—3462no—
188C2×D4—3468no—
189C2×D4—3464no—
190C2×D4—3464no—
191C2×D4—3462no—
192C2×D4—1732no—
193C2×D4C22⋊D17312yes—
194C2×D4C22×D17312yes—
195C2×D4C22×D17311yescharacteristic
196C22×C4—3464no—
197C22×C4—3462no—
198C22×C4C22⋊D17312yes—
199C22×C4C22×D17311yescharacteristic
200SD16—3468no—
201D8—3468no—
202M16—3462no—
203M16—1734no—
204C2×C8—3462no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D17312yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—3464no—
213C23—3462no—
214C23C2×C22⋊D17312yes—
215C23C2×C22⋊D17311yescharacteristic
216Q8—22no—
217Q8—3464no—
218Q8—3462no—
219Q8—1734no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—3464no—
227D4—3468no—
228D4—3464no—
229D4—3464no—
230D4—3462no—
231D4—1734no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—3468no—
236C2×C4—3464no—
237C2×C4—3462no—
238C2×C4—3464no—
239C2×C4—3464no—
240C2×C4D4×D17312yes—
241C2×C4D4×D17312yes—
242C2×C4C2×C22⋊D17312yes—
243C2×C4C2×C22⋊D17311yescharacteristic
244C2×C4C23×D17311yescharacteristic
245C8—3464no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—3464no—
253C22—13842no—
254C22—13844no—
255C22#1914964712yes—
256C22C22×C22⋊D17311yescharacteristic
257C4—22no—
258C4—42no—
259C4—3464no—
260C4#1914964711yescharacteristic
261C4#1914964711yescharacteristic
262C4C2×D4×D17312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—6924no—
268C2C2×#1914964711yescharacteristic
subgroupG/H
centerC2C2×#19149647
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#535×C173C2
socleC2×C173C2×#120
hypercenter#535D173
hypocenterC173#3021
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. C2×#19149705C2semidirect
3. #26011663C2semidirect
4. #26011658C2semidirect
5. #26011509C2semidirect
6. #26011475C2semidirect
7. #535×C173C2semidirect
8. #19149727C22semidirect
12. #19149724C22semidirect
17. #19149705C22semidirect
20. C2×#14583810C22semidirect
21. #19149607C22semidirect
22. #19149598C22semidirect
23. #19149596C22semidirect
25. C22×D4×C173C22semidirect
26. #127×C173C22semidirect
27. #124×C173C22semidirect
30. #117×C173C22semidirect
37. #14583826D4semidirect
42. C2×Dic346D4semidirect
45. C2×D692D4semidirect
134. C173#3021semidirect
138. #535D173semidirect
152. C22×D4C2×D173semidirect
158. #127C2×D173semidirect
159. #124C2×D173semidirect
162. #117C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
C2×#191497052
#260116632
#260116582
#260115092
#260114752
#535×C1732
#3021173