Finite Group Databasebeta

← #36364168

Subgroups — #36364168

Order 22144.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#26012046C211yesmaximal, characteristic, complemented
2#26012026C211yesmaximal, characteristic
3#26011901C211yesmaximal, characteristic, complemented
4#26011883C211yesmaximal, characteristic, complemented
5#26011851C211yesmaximal, characteristic, complemented
6#26011748C212yesmaximal, complemented
7#26011740C212yesmaximal
8#26011583C212yesmaximal, complemented
9#26011576C212yesmaximal, complemented
10#26011443C211yesmaximal, characteristic, complemented
11#491×C173C211yesmaximal, characteristic, complemented
12C2×#14583816C2212yes—
13C2×#14583813C2212yes—
14C2×#14583809C2212yes—
15#19149679C2211yescharacteristic
16#19149674—21no—
17#19149670—21no—
18#19149667C2211yescharacteristic
19#19149664—21no—
20#19149659—21no—
21#19149659C2211yescharacteristic
22#19149655C2212yes—
23#19149654C2211yescharacteristic
24Q8×Dic173C2212yes—
25#19149644—21no—
26#19149644C2211yescharacteristic
27#19149641—21no—
28#19149641C2211yescharacteristic
29#19149637—21no—
30#19149637—21no—
31C42×D173C2211yescharacteristic
32#19149612C2212yes—
33#19149610C2212yes—
34#19149604C2212yes—
35#19149601C2212yes—
36#19149590C2211yescharacteristic
37C8×Dic173C2212yes—
38C4⋊Q8×C173C2211yescharacteristic
39#125×C173C2211yescharacteristic
40#103×C173C2212yes—
41#103×C173C2212yes—
42#97×C173C2211yescharacteristic
43C2×Dic346—21no—
44C2×Dic346C2311yescharacteristic
45C2×D692—21no—
46C2×D692C2311yescharacteristic
47C2×C4×D173—21no—
48C2×C4×D173D411yescharacteristic
49C2×C4×D173D411yescharacteristic
50C2×C4×D173C2311yescharacteristic
51#14583816—24no—
52#14583813—24no—
53#14583809—24no—
54C2×#13063436C2312yes—
55C4⋊Dic173—21no—
56C4⋊Dic173—22no—
57C4⋊Dic173—21no—
58C4⋊Dic173C2312yes—
59#14583805—22no—
60#14583805—21no—
61#14583803—22no—
62#14583803—21no—
63#14583803—21no—
64#14583803—21no—
65#14583803—21no—
66#14583803—21no—
67C4×Dic173—21no—
68C4×Dic173—22no—
69C4×Dic173D411yescharacteristic
70C4×Dic173D411yescharacteristic
71C4×Dic173C2311yescharacteristic
72C2×Q8×C173C2312yes—
73C2×C8×C173C2312yes—
74C4⋊C4×C173—21no—
75C4⋊C4×C173—21no—
76C4⋊C4×C173—21no—
77C4⋊C4×C173—21no—
78C4⋊C4×C173C2311yescharacteristic
79C4⋊C4×C173C2311yescharacteristic
80C42×C173C2311yescharacteristic
81C22×D173—21no—
82C22×D173C2×D411yescharacteristic
83Dic346—22no—
84Dic346—42no—
85D692—22no—
86D692—42no—
87C2×Dic173—22no—
88C2×Dic173—21no—
89C2×Dic173C2×D411yescharacteristic
90C2×Dic173C2×D411yescharacteristic
91C2×Dic173C2×D411yescharacteristic
92C4×D173—22no—
93C4×D173—22no—
94C4×D173—41no—
95C4×D173—42no—
96#13063436#4114yes—
97Q8×C173—22no—
98Q8×C173—22no—
99C2×C4×C173—21no—
100C2×C4×C173—21no—
101C2×C4×C173—22no—
102C2×C4×C173C2×D411yescharacteristic
103C2×C4×C173C2×D411yescharacteristic
104C2×C4×C173C2411yescharacteristic
105C8×C173—22no—
106C22×C173C22×D411yescharacteristic
107C2×D173—21no—
108C2×D173—21no—
109C2×D173—21no—
110C2×D173—41no—
111C2×D173—41no—
112C2×D173—41no—
113C4×C173—21no—
114C4×C173—21no—
115C4×C173—42no—
116C4×C173—41no—
117C4×C173—41no—
118C4×C173C2×#4112yes—
119Dic173—21no—
120Dic173—21no—
121Dic173—41no—
122Dic173—42no—
123Dic173#13712yes—
124C2×C173#53211yescharacteristic
125C2×C173C2×#13711yescharacteristic
126C2×C173#57511yescharacteristic
127D173—41no—
128D173—81no—
129C173#317211yescharacteristic, complemented, Sylow
130#3172—1731nomaximal, Sylow
131#533—1731no—
132#529—1731no—
133#501—1731no—
134#497—1731no—
135#491D17311yescharacteristic, complemented
136#489—1731no—
137#470—1732no—
138#468—1732no—
139C4×Q16—1732no—
140C4×SD16—1732no—
141C4×M16—1731no—
142C2×Q16—1732no—
143C2×SD16—1732no—
144C2×M16—1732no—
145C4⋊Q8—1731no—
146C4⋊Q8C2×D17311yescharacteristic
147#126—3461no—
148#125—3461no—
149#125C2×D17311yescharacteristic
150#124—1731no—
151#123—3461no—
152C22⋊Q8—3461no—
153C22⋊Q8—1732no—
154C22⋊Q8—1731no—
155#121—1731no—
156C4×Q8—3461no—
157C4×Q8—1732no—
158C4×Q8—1731no—
159C4×D4—3461no—
160C4×D4—1731no—
161#117—3461no—
162#117—3461no—
163C2×C42—1731no—
164#107—1732no—
165#106—1732no—
166#103—1732no—
167#103C2×D17312yes—
168#103C2×D17312yes—
169#102—1732no—
170#97—1731no—
171#97C2×D17311yescharacteristic
172C4×C8—1732no—
173C2×Q8—3461no—
174C2×Q8—1731no—
175C2×Q8C22×D17312yes—
176C2×D4—3461no—
177C2×D4—1731no—
178C22×C4—3461no—
179C22×C4—1731no—
180C22×C4—1731no—
181C22×C4—1731no—
182Q16—3464no—
183SD16—3464no—
184M16—3464no—
185C2×C8—1732no—
186C2×C8C22×D17312yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—3461no—
192C4⋊C4—3462no—
193C4⋊C4—3462no—
194C4⋊C4—3461no—
195C4⋊C4—3461no—
196C4⋊C4—1732no—
197C4⋊C4C22×D17311yescharacteristic
198C4⋊C4C22×D17311yescharacteristic
199C22⋊C4—3462no—
200C22⋊C4—3461no—
201C22⋊C4—3461no—
202C22⋊C4—3461no—
203C22⋊C4—3461no—
204C22⋊C4—3461no—
205C42—3461no—
206C42—3462no—
207C42—1731no—
208C42—1731no—
209C42—1731no—
210C42C22×D17311yescharacteristic
211C23—3461no—
212C23—1731no—
213Q8—22no—
214Q8—22no—
215Q8—3462no—
216Q8—6922no—
217D4—3462no—
218D4—6922no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—3462no—
223C2×C4—3462no—
224C2×C4—3461no—
225C2×C4—3462no—
226C2×C4—1731no—
227C2×C4—1731no—
228C2×C4—1731no—
229C2×C4—6921no—
230C2×C4—6922no—
231C2×C4D4×D17311yescharacteristic
232C2×C4D4×D17311yescharacteristic
233C2×C4C23×D17311yescharacteristic
234C8—22no—
235C8—1734no—
236C22—3461no—
237C22—3461no—
238C22—3461no—
239C22—6921no—
240C22—6921no—
241C22—6921no—
242C22C2×D4×D17311yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—3461no—
249C4—3461no—
250C4—1732no—
251C4—6922no—
252C4—6921no—
253C4#41×D17312yes—
254C2—13841no—
255C2—6921no—
256C2#2601204011yescharacteristic
257C2#137×D17311yescharacteristic
258C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#491×C173C2
socleC22×C173C22×D4
prime radical#19149654C22
hypercenter#491D173
hypocenterC173#3172
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #26012046C2semidirect
3. #26011901C2semidirect
4. #26011883C2semidirect
5. #26011851C2semidirect
6. #26011748C2semidirect
8. #26011583C2semidirect
9. #26011576C2semidirect
10. #26011443C2semidirect
11. #491×C173C2semidirect
129. C173#3172semidirect
135. #491D173semidirect

Maximal subgroups

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

subgroupindex
#260120462
#260120262
#260119012
#260118832
#260118512
#260117482
#260117402
#260115832
#260115762
#260114432
#491×C1732
#3172173