Finite Group Databasebeta

← #36352067

Subgroups — #36352067

Order 22144.

334 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 85 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#26011455C211yesmaximal, characteristic, complemented
2C2×#19149586C211yesmaximal, characteristic, complemented
3#26011228C212yesmaximal, complemented
4#26011225C212yesmaximal, complemented
5#525×C173C211yesmaximal, characteristic
6C23×Dic173—21no—
7C2×#14583810—21no—
8C22⋊#13063436—21no—
9C22⋊#13063436C2212yes—
10#19149586—21no—
11#19149586C2212yes—
12C2×#41×C173C411yescharacteristic, complemented
13C23×C4×C173C2211yescharacteristic
14#123×C173—21no—
15C22⋊Q8×C173C411yescharacteristic, complemented
16C22⋊Q8×C173C2211yescharacteristic
17#121×C173C411yescharacteristic, complemented
18#121×C173C2211yescharacteristic
19#120×C173—21no—
20C4×D4×C173—21no—
21C4×D4×C173—21no—
22#117×C173C411yescharacteristic, complemented
23C22×Dic173—24no—
24C22×Dic173—22no—
25C22×Dic173—42no—
26C22×Dic173—42no—
27#14583810—42no—
28C2×#13063436—22no—
29C24×C173D411yescharacteristic
30#41×C173—42no—
31C2×Q8×C173C2×C411yescharacteristic, complemented
32C2×D4×C173—21no—
33C2×D4×C173—21no—
34C2×D4×C173—21no—
35C2×D4×C173C2×C411yescharacteristic, complemented
36C22×C4×C173—21no—
37C22×C4×C173—42no—
38C22×C4×C173C2×C411yescharacteristic
39C22×C4×C173C2×C411yescharacteristic
40C22×C4×C173D411yescharacteristic
41C22×C4×C173D411yescharacteristic
42C22×C4×C173D411yescharacteristic
43C22×C4×C173C2311yescharacteristic
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173C2×C411yescharacteristic, complemented
47C4⋊C4×C173C2×C411yescharacteristic, complemented
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C22⋊C4×C173—21no—
51C22⋊C4×C173—21no—
52C22⋊C4×C173—21no—
53C42×C173—21no—
54C2×Dic173—24no—
55C2×Dic173—48no—
56C2×Dic173—42no—
57C2×Dic173—42no—
58#13063436—42no—
59C23×C173—22no—
60C23×C173—22no—
61C23×C173—21no—
62C23×C173—21no—
63C23×C173—21no—
64C23×C173C22⋊C411yescharacteristic
65C23×C173C22⋊C411yescharacteristic
66C23×C173C2×D411yescharacteristic
67Q8×C173—41no—
68D4×C173—41no—
69D4×C173—42no—
70D4×C173—42no—
71D4×C173—41no—
72D4×C173—41no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—21no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173—42no—
83C2×C4×C173C22⋊C411yescharacteristic
84C2×C4×C173C22⋊C411yescharacteristic
85C2×C4×C173C22×C411yescharacteristic
86C2×C4×C173C2×D411yescharacteristic
87C22×C173—24no—
88C22×C173—24no—
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173#9912yes—
99C22×C173#10412yes—
100C22×C173C2×C22⋊C411yescharacteristic
101C22×C173AGL(1,ℤ/8)11yescharacteristic
102C22×C173#13711yescharacteristic
103C4×C173—21no—
104C4×C173—21no—
105C4×C173—21no—
106C4×C173—21no—
107C4×C173—41no—
108C4×C173—41no—
109C4×C173—41no—
110Dic173—44no—
111C2×C173—21no—
112C2×C173—22no—
113C2×C173—22no—
114C2×C173—21no—
115C2×C173—81no—
116C2×C173C2×#9911yescharacteristic
117C2×C173#41711yescharacteristic
118C2×C173C2×#10411yescharacteristic
119C173#150411yescharacteristic, complemented, Sylow
120#1504—1731nomaximal, Sylow
121#525D17311yescharacteristic
122#407—1731no—
123C2×#95—1731no—
124#328—1732no—
125#327—1732no—
126C2×#41Dic17311yescharacteristic, complemented
127C23×C4—3461no—
128C23×C4C2×D17311yescharacteristic
129C2×M16—3461no—
130#123—21no—
131C22⋊Q8Dic17311yescharacteristic, complemented
132C22⋊Q8C2×D17311yescharacteristic
133#121Dic17311yescharacteristic, complemented
134#121C2×D17311yescharacteristic
135#120—21no—
136C4×D4—21no—
137C4×D4—21no—
138#117Dic17311yescharacteristic, complemented
139C22⋊C8—3461no—
140C22⋊C8—1732no—
141#95—3461no—
142#95—1732no—
143C24C22⋊D17311yescharacteristic
144#41—42no—
145C2×Q8C2×Dic17311yescharacteristic, complemented
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C2×Dic17311yescharacteristic, complemented
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—3464no—
153C22×C4—3462no—
154C22×C4—6922no—
155C22×C4—6922no—
156C22×C4C2×Dic17311yescharacteristic
157C22×C4C2×Dic17311yescharacteristic
158C22×C4C22⋊D17311yescharacteristic
159C22×C4C22⋊D17311yescharacteristic
160C22×C4C22⋊D17311yescharacteristic
161C22×C4C22×D17311yescharacteristic
162M16—6922no—
163C2×C8—3462no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4C2×Dic17311yescharacteristic, complemented
167C4⋊C4C2×Dic17311yescharacteristic, complemented
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C42—21no—
174C23—22no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23C22⋊Dic17311yescharacteristic
180C23C22⋊Dic17311yescharacteristic
181C23C2×C22⋊D17311yescharacteristic
182Q8—41no—
183D4—41no—
184D4—42no—
185D4—42no—
186D4—41no—
187D4—41no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—3464no—
199C2×C4—6928no—
200C2×C4—6922no—
201C2×C4—6922no—
202C2×C4C22⋊Dic17311yescharacteristic
203C2×C4C22⋊Dic17311yescharacteristic
204C2×C4C22×Dic17311yescharacteristic
205C2×C4C2×C22⋊D17311yescharacteristic
206C8—6922no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22#1914959412yes—
219C22#1914960612yes—
220C22C2×C22⋊Dic17311yescharacteristic
221C22#1914970411yescharacteristic
222C22#1914970911yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—6924no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2—21no—
235C2—81no—
236C2C2×#1914959411yescharacteristic
237C2#2601149211yescharacteristic
238C2C2×#1914960611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radical#525×C173C2
socleC22×C173C2×C22⋊C4
cosocleC22×C4×C173C23
hypercenter#525D173
hypocenterC173#1504
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #26011455C2semidirect
2. C2×#19149586C2semidirect
3. #26011228C2semidirect
4. #26011225C2semidirect
12. C2×#41×C173C4semidirect
15. C22⋊Q8×C173C4semidirect
17. #121×C173C4semidirect
22. #117×C173C4semidirect
31. C2×Q8×C173C2×C4semidirect
35. C2×D4×C173C2×C4semidirect
46. C4⋊C4×C173C2×C4semidirect
47. C4⋊C4×C173C2×C4semidirect
119. C173#1504semidirect
126. C2×#41Dic173semidirect
131. C22⋊Q8Dic173semidirect
133. #121Dic173semidirect
138. #117Dic173semidirect
145. C2×Q8C2×Dic173semidirect
149. C2×D4C2×Dic173semidirect
166. C4⋊C4C2×Dic173semidirect
167. C4⋊C4C2×Dic173semidirect

Maximal subgroups

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

subgroupindex
#260114552
C2×#191495862
#260112282
#260112252
#525×C1732
#1504173