Finite Group Databasebeta

← #36352064

Subgroups — #36352064

Order 22144.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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#26012001C211yesmaximal, characteristic
2#26011453C211yesmaximal, characteristic, complemented
3#95×D173C211yesmaximal, characteristic, complemented
4#26011229C211yesmaximal, characteristic, complemented
5#26011227C211yesmaximal, characteristic, complemented
6#26011225C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic, complemented
8C2×#14583825C411yescharacteristic, complemented
9C22×C4×D173—21no—
10C22×C4×D173C2211yescharacteristic
11C2×#14583809—21no—
12#19149661—21no—
13#19149659C411yescharacteristic, complemented
14#19149658C2211yescharacteristic
15#19149650C411yescharacteristic, complemented
16#19149647—21no—
17D4×Dic173—21no—
18#19149639—21no—
19#19149637C411yescharacteristic, complemented
20C22⋊#13063436C2211yescharacteristic
21#19149591—21no—
22#19149586—21no—
23#19149586C2211yescharacteristic
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28#14583825—42no—
29C2×Dic346C2×C411yescharacteristic, complemented
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C22×Dic173—21no—
34C22×Dic173D411yescharacteristic
35C2×C4×D173—24no—
36C2×C4×D173—42no—
37C2×C4×D173—42no—
38C2×C4×D173—42no—
39C2×C4×D173C2×C411yescharacteristic
40C2×C4×D173C2×C411yescharacteristic
41C2×C4×D173D411yescharacteristic
42C2×C4×D173D411yescharacteristic
43#14583809—42no—
44C2×#13063436—21no—
45C4⋊Dic173—21no—
46C4⋊Dic173C2×C411yescharacteristic, complemented
47#14583805—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50#14583803—21no—
51#14583803—21no—
52C4×Dic173—21no—
53C2×D4×C173—21no—
54C2×D4×C173C2×C411yescharacteristic, complemented
55C22×C4×C173—21no—
56C22×C4×C173C2311yescharacteristic
57C2×C8×C173—21no—
58C4⋊C4×C173C2×C411yescharacteristic, complemented
59C22⋊C4×C173—21no—
60C22×D173—21no—
61C22×D173—21no—
62C22×D173—22no—
63C22×D173—22no—
64C22×D173C22⋊C411yescharacteristic
65C22×D173C22⋊C411yescharacteristic
66Dic346—41no—
67C22⋊D173—42no—
68C22⋊D173—42no—
69C2×Dic173—22no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—41no—
73C2×Dic173—42no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76C2×Dic173C22⋊C411yescharacteristic
77C2×Dic173C22⋊C411yescharacteristic
78C4×D173—22no—
79C4×D173—22no—
80C4×D173—41no—
81C4×D173—41no—
82C4×D173—48no—
83#13063436—41no—
84C23×C173—21no—
85C23×C173C2×D411yescharacteristic
86D4×C173—41no—
87D4×C173—41no—
88D4×C173—41no—
89C2×C4×C173—22no—
90C2×C4×C173—21no—
91C2×C4×C173—41no—
92C2×C4×C173—41no—
93C2×C4×C173—41no—
94C2×C4×C173C22×C411yescharacteristic
95C2×C4×C173C2×D411yescharacteristic
96C8×C173—41no—
97C22×C173—21no—
98C22×C173—21no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—41no—
102C22×C173C2×C22⋊C411yescharacteristic
103C22×C173AGL(1,ℤ/8)11yescharacteristic
104C22×C173#13711yescharacteristic
105C2×D173—24no—
106C2×D173—24no—
107C2×D173—21no—
108C2×D173—21no—
109C2×D173—21no—
110C2×D173—21no—
111C2×D173#9912yes—
112C2×D173#10412yes—
113C4×C173—21no—
114C4×C173—21no—
115C4×C173—42no—
116C4×C173—41no—
117Dic173—21no—
118Dic173—21no—
119Dic173—42no—
120Dic173—41no—
121Dic173—41no—
122C2×C173—21no—
123C2×C173—21no—
124C2×C173—81no—
125C2×C173C2×#9911yescharacteristic
126C2×C173#41711yescharacteristic
127C2×C173C2×#10411yescharacteristic
128D173—22no—
129D173—22no—
130C173#150411yescharacteristic, complemented, Sylow
131#1504—1731nomaximal, Sylow
132#525—1731no—
133#407—1731no—
134C2×#95—1731no—
135#328—1731no—
136#328—1731no—
137#327—1731no—
138#327D17311yescharacteristic, complemented
139C2×#41—1731no—
140C23×C4—3461no—
141C23×C4—1731no—
142C2×M16—3461no—
143#123—3461no—
144C22⋊Q8—1731no—
145C22⋊Q8—1731no—
146#121—1731no—
147#121C2×D17311yescharacteristic
148#120—3461no—
149C4×D4—3461no—
150C4×D4—3461no—
151#117—1731no—
152C22⋊C8—3461no—
153C22⋊C8—1731no—
154C22⋊C8C2×D17311yescharacteristic
155#95—3461no—
156#95—1731no—
157#95C2×D17311yescharacteristic
158C24—1731no—
159#41—6922no—
160C2×Q8—1731no—
161C2×D4—21no—
162C2×D4—3461no—
163C2×D4—3461no—
164C2×D4C4×D17311yescharacteristic, complemented
165C22×C4—21no—
166C22×C4—3464no—
167C22×C4—3461no—
168C22×C4—3461no—
169C22×C4—1731no—
170C22×C4—1731no—
171C22×C4—1731no—
172C22×C4—1731no—
173C22×C4—1731no—
174C22×C4—6922no—
175C22×C4—6922no—
176C22×C4—6922no—
177C22×C4C22×D17311yescharacteristic
178M16—6922no—
179C2×C8—21no—
180C2×C8—3461no—
181C4⋊C4—3461no—
182C4⋊C4—3461no—
183C4⋊C4—1731no—
184C4⋊C4C4×D17311yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3461no—
187C22⋊C4—3461no—
188C22⋊C4—3461no—
189C22⋊C4—3461no—
190C42—3461no—
191C23—21no—
192C23—3462no—
193C23—3462no—
194C23—3461no—
195C23—3461no—
196C23—1731no—
197C23—1731no—
198C23D4×D17311yescharacteristic
199Q8—6921no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—6922no—
204D4—6922no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—3462no—
211C2×C4—3462no—
212C2×C4—3462no—
213C2×C4—3461no—
214C2×C4—3461no—
215C2×C4—1731no—
216C2×C4—1731no—
217C2×C4—6921no—
218C2×C4—6921no—
219C2×C4—6928no—
220C2×C4—6921no—
221C2×C4—6922no—
222C2×C4—6921no—
223C2×C4—6921no—
224C2×C4C2×C4×D17311yescharacteristic
225C2×C4D4×D17311yescharacteristic
226C8—41no—
227C8—6921no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—3464no—
234C22—3464no—
235C22—3461no—
236C22—3461no—
237C22—3461no—
238C22—3461no—
239C22—1732no—
240C22—1732no—
241C22C22⋊C4×D17311yescharacteristic
242C22#1914970311yescharacteristic
243C22#1914970811yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—3461no—
249C4—3461no—
250C4—6922no—
251C4—6921no—
252C4—6921no—
253C2—21no—
254C2—21no—
255C2—81no—
256C2—3462no—
257C2—3462no—
258C2#99×D17311yescharacteristic
259C2#2601148911yescharacteristic
260C2#104×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012001C2
cosocleC22×C4×C173C23
hypercenter#327D173
hypocenterC173#1504
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26011453C2semidirect
3. #95×D173C2semidirect
4. #26011229C2semidirect
5. #26011227C2semidirect
6. #26011225C2semidirect
7. #327×C173C2semidirect
8. C2×#14583825C4semidirect
13. #19149659C4semidirect
15. #19149650C4semidirect
19. #19149637C4semidirect
29. C2×Dic346C2×C4semidirect
46. C4⋊Dic173C2×C4semidirect
54. C2×D4×C173C2×C4semidirect
58. C4⋊C4×C173C2×C4semidirect
130. C173#1504semidirect
138. #327D173semidirect
164. C2×D4C4×D173semidirect
184. C4⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
#260114532
#95×D1732
#260112292
#260112272
#260112252
#327×C1732
#1504173