Finite Group Databasebeta

← #323×D173 #36351868

Subgroups — #323×D173 #36351868

Order 22144.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×C22⋊C4×D173C211yesmaximal, characteristic
2C22⋊C8×D173C212yesmaximal, complemented
3#26011215C211yesmaximal, characteristic, complemented
4#26011214C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic, complemented
6C24×D173C411yescharacteristic
7C22×C4×D173C411yescharacteristic
8C22×C4×D173C2211yescharacteristic
9C2×C8×D173—22no—
10C2×C22⋊Dic173C411yescharacteristic
11C2×#14583803C411yescharacteristic
12C2×#14583803C2211yescharacteristic
13C22⋊C4×D173—22no—
14C22⋊C4×D173—22no—
15C22⋊#13063436C2212yescomplemented
16#19149591—22no—
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yescomplemented
19C23×D173—21no—
20C23×D173—21no—
21C23×D173—22no—
22C23×D173—42no—
23C23×D173—42no—
24C23×D173C814yescomplemented
25C23×D173C2×C411yescharacteristic
26C23×D173C2×C411yescharacteristic
27C22×Dic173C2×C411yescharacteristic
28C22×Dic173C2×C411yescharacteristic
29C2×C4×D173—21no—
30C2×C4×D173—21no—
31C2×C4×D173—42no—
32C2×C4×D173—42no—
33C2×C4×D173D412yes—
34C2×C4×D173D412yes—
35C2×#13063436—22no—
36C8×D173—44no—
37C22⋊Dic173—22no—
38#14583803—22no—
39#14583803—22no—
40C24×C173C2×C411yescharacteristic
41C22×C4×C173C2×C411yescharacteristic
42C22×C4×C173C2311yescharacteristic
43C2×C8×C173—22no—
44C22⋊C4×C173—22no—
45C22×D173—22no—
46C22×D173—22no—
47C22×D173—21no—
48C22×D173—21no—
49C22×D173—24no—
50C22×D173—24no—
51C22×D173—42no—
52C22×D173—42no—
53C22×D173—48no—
54C22×D173—48no—
55C22×D173—42no—
56C22×D173—42no—
57C22×D173—41no—
58C22×D173—41no—
59C22×D173C22⋊C411yescharacteristic
60C22×D173C22⋊C411yescharacteristic
61C22×D173C2×C812yes—
62C22×D173C2×C812yescomplemented
63C22×D173M1612yes—
64C2×Dic173—21no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173C22⋊C412yes—
68C4×D173—24no—
69C4×D173—42no—
70C4×D173—42no—
71C4×D173—42no—
72#13063436—42no—
73C23×C173—21no—
74C23×C173—21no—
75C23×C173—41no—
76C23×C173—41no—
77C23×C173C2×C812yescomplemented
78C23×C173C22×C411yescharacteristic
79C2×C4×C173—21no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173C2×D412yes—
83C8×C173—42no—
84C22×C173—21no—
85C22×C173—21no—
86C22×C173—22no—
87C22×C173—42no—
88C22×C173—42no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173C2×C22⋊C411yescharacteristic
92C22×C173C22×C811yescharacteristic
93C22×C173C2×M1611yescharacteristic
94C2×D173—21no—
95C2×D173—21no—
96C2×D173—24no—
97C2×D173—24no—
98C2×D173—22no—
99C2×D173—48no—
100C2×D173—42no—
101C2×D173—41no—
102C2×D173—41no—
103C2×D173—48no—
104C2×D173—42no—
105C2×D173C22⋊C812yes—
106C2×D173C22⋊C812yes—
107C2×D173#9912yes—
108C2×D173#10012yes—
109C4×C173—22no—
110C4×C173—41no—
111Dic173—22no—
112Dic173—41no—
113C2×C173—21no—
114C2×C173—21no—
115C2×C173—42no—
116C2×C173C2×C22⋊C811yescharacteristic
117C2×C173C2×#9911yescharacteristic
118C2×C173C2×#10011yescharacteristic
119D173—22no—
120D173—42no—
121D173#32314yesdirect product factor
122C173C2×#32311yescharacteristic, complemented, Sylow
123C2×#323—1731nomaximal, Sylow
124C22×C22⋊C4—1731no—
125C2×C22⋊C8—1732no—
126#323—1731no—
127#323—1732no—
128#323D17311yescharacteristic, direct product factor
129C25—1731no—
130C23×C4—1731no—
131C23×C4—1731no—
132C22×C8—3462no—
133C2×C22⋊C4—3462no—
134C2×C22⋊C4—3462no—
135C2×C22⋊C4—1731no—
136C2×C22⋊C4—1731no—
137C2×C22⋊C4—1731no—
138C2×C22⋊C4C2×D17311yescharacteristic
139C22⋊C8—3462no—
140C22⋊C8—1732no—
141C22⋊C8C2×D17312yescomplemented
142C24—3461no—
143C24—3461no—
144C24—3462no—
145C24—1731no—
146C24—1731no—
147C24—1734no—
148C24—6922no—
149C24—6922no—
150C24C4×D17311yescharacteristic
151C22×C4—3461no—
152C22×C4—3461no—
153C22×C4—1732no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4—1732no—
157C22×C4—6922no—
158C22×C4—6922no—
159C22×C4C4×D17311yescharacteristic
160C22×C4C22×D17311yescharacteristic
161C2×C8—22no—
162C2×C8—3462no—
163C2×C8—6924no—
164C22⋊C4—22no—
165C22⋊C4—3462no—
166C22⋊C4—3462no—
167C22⋊C4—3462no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—3462no—
173C23—3462no—
174C23—3464no—
175C23—3464no—
176C23—3461no—
177C23—3461no—
178C23—1731no—
179C23—1732no—
180C23—1732no—
181C23—1732no—
182C23—1731no—
183C23—6921no—
184C23—6921no—
185C23—6928no—
186C23—6928no—
187C23—6922no—
188C23—6922no—
189C23—6922no—
190C23—6922no—
191C23C8×D17312yescomplemented
192C23C2×C4×D17311yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—3464no—
197C2×C4—3461no—
198C2×C4—1732no—
199C2×C4—6922no—
200C2×C4—6922no—
201C2×C4—6922no—
202C2×C4—6921no—
203C2×C4—6921no—
204C2×C4D4×D17312yes—
205C8—42no—
206C8—6922no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—3464no—
215C22—3464no—
216C22—3461no—
217C22—3461no—
218C22—3462no—
219C22—1732no—
220C22—1732no—
221C22—1732no—
222C22—1732no—
223C22—6922no—
224C22—6922no—
225C22—6928no—
226C22—6928no—
227C22—6921no—
228C22—6921no—
229C22C22⋊C4×D17311yescharacteristic
230C22C2×C8×D17311yescharacteristic
231C22M16×D17311yescharacteristic
232C4—22no—
233C4—41no—
234C4—3462no—
235C4—6921no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—3462no—
240C2—1734no—
241C2—6922no—
242C2C22⋊C8×D17311yescharacteristic
243C2#99×D17311yescharacteristic
244C2#100×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC22×C173C22×C8
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC24×D173C4
cosocleC22×C4×C173C23
hypercenter#323D173
hypocenterC173C2×#323
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D173C2semidirect
3. #26011215C2semidirect
4. #26011214C2semidirect
5. #323×C173C2semidirect
15. C22⋊#13063436C22semidirect
18. C22⋊C8×C173C22semidirect
24. C23×D173C8semidirect
62. C22×D173C2×C8semidirect
77. C23×C173C2×C8semidirect
121. D173#323direct (N × H)
122. C173C2×#323semidirect
128. #323D173direct (N × H)
141. C22⋊C8C2×D173semidirect
191. C23C8×D173semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1732
C22⋊C8×D1732
#260112152
#260112142
#323×C1732
C2×#323173