Finite Group Databasebeta

← #36351882

Subgroups — #36351882

Order 22144.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×#19149639C211yesmaximal, characteristic
2#26011541C212yesmaximal, complemented
3#26011215C211yesmaximal, characteristic, complemented
4#26011214C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173C411yescharacteristic
7C23×Dic173C411yescharacteristic
8C22×C4×D173C411yescharacteristic
9#19149639—22no—
10#19149639—22no—
11C2×#14583805C411yescharacteristic
12C2×#14583803C2211yescharacteristic
13C2×C4×Dic173C2211yescharacteristic
14#19149608—22no—
15C22⋊#13063436C2212yes—
16C8×Dic173—22no—
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C23×D173C2×C411yescharacteristic
20C2×C22⋊D173—22no—
21C2×C22⋊D173—22no—
22C2×C22⋊D173C814yescomplemented
23C22×Dic173—21no—
24C22×Dic173—21no—
25C22×Dic173—42no—
26C22×Dic173C2×C411yescharacteristic
27C22×Dic173C2×C411yescharacteristic
28C22×Dic173C2×C411yescharacteristic
29C2×C4×D173—21no—
30C2×C4×D173—21no—
31C2×C4×D173—42no—
32C2×#13063436—22no—
33#14583805—22no—
34#14583803—22no—
35C4×Dic173D412yes—
36C4×Dic173D412yes—
37C24×C173C2×C411yescharacteristic
38C22×C4×C173C2×C411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C2×C8×C173—22no—
41C22⋊C4×C173—22no—
42C22×D173—21no—
43C22×D173—21no—
44C22×D173—41no—
45C22×D173—41no—
46C22×D173C2×C812yes—
47C22⋊D173—28no—
48C2×Dic173—22no—
49C2×Dic173—21no—
50C2×Dic173—42no—
51C2×Dic173—42no—
52C2×Dic173—41no—
53C2×Dic173C22⋊C411yescharacteristic
54C2×Dic173C22⋊C412yes—
55C2×Dic173C22⋊C411yescharacteristic
56C2×Dic173C2×C812yes—
57C2×Dic173M1612yes—
58C4×D173—42no—
59C4×D173—42no—
60#13063436—42no—
61C23×C173—21no—
62C23×C173—21no—
63C23×C173—41no—
64C23×C173—41no—
65C23×C173C2×C812yes—
66C23×C173C22×C411yescharacteristic
67C2×C4×C173—21no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173C2×D412yes—
71C8×C173—42no—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173—22no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173C2×C22⋊C411yescharacteristic
80C22×C173C22×C811yescharacteristic
81C22×C173C2×M1611yescharacteristic
82C2×D173—22no—
83C2×D173—42no—
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—42no—
87C4×C173—22no—
88C4×C173—41no—
89Dic173—21no—
90Dic173—41no—
91Dic173C22⋊C814yes—
92Dic173#10012yes—
93C2×C173—21no—
94C2×C173—21no—
95C2×C173—42no—
96C2×C173C2×C22⋊C811yescharacteristic
97C2×C173#41011yescharacteristic
98C2×C173C2×#10011yescharacteristic
99D173—42no—
100C173#146011yescharacteristic, complemented, Sylow
101#1460—1731nomaximal, Sylow
102C2×C4×D4—1731no—
103#431—1732no—
104#323—1731no—
105#323—1732no—
106#323D17311yescharacteristic, complemented
107C22×D4—1731no—
108C23×C4—1731no—
109C23×C4—1731no—
110C4×D4—3462no—
111C4×D4—3462no—
112C2×C4⋊C4—1731no—
113C2×C22⋊C4—1731no—
114C2×C22⋊C4C2×D17311yescharacteristic
115C2×C42—1731no—
116C4⋊C8—3462no—
117C22⋊C8—1732no—
118C22⋊C8C2×D17312yes—
119C4×C8—3462no—
120C24—1731no—
121C24C4×D17311yescharacteristic
122C2×D4—3462no—
123C2×D4—3462no—
124C2×D4—1734no—
125C22×C4—3461no—
126C22×C4—3461no—
127C22×C4—3461no—
128C22×C4—3461no—
129C22×C4—1731no—
130C22×C4—1731no—
131C22×C4—1731no—
132C22×C4—6922no—
133C22×C4—6922no—
134C22×C4C4×D17311yescharacteristic
135C22×C4C22×D17311yescharacteristic
136C2×C8—22no—
137C2×C8—3462no—
138C4⋊C4—3462no—
139C22⋊C4—22no—
140C22⋊C4—3462no—
141C42—1732no—
142C42—1732no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—3461no—
148C23—3461no—
149C23—1732no—
150C23—6921no—
151C23—6921no—
152C23C8×D17312yes—
153C23C2×C4×D17311yescharacteristic
154D4—3468no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—3462no—
159C2×C4—3461no—
160C2×C4—1731no—
161C2×C4—1732no—
162C2×C4—1731no—
163C2×C4—1732no—
164C2×C4—1732no—
165C2×C4—6922no—
166C2×C4—6922no—
167C2×C4—6922no—
168C2×C4—6922no—
169C2×C4—6921no—
170C2×C4D4×D17312yes—
171C8—42no—
172C8—6922no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—3462no—
181C22—6922no—
182C22—6922no—
183C22—6921no—
184C22—6921no—
185C22C22⋊C4×D17311yescharacteristic
186C22C2×C8×D17311yescharacteristic
187C22M16×D17311yescharacteristic
188C4—22no—
189C4—41no—
190C4—3461no—
191C4—1734no—
192C4—1732no—
193C4—6921no—
194C2—21no—
195C2—21no—
196C2—42no—
197C2—6922no—
198C2C22⋊C8×D17311yescharacteristic
199C2#2601146311yescharacteristic
200C2#100×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC22×C173C22×C8
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C4
cosocleC22×C4×C173C23
hypercenter#323D173
hypocenterC173#1460
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. #26011541C2semidirect
3. #26011215C2semidirect
4. #26011214C2semidirect
5. #323×C173C2semidirect
22. C2×C22⋊D173C8semidirect
100. C173#1460semidirect
106. #323D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496392
#260115412
#260112152
#260112142
#323×C1732
#1460173