Finite Group Databasebeta

← #36351880

Subgroups — #36351880

Order 22144.

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

normal subgroups only

This group has 332 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×#19149641C211yesmaximal, characteristic
2C22⋊C8×D173C212yesmaximal, complemented
3#26011217C211yesmaximal, characteristic, complemented
4#26011214C212yesmaximal, complemented
5#324×C173C211yesmaximal, characteristic, complemented
6C22×D692C411yescharacteristic
7C22×C4×D173C411yescharacteristic
8C22×C4×D173C2211yescharacteristic
9C2×C8×D173—22no—
10#19149641—22no—
11#19149641—22no—
12C2×#14583803C411yescharacteristic
13C2×#14583803C2211yescharacteristic
14C2×C4×Dic173C411yescharacteristic
15C22⋊#13063436C2212yes—
16#19149591—22no—
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C23×D173C2×C411yescharacteristic
20C23×D173C2×C411yescharacteristic
21C2×D692—22no—
22C2×D692—22no—
23C2×D692C814yescomplemented
24C22×Dic173C2×C411yescharacteristic
25C22×Dic173C2×C411yescharacteristic
26C2×C4×D173—21no—
27C2×C4×D173—21no—
28C2×C4×D173—42no—
29C2×C4×D173—42no—
30C2×C4×D173D412yes—
31C2×C4×D173D412yes—
32C2×#13063436—22no—
33C8×D173—44no—
34#14583803—22no—
35#14583803—22no—
36C4×Dic173—21no—
37C4×Dic173—21no—
38C22×C4×C173C2×C411yescharacteristic
39C22×C4×C173C2×C411yescharacteristic
40C22×C4×C173C2311yescharacteristic
41C2×C8×C173—22no—
42C4⋊C4×C173—22no—
43C22×D173—21no—
44C22×D173—21no—
45C22×D173—41no—
46C22×D173—41no—
47C22×D173—41no—
48C22×D173—41no—
49C22×D173C22⋊C412yes—
50C22×D173C2×C812yes—
51C22×D173C2×C812yescomplemented
52C22×D173M1612yes—
53D692—28no—
54C2×Dic173—22no—
55C2×Dic173—21no—
56C2×Dic173—41no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C22⋊C411yescharacteristic
59C4×D173—24no—
60C4×D173—42no—
61C4×D173—42no—
62C4×D173—42no—
63#13063436—42no—
64C23×C173C22×C411yescharacteristic
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173C2×C812yescomplemented
71C2×C4×C173C2×D412yes—
72C8×C173—42no—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173C2×C22⋊C411yescharacteristic
76C22×C173C22×C811yescharacteristic
77C22×C173C2×M1611yescharacteristic
78C2×D173—22no—
79C2×D173—22no—
80C2×D173—22no—
81C2×D173—42no—
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—42no—
87C2×D173C22⋊C814yes—
88C4×C173—22no—
89C4×C173—22no—
90C4×C173—41no—
91Dic173—21no—
92Dic173—41no—
93Dic173#9912yes—
94C2×C173—21no—
95C2×C173—21no—
96C2×C173C2×C22⋊C811yescharacteristic
97C2×C173C2×#9911yescharacteristic
98C2×C173#41311yescharacteristic
99D173—24no—
100D173—42no—
101C173#145911yescharacteristic, complemented, Sylow
102#1459—1731nomaximal, Sylow
103C2×C4×D4—1731no—
104C2×C22⋊C8—1732no—
105#324—1731no—
106#324D17311yescharacteristic, complemented
107#323—1732no—
108C22×D4—1731no—
109C23×C4—1731no—
110C23×C4—1731no—
111C22×C8—3462no—
112C4×D4—3462no—
113C4×D4—3462no—
114C2×C4⋊C4C2×D17311yescharacteristic
115C2×C22⋊C4—1731no—
116C2×C22⋊C4—1731no—
117C2×C42—1731no—
118C22⋊C8—3462no—
119C22⋊C8—1732no—
120C22⋊C8C2×D17312yes—
121C24—1731no—
122C24—1731no—
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—1734no—
126C22×C4—3461no—
127C22×C4—3461no—
128C22×C4—1731no—
129C22×C4—1732no—
130C22×C4—1731no—
131C22×C4—1732no—
132C22×C4—6922no—
133C22×C4—6922no—
134C22×C4C4×D17311yescharacteristic
135C22×C4C4×D17311yescharacteristic
136C22×C4C22×D17311yescharacteristic
137C2×C8—22no—
138C2×C8—3462no—
139C2×C8—6924no—
140C4⋊C4—22no—
141C22⋊C4—3462no—
142C22⋊C4—3462no—
143C42—3461no—
144C42—3461no—
145C23—3461no—
146C23—3461no—
147C23—1732no—
148C23—1732no—
149C23—1732no—
150C23—1732no—
151C23—6921no—
152C23—6921no—
153C23—6921no—
154C23—6921no—
155C23C2×C4×D17311yescharacteristic
156D4—3468no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3464no—
163C2×C4—3462no—
164C2×C4—3461no—
165C2×C4—1731no—
166C2×C4—1731no—
167C2×C4—6922no—
168C2×C4—6922no—
169C2×C4—6922no—
170C2×C4—6921no—
171C2×C4C8×D17312yescomplemented
172C2×C4D4×D17312yes—
173C8—42no—
174C8—6922no—
175C22—21no—
176C22—21no—
177C22—3462no—
178C22—3462no—
179C22—3462no—
180C22—1734no—
181C22—6922no—
182C22—6922no—
183C22—6922no—
184C22—6922no—
185C22—6921no—
186C22—6921no—
187C22C22⋊C4×D17311yescharacteristic
188C22C2×C8×D17311yescharacteristic
189C22M16×D17311yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—3461no—
194C4—1732no—
195C4—6921no—
196C2—21no—
197C2—21no—
198C2—3464no—
199C2—6922no—
200C2C22⋊C8×D17311yescharacteristic
201C2#99×D17311yescharacteristic
202C2#2601147411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC22×C173C22×C8
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×D692C4
cosocleC22×C4×C173C23
hypercenter#324D173
hypocenterC173#1459
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D173C2semidirect
3. #26011217C2semidirect
4. #26011214C2semidirect
5. #324×C173C2semidirect
23. C2×D692C8semidirect
51. C22×D173C2×C8semidirect
70. C2×C4×C173C2×C8semidirect
101. C173#1459semidirect
106. #324D173semidirect
171. C2×C4C8×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496412
C22⋊C8×D1732
#260112172
#260112142
#324×C1732
#1459173