Finite Group Databasebeta

← #36353489

Subgroups — #36353489

Order 22144.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14583826C211yesmaximal, characteristic
2C2×#19149597C212yesmaximal, complemented
3C2×#19149595C212yesmaximal, complemented
4#26011455C211yesmaximal, characteristic, complemented
5#407×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—22no—
7C2×#14583826—22no—
8C2×#14583826—24no—
9C2×#14583826—24no—
10C22×Dic346C2211yescharacteristic
11C22×D692C2211yescharacteristic
12C22×C22⋊D173C412yes—
13C22×C4×D173C412yes—
14C2×#14583810C2212yes—
15#19149597—24no—
16#19149595—24no—
17C22⋊#13063436—22no—
18C23×C4×C173C2211yescharacteristic
19C2×M16×C173C2212yes—
20C22⋊C8×C173—22no—
21C23×D173C2×C412yes—
22#14583826—216no—
23#14583826—216no—
24C2×Dic346—22no—
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346D414yes—
28C2×D692—22no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692D414yes—
32C2×C22⋊D173—24no—
33C2×C22⋊D173—24no—
34C2×C22⋊D173—24no—
35C22×Dic173C2×C412yes—
36C2×C4×D173—24no—
37C2×C4×D173—24no—
38C2×C4×D173—24no—
39#14583810—22no—
40#14583810—22no—
41C2×#13063436—22no—
42C24×C173C2×C411yescharacteristic
43C22×C4×C173—22no—
44C22×C4×C173—22no—
45C22×C4×C173C2×C411yescharacteristic
46C22×C4×C173D412yes—
47C22×C4×C173D412yes—
48C22×C4×C173C2311yescharacteristic
49M16×C173—22no—
50M16×C173—22no—
51C2×C8×C173—22no—
52C22×D173—22no—
53C22×D173—22no—
54C22×D173—44no—
55C22×D173C22⋊C414yes—
56Dic346—24no—
57Dic346—24no—
58D692—24no—
59D692—24no—
60C22⋊D173—216no—
61C2×Dic173—22no—
62C2×Dic173—22no—
63C2×Dic173C22⋊C414yes—
64C4×D173—216no—
65#13063436—42no—
66C23×C173—21no—
67C23×C173—21no—
68C23×C173—42no—
69C23×C173C22⋊C412yes—
70C23×C173C22×C411yescharacteristic
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173—24no—
74C2×C4×C173—24no—
75C2×C4×C173C22⋊C412yes—
76C2×C4×C173C2×D412yes—
77C2×C4×C173C2×D412yes—
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C8×C173—42no—
81C22×C173—22no—
82C22×C173—22no—
83C22×C173—24no—
84C22×C173—42no—
85C22×C173—42no—
86C22×C173C2×C22⋊C411yescharacteristic
87C22×C173C2×C22⋊C411yescharacteristic
88C22×C173C2×C22⋊C411yescharacteristic
89C2×D173—24no—
90C2×D173—44no—
91C2×D173—44no—
92C2×D173—44no—
93C4×C173—22no—
94C4×C173#12012yes—
95C4×C173#12012yes—
96Dic173—24no—
97C2×C173—21no—
98C2×C173—21no—
99C2×C173—42no—
100C2×C173#37911yescharacteristic
101C2×C173#41312yes—
102D173—44no—
103C173#178511yescharacteristic, complemented, Sylow
104#1785—1731nomaximal, Sylow
105C22×#41—1731no—
106C2×#101—1732no—
107C2×#100—1732no—
108#407—1731no—
109#407D17311yescharacteristic, complemented
110C2×#41—3464no—
111C2×#41—3464no—
112C2×#41—3462no—
113C2×#41—3462no—
114C22×Q8—1731no—
115C22×D4—1732no—
116C22×D4—1731no—
117C23×C4—1732no—
118C23×C4C2×D17311yescharacteristic
119C2×M16—1732no—
120C2×M16C2×D17312yes—
121#101—3464no—
122#100—3464no—
123C22⋊C8—22no—
124C22⋊C8—3462no—
125C24—1732no—
126C24C4×D17311yescharacteristic
127#41—34616no—
128#41—34616no—
129C2×Q8—3462no—
130C2×Q8—3461no—
131C2×Q8—3461no—
132C2×Q8—1734no—
133C2×D4—3464no—
134C2×D4—3464no—
135C2×D4—3464no—
136C2×D4—3462no—
137C2×D4—3461no—
138C2×D4—3461no—
139C2×D4—1734no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—3464no—
143C22×C4—3464no—
144C22×C4—3464no—
145C22×C4—1732no—
146C22×C4C4×D17311yescharacteristic
147C22×C4C22⋊D17312yes—
148C22×C4D69212yes—
149C22×C4C22×D17311yescharacteristic
150M16—22no—
151M16—22no—
152M16—3462no—
153M16—3462no—
154C2×C8—22no—
155C2×C8—3462no—
156C23—21no—
157C23—21no—
158C23—42no—
159C23—3462no—
160C23—3462no—
161C23—1734no—
162C23—6924no—
163C23#1458380312yes—
164C23C2×C4×D17311yescharacteristic
165Q8—3464no—
166Q8—3464no—
167D4—34616no—
168D4—3464no—
169D4—3464no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—34616no—
175C2×C4—3462no—
176C2×C4—3462no—
177C2×C4—1734no—
178C2×C4#1458380312yes—
179C2×C4D4×D17312yes—
180C2×C4D4×D17312yes—
181C2×C4C2×C22⋊D17311yescharacteristic
182C2×C4C2×D69211yescharacteristic
183C8—42no—
184C8—6922no—
185C22—22no—
186C22—24no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—3464no—
191C22—6924no—
192C22—6924no—
193C22—6924no—
194C22C22⋊C4×D17311yescharacteristic
195C22C22⋊C4×D17311yescharacteristic
196C22C2×#1458380311yescharacteristic
197C4—22no—
198C4—3464no—
199C4#1914964712yes—
200C4#1914964912yes—
201C2—21no—
202C2—21no—
203C2—42no—
204C2—6924no—
205C2#2601134611yescharacteristic
206C2#2601147312yes—
subgroupG/H
centerC22C2×#14583803
commutatorC23×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#407×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×#14583826C2
cosocleC22×C4×C173C23
hypercenter#407D173
hypocenterC173#1785
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#19149597C2semidirect
3. C2×#19149595C2semidirect
4. #26011455C2semidirect
5. #407×C173C2semidirect
103. C173#1785semidirect
109. #407D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838262
C2×#191495972
C2×#191495952
#260114552
#407×C1732
#1785173