Finite Group Databasebeta

← #36353480

Subgroups — #36353480

Order 22144.

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

normal subgroups only

This group has 612 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×#19149593C214yesmaximal, complemented
3#26011354C211yesmaximal, characteristic, complemented
4#381×C173C211yesmaximal, characteristic, complemented
5C2×#14583826—22no—
6C2×#14583826—22no—
7C2×#14583826—28no—
8C22×Dic346C411yescharacteristic, complemented
9C22×D692C411yescharacteristic, complemented
10C22×C22⋊D173C2212yes—
11C22×C4×D173C412yescomplemented
12C2×C22⋊Dic173C2212yes—
13#19149593—28no—
14#19149586—22no—
15C23×C4×C173C2211yescharacteristic
16C2×C22⋊C4×C173C2212yes—
17#95×C173—22no—
18C23×D173C2×C412yes—
19#14583826—216no—
20#14583826—216no—
21C2×Dic346—24no—
22C2×Dic346—22no—
23C2×D692—24no—
24C2×D692—22no—
25C2×C22⋊D173—24no—
26C2×C22⋊D173—22no—
27C2×C22⋊D173—22no—
28C2×C22⋊D173D418yes—
29C22×Dic173—22no—
30C22×Dic173C2×C412yes—
31C2×C4×D173—24no—
32C2×C4×D173—24no—
33C2×C4×D173—24no—
34C22⋊Dic173—24no—
35C24×C173C2311yescharacteristic
36C22×C4×C173—24no—
37C22×C4×C173—22no—
38C22×C4×C173C2×C411yescharacteristic
39C22×C4×C173C2×C411yescharacteristic
40C22×C4×C173D412yes—
41C22×C4×C173D412yes—
42C22⋊C4×C173—24no—
43C22×D173—22no—
44C22×D173—22no—
45C22×D173—44no—
46C22×D173C22⋊C414yes—
47Dic346—28no—
48D692—28no—
49C22⋊D173—28no—
50C22⋊D173—28no—
51C2×Dic173—22no—
52C2×Dic173—22no—
53C2×Dic173—44no—
54C2×Dic173—42no—
55C2×Dic173C22⋊C414yes—
56C4×D173—216no—
57C23×C173—42no—
58C23×C173C22×C411yescharacteristic
59C23×C173C2×D414yes—
60C23×C173C2×D411yescharacteristic
61C23×C173C2×D411yescharacteristic
62C2×C4×C173—22no—
63C2×C4×C173—22no—
64C2×C4×C173—24no—
65C2×C4×C173—24no—
66C2×C4×C173—44no—
67C2×C4×C173—42no—
68C2×C4×C173C22⋊C412yes—
69C2×C4×C173C22⋊C412yes—
70C22×C173—22no—
71C22×C173—22no—
72C22×C173—22no—
73C22×C173—44no—
74C22×C173C2×C22⋊C412yes—
75C22×C173C2×C22⋊C411yescharacteristic
76C22×C173#12012yes—
77C22×C173#12012yes—
78C2×D173—24no—
79C2×D173—44no—
80C2×D173—44no—
81C2×D173—44no—
82C4×C173—22no—
83C4×C173—22no—
84C4×C173—82no—
85Dic173—24no—
86Dic173—82no—
87C2×C173—22no—
88C2×C173—22no—
89C2×C173—22no—
90C2×C173#37911yescharacteristic
91C2×C173#41012yes—
92D173—44no—
93C173#178211yescharacteristic, complemented, Sylow
94#1782—1731nomaximal, Sylow
95C22×#41—1731no—
96C2×#99—1734no—
97#381—1731no—
98#381D17311yescharacteristic, complemented
99C2×#41—3468no—
100C2×#41—3462no—
101C2×#41—3462no—
102C22×Q8—1731no—
103C22×D4—1732no—
104C22×D4—1731no—
105C23×C4—1732no—
106C23×C4C2×D17311yescharacteristic
107C2×C22⋊C4—1732no—
108C2×C22⋊C4C2×D17312yes—
109#99—3468no—
110#95—22no—
111#95—3462no—
112C24—1732no—
113C24C22×D17311yescharacteristic
114#41—34616no—
115#41—34616no—
116C2×Q8—3464no—
117C2×Q8—3462no—
118C2×D4—3464no—
119C2×D4—3464no—
120C2×D4—3462no—
121C2×D4—3462no—
122C2×D4—3462no—
123C2×D4—1738no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—3464no—
127C22×C4—3464no—
128C22×C4—3464no—
129C22×C4—3462no—
130C22×C4—1732no—
131C22×C4C4×D17311yescharacteristic
132C22×C4C4×D17311yescharacteristic
133C22×C4C22⋊D17312yes—
134C22×C4D69212yes—
135C22⋊C4—24no—
136C22⋊C4—3464no—
137C23—42no—
138C23—3462no—
139C23—3462no—
140C23—1734no—
141C23—6924no—
142C23C2×C4×D17311yescharacteristic
143C23D4×D17314yes—
144C23C2×C22⋊D17311yescharacteristic
145C23C2×D69211yescharacteristic
146Q8—3468no—
147D4—3468no—
148D4—3468no—
149D4—3468no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—34616no—
157C2×C4—3462no—
158C2×C4—3462no—
159C2×C4—1734no—
160C2×C4—6924no—
161C2×C4—6922no—
162C2×C4#1458380312yes—
163C2×C4#1458380312yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—3464no—
169C22—6924no—
170C22—6924no—
171C22—6924no—
172C22C22⋊C4×D17312yes—
173C22C2×#1458380311yescharacteristic
174C22#1914964712yes—
175C22#1914964912yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—3464no—
180C4—13842no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—6924no—
185C2#2601134611yescharacteristic
186C2#2601146412yes—
subgroupG/H
centerC22C2×#14583803
commutatorC23×C173C22×C4
FrattiniC24C22×D173
Fitting#381×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×#14583826C2
cosocleC24×C173C23
hypercenter#381D173
hypocenterC173#1782
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#19149593C2semidirect
3. #26011354C2semidirect
4. #381×C173C2semidirect
8. C22×Dic346C4semidirect
9. C22×D692C4semidirect
11. C22×C4×D173C4semidirect
93. C173#1782semidirect
98. #381D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838262
C2×#191495932
#260113542
#381×C1732
#1782173