Finite Group Databasebeta

← #36368841

Subgroups — #36368841

Order 22144.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#26012462C211yesmaximal, characteristic
2D173⋊(C22⋊C8)C211yesmaximal, characteristic
3#26012336C214yesmaximal, complemented
4C2×#19149659C211yesmaximal, characteristic
5C2×#14583836—21no—
6C2×D173⋊C8—21no—
7#19149736—21no—
8#19149736—21no—
9C22⋊#13063443C2212yes—
10C22⋊#13063443C2212yes—
11C22×Dic346C411yescharacteristic
12C22×C4×D173C2211yescharacteristic
13#19149659—21no—
14#19149659—21no—
15#19149659—21no—
16#19149659—21no—
17C2×C4⋊Dic173C2212yes—
18C2×#14583803C412yes—
19C2×C4⋊C4×C173C411yescharacteristic
20#14583836—42no—
21D173⋊C8—42no—
22C2×#13063443—22no—
23C2×#13063443—22no—
24C23×D173C2×C411yescharacteristic
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346—42no—
28C22×Dic173C2×C412yes—
29C22×Dic173C2311yescharacteristic
30C2×C4×D173—42no—
31C2×C4×D173D411yescharacteristic
32C2×C4×D173D411yescharacteristic
33C2×C4×D173D411yescharacteristic
34C2×C4×D173D411yescharacteristic
35C4⋊Dic173—22no—
36C4⋊Dic173—22no—
37#14583803—22no—
38#14583803—22no—
39C22×C4×C173C2×C412yes—
40C22×C4×C173C2×C411yescharacteristic
41C4⋊C4×C173—21no—
42C4⋊C4×C173—21no—
43#13063443—42no—
44#13063443—42no—
45C22×D173—22no—
46C22×D173—22no—
47C22×D173—41no—
48C22×D173C22⋊C411yescharacteristic
49C22×D173C22⋊C411yescharacteristic
50Dic346—42no—
51Dic346—42no—
52C2×Dic173—22no—
53C2×Dic173—41no—
54C2×Dic173—42no—
55C2×Dic173C2×D411yescharacteristic
56C2×Dic173C2×D411yescharacteristic
57C4×D173—22no—
58C4×D173—22no—
59C4×D173—41no—
60C4×D173—41no—
61C23×C173C22×C411yescharacteristic
62C2×C4×C173—22no—
63C2×C4×C173—22no—
64C2×C4×C173—24no—
65C2×C4×C173C22⋊C411yescharacteristic
66C2×C4×C173C22⋊C411yescharacteristic
67C22×C173—21no—
68C22×C173—21no—
69C22×C173C2×C22⋊C411yescharacteristic
70C22×C173#13112yes—
71C2×D173—21no—
72C2×D173—21no—
73C2×D173—21no—
74C2×D173—21no—
75C2×D173—21no—
76C2×D173—44no—
77C2×D173#10112yes—
78C4×C173—21no—
79C4×C173—42no—
80C4×C173#9912yes—
81Dic173—21no—
82Dic173—21no—
83Dic173—42no—
84C2×C173—22no—
85C2×C173#40811yescharacteristic
86C2×C173C2×#9911yescharacteristic
87C2×C173C2×#10111yescharacteristic
88D173—22no—
89D173—41no—
90C173#147211yescharacteristic, complemented, Sylow
91#1472—1731nomaximal, Sylow
92C2×C22⋊Q8—1731no—
93#407—1731no—
94C2×C22⋊C8—1731no—
95#324—1734no—
96C22×Q8—1731no—
97C23×C4—1731no—
98C2×M16—3461no—
99C22×C8—3461no—
100C22⋊Q8—3461no—
101C22⋊Q8—3461no—
102C22⋊Q8—3461no—
103C22⋊Q8—3461no—
104C2×C4⋊C4—1732no—
105C2×C4⋊C4#1062192411yescharacteristic
106C2×C22⋊C4—1732no—
107C22⋊C8—3461no—
108C22⋊C8—3461no—
109C22⋊C8—1732no—
110C22⋊C8—1732no—
111C24—1731no—
112C2×Q8—3461no—
113C2×Q8—3461no—
114C2×Q8—6922no—
115C22×C4—1731no—
116C22×C4—1731no—
117C22×C4—1732no—
118C22×C4—1731no—
119C22×C4—1731no—
120C22×C4—1731no—
121C22×C4—6922no—
122C22×C4C2×#1062192412yes—
123C22×C4C2×#1062192411yescharacteristic
124M16—6922no—
125C2×C8—3462no—
126C2×C8—3462no—
127C2×C8—6922no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—3462no—
131C4⋊C4—3462no—
132C22⋊C4—3462no—
133C22⋊C4—3462no—
134C23—3462no—
135C23—3462no—
136C23—1731no—
137C23—1731no—
138C23—6921no—
139C23C22×#1062192411yescharacteristic
140Q8—6922no—
141Q8—6922no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—3462no—
146C2×C4—3462no—
147C2×C4—3462no—
148C2×C4—1731no—
149C2×C4—1731no—
150C2×C4—6921no—
151C2×C4—6921no—
152C2×C4—6921no—
153C2×C4—6922no—
154C2×C4C22⋊#1062192411yescharacteristic
155C2×C4C22⋊#1062192411yescharacteristic
156C8—6922no—
157C8—6922no—
158C22—21no—
159C22—21no—
160C22—3461no—
161C22—3461no—
162C22—3461no—
163C22—3461no—
164C22—3461no—
165C22—1732no—
166C22—6924no—
167C22C2×C22⋊#1062192411yescharacteristic
168C22#1914976612yes—
169C4—21no—
170C4—42no—
171C4—3461no—
172C4—3461no—
173C4—6922no—
174C4#1914973812yes—
175C2—22no—
176C2—3462no—
177C2—6921no—
178C2#2601246511yescharacteristic
179C2C2×#1914973811yescharacteristic
180C2D173⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173C2×C4
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1472
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012336C2semidirect
90. C173#1472semidirect

Maximal subgroups

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

subgroupindex
#260124622
D173⋊(C22⋊C8)2
#260123362
C2×#191496592
#1472173