Finite Group Databasebeta

← #36368797

Subgroups — #36368797

Order 22144.

264 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 76 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#26012508C212yesmaximal, complemented
2#26012336C212yesmaximal, complemented
3#26012335C212yesmaximal
4C2×#19149637C211yesmaximal, characteristic
5#19149746—22no—
6C22⋊#13063443C2214yes—
7C4×#13063443—22no—
8C22×C4×D173C411yescharacteristic
9#19149637—22no—
10#19149637—22no—
11C2×C4⋊Dic173C2211yescharacteristic
12C2×#14583803C411yescharacteristic
13C2×#14583803C2211yescharacteristic
14C2×C4×Dic173C411yescharacteristic
15C2×C4×Dic173C2211yescharacteristic
16C2×C4⋊C4×C173C411yescharacteristic
17C2×#13063443—24no—
18C23×D173C2×C411yescharacteristic
19C22×Dic173C2×C411yescharacteristic
20C22×Dic173C2×C411yescharacteristic
21C22×Dic173C2311yescharacteristic
22C2×C4×D173—21no—
23C2×C4×D173—21no—
24C2×C4×D173—22no—
25C2×C4×D173C814yescomplemented
26C4⋊Dic173—22no—
27#14583803—22no—
28#14583803—22no—
29C4×Dic173—21no—
30C4×Dic173—21no—
31C4×Dic173D412yes—
32C4×Dic173D412yes—
33C22×C4×C173C2×C411yescharacteristic
34C22×C4×C173C2×C411yescharacteristic
35C22×C4×C173C2×C411yescharacteristic
36C4⋊C4×C173—22no—
37#13063443—44no—
38C22×D173—21no—
39C22×D173—21no—
40C22×D173—41no—
41C22×D173—41no—
42C22×D173C2×C812yes—
43C2×Dic173—21no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46C2×Dic173C22⋊C411yescharacteristic
47C2×Dic173C22⋊C411yescharacteristic
48C2×Dic173C2×C812yes—
49C2×Dic173M1612yes—
50C2×Dic173C2×D412yes—
51C4×D173—24no—
52C4×D173—24no—
53C23×C173C22×C411yescharacteristic
54C2×C4×C173—21no—
55C2×C4×C173—21no—
56C2×C4×C173—21no—
57C2×C4×C173—41no—
58C2×C4×C173—41no—
59C2×C4×C173C22⋊C412yes—
60C2×C4×C173C2×C812yes—
61C22×C173—21no—
62C22×C173—21no—
63C22×C173C2×C22⋊C411yescharacteristic
64C22×C173C22×C811yescharacteristic
65C22×C173C2×M1611yescharacteristic
66C2×D173—22no—
67C2×D173—42no—
68C2×D173—41no—
69C2×D173—41no—
70C2×D173—42no—
71C4×C173—22no—
72C4×C173—22no—
73C4×C173—41no—
74Dic173—22no—
75Dic173—41no—
76Dic173C22⋊C812yes—
77Dic173C22⋊C812yes—
78C2×C173—21no—
79C2×C173—21no—
80C2×C173C2×C22⋊C811yescharacteristic
81C2×C173#41011yescharacteristic
82C2×C173#41311yescharacteristic
83D173—42no—
84C173#145811yescharacteristic, complemented, Sylow
85#1458—1731nomaximal, Sylow
86C2×#117—1731no—
87#431—1732no—
88#324—1732no—
89#323—1732no—
90C23×C4—1731no—
91#117—3462no—
92#117—3462no—
93C2×C4⋊C4—1731no—
94C2×C4⋊C4#1062192411yescharacteristic
95C2×C22⋊C4—1731no—
96C2×C22⋊C4—1731no—
97C2×C42—1731no—
98C2×C42—1731no—
99C4⋊C8—3462no—
100C22⋊C8—1734no—
101C4×C8—3462no—
102C24—1731no—
103C22×C4—3461no—
104C22×C4—3461no—
105C22×C4—3462no—
106C22×C4—1734no—
107C22×C4—1731no—
108C22×C4—1731no—
109C22×C4—1731no—
110C22×C4C2×#1062192411yescharacteristic
111C22×C4C2×#1062192411yescharacteristic
112C22×C4C2×#1062192411yescharacteristic
113C2×C8—3464no—
114C4⋊C4—22no—
115C4⋊C4—3462no—
116C22⋊C4—3462no—
117C22⋊C4—3462no—
118C42—3461no—
119C42—3461no—
120C42—1732no—
121C42—1732no—
122C23—3461no—
123C23—3461no—
124C23—1732no—
125C23—6921no—
126C23—6921no—
127C23C22×#1062192411yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—3464no—
134C2×C4—3464no—
135C2×C4—3461no—
136C2×C4—1732no—
137C2×C4—1731no—
138C2×C4—1731no—
139C2×C4—1732no—
140C2×C4—1732no—
141C2×C4—6921no—
142C2×C4—6921no—
143C2×C4C22⋊#1062192412yes—
144C2×C4D173⋊C812yes—
145C8—6924no—
146C22—21no—
147C22—21no—
148C22—3462no—
149C22—6922no—
150C22—6922no—
151C22—6921no—
152C22—6921no—
153C22C2×C22⋊#1062192411yescharacteristic
154C22C2×D173⋊C811yescharacteristic
155C22D173⋊M1611yescharacteristic
156C4—22no—
157C4—22no—
158C4—41no—
159C4—3462no—
160C4—1732no—
161C4—1732no—
162C4—6921no—
163C2—21no—
164C2—21no—
165C2—6922no—
166C2D173⋊(C22⋊C8)11yescharacteristic
167C2#2601247211yescharacteristic
168C2#2601247811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC22×C173C22×C8
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173C2×C4
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1458
LCSG ▷ C22×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
1. #26012508C2semidirect
2. #26012336C2semidirect
25. C2×C4×D173C8semidirect
84. C173#1458semidirect

Maximal subgroups

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

subgroupindex
#260125082
#260123362
#260123352
C2×#191496372
#1458173