Finite Group Databasebeta

← #36369043

Subgroups — #36369043

Order 22144.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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#26012513C212yesmaximal, complemented
2#26012347C212yesmaximal, complemented
3#26012346C212yesmaximal, complemented
4#124×D173C211yesmaximal, characteristic
5C4⋊#13063443—22no—
6#19149736—22no—
7#19149734C2214yescomplemented
8C2×Q8×D173C411yescharacteristic
9C2×D4×D173C411yescharacteristic
10#19149668C2211yescharacteristic
11#19149667C411yescharacteristic
12#19149666C2211yescharacteristic
13#19149665—22no—
14C22⋊C4×D173—22no—
15C42×D173C2211yescharacteristic
16#124×C173C411yescharacteristic
17C2×#13063443—24no—
18C23×D173—21no—
19C2×Dic346—21no—
20C2×Dic346C2×C411yescharacteristic
21Q8×D173—42no—
22C2×D692C2×C411yescharacteristic
23C2×C22⋊D173—21no—
24D4×D173—42no—
25C2×C4×D173—21no—
26C2×C4×D173C2×C411yescharacteristic
27C2×C4×D173D412yes—
28C22⋊Dic173—22no—
29#14583803—22no—
30#14583803—22no—
31C4×Dic173D412yes—
32C4×Dic173C2311yescharacteristic
33C2×Q8×C173C2×C411yescharacteristic
34C2×D4×C173C2×C411yescharacteristic
35C22⋊C4×C173—22no—
36C42×C173C2×C411yescharacteristic
37#13063443—44no—
38C22×D173—21no—
39C22×D173—44no—
40C22×D173—42no—
41C22×D173C22⋊C411yescharacteristic
42Dic346—42no—
43Dic346—41no—
44D692—41no—
45C22⋊D173—42no—
46C2×Dic173—21no—
47C2×Dic173C22⋊C411yescharacteristic
48C2×Dic173C2×D412yes—
49C4×D173—22no—
50C4×D173—42no—
51C4×D173—42no—
52C23×C173—21no—
53Q8×C173—41no—
54D4×C173—41no—
55C2×C4×C173—21no—
56C2×C4×C173C22⋊C412yes—
57C2×C4×C173C22×C411yescharacteristic
58C22×C173—41no—
59C22×C173—42no—
60C22×C173C2×C22⋊C411yescharacteristic
61C2×D173—22no—
62C2×D173—41no—
63C2×D173—42no—
64C2×D173—84no—
65C2×D173#10012yes—
66C4×C173—21no—
67C4×C173—22no—
68C4×C173—41no—
69Dic173—21no—
70Dic173—22no—
71Dic173—41no—
72C2×C173—81no—
73C2×C173C2×#10011yescharacteristic
74C2×C173#42112yes—
75D173—22no—
76D173—81no—
77C173#153311yescharacteristic, complemented, Sylow
78#1533—1731nomaximal, Sylow
79C2×#124—1731no—
80#432—1732no—
81#329—1732no—
82#329—1732no—
83C22×Q8—1731no—
84C22×D4—1731no—
85#124—3462no—
86#124—1731no—
87#124—1731no—
88#124—1731no—
89#124#1062192411yescharacteristic
90C2×C22⋊C4—3462no—
91C2×C42—1731no—
92C4⋊C8—3462no—
93C22⋊C8—3462no—
94#97—1734no—
95C24—3461no—
96C2×Q8—3461no—
97C2×Q8—1731no—
98C2×Q8—6922no—
99C2×Q8C2×#1062192411yescharacteristic
100C2×D4—3461no—
101C2×D4—1731no—
102C2×D4—6922no—
103C2×D4C2×#1062192411yescharacteristic
104C22×C4—3461no—
105C22×C4—1731no—
106C22×C4—1732no—
107C2×C8—3464no—
108C22⋊C4—22no—
109C22⋊C4—3462no—
110C22⋊C4—3462no—
111C22⋊C4—3462no—
112C42—1732no—
113C42—1731no—
114C42C2×#1062192411yescharacteristic
115C23—21no—
116C23—3461no—
117C23—1731no—
118C23—6924no—
119C23—6922no—
120Q8—41no—
121Q8—6921no—
122Q8—6922no—
123D4—41no—
124D4—6921no—
125D4—6922no—
126C2×C4—21no—
127C2×C4—3462no—
128C2×C4—3461no—
129C2×C4—1731no—
130C2×C4—1732no—
131C2×C4—6922no—
132C2×C4—6922no—
133C2×C4C22⋊#1062192412yes—
134C2×C4C22×#1062192411yescharacteristic
135C8—6924no—
136C22—41no—
137C22—42no—
138C22—3462no—
139C22—13844no—
140C22—1732no—
141C22—6921no—
142C22—6922no—
143C22C2×C22⋊#1062192411yescharacteristic
144C4—21no—
145C4—22no—
146C4—41no—
147C4—3461no—
148C4—3462no—
149C4—6921no—
150C2—81no—
151C2—3462no—
152C2—13841no—
153C2D173⋊#10011yescharacteristic
154C2#2601248812yes—
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C4
cosocleC4×Dic173C23
hypercenter#124#10621924
hypocenterC173#1533
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012513C2semidirect
2. #26012347C2semidirect
3. #26012346C2semidirect
7. #19149734C22semidirect
77. C173#1533semidirect

Maximal subgroups

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

subgroupindex
#260125132
#260123472
#260123462
#124×D1732
#1533173