Finite Group Databasebeta

← #36369845

Subgroups — #36369845

Order 22144.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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#26012470C212yesmaximal
2#26012397C214yesmaximal, complemented
3#26012065C211yesmaximal, characteristic
4#19149758—22no—
5#19149738C2214yes—
6#19149737—22no—
7C2×#14583826C2211yescharacteristic
8#19149677C412yescomplemented
9#19149668C2212yes—
10#19149662—22no—
11#19149655—22no—
12#19149638C411yescharacteristic, complemented
13C22×Q8×C173C411yescharacteristic, complemented
14C4⋊#10621924—42no—
15C22⋊#10621924—24no—
16C4×#10621924—42no—
17#14583826—42no—
18C2×Dic346C2×C411yescharacteristic
19C2×D692C2311yescharacteristic
20C2×C22⋊D173D412yes—
21C2×C4×D173D412yes—
22C4⋊Dic173—21no—
23#14583805—24no—
24C22⋊Dic173—21no—
25#14583803—24no—
26C4×Dic173C2×C412yes—
27C2×Q8×C173—21no—
28C2×Q8×C173—42no—
29C2×Q8×C173C2×C412yes—
30C22×C4×C173—21no—
31C22×C4×C173C2×C411yescharacteristic
32C2×#10621924—44no—
33C22×D173C2×D412yes—
34Dic346—41no—
35D692—41no—
36C22⋊D173—42no—
37C2×Dic173—22no—
38C2×Dic173C22⋊C412yes—
39C4×D173—22no—
40C4×D173—22no—
41C23×C173C22⋊C411yescharacteristic
42Q8×C173—42no—
43Q8×C173—42no—
44C2×C4×C173—22no—
45C2×C4×C173—22no—
46C2×C4×C173—42no—
47C2×C4×C173C22⋊C411yescharacteristic
48C2×C4×C173C22×C411yescharacteristic
49C22×C173—21no—
50C22×C173—41no—
51C22×C173C2×C22⋊C411yescharacteristic
52C2×D173—22no—
53C2×D173—42no—
54#10621924—84no—
55C4×C173—21no—
56C4×C173—42no—
57C4×C173#9912yes—
58Dic173—22no—
59Dic173—42no—
60C2×C173—21no—
61C2×C173—41no—
62C2×C173C2×#9911yescharacteristic
63D173—42no—
64C173#212911yescharacteristic, complemented, Sylow
65#2129—1731nomaximal, Sylow
66#536—1731no—
67#410—1732no—
68#354—1734no—
69C2×#41—1731no—
70C22×Q8#1062192411yescharacteristic, complemented
71C4⋊Q8—1732no—
72#124—1732no—
73#123—3462no—
74C22⋊Q8—3462no—
75#117—3462no—
76#117—1731no—
77#99—3462no—
78#99—1734no—
79#41—6922no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8—1731no—
83C2×Q8C2×#1062192412yes—
84C2×D4—1732no—
85C2×D4—1731no—
86C22×C4—21no—
87C22×C4—1732no—
88C22×C4C2×#1062192411yescharacteristic
89C4⋊C4—3464no—
90C4⋊C4—3461no—
91C4⋊C4—6922no—
92C22⋊C4—3464no—
93C22⋊C4—3461no—
94C22⋊C4—3464no—
95C42—1732no—
96C42—6922no—
97C23—1732no—
98C23C22⋊#1062192411yescharacteristic
99Q8—42no—
100Q8—42no—
101Q8—6921no—
102D4—6921no—
103D4—6922no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—3462no—
108C2×C4—3462no—
109C2×C4—3462no—
110C2×C4—1732no—
111C2×C4—6924no—
112C2×C4C22⋊#1062192411yescharacteristic
113C2×C4C22×#1062192411yescharacteristic
114C22—21no—
115C22—41no—
116C22—3462no—
117C22—6922no—
118C22C2×C22⋊#1062192411yescharacteristic
119C4—21no—
120C4—42no—
121C4—3462no—
122C4—13844no—
123C4—6922no—
124C4#1914973712yes—
125C2—21no—
126C2—41no—
127C2—6922no—
128C2C2×#1914973711yescharacteristic
subgroupG/H
centerC2C2×#19149737
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×Q8×C173C4
socleC2×C173C2×#99
prime radicalC2×#14583826C22
cosocleC2×D692C23
hypercenterC22×Q8#10621924
hypocenterC173#2129
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #26012397C2semidirect
8. #19149677C4semidirect
12. #19149638C4semidirect
13. C22×Q8×C173C4semidirect
64. C173#2129semidirect
70. C22×Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124702
#260123972
#260120652
#2129173