Finite Group Databasebeta

← #36369864

Subgroups — #36369864

Order 22144.

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

normal subgroups only

This group has 262 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#26012477C212yesmaximal, complemented
2#26012400C214yesmaximal, complemented
3#26012060C211yesmaximal, characteristic
4D173⋊M16—22no—
5#19149740C2214yescomplemented
6#19149739—22no—
7C2×#14583826C2211yescharacteristic
8C22×D692C411yescharacteristic
9#19149676C412yes—
10#19149667C2212yes—
11#19149654—22no—
12#19149649—22no—
13#117×C173C411yescharacteristic
14#14583836—42no—
15#14583835—24no—
16D173⋊C8—42no—
17C23×D173—21no—
18#14583826—42no—
19C2×Dic346C2311yescharacteristic
20C2×D692—24no—
21C2×D692—21no—
22C2×D692—42no—
23C2×D692C2×C412yes—
24C2×D692C2×C411yescharacteristic
25C2×C22⋊D173D412yes—
26C2×C4×D173D412yes—
27#14583803—24no—
28C22×C4×C173C2×C411yescharacteristic
29C4⋊C4×C173—21no—
30C22⋊C4×C173—21no—
31C42×C173C2×C412yes—
32#13063443—44no—
33C22×D173—22no—
34C22×D173—42no—
35C22×D173—82no—
36C22×D173C22⋊C412yes—
37Dic346—41no—
38D692—41no—
39D692—42no—
40D692—42no—
41D692—84no—
42C22⋊D173—42no—
43C2×Dic173C2×D412yes—
44C4×D173—22no—
45C4×D173—22no—
46C23×C173C22⋊C411yescharacteristic
47C2×C4×C173—22no—
48C2×C4×C173—22no—
49C2×C4×C173C22⋊C411yescharacteristic
50C2×C4×C173C22×C411yescharacteristic
51C22×C173—21no—
52C22×C173—41no—
53C22×C173C2×C22⋊C411yescharacteristic
54C2×D173—22no—
55C2×D173—42no—
56C2×D173—44no—
57C2×D173—42no—
58C2×D173—84no—
59C4×C173—21no—
60C4×C173—42no—
61C4×C173#9912yes—
62Dic173—22no—
63C2×C173—21no—
64C2×C173—41no—
65C2×C173C2×#9911yescharacteristic
66D173—42no—
67D173—82no—
68C173#213411yescharacteristic, complemented, Sylow
69#2134—1731nomaximal, Sylow
70#535—1731no—
71#413—1732no—
72#355—1734no—
73C2×#41—1731no—
74C22×D4—1731no—
75C2×M16—3462no—
76#127—1732no—
77#124—1732no—
78#121—3462no—
79#120—3462no—
80#117#1062192411yescharacteristic
81#101—1734no—
82#100—3462no—
83C24—3461no—
84#41—6922no—
85C2×Q8—1731no—
86C2×D4—3464no—
87C2×D4—3461no—
88C2×D4—1732no—
89C2×D4—1732no—
90C2×D4—1731no—
91C2×D4—6922no—
92C22×C4—1732no—
93C22×C4C2×#1062192411yescharacteristic
94M16—3464no—
95M16—6922no—
96C2×C8—6922no—
97C4⋊C4—21no—
98C22⋊C4—21no—
99C22⋊C4—3464no—
100C42C2×#1062192412yes—
101C23—3462no—
102C23—13842no—
103C23—1732no—
104C23—6922no—
105C23C22⋊#1062192411yescharacteristic
106Q8—6921no—
107D4—13844no—
108D4—6921no—
109D4—6922no—
110D4—6922no—
111D4—6922no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—3462no—
115C2×C4—3462no—
116C2×C4—1732no—
117C2×C4C22⋊#1062192411yescharacteristic
118C2×C4C22×#1062192411yescharacteristic
119C8—6924no—
120C22—21no—
121C22—41no—
122C22—3462no—
123C22—13844no—
124C22—6924no—
125C22—6922no—
126C22—6922no—
127C22C2×C22⋊#1062192411yescharacteristic
128C4—21no—
129C4—42no—
130C4—3462no—
131C4#1914973812yes—
132C2—21no—
133C2—41no—
134C2—13842no—
135C2—6922no—
136C2C2×#1914973811yescharacteristic
subgroupG/H
centerC2C2×#19149738
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#117×C173C4
socleC2×C173C2×#99
prime radical#26012060C2
cosocleC2×Dic346C23
hypercenter#117#10621924
hypocenterC173#2134
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #26012477C2semidirect
2. #26012400C2semidirect
5. #19149740C22semidirect
68. C173#2134semidirect

Maximal subgroups

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

subgroupindex
#260124772
#260124002
#260120602
#2134173