Finite Group Databasebeta

← #36369865

Subgroups — #36369865

Order 22144.

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

normal subgroups only

This group has 230 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
2#26012401C214yesmaximal, complemented
3#26012065C211yesmaximal, characteristic
4D173⋊M16—22no—
5#19149740C2214yes—
6#19149739—22no—
7C2×#14583826C2211yescharacteristic
8#19149677C2212yes—
9#19149668C412yes—
10#19149662—22no—
11#19149655—22no—
12#19149638C411yescharacteristic
13C22×Q8×C173C411yescharacteristic
14#14583836—42no—
15#14583835—24no—
16D173⋊C8—42no—
17#14583826—42no—
18C2×Dic346C2311yescharacteristic
19C2×D692C2×C411yescharacteristic
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
32#13063443—44no—
33C22×D173C22⋊C412yes—
34Dic346—41no—
35D692—41no—
36C22⋊D173—42no—
37C2×Dic173—22no—
38C2×Dic173C2×D412yes—
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—
54C4×C173—21no—
55C4×C173—42no—
56C4×C173#9912yes—
57Dic173—22no—
58Dic173—42no—
59C2×C173—21no—
60C2×C173—41no—
61C2×C173C2×#9911yescharacteristic
62D173—42no—
63C173#213511yescharacteristic, complemented, Sylow
64#2135—1731nomaximal, Sylow
65#536—1731no—
66#413—1732no—
67#356—1734no—
68C2×#41—1731no—
69C22×Q8#1062192411yescharacteristic
70C2×M16—3462no—
71C4⋊Q8—1732no—
72#124—1732no—
73#123—3462no—
74C22⋊Q8—3462no—
75#117—1731no—
76#101—1734no—
77#100—3462no—
78#41—6922no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8—1731no—
82C2×Q8C2×#1062192412yes—
83C2×D4—1732no—
84C2×D4—1731no—
85C22×C4—21no—
86C22×C4—1732no—
87C22×C4C2×#1062192411yescharacteristic
88M16—3464no—
89M16—6922no—
90C2×C8—6922no—
91C4⋊C4—3464no—
92C4⋊C4—3461no—
93C22⋊C4—3464no—
94C22⋊C4—3461no—
95C42—1732no—
96C23—1732no—
97C23C22⋊#1062192411yescharacteristic
98Q8—42no—
99Q8—42no—
100Q8—6921no—
101D4—6921no—
102D4—6922no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—42no—
106C2×C4—3462no—
107C2×C4—3462no—
108C2×C4—3462no—
109C2×C4—1732no—
110C2×C4C22⋊#1062192411yescharacteristic
111C2×C4C22×#1062192411yescharacteristic
112C8—6924no—
113C22—21no—
114C22—41no—
115C22—3462no—
116C22—6922no—
117C22C2×C22⋊#1062192411yescharacteristic
118C4—21no—
119C4—42no—
120C4—3462no—
121C4—6922no—
122C4#1914973712yes—
123C2—21no—
124C2—41no—
125C2—6922no—
126C2C2×#1914973711yescharacteristic
subgroupG/H
centerC2C2×#19149737
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×Q8×C173C4
socleC2×C173C2×#99
prime radicalC2×#14583826C22
cosocleC2×Dic346C23
hypercenterC22×Q8#10621924
hypocenterC173#2135
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. #26012401C2semidirect
63. C173#2135semidirect

Maximal subgroups

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

subgroupindex
#260124772
#260124012
#260120652
#2135173