Finite Group Databasebeta

← #36352275

Subgroups — #36352275

Order 22144.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#127×D173C211yesmaximal, characteristic
2#26011547C212yesmaximal, complemented
3#26011235C211yesmaximal, characteristic, complemented
4#26011234C212yesmaximal, complemented
5#331×C173C211yesmaximal, characteristic, complemented
6C2×D4×D173—22no—
7C2×D4×D173C412yes—
8#19149676C2211yescharacteristic
9#19149675—22no—
10#19149675C412yes—
11C42×D173C2211yescharacteristic
12C4⋊#13063436C2212yescomplemented
13#19149591—22no—
14#19149589—22no—
15#127×C173C2211yescharacteristic
16C4⋊C8×C173C2212yescomplemented
17C23×D173—22no—
18C2×D692—22no—
19C2×D692C2×C412yes—
20C2×C22⋊D173—22no—
21C2×C22⋊D173—24no—
22D4×D173—44no—
23D4×D173—44no—
24D4×D173—44no—
25C2×C4×D173C2×C411yescharacteristic
26C2×C4×D173D412yes—
27C2×#13063436—22no—
28C4×Dic173C2×C411yescharacteristic
29C4×Dic173D412yes—
30C2×D4×C173—22no—
31C2×D4×C173C2×C412yes—
32C2×C8×C173—22no—
33C42×C173C2311yescharacteristic
34C22×D173—22no—
35C22×D173—48no—
36C22×D173—44no—
37C22×D173C22⋊C411yescharacteristic
38D692—42no—
39D692—42no—
40D692—42no—
41C22⋊D173—44no—
42C22⋊D173—48no—
43C2×Dic173C22⋊C411yescharacteristic
44C2×Dic173C22⋊C412yes—
45C4×D173—22no—
46C4×D173—22no—
47C4×D173—22no—
48#13063436—42no—
49C23×C173—22no—
50D4×C173—42no—
51D4×C173—42no—
52D4×C173—42no—
53C2×C4×C173C22×C411yescharacteristic
54C2×C4×C173C2×D412yes—
55C8×C173—42no—
56C22×C173—42no—
57C22×C173—44no—
58C22×C173C2×C22⋊C411yescharacteristic
59C2×D173—22no—
60C2×D173—42no—
61C2×D173—44no—
62C2×D173—88no—
63C2×D173#10012yes—
64C4×C173—21no—
65C4×C173AGL(1,ℤ/8)12yes—
66C4×C173AGL(1,ℤ/8)12yes—
67Dic173—21no—
68Dic173—22no—
69C2×C173—82no—
70C2×C173C2×#10011yescharacteristic
71C2×C173#41812yes—
72D173—22no—
73D173—82no—
74C173#155511yescharacteristic, complemented, Sylow
75#1555—1731nomaximal, Sylow
76C2×#127—1731no—
77#432—1732no—
78#331—1731no—
79#331—1732no—
80#331D17311yescharacteristic, complemented
81C22×D4—3462no—
82C22×D4—1732no—
83#127—3462no—
84#127—1732no—
85#127—1731no—
86#127C2×D17311yescharacteristic
87C2×C42—1731no—
88C4⋊C8—1732no—
89C4⋊C8C2×D17312yescomplemented
90C22⋊C8—3462no—
91#97—3462no—
92C24—3462no—
93C2×D4—22no—
94C2×D4—3462no—
95C2×D4—3464no—
96C2×D4—3462no—
97C2×D4—1732no—
98C2×D4—6924no—
99C2×D4—6924no—
100C2×D4—6924no—
101C2×D4C4×D17312yes—
102C22×C4—1731no—
103C22×C4—1732no—
104C2×C8—22no—
105C2×C8—3462no—
106C42—1731no—
107C42—1732no—
108C42C22×D17311yescharacteristic
109C23—22no—
110C23—3462no—
111C23—1731no—
112C23—6928no—
113C23—6924no—
114D4—42no—
115D4—42no—
116D4—42no—
117D4—6922no—
118D4—6924no—
119D4—6928no—
120D4—6922no—
121D4—6922no—
122C2×C4—3462no—
123C2×C4—3462no—
124C2×C4—3462no—
125C2×C4—1731no—
126C2×C4—1732no—
127C2×C4C2×C4×D17311yescharacteristic
128C2×C4D4×D17312yes—
129C8—42no—
130C8—6922no—
131C22—42no—
132C22—44no—
133C22—3462no—
134C22—13848no—
135C22—1732no—
136C22—6922no—
137C22—6924no—
138C22C22⋊C4×D17311yescharacteristic
139C4—21no—
140C4—3461no—
141C4—3462no—
142C4#1914970312yes—
143C4#1914970612yes—
144C2—82no—
145C2—3462no—
146C2—13842no—
147C2#100×D17311yescharacteristic
148C2#2601149412yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
Fitting#331×C173C2
socleC22×C173C2×C22⋊C4
prime radical#127×D173C2
cosocleC42×C173C23
hypercenter#331D173
hypocenterC173#1555
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26011547C2semidirect
3. #26011235C2semidirect
4. #26011234C2semidirect
5. #331×C173C2semidirect
12. C4⋊#13063436C22semidirect
16. C4⋊C8×C173C22semidirect
74. C173#1555semidirect
80. #331D173semidirect
89. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#127×D1732
#260115472
#260112352
#260112342
#331×C1732
#1555173