Finite Group Databasebeta

← #36352110

Subgroups — #36352110

Order 22144.

252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 77 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#26011548C211yesmaximal, characteristic, complemented
2#26011359C211yesmaximal, characteristic, complemented
3#26011225C214yesmaximal, complemented
4#532×C173C211yesmaximal, characteristic
5C4⋊#13063436—21no—
6C22⋊#13063436C2212yes—
7#19149590—21no—
8#19149586—21no—
9#19149586—21no—
10#19149586C2212yes—
11C2×#41×C173—21no—
12C4⋊Q8×C173C411yescharacteristic, complemented
13#127×C173C411yescharacteristic, complemented
14#124×C173—21no—
15#121×C173—21no—
16#121×C173C2212yes—
17C4×D4×C173—21no—
18C4×D4×C173C412yescomplemented
19C2×C42×C173C2211yescharacteristic
20C22×Dic173—22no—
21C2×#13063436—22no—
22#41×C173—44no—
23C2×Q8×C173—21no—
24C2×D4×C173—22no—
25C2×D4×C173—21no—
26C2×D4×C173—21no—
27C2×D4×C173C2×C412yes—
28C22×C4×C173—22no—
29C22×C4×C173D411yescharacteristic
30C22×C4×C173D411yescharacteristic
31C22×C4×C173C2311yescharacteristic
32C4⋊C4×C173—21no—
33C4⋊C4×C173C2×C412yes—
34C22⋊C4×C173—22no—
35C22⋊C4×C173—22no—
36C42×C173C2×C411yescharacteristic
37C42×C173C2×C411yescharacteristic
38C42×C173D411yescharacteristic
39C42×C173D411yescharacteristic
40C2×Dic173—42no—
41C2×Dic173—42no—
42C2×Dic173—42no—
43#13063436—42no—
44C23×C173—22no—
45C23×C173C2×D411yescharacteristic
46Q8×C173—42no—
47D4×C173—42no—
48D4×C173—42no—
49D4×C173—42no—
50D4×C173—44no—
51C2×C4×C173—22no—
52C2×C4×C173—22no—
53C2×C4×C173—41no—
54C2×C4×C173—41no—
55C2×C4×C173—44no—
56C2×C4×C173C22⋊C411yescharacteristic
57C2×C4×C173C22⋊C411yescharacteristic
58C2×C4×C173C22⋊C411yescharacteristic
59C2×C4×C173C22⋊C411yescharacteristic
60C2×C4×C173C22×C411yescharacteristic
61C2×C4×C173C2×D411yescharacteristic
62C22×C173—21no—
63C22×C173—21no—
64C22×C173—42no—
65C22×C173—42no—
66C22×C173—42no—
67C22×C173C2×C22⋊C411yescharacteristic
68C22×C173AGL(1,ℤ/8)12yes—
69C4×C173—21no—
70C4×C173—21no—
71C4×C173—21no—
72C4×C173—21no—
73C4×C173—21no—
74C4×C173—21no—
75C4×C173—42no—
76Dic173—82no—
77C2×C173—22no—
78C2×C173—82no—
79C2×C173#41011yescharacteristic
80C2×C173#41811yescharacteristic
81C2×C173#42111yescharacteristic
82C173#151411yescharacteristic, complemented, Sylow
83#1514—1731nomaximal, Sylow
84#532D17311yescharacteristic
85#432—1731no—
86#383—1731no—
87#327—1734no—
88C2×#41—21no—
89C4⋊Q8Dic17311yescharacteristic, complemented
90#127Dic17311yescharacteristic, complemented
91#124—21no—
92#121—21no—
93#121C2×D17312yes—
94C4×D4—21no—
95C4×D4Dic17312yescomplemented
96C2×C42C2×D17311yescharacteristic
97C4⋊C8—3461no—
98C22⋊C8—1732no—
99#97—3461no—
100#95—3461no—
101#95—3461no—
102#95—1732no—
103#41—44no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4C2×Dic17312yes—
109C22×C4—22no—
110C22×C4—3462no—
111C22×C4C22⋊D17311yescharacteristic
112C22×C4C22⋊D17311yescharacteristic
113C22×C4C22×D17311yescharacteristic
114C2×C8—3462no—
115C4⋊C4—21no—
116C4⋊C4C2×Dic17312yes—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C42C2×Dic17311yescharacteristic
120C42C2×Dic17311yescharacteristic
121C42C22⋊D17311yescharacteristic
122C42C22⋊D17311yescharacteristic
123C23—22no—
124C23C2×C22⋊D17311yescharacteristic
125Q8—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—44no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—6922no—
136C2×C4—6922no—
137C2×C4—6922no—
138C2×C4C22⋊Dic17311yescharacteristic
139C2×C4C22⋊Dic17311yescharacteristic
140C2×C4C22⋊Dic17311yescharacteristic
141C2×C4C22⋊Dic17311yescharacteristic
142C2×C4C22×Dic17311yescharacteristic
143C2×C4C2×C22⋊D17311yescharacteristic
144C8—6922no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22C2×C22⋊Dic17311yescharacteristic
151C22#1914970512yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—13842no—
160C2—22no—
161C2—82no—
162C2#2601146511yescharacteristic
163C2#2601149611yescharacteristic
164C2#2601150811yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radical#532×C173C2
socleC22×C173C2×C22⋊C4
cosocleC22×C4×C173C23
hypercenter#532D173
hypocenterC173#1514
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26011548C2semidirect
2. #26011359C2semidirect
3. #26011225C2semidirect
12. C4⋊Q8×C173C4semidirect
13. #127×C173C4semidirect
18. C4×D4×C173C4semidirect
82. C173#1514semidirect
89. C4⋊Q8Dic173semidirect
90. #127Dic173semidirect
95. C4×D4Dic173semidirect

Maximal subgroups

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

subgroupindex
#260115482
#260113592
#260112252
#532×C1732
#1514173