Finite Group Databasebeta

← #36352109

Subgroups — #36352109

Order 22144.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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#26012041C211yesmaximal, characteristic
2#26011548C211yesmaximal, characteristic, complemented
3#26011224C214yesmaximal, complemented
4#383×C173C211yesmaximal, characteristic, complemented
5C2×#14583826—21no—
6#19149679C411yescharacteristic, complemented
7#19149676C411yescharacteristic, complemented
8#19149667—21no—
9#19149653—21no—
10#19149653C2212yes—
11C4×D692—21no—
12C4×D692C412yescomplemented
13C4⋊#13063436—21no—
14C22⋊#13063436C2212yes—
15#19149590—21no—
16C2×C42×C173C2211yescharacteristic
17#95×C173—21no—
18#95×C173—21no—
19#95×C173C2212yes—
20#14583826—44no—
21C2×Dic346—21no—
22C2×D692—21no—
23C2×D692—21no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—22no—
26C2×C4×D173—22no—
27C2×#13063436—22no—
28C4⋊Dic173—21no—
29C4⋊Dic173C2×C412yes—
30#14583803—22no—
31#14583803—22no—
32C22×C4×C173—22no—
33C22×C4×C173D411yescharacteristic
34C22×C4×C173D411yescharacteristic
35C22×C4×C173C2311yescharacteristic
36C42×C173C2×C411yescharacteristic
37C42×C173C2×C411yescharacteristic
38C42×C173D411yescharacteristic
39C42×C173D411yescharacteristic
40C22×D173—22no—
41Dic346—42no—
42D692—42no—
43D692—42no—
44D692—42no—
45C22⋊D173—44no—
46C2×Dic173—22no—
47C4×D173—44no—
48#13063436—42no—
49C23×C173C2×D411yescharacteristic
50C2×C4×C173—22no—
51C2×C4×C173—41no—
52C2×C4×C173—41no—
53C2×C4×C173—42no—
54C2×C4×C173—42no—
55C2×C4×C173—42no—
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×C173C2×C22⋊C411yescharacteristic
65C22×C173AGL(1,ℤ/8)12yes—
66C2×D173—42no—
67C2×D173—42no—
68C2×D173—42no—
69C4×C173—21no—
70C4×C173—21no—
71C4×C173—21no—
72C4×C173—21no—
73C4×C173—21no—
74C4×C173—21no—
75C4×C173—82no—
76Dic173—42no—
77C2×C173—22no—
78C2×C173#41011yescharacteristic
79C2×C173#41811yescharacteristic
80C2×C173#42111yescharacteristic
81D173—82no—
82C173#151411yescharacteristic, complemented, Sylow
83#1514—1731nomaximal, Sylow
84#532—1731no—
85#432—1731no—
86#383D17311yescharacteristic, complemented
87#327—1734no—
88C2×#41—3461no—
89C4⋊Q8—1731no—
90#127—1731no—
91#124—3461no—
92#121—3461no—
93#121—1732no—
94C4×D4—3461no—
95C4×D4—1732no—
96C2×C42C2×D17311yescharacteristic
97C4⋊C8—3461no—
98C22⋊C8—1732no—
99#97—3461no—
100#95—21no—
101#95—21no—
102#95C2×D17312yes—
103#41—6924no—
104C2×Q8—3461no—
105C2×D4—3462no—
106C2×D4—3461no—
107C2×D4—3461no—
108C2×D4—1732no—
109C22×C4—22no—
110C22×C4—3462no—
111C22×C4C22⋊D17311yescharacteristic
112C22×C4C22⋊D17311yescharacteristic
113C22×C4C22×D17311yescharacteristic
114C2×C8—3462no—
115C4⋊C4—3461no—
116C4⋊C4—1732no—
117C22⋊C4—3462no—
118C22⋊C4—3462no—
119C42C4×D17311yescharacteristic
120C42C4×D17311yescharacteristic
121C42D69211yescharacteristic
122C42D69211yescharacteristic
123C23—3462no—
124C23C2×C22⋊D17311yescharacteristic
125Q8—6922no—
126D4—6922no—
127D4—6922no—
128D4—6922no—
129D4—6924no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—3462no—
137C2×C4—6924no—
138C2×C4#1458380311yescharacteristic
139C2×C4#1458380311yescharacteristic
140C2×C4#1458380311yescharacteristic
141C2×C4#1458380311yescharacteristic
142C2×C4C2×C4×D17311yescharacteristic
143C2×C4C2×D69211yescharacteristic
144C8—6922no—
145C22—21no—
146C22—21no—
147C22—6922no—
148C22—6922no—
149C22—6922no—
150C22C2×#1458380311yescharacteristic
151C22#1914970412yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—82no—
159C4—6922no—
160C2—22no—
161C2—13842no—
162C2#2601146411yescharacteristic
163C2#2601149511yescharacteristic
164C2#2601150911yescharacteristic
subgroupG/H
centerC22C2×#14583803
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#383×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012041C2
cosocleC22×C4×C173C23
hypercenter#383D173
hypocenterC173#1514
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #383

Complements

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

NH ≅ G/Ntype
2. #26011548C2semidirect
3. #26011224C2semidirect
4. #383×C173C2semidirect
6. #19149679C4semidirect
7. #19149676C4semidirect
12. C4×D692C4semidirect
82. C173#1514semidirect
86. #383D173semidirect

Maximal subgroups

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

subgroupindex
#260120412
#260115482
#260112242
#383×C1732
#1514173