Finite Group Databasebeta

← #36368954

Subgroups — #36368954

Order 22144.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#26012513C211yesmaximal, characteristic, complemented
2#26012427C211yesmaximal, characteristic, complemented
3#26012343C214yesmaximal, complemented
4#26012042C211yesmaximal, characteristic
5C2×C4×#10621924—21no—
6C4⋊#13063443—21no—
7#19149736C2212yes—
8#19149734—21no—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4C2212yes—
11C2×#14583827—21no—
12#19149676C411yescharacteristic, complemented
13#19149668—21no—
14#19149654—21no—
15#19149654C2212yes—
16#19149641—21no—
17#19149641C412yescomplemented
18C42×D173C2211yescharacteristic
19C4⋊Q8×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21C2×#13063443—22no—
22C4×#10621924—24no—
23#14583827—44no—
24C2×D692—22no—
25C2×D692—21no—
26C2×D692—21no—
27C2×D692C2×C412yes—
28C2×C4×D173—22no—
29C2×C4×D173D411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C2×C4×D173C2311yescharacteristic
32#14583803—22no—
33#14583803—22no—
34C4×Dic173C2×C411yescharacteristic
35C4×Dic173D411yescharacteristic
36C4×Dic173D411yescharacteristic
37C2×Q8×C173—21no—
38C4⋊C4×C173—21no—
39C4⋊C4×C173C2×C412yes—
40C42×C173C2×C411yescharacteristic
41C2×#10621924—24no—
42C2×#10621924—42no—
43C2×#10621924—42no—
44#13063443—42no—
45C22×D173—22no—
46C22×D173C2×D411yescharacteristic
47D692—42no—
48D692—42no—
49D692—44no—
50D692—42no—
51C2×Dic173C22⋊C411yescharacteristic
52C2×Dic173C22⋊C411yescharacteristic
53C2×Dic173C2×D411yescharacteristic
54C4×D173—22no—
55C4×D173—22no—
56C4×D173—41no—
57C4×D173—44no—
58Q8×C173—42no—
59C2×C4×C173—22no—
60C2×C4×C173C22⋊C411yescharacteristic
61C2×C4×C173C22⋊C411yescharacteristic
62C2×C4×C173C22×C411yescharacteristic
63C22×C173C2×C22⋊C411yescharacteristic
64C2×D173—21no—
65C2×D173—21no—
66C2×D173—42no—
67C2×D173—42no—
68C2×D173—42no—
69C2×D173AGL(1,ℤ/8)12yes—
70#10621924—44no—
71C4×C173—21no—
72C4×C173—21no—
73C4×C173—42no—
74C4×C173#10412yes—
75Dic173—21no—
76Dic173—21no—
77Dic173—21no—
78C2×C173#41011yescharacteristic
79C2×C173#41811yescharacteristic
80C2×C173C2×#10411yescharacteristic
81D173—22no—
82D173—82no—
83C173#150511yescharacteristic, complemented, Sylow
84#1505—1731nomaximal, Sylow
85#532—1731no—
86#432—1731no—
87#376—1731no—
88#327—1734no—
89C2×#41—3461no—
90C4⋊Q8#1062192411yescharacteristic, complemented
91#127—1731no—
92#124—3461no—
93#121—3461no—
94#121—1732no—
95C4×D4—3461no—
96C4×D4—1732no—
97C2×C42—3461no—
98C2×C42—1731no—
99C4⋊C8—3461no—
100C22⋊C8—1732no—
101#97—3461no—
102#95—3461no—
103#95—1732no—
104#41—6924no—
105C2×Q8—21no—
106C2×D4—3462no—
107C2×D4—3461no—
108C2×D4—3461no—
109C2×D4—1732no—
110C22×C4—3462no—
111C22×C4—3462no—
112C22×C4—1731no—
113C22×C4—1731no—
114C22×C4—1731no—
115C2×C8—3462no—
116C4⋊C4—21no—
117C4⋊C4C2×#1062192412yes—
118C22⋊C4—3462no—
119C22⋊C4—3462no—
120C42—3464no—
121C42—1731no—
122C42—1731no—
123C42—1731no—
124C42C2×#1062192411yescharacteristic
125C23—3462no—
126C23—1731no—
127Q8—42no—
128D4—6922no—
129D4—6922no—
130D4—6924no—
131D4—6922no—
132C2×C4—22no—
133C2×C4—3462no—
134C2×C4—3462no—
135C2×C4—3464no—
136C2×C4—1731no—
137C2×C4—1731no—
138C2×C4—1731no—
139C2×C4—6921no—
140C2×C4—6924no—
141C2×C4—6922no—
142C2×C4—6922no—
143C2×C4C22⋊#1062192411yescharacteristic
144C2×C4C22⋊#1062192411yescharacteristic
145C2×C4C22×#1062192411yescharacteristic
146C8—6922no—
147C22—3461no—
148C22—3461no—
149C22—1732no—
150C22—6922no—
151C22—6922no—
152C22—6922no—
153C22C2×C22⋊#1062192411yescharacteristic
154C4—21no—
155C4—21no—
156C4—42no—
157C4—3461no—
158C4—3461no—
159C4—3461no—
160C4—6924no—
161C4#1914974412yes—
162C2—3462no—
163C2—13842no—
164C2#2601247211yescharacteristic
165C2#2601248311yescharacteristic
166C2C2×#1914974411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012042C2
cosocleC2×C4×D173C23
hypercenterC4⋊Q8#10621924
hypocenterC173#1505
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26012513C2semidirect
2. #26012427C2semidirect
3. #26012343C2semidirect
12. #19149676C4semidirect
17. #19149641C4semidirect
19. C4⋊Q8×C173C4semidirect
83. C173#1505semidirect
90. C4⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125132
#260124272
#260123432
#260120422
#1505173