Finite Group Databasebeta

← #36368984

Subgroups — #36368984

Order 22144.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#26012514C211yesmaximal, characteristic, complemented
2#26012436C211yesmaximal, characteristic, complemented
3#26012343C214yesmaximal, complemented
4#26012042C211yesmaximal, characteristic
5#19149746—21no—
6#19149736C2212yes—
7#19149734—21no—
8(C2×#10621924)⋊C4—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—
22#14583827—44no—
23C2×D692—22no—
24C2×D692—21no—
25C2×D692—21no—
26C2×D692C2×C412yes—
27C2×C4×D173—22no—
28C2×C4×D173D411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C2×C4×D173C2311yescharacteristic
31#14583803—22no—
32#14583803—22no—
33C4×Dic173C2×C411yescharacteristic
34C4×Dic173D411yescharacteristic
35C4×Dic173D411yescharacteristic
36C2×Q8×C173—21no—
37C4⋊C4×C173—21no—
38C4⋊C4×C173C2×C412yes—
39C42×C173C2×C411yescharacteristic
40C2×#10621924—42no—
41C2×#10621924—42no—
42C2×#10621924—42no—
43#13063443—42no—
44C22×D173—22no—
45C22×D173C2×D411yescharacteristic
46D692—42no—
47D692—42no—
48D692—42no—
49D692—44no—
50C2×Dic173C22⋊C411yescharacteristic
51C2×Dic173C22⋊C411yescharacteristic
52C2×Dic173C2×D411yescharacteristic
53C4×D173—22no—
54C4×D173—41no—
55C4×D173—41no—
56C4×D173—44no—
57Q8×C173—42no—
58C2×C4×C173—22no—
59C2×C4×C173C22⋊C411yescharacteristic
60C2×C4×C173C22⋊C411yescharacteristic
61C2×C4×C173C22×C411yescharacteristic
62C22×C173C2×C22⋊C411yescharacteristic
63C2×D173—21no—
64C2×D173—21no—
65C2×D173—42no—
66C2×D173—42no—
67C2×D173—42no—
68C2×D173AGL(1,ℤ/8)12yes—
69#10621924—82no—
70C4×C173—21no—
71C4×C173—21no—
72C4×C173—21no—
73C4×C173—42no—
74Dic173—21no—
75Dic173—21no—
76Dic173—21no—
77C2×C173#41011yescharacteristic
78C2×C173#41811yescharacteristic
79C2×C173#42111yescharacteristic
80D173—22no—
81D173—82no—
82C173#151411yescharacteristic, complemented, Sylow
83#1514—1731nomaximal, Sylow
84#532—1731no—
85#432—1731no—
86#383—1731no—
87#327—1734no—
88C2×#41—3461no—
89C4⋊Q8#1062192411yescharacteristic, complemented
90#127—1731no—
91#124—3461no—
92#121—3461no—
93#121—1732no—
94C4×D4—3461no—
95C4×D4—1732no—
96C2×C42—1731no—
97C4⋊C8—3461no—
98C22⋊C8—1732no—
99#97—3461no—
100#95—3461no—
101#95—3461no—
102#95—1732no—
103#41—6924no—
104C2×Q8—21no—
105C2×D4—3462no—
106C2×D4—3461no—
107C2×D4—3461no—
108C2×D4—1732no—
109C22×C4—3462no—
110C22×C4—3462no—
111C22×C4—1731no—
112C22×C4—1731no—
113C22×C4—1731no—
114C2×C8—3462no—
115C4⋊C4—21no—
116C4⋊C4C2×#1062192412yes—
117C22⋊C4—3462no—
118C22⋊C4—3462no—
119C42—1731no—
120C42—1731no—
121C42—1731no—
122C42C2×#1062192411yescharacteristic
123C23—3462no—
124C23—1731no—
125Q8—42no—
126D4—6922no—
127D4—6922no—
128D4—6922no—
129D4—6924no—
130C2×C4—22no—
131C2×C4—3462no—
132C2×C4—1731no—
133C2×C4—1731no—
134C2×C4—1731no—
135C2×C4—6921no—
136C2×C4—6921no—
137C2×C4—6924no—
138C2×C4—6922no—
139C2×C4—6922no—
140C2×C4—6922no—
141C2×C4C22⋊#1062192411yescharacteristic
142C2×C4C22⋊#1062192411yescharacteristic
143C2×C4C22×#1062192411yescharacteristic
144C8—6922no—
145C22—3461no—
146C22—3461no—
147C22—1732no—
148C22—6922no—
149C22—6922no—
150C22—6922no—
151C22C2×C22⋊#1062192411yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—3461no—
157C4—3461no—
158C4—3461no—
159C4—13842no—
160C2—3462no—
161C2—13842no—
162C2#2601247211yescharacteristic
163C2#2601248311yescharacteristic
164C2#2601249011yescharacteristic
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#1514
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. #26012514C2semidirect
2. #26012436C2semidirect
3. #26012343C2semidirect
12. #19149676C4semidirect
17. #19149641C4semidirect
19. C4⋊Q8×C173C4semidirect
82. C173#1514semidirect
89. C4⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125142
#260124362
#260123432
#260120422
#1514173