Finite Group Databasebeta

← #37549142

Subgroups — #37549142

Order 23168.

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#26042302C211yesmaximal, characteristic, complemented
2#26042224C211yesmaximal, characteristic, complemented
3#26042131C214yesmaximal, complemented
4#26041830C211yesmaximal, characteristic
5#19153883—21no—
6#19153873C2212yes—
7#19153871—21no—
8(C2×#10624382)⋊C4—21no—
9(C2×#10624382)⋊C4—21no—
10(C2×#10624382)⋊C4C2212yes—
11C2×#15668450—21no—
12#19153813C411yescharacteristic, complemented
13#19153805—21no—
14#19153791—21no—
15#19153791C2212yes—
16#19153778—21no—
17#19153778C412yescomplemented
18C42×D181C2211yescharacteristic
19C4⋊Q8×C181C411yescharacteristic, complemented
20C22×#10624382—22no—
21C2×#13089685—22no—
22#15668450—44no—
23C2×D724—22no—
24C2×D724—21no—
25C2×D724—21no—
26C2×D724C2×C412yes—
27C2×C4×D181—22no—
28C2×C4×D181D411yescharacteristic
29C2×C4×D181D411yescharacteristic
30C2×C4×D181C2311yescharacteristic
31#15668426—22no—
32#15668426—22no—
33C4×Dic181C2×C411yescharacteristic
34C4×Dic181D411yescharacteristic
35C4×Dic181D411yescharacteristic
36C2×Q8×C181—21no—
37C4⋊C4×C181—21no—
38C4⋊C4×C181C2×C412yes—
39C42×C181C2×C411yescharacteristic
40C2×#10624382—42no—
41C2×#10624382—42no—
42C2×#10624382—42no—
43#13089685—42no—
44C22×D181—22no—
45C22×D181C2×D411yescharacteristic
46D724—42no—
47D724—42no—
48D724—42no—
49D724—44no—
50C2×Dic181C22⋊C411yescharacteristic
51C2×Dic181C22⋊C411yescharacteristic
52C2×Dic181C2×D411yescharacteristic
53C4×D181—22no—
54C4×D181—41no—
55C4×D181—41no—
56C4×D181—44no—
57Q8×C181—42no—
58C2×C4×C181—22no—
59C2×C4×C181C22⋊C411yescharacteristic
60C2×C4×C181C22⋊C411yescharacteristic
61C2×C4×C181C22×C411yescharacteristic
62C22×C181C2×C22⋊C411yescharacteristic
63C2×D181—21no—
64C2×D181—21no—
65C2×D181—42no—
66C2×D181—42no—
67C2×D181—42no—
68C2×D181AGL(1,ℤ/8)12yes—
69#10624382—82no—
70C4×C181—21no—
71C4×C181—21no—
72C4×C181—21no—
73C4×C181—42no—
74Dic181—21no—
75Dic181—21no—
76Dic181—21no—
77C2×C181#41011yescharacteristic
78C2×C181#41811yescharacteristic
79C2×C181#42111yescharacteristic
80D181—22no—
81D181—82no—
82C181#151411yescharacteristic, complemented, Sylow
83#1514—1811nomaximal, Sylow
84#532—1811no—
85#432—1811no—
86#383—1811no—
87#327—1814no—
88C2×#41—3621no—
89C4⋊Q8#1062438211yescharacteristic, complemented
90#127—1811no—
91#124—3621no—
92#121—3621no—
93#121—1812no—
94C4×D4—3621no—
95C4×D4—1812no—
96C2×C42—1811no—
97C4⋊C8—3621no—
98C22⋊C8—1812no—
99#97—3621no—
100#95—3621no—
101#95—3621no—
102#95—1812no—
103#41—7244no—
104C2×Q8—21no—
105C2×D4—3622no—
106C2×D4—3621no—
107C2×D4—3621no—
108C2×D4—1812no—
109C22×C4—3622no—
110C22×C4—3622no—
111C22×C4—1811no—
112C22×C4—1811no—
113C22×C4—1811no—
114C2×C8—3622no—
115C4⋊C4—21no—
116C4⋊C4C2×#1062438212yes—
117C22⋊C4—3622no—
118C22⋊C4—3622no—
119C42—1811no—
120C42—1811no—
121C42—1811no—
122C42C2×#1062438211yescharacteristic
123C23—3622no—
124C23—1811no—
125Q8—42no—
126D4—7242no—
127D4—7242no—
128D4—7242no—
129D4—7244no—
130C2×C4—22no—
131C2×C4—3622no—
132C2×C4—1811no—
133C2×C4—1811no—
134C2×C4—1811no—
135C2×C4—7241no—
136C2×C4—7241no—
137C2×C4—7244no—
138C2×C4—7242no—
139C2×C4—7242no—
140C2×C4—7242no—
141C2×C4C22⋊#1062438211yescharacteristic
142C2×C4C22⋊#1062438211yescharacteristic
143C2×C4C22×#1062438211yescharacteristic
144C8—7242no—
145C22—3621no—
146C22—3621no—
147C22—1812no—
148C22—7242no—
149C22—7242no—
150C22—7242no—
151C22C2×C22⋊#1062438211yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—3621no—
157C4—3621no—
158C4—3621no—
159C4—14482no—
160C2—3622no—
161C2—14482no—
162C2#2604226011yescharacteristic
163C2#2604227111yescharacteristic
164C2#2604227811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4⋊Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041830C2
cosocleC2×C4×D181C23
hypercenterC4⋊Q8#10624382
hypocenterC181#1514
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26042302C2semidirect
2. #26042224C2semidirect
3. #26042131C2semidirect
12. #19153813C4semidirect
17. #19153778C4semidirect
19. C4⋊Q8×C181C4semidirect
82. C181#1514semidirect
89. C4⋊Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423022
#260422242
#260421312
#260418302
#1514181