Finite Group Databasebeta

← #37535739

Subgroups — #37535739

Order 23168.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 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#26042021C211yesmaximal, characteristic, complemented
2#26041454C212yesmaximal, complemented
3#26041451C211yesmaximal, characteristic, complemented
4#26041075C212yesmaximal, complemented
5#355×C181C211yesmaximal, characteristic, complemented
6#19153864C2211yescharacteristic, complemented
7#19153844—22no—
8#19153842—21no—
9#19153805C2212yes—
10#19153804—21no—
11#19153802—22no—
12#19153786—22no—
13#19153784—21no—
14#19153744—21no—
15#19153742—22no—
16#19153735C2211yescharacteristic, complemented
17#124×C181C2211yescharacteristic
18#101×C181C2212yescomplemented
19C23×D181—21no—
20#15668450—41no—
21#15668449—22no—
22#15668449—21no—
23C2×Dic362—21no—
24Q8×D181—41no—
25C2×D724—21no—
26C2×D724D412yes—
27C2×C22⋊D181—22no—
28#15668438—42no—
29#15668437—41no—
30D1448—42no—
31#15668434—41no—
32#15668433—21no—
33C22⋊Dic181—21no—
34#15668426—22no—
35#15668426—22no—
36#15668426—22no—
37C4×Dic181D412yes—
38C2×Q8×C181C2311yescharacteristic
39C2×D4×C181D411yescharacteristic
40M16×C181—22no—
41C22⋊C4×C181—22no—
42C42×C181D411yescharacteristic
43C22×D181—22no—
44C22×D181—82no—
45C22×D181—81no—
46Dic362—21no—
47Dic362—22no—
48Dic362—81no—
49D724—21no—
50D724—22no—
51D724—42no—
52D724—81no—
53C22⋊D181—41no—
54C22⋊D181—82no—
55C2×Dic181—22no—
56C4×D181—42no—
57C4×D181—41no—
58#13089678—41no—
59C23×C181—21no—
60Q8×C181—41no—
61D4×C181—41no—
62C2×C4×C181—21no—
63C2×C4×C181C2×D412yes—
64C2×C4×C181C2×D411yescharacteristic
65C8×C181—42no—
66C22×C181—42no—
67C22×C181—41no—
68C22×C181#12011yescharacteristic
69C2×D181—42no—
70C2×D181—42no—
71C2×D181—41no—
72C2×D181—42no—
73C2×D181—84no—
74C4×C181—22no—
75C4×C181—21no—
76C4×C181—41no—
77Dic181—42no—
78Dic181—41no—
79C2×C181—21no—
80C2×C181—81no—
81C2×C181B(4,𝔽2)11yescharacteristic
82D181—81no—
83D181—82no—
84C181#220211yescharacteristic, complemented, Sylow
85#2202—1811nomaximal, Sylow
86H(3,𝔽4)—1811no—
87#454—1811no—
88#454—1812no—
89#355—1812no—
90#355D18111yescharacteristic, complemented
912−1+4—1811no—
92AGL(1,ℤ/8)—3622no—
93AGL(1,ℤ/8)—3621no—
94#124—3622no—
95#124—3621no—
96#124—1812no—
97#124C2×D18111yescharacteristic
98#120—3622no—
99#120—3621no—
100#104—3622no—
101#104—3621no—
102#101—1811no—
103#101C2×D18112yescomplemented
104C24—3621no—
105#41—3622no—
106#41—3621no—
107#41—7241no—
108C2×Q8—3621no—
109C2×Q8—7241no—
110C2×Q8C22×D18111yescharacteristic
111C2×D4—3621no—
112C2×D4—3622no—
113C2×D4—1812no—
114C2×D4D72411yescharacteristic
115SD16—7242no—
116SD16—7241no—
117D8—7242no—
118D8—7241no—
119M16—22no—
120M16—3621no—
121C22⋊C4—22no—
122C22⋊C4—3622no—
123C22⋊C4—3621no—
124C22⋊C4—3622no—
125C22⋊C4—3622no—
126C42—1812no—
127C42D72411yescharacteristic
128C23—21no—
129C23—3622no—
130C23—14482no—
131C23—14481no—
132Q8—41no—
133Q8—3622no—
134Q8—3621no—
135Q8—14481no—
136D4—41no—
137D4—3622no—
138D4—3621no—
139D4—14481no—
140D4—14482no—
141D4—7242no—
142D4—7241no—
143C2×C4—21no—
144C2×C4—3622no—
145C2×C4—7242no—
146C2×C4—7241no—
147C2×C4D4×D18112yes—
148C2×C4C2×D72411yescharacteristic
149C8—42no—
150C8—7241no—
151C22—42no—
152C22—41no—
153C22—14484no—
154C22—7242no—
155C22—7242no—
156C22—7241no—
157C22—7242no—
158C22#1915378611yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—7242no—
163C4—7241no—
164C2—21no—
165C2—81no—
166C2—14482no—
167C2—14481no—
168C2#2604146711yescharacteristic
subgroupG/H
centerC2#26041467
commutator, cosocleC2×Q8×C181C23
FrattiniC2×Q8C22×D181
Fitting#355×C181C2
socleC2×C181B(4,𝔽2)
hypercenter#355D181
hypocenterC181#2202
LCSG ▷ C2×Q8×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #26042021C2semidirect
2. #26041454C2semidirect
3. #26041451C2semidirect
4. #26041075C2semidirect
5. #355×C181C2semidirect
6. #19153864C22semidirect
16. #19153735C22semidirect
18. #101×C181C22semidirect
84. C181#2202semidirect
90. #355D181semidirect
103. #101C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420212
#260414542
#260414512
#260410752
#355×C1812
#2202181