Finite Group Databasebeta

← #37535669

Subgroups — #37535669

Order 23168.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#26042079C211yesmaximal, characteristic, complemented
2#26041717C212yesmaximal, complemented
3#26041532C211yesmaximal, characteristic, complemented
4#26041090C212yesmaximal, complemented
5#349×C181C211yesmaximal, characteristic, complemented
6C2×#15668449—21no—
7#19153849—22no—
8#19153844—22no—
9#19153838—21no—
10#19153838C2212yes—
11C2×D1448C2211yescharacteristic, complemented
12#19153790—21no—
13#19153761—22no—
14D2896—22no—
15#19153748C2211yescharacteristic, complemented
16#19153738—21no—
17C2×M16×C181C2211yescharacteristic
18M32×C181C2212yescomplemented
19#15668449—22no—
20#15668449—22no—
21#15668449—41no—
22#15668449—41no—
23C2×Dic362—21no—
24C2×D724—21no—
25C2×D724—21no—
26C2×C22⋊D181—41no—
27C2×C22⋊D181—41no—
28C2×C4×D181—41no—
29Dic724—21no—
30Dic724D412yescomplemented
31#15668438—22no—
32#15668438—22no—
33D1448—21no—
34D1448—22no—
35D1448D412yescomplemented
36C4⋊Dic181—21no—
37#15668426—41no—
38C22×C4×C181D411yescharacteristic
39M16×C181—22no—
40C2×C8×C181D411yescharacteristic
41C2×C8×C181C2311yescharacteristic
42C16×C181—22no—
43C22×D181—41no—
44C22×D181—41no—
45Dic362—22no—
46Dic362—41no—
47D724—22no—
48D724—41no—
49D724—41no—
50D724—41no—
51C22⋊D181—42no—
52C22⋊D181—42no—
53C22⋊D181—82no—
54C2×Dic181—41no—
55C2×Dic181—41no—
56C4×D181—42no—
57C4×D181—42no—
58C23×C181D811yescharacteristic
59C2×C4×C181—21no—
60C2×C4×C181—21no—
61C2×C4×C181D811yescharacteristic
62C2×C4×C181C2×D411yescharacteristic
63C8×C181—21no—
64C8×C181C2×D412yes—
65C22×C181—21no—
66C22×C181—41no—
67C22×C181C2×D811yescharacteristic
68C2×D181—42no—
69C2×D181—82no—
70C2×D181—81no—
71C2×D181—81no—
72C2×D181—81no—
73C4×C181—21no—
74C4×C181#12011yescharacteristic
75C4×C181AGL(1,ℤ/8)11yescharacteristic
76Dic181—42no—
77Dic181—81no—
78C2×C181—21no—
79C2×C181—41no—
80C2×C181#44711yescharacteristic
81D181—82no—
82D181—161no—
83C181#219011yescharacteristic, complemented, Sylow
84#2190—1811nomaximal, Sylow
85#575—1811no—
86#509—1812no—
87#469—1811no—
88#360—1812no—
89#349D18111yescharacteristic, complemented
90C2×#41—3621no—
91#137—3622no—
92AGL(1,ℤ/8)—3622no—
93#135—3621no—
94#135—1812no—
95C2×D8—1811no—
96C2×M16C2×D18111yescharacteristic
97#121—3621no—
98SD32—3622no—
99D16—3622no—
100M32C2×D18112yescomplemented
101#106—1811no—
102#102—3621no—
103#41—3622no—
104#41—3622no—
105#41—7241no—
106#41—7241no—
107C2×Q8—3621no—
108C2×D4—3621no—
109C2×D4—3621no—
110C2×D4—7241no—
111C2×D4—7241no—
112C22×C4—7241no—
113C22×C4D72411yescharacteristic
114Q16—3621no—
115Q16—1812no—
116SD16—3622no—
117SD16—3622no—
118D8—3621no—
119D8—3622no—
120D8—1812no—
121M16—22no—
122C2×C8D72411yescharacteristic
123C2×C8C22×D18111yescharacteristic
124C4⋊C4—3621no—
125C22⋊C4—7241no—
126C16—22no—
127C23—7241no—
128C23—7241no—
129C23D144811yescharacteristic
130Q8—3622no—
131Q8—7241no—
132D4—3622no—
133D4—14482no—
134D4—7241no—
135D4—7242no—
136D4—7242no—
137D4—7241no—
138D4—7241no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—7242no—
142C2×C4—7241no—
143C2×C4—7241no—
144C2×C4—7242no—
145C2×C4D144811yescharacteristic
146C2×C4C2×D72411yescharacteristic
147C8—21no—
148C8D4×D18112yes—
149C22—21no—
150C22—41no—
151C22—14481no—
152C22—14481no—
153C22—14482no—
154C22—14481no—
155C22—7242no—
156C22C2×D144811yescharacteristic
157C4—21no—
158C4—14481no—
159C4—7242no—
160C4#1915378611yescharacteristic
161C4#1915384411yescharacteristic
162C2—21no—
163C2—41no—
164C2—28961no—
165C2—14482no—
166C2#2604141511yescharacteristic
subgroupG/H
centerC2#26041415
commutator, cosocleC2×C8×C181C23
FrattiniC2×C8C22×D181
Fitting#349×C181C2
socleC2×C181#447
hypercenter#349D181
hypocenterC181#2190
LCSG ▷ C2×C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #26042079C2semidirect
2. #26041717C2semidirect
3. #26041532C2semidirect
4. #26041090C2semidirect
5. #349×C181C2semidirect
11. C2×D1448C22semidirect
15. #19153748C22semidirect
18. M32×C181C22semidirect
30. Dic724D4semidirect
35. D1448D4semidirect
83. C181#2190semidirect
89. #349D181semidirect
100. M32C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420792
#260417172
#260415322
#260410902
#349×C1812
#2190181