Finite Group Databasebeta

← #37533057

Subgroups — #37533057

Order 23168.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#26042033C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041648C211yesmaximal, characteristic, complemented
4#26041639C211yesmaximal, characteristic
5#26041020C211yesmaximal, characteristic, complemented
6#26041018C211yesmaximal, characteristic
7#330×C181C211yesmaximal, characteristic, complemented
8#19153816C2211yescharacteristic
9#19153815—21no—
10#19153811—21no—
11#19153811—21no—
12#19153808C2211yescharacteristic
13#19153804C2211yescharacteristic
14#19153801—21no—
15#19153796—21no—
16#19153794—21no—
17#19153749—21no—
18#19153747—21no—
19#19153741—21no—
20#19153741—21no—
21#19153741—21no—
22#19153738—21no—
23#19153727C2211yescharacteristic
24#125×C181C2211yescharacteristic
25#97×C181C2211yescharacteristic
26#97×C181C2211yescharacteristic
27C2×Dic362—21no—
28C2×Dic362—21no—
29C2×Dic362D411yescharacteristic
30C2×D724D411yescharacteristic
31C2×C4×D181—21no—
32C2×#13089678—21no—
33C4⋊Dic181—21no—
34C4⋊Dic181—21no—
35C4⋊Dic181—21no—
36C4⋊Dic181D411yescharacteristic
37C4⋊Dic181D411yescharacteristic
38#15668428—21no—
39#15668428—21no—
40#15668428—21no—
41#15668426—21no—
42#15668426—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46C4×Dic181—21no—
47C2×C8×C181—21no—
48C2×C8×C181—21no—
49C4⋊C4×C181—21no—
50C4⋊C4×C181—21no—
51C4⋊C4×C181D411yescharacteristic
52C4⋊C4×C181D411yescharacteristic
53C42×C181C2311yescharacteristic
54C22×D181—21no—
55Dic362—41no—
56Dic362—42no—
57Dic362—81no—
58D724—41no—
59C2×Dic181—21no—
60C2×Dic181—21no—
61C2×Dic181—21no—
62C2×Dic181—41no—
63C4×D181—81no—
64#13089678—22no—
65C2×C4×C181—21no—
66C2×C4×C181—21no—
67C2×C4×C181C2×D411yescharacteristic
68C2×C4×C181C2×D411yescharacteristic
69C2×C4×C181C2×D411yescharacteristic
70C8×C181—41no—
71C8×C181—41no—
72C22×C181#12011yescharacteristic
73C2×D181—41no—
74C2×D181—41no—
75C2×D181—41no—
76C4×C181—21no—
77C4×C181—21no—
78C4×C181—21no—
79C4×C181—41no—
80C4×C181—41no—
81Dic181—41no—
82Dic181—41no—
83Dic181—41no—
84Dic181—81no—
85C2×C181#45411yescharacteristic
86C2×C181#45511yescharacteristic
87C2×C181#45611yescharacteristic
88D181—81no—
89C181#166411yescharacteristic, complemented, Sylow
90#1664—1811nomaximal, Sylow
91#563—1811no—
92#501—1811no—
93#491—1811no—
94#489—1811no—
95#330—1811no—
96#330D18111yescharacteristic, complemented
97#329—1811no—
98C4⋊Q8—3621no—
99C4⋊Q8—1811no—
100#126—3621no—
101#126—3621no—
102#125—1811no—
103#125C2×D18111yescharacteristic
104#124—1811no—
105#123—3621no—
106C22⋊Q8—3621no—
107C22⋊Q8—3621no—
108#107—3621no—
109#106—3621no—
110#103—3621no—
111#103—3621no—
112#103—3621no—
113#102—3621no—
114#97—1811no—
115#97C2×D18111yescharacteristic
116#97C2×D18111yescharacteristic
117C2×Q8—3621no—
118C2×Q8—3621no—
119C2×Q8—1811no—
120C2×D4—1811no—
121C22×C4—3621no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—3621no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3621no—
128C4⋊C4—3621no—
129C4⋊C4—3621no—
130C4⋊C4—3621no—
131C4⋊C4—3621no—
132C4⋊C4—3621no—
133C4⋊C4—1811no—
134C4⋊C4—1811no—
135C4⋊C4D72411yescharacteristic
136C4⋊C4D72411yescharacteristic
137C22⋊C4—3621no—
138C22⋊C4—3621no—
139C22⋊C4—3621no—
140C22⋊C4—3621no—
141C22⋊C4—3621no—
142C42—3621no—
143C42C22×D18111yescharacteristic
144C23—3621no—
145Q8—14481no—
146Q8—7242no—
147Q8—7241no—
148D4—7241no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—3621no—
152C2×C4—3621no—
153C2×C4—3621no—
154C2×C4—14481no—
155C2×C4—7241no—
156C2×C4D4×D18111yescharacteristic
157C2×C4D4×D18111yescharacteristic
158C2×C4C2×D72411yescharacteristic
159C8—41no—
160C8—41no—
161C8—3622no—
162C22—7241no—
163C22—7241no—
164C22—7241no—
165C22#1915378611yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—14481no—
172C4—7241no—
173C4—7241no—
174C4—7241no—
175C2—14481no—
176C2#2604145311yescharacteristic
177C2#2604145811yescharacteristic
178C2#2604146411yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#330×C181C2
socleC22×C181#120
prime radicalC2×D724D4
hypercenter#330D181
hypocenterC181#1664
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #26041689C2semidirect
3. #26041648C2semidirect
5. #26041020C2semidirect
7. #330×C181C2semidirect
89. C181#1664semidirect
96. #330D181semidirect

Maximal subgroups

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

subgroupindex
#260420332
#260416892
#260416482
#260416392
#260410202
#260410182
#330×C1812
#1664181