Finite Group Databasebeta

← #37532911

Subgroups — #37532911

Order 23168.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#26041949C211yesmaximal, characteristic
2#26041556C211yesmaximal, characteristic, complemented
3#26041486C211yesmaximal, characteristic, complemented
4#26041023C211yesmaximal, characteristic, complemented
5#26041011C211yesmaximal, characteristic, complemented
6#26041008C211yesmaximal, characteristic, complemented
7#486×C181C211yesmaximal, characteristic, complemented
8C2×#15668448—21no—
9C2×#15668439—21no—
10#19153812—21no—
11#19153788—22no—
12#19153787—21no—
13C4×Dic362C2211yescharacteristic
14C4×D724C2211yescharacteristic
15D4×Dic181—21no—
16#19153749—21no—
17C4⋊#13089678C2211yescharacteristic
18C4⋊#13089678C2211yescharacteristic
19#19153740—21no—
20#19153736—21no—
21C4⋊Q8×C181C2211yescharacteristic
22#127×C181C2211yescharacteristic
23#102×C181—22no—
24C4×C8×C181C2211yescharacteristic
25#15668448—42no—
26#15668448—42no—
27C2×Dic362D411yescharacteristic
28C2×D724D411yescharacteristic
29C2×C22⋊D181—22no—
30C2×C22⋊D181—22no—
31C22×Dic181—22no—
32C2×C4×D181—21no—
33#15668439—42no—
34C2×#13089678—21no—
35C2×#13089678—21no—
36C4⋊Dic181D411yescharacteristic
37C4⋊Dic181D411yescharacteristic
38#15668428—21no—
39C22⋊Dic181—22no—
40#15668426—21no—
41C4×Dic181—21no—
42C2×Q8×C181—21no—
43C2×D4×C181—21no—
44C2×D4×C181—21no—
45C2×D4×C181D412yes—
46C2×C8×C181—21no—
47C4⋊C4×C181—21no—
48C4⋊C4×C181—21no—
49C42×C181C2311yescharacteristic
50C22×D181—21no—
51Dic362—21no—
52Dic362D812yes—
53D724—21no—
54D724—21no—
55C22⋊D181—44no—
56C22⋊D181—82no—
57C2×Dic181—21no—
58C2×Dic181—21no—
59C2×Dic181—21no—
60C2×Dic181—44no—
61C4×D181—41no—
62C4×D181—41no—
63#13089678—41no—
64#13089678—41no—
65C23×C181—22no—
66Q8×C181—41no—
67Q8×C181—41no—
68D4×C181—42no—
69D4×C181—41no—
70D4×C181—41no—
71D4×C181—41no—
72D4×C181—41no—
73C2×C4×C181—41no—
74C2×C4×C181C2×D411yescharacteristic
75C2×C4×C181C2×D411yescharacteristic
76C2×C4×C181C2×D411yescharacteristic
77C8×C181—41no—
78C22×C181—42no—
79C22×C181—42no—
80C22×C181—42no—
81C22×C181#12011yescharacteristic
82C2×D181—41no—
83C2×D181—41no—
84C2×D181—41no—
85C4×C181—21no—
86C4×C181—81no—
87C4×C181C2×D811yescharacteristic
88C4×C181#13511yescharacteristic
89C4×C181AGL(1,ℤ/8)11yescharacteristic
90C4×C181#13711yescharacteristic
91Dic181—22no—
92Dic181—41no—
93Dic181—41no—
94C2×C181—82no—
95C2×C181#44711yescharacteristic
96C2×C181#45211yescharacteristic
97C2×C181#45511yescharacteristic
98D181—81no—
99C181#164111yescharacteristic, complemented, Sylow
100#1641—1811nomaximal, Sylow
101#550—1811no—
102#486D18111yescharacteristic, complemented
103#474—1811no—
104#462—1811no—
105#331—1811no—
106Q8⋊C8—1811no—
107D4⋊C8—1811no—
108C2×#41—3621no—
109C2×Q16—3621no—
110C4⋊Q8C2×D18111yescharacteristic
111#127—3621no—
112#127C2×D18111yescharacteristic
113#121—3622no—
114#121—3621no—
115C4×Q8—1811no—
116C4×D4—3621no—
117C4×D4—1811no—
118#107—3621no—
119C4⋊C8—1811no—
120C4⋊C8—1811no—
121#103—3621no—
122#102—22no—
123#102—3621no—
124C4×C8C2×D18111yescharacteristic
125#41—7242no—
126#41—7242no—
127C2×Q8—21no—
128C2×Q8—1811no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3622no—
132C2×D4—3622no—
133C2×D4—1811no—
134C2×D4C22⋊D18112yes—
135C22×C4—3622no—
136C22×C4—3621no—
137Q16—7242no—
138C2×C8—21no—
139C2×C8—3621no—
140C2×C8—3621no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—3621no—
144C4⋊C4—1811no—
145C4⋊C4—1811no—
146C22⋊C4—3622no—
147C22⋊C4—3621no—
148C42—3621no—
149C42C22×D18111yescharacteristic
150C23—22no—
151C23—3621no—
152Q8—41no—
153Q8—41no—
154Q8—3621no—
155Q8—1812no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—3621no—
162D4—3621no—
163D4—14482no—
164D4—7244no—
165C2×C4—41no—
166C2×C4—3621no—
167C2×C4—3621no—
168C2×C4—3621no—
169C2×C4—7244no—
170C2×C4—7241no—
171C2×C4—7241no—
172C2×C4D4×D18111yescharacteristic
173C2×C4D4×D18111yescharacteristic
174C2×C4C2×C22⋊D18111yescharacteristic
175C8—41no—
176C8—7241no—
177C8—7241no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—7241no—
182C22—7241no—
183C22—7241no—
184C22#1915378411yescharacteristic
185C4—21no—
186C4—81no—
187C4—3622no—
188C4—7241no—
189C4—7241no—
190C4D8×D18111yescharacteristic
191C4#1915383711yescharacteristic
192C4#1915384011yescharacteristic
193C4#1915384511yescharacteristic
194C2—82no—
195C2—14481no—
196C2#2604141111yescharacteristic
197C2#2604143811yescharacteristic
198C2#2604145611yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#486×C181C2
socleC22×C181#120
prime radical#26041949C2
hypercenter#486D181
hypocenterC181#1641
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #26041556C2semidirect
3. #26041486C2semidirect
4. #26041023C2semidirect
5. #26041011C2semidirect
6. #26041008C2semidirect
7. #486×C181C2semidirect
99. C181#1641semidirect
102. #486D181semidirect

Maximal subgroups

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

subgroupindex
#260419492
#260415562
#260414862
#260410232
#260410112
#260410082
#486×C1812
#1641181