Finite Group Databasebeta

← #37532644

Subgroups — #37532644

Order 23168.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#26041429C212yesmaximal, complemented
2#26041188C211yesmaximal, characteristic
3#26041016C212yesmaximal, complemented
4#26041003C211yesmaximal, characteristic, complemented
5#543×C181C211yesmaximal, characteristic, complemented
6C22×D724C2211yescharacteristic
7C22×C22⋊D181—21no—
8C2×#15668436—22no—
9C2×#15668426—21no—
10#19153736—22no—
11C22⋊#13089678C2212yes—
12#19153723C2211yescharacteristic
13C22⋊Q8×C181—22no—
14C22⋊Q8×C181—21no—
15C22⋊Q8×C181—22no—
16C22⋊Q8×C181C2212yes—
17C2×C22⋊C4×C181—21no—
18C2×C22⋊C4×C181C2211yescharacteristic
19C23×D181—21no—
20C2×D724—22no—
21C2×D724—42no—
22C2×C22⋊D181—42no—
23C2×C22⋊D181—42no—
24C2×C22⋊D181—42no—
25C22×Dic181—21no—
26#15668436—44no—
27C2×#13089678—22no—
28#15668426—42no—
29C24×C181D411yescharacteristic
30C2×Q8×C181—21no—
31C2×Q8×C181D412yescomplemented
32C22×C4×C181—21no—
33C22×C4×C181—21no—
34C22×C4×C181D411yescharacteristic
35C22×C4×C181C2311yescharacteristic
36C4⋊C4×C181—21no—
37C4⋊C4×C181—22no—
38C4⋊C4×C181—22no—
39C4⋊C4×C181D412yescomplemented
40C22⋊C4×C181—22no—
41C22⋊C4×C181—22no—
42C22⋊C4×C181—22no—
43C22×D181—22no—
44C22×D181—41no—
45C22×D181—41no—
46C22×D181—41no—
47C22×D181—41no—
48C22×D181—41no—
49C22×D181—41no—
50D724—24no—
51D724—42no—
52C22⋊D181—48no—
53C2×Dic181—22no—
54C2×Dic181—41no—
55C2×Dic181—41no—
56#13089678—42no—
57C23×C181—21no—
58C23×C181—21no—
59C23×C181—41no—
60C23×C181—41no—
61C23×C181SD1612yescomplemented
62C23×C181C2×D411yescharacteristic
63Q8×C181—42no—
64Q8×C181—81no—
65C2×C4×C181—22no—
66C2×C4×C181—21no—
67C2×C4×C181—22no—
68C2×C4×C181—41no—
69C2×C4×C181—41no—
70C2×C4×C181—81no—
71C2×C4×C181—81no—
72C2×C4×C181C2×D412yes—
73C22×C181—21no—
74C22×C181—21no—
75C22×C181—22no—
76C22×C181—42no—
77C22×C181—41no—
78C22×C181—41no—
79C22×C181—81no—
80C22×C181#12011yescharacteristic
81C22×C181C2×SD1611yescharacteristic
82C22×C181#13711yescharacteristic
83C2×D181—42no—
84C2×D181—42no—
85C2×D181—42no—
86C2×D181—81no—
87C2×D181—81no—
88C2×D181—81no—
89C2×D181—81no—
90C4×C181—22no—
91C4×C181—41no—
92C4×C181—42no—
93C4×C181—42no—
94Dic181—42no—
95C2×C181—21no—
96C2×C181—21no—
97C2×C181—42no—
98C2×C181#44811yescharacteristic
99C2×C181#45311yescharacteristic
100C2×C181B(4,𝔽2)11yescharacteristic
101D181—82no—
102C181#159611yescharacteristic, complemented, Sylow
103#1596—1811nomaximal, Sylow
104#543D18111yescharacteristic, complemented
105#450—1812no—
106#392—1811no—
107#328—1812no—
108#323—1811no—
109C22×D4—3621no—
110C22×D4—1811no—
111C2×SD16—3622no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D18112yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—3621no—
118C2×C22⋊C4C2×D18111yescharacteristic
119#102—3622no—
120C22⋊C8—1812no—
121#95—1811no—
122C24—3621no—
123C24C22⋊D18111yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D18112yescomplemented
126C2×D4—3622no—
127C2×D4—7242no—
128C2×D4—7242no—
129C2×D4—7242no—
130C2×D4—7242no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—3621no—
134C22×C4C22⋊D18111yescharacteristic
135C22×C4C22×D18111yescharacteristic
136SD16—7244no—
137C2×C8—3622no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D18112yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—7242no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—3622no—
151C23—7241no—
152C23—7241no—
153C23—7241no—
154C23—7241no—
155C23—7241no—
156C23—7241no—
157C23#1566843612yescomplemented
158C23C2×C22⋊D18111yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—3624no—
162D4—7242no—
163D4—7248no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—3622no—
172C2×C4—7241no—
173C2×C4—7241no—
174C2×C4C2×C22⋊D18112yes—
175C8—7242no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—14481no—
184C22—14481no—
185C22—14481no—
186C22—14481no—
187C22—7242no—
188C22—7242no—
189C22—7242no—
190C22#1915378511yescharacteristic
191C22C2×#1566843611yescharacteristic
192C22#1915384711yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—7242no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—14482no—
202C2#2604141811yescharacteristic
203C2#2604144711yescharacteristic
204C2#2604146611yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#543×C181C2
socleC22×C181#120
prime radical#26041188C2
hypercenter#543D181
hypocenterC181#1596
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #26041429C2semidirect
3. #26041016C2semidirect
4. #26041003C2semidirect
5. #543×C181C2semidirect
31. C2×Q8×C181D4semidirect
39. C4⋊C4×C181D4semidirect
61. C23×C181SD16semidirect
102. C181#1596semidirect
104. #543D181semidirect
125. C2×Q8C22⋊D181semidirect
141. C4⋊C4C22⋊D181semidirect
157. C23#15668436semidirect

Maximal subgroups

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

subgroupindex
#260414292
#260411882
#260410162
#260410032
#543×C1812
#1596181