Finite Group Databasebeta

← #37532769

Subgroups — #37532769

Order 23168.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#26041612C211yesmaximal, characteristic, complemented
2#26041595C211yesmaximal, characteristic, complemented
3#26041215C211yesmaximal, characteristic, complemented
4#26041016C211yesmaximal, characteristic, complemented
5#26041013C211yesmaximal, characteristic, complemented
6#26041003C211yesmaximal, characteristic, complemented
7#538×C181C211yesmaximal, characteristic
8C2×C4⋊Dic181C2211yescharacteristic
9C2×#15668428—21no—
10C2×C22⋊Dic181—21no—
11#19153749—21no—
12#19153747—21no—
13#19153740—21no—
14#19153737—21no—
15C22⋊#13089678C2211yescharacteristic
16C22⋊#13089678C2211yescharacteristic
17#19153723C2211yescharacteristic
18#126×C181—21no—
19#124×C181—21no—
20#123×C181—21no—
21C22⋊Q8×C181C2211yescharacteristic
22#121×C181—21no—
23#121×C181C2211yescharacteristic
24#120×C181—21no—
25C4×D4×C181—21no—
26C2×C22⋊C4×C181C2211yescharacteristic
27C22×Dic181—21no—
28C22×Dic181—21no—
29C2×#13089678—21no—
30C2×#13089678—21no—
31C4⋊Dic181—21no—
32C4⋊Dic181—21no—
33#15668428—42no—
34C22⋊Dic181—42no—
35C24×C181D411yescharacteristic
36C2×Q8×C181D411yescharacteristic
37C2×D4×C181—21no—
38C2×D4×C181—21no—
39C2×D4×C181—21no—
40C2×D4×C181D411yescharacteristic
41C22×C4×C181—21no—
42C22×C4×C181D411yescharacteristic
43C22×C4×C181C2311yescharacteristic
44C4⋊C4×C181—21no—
45C4⋊C4×C181—21no—
46C4⋊C4×C181D411yescharacteristic
47C4⋊C4×C181D411yescharacteristic
48C22⋊C4×C181—21no—
49C22⋊C4×C181—21no—
50C22⋊C4×C181—21no—
51C22⋊C4×C181—21no—
52C22⋊C4×C181—21no—
53C22⋊C4×C181—21no—
54C22⋊C4×C181—21no—
55C42×C181—21no—
56C2×Dic181—22no—
57C2×Dic181—41no—
58C2×Dic181—41no—
59C2×Dic181—41no—
60C2×Dic181—41no—
61C2×Dic181—41no—
62#13089678—41no—
63#13089678—41no—
64C23×C181—21no—
65C23×C181—21no—
66C23×C181—21no—
67C23×C181—21no—
68C23×C181—41no—
69C23×C181—41no—
70C23×C181C2×D411yescharacteristic
71Q8×C181—41no—
72D4×C181—41no—
73D4×C181—41no—
74D4×C181—41no—
75D4×C181—81no—
76D4×C181—81no—
77C2×C4×C181—21no—
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181—41no—
82C2×C4×C181—41no—
83C2×C4×C181—81no—
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C22×C181—21no—
87C22×C181—21no—
88C22×C181—42no—
89C22×C181—41no—
90C22×C181—41no—
91C22×C181—41no—
92C22×C181—41no—
93C22×C181—41no—
94C22×C181—41no—
95C22×C181—81no—
96C22×C181#12011yescharacteristic
97C22×C181#13511yescharacteristic
98C22×C181#13711yescharacteristic
99C4×C181—21no—
100C4×C181—21no—
101C4×C181—41no—
102C4×C181—41no—
103C4×C181—41no—
104C4×C181—41no—
105Dic181—42no—
106Dic181—81no—
107C2×C181—21no—
108C2×C181—21no—
109C2×C181—81no—
110C2×C181—81no—
111C2×C181#45211yescharacteristic
112C2×C181#45311yescharacteristic
113C2×C181#45811yescharacteristic
114C181#161811yescharacteristic, complemented, Sylow
115#1618—1811nomaximal, Sylow
116#538D18111yescharacteristic
117#484—1811no—
118#481—1811no—
119#399—1811no—
120#328—1811no—
121#327—1811no—
122#323—1811no—
123#126—21no—
124#124—21no—
125#123—21no—
126C22⋊Q8C2×D18111yescharacteristic
127#121—21no—
128#121C2×D18111yescharacteristic
129#120—21no—
130C4×D4—21no—
131C2×C4⋊C4—3621no—
132C2×C4⋊C4—1811no—
133C2×C22⋊C4—3621no—
134C2×C22⋊C4C2×D18111yescharacteristic
135#107—3621no—
136#106—3621no—
137#103—3621no—
138#102—3621no—
139C22⋊C8—1811no—
140C22⋊C8—1811no—
141#95—1811no—
142C24C22⋊D18111yescharacteristic
143C2×Q8C22⋊D18111yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C22⋊D18111yescharacteristic
148C22×C4—21no—
149C22×C4—3621no—
150C22×C4—3621no—
151C22×C4C22⋊D18111yescharacteristic
152C22×C4C22×D18111yescharacteristic
153C2×C8—3621no—
154C2×C8—3621no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3621no—
158C4⋊C4—3621no—
159C4⋊C4—7242no—
160C4⋊C4C22⋊D18111yescharacteristic
161C4⋊C4C22⋊D18111yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—7242no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×C22⋊D18111yescharacteristic
178Q8—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—3622no—
192C2×C4—7241no—
193C2×C4—7241no—
194C2×C4—7241no—
195C2×C4—7241no—
196C2×C4—7241no—
197C2×C4C2×C22⋊D18111yescharacteristic
198C2×C4C2×C22⋊D18111yescharacteristic
199C8—7241no—
200C8—7241no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22#1915378511yescharacteristic
212C22#1915383611yescharacteristic
213C22#1915384611yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—14481no—
221C4—7242no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2604144111yescharacteristic
227C2#2604144711yescharacteristic
228C2#2604146911yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting, prime radical#538×C181C2
socleC22×C181#120
hypercenter#538D181
hypocenterC181#1618
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #26041612C2semidirect
2. #26041595C2semidirect
3. #26041215C2semidirect
4. #26041016C2semidirect
5. #26041013C2semidirect
6. #26041003C2semidirect
114. C181#1618semidirect

Maximal subgroups

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

subgroupindex
#260416122
#260415952
#260412152
#260410162
#260410132
#260410032
#538×C1812
#1618181