Finite Group Databasebeta

← #37532772

Subgroups — #37532772

Order 23168.

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

normal subgroups only

This group has 240 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#26041869C211yesmaximal, characteristic
2#26041612C211yesmaximal, characteristic, complemented
3#26041598C211yesmaximal, characteristic, complemented
4#26041216C211yesmaximal, characteristic, complemented
5#26041012C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic, complemented
7#328×C181C211yesmaximal, characteristic, complemented
8#19153811—21no—
9#19153805—21no—
10#19153801—21no—
11#19153791—21no—
12#19153790C2211yescharacteristic
13#19153786—21no—
14#19153776—21no—
15C2×C4⋊Dic181—21no—
16C2×C4⋊Dic181C2211yescharacteristic
17C2×#15668426—21no—
18C2×#15668426C2211yescharacteristic
19#19153749—21no—
20#19153748—21no—
21#19153740—21no—
22#19153738—21no—
23C22⋊#13089678C2211yescharacteristic
24C22⋊Q8×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×C181C2211yescharacteristic
27C23×D181D411yescharacteristic
28C2×D724—21no—
29C2×D724—21no—
30C2×D724D411yescharacteristic
31C2×C22⋊D181—21no—
32C22×Dic181—21no—
33C22×Dic181D411yescharacteristic
34C2×C4×D181—21no—
35C2×#13089678—21no—
36C4⋊Dic181—21no—
37C4⋊Dic181—21no—
38C4⋊Dic181—42no—
39C4⋊Dic181D411yescharacteristic
40#15668428—21no—
41#15668426—21no—
42#15668426—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—42no—
48C4×Dic181—21no—
49C2×Q8×C181D411yescharacteristic
50C22×C4×C181—21no—
51C22×C4×C181C2311yescharacteristic
52C2×C8×C181—21no—
53C4⋊C4×C181—21no—
54C4⋊C4×C181D411yescharacteristic
55C22⋊C4×C181—21no—
56C22×D181—21no—
57C22×D181—21no—
58C22×D181—21no—
59C22×D181—21no—
60C22×D181—41no—
61C22×D181—41no—
62D724—41no—
63D724—81no—
64D724—81no—
65C22⋊D181—41no—
66C22⋊D181—41no—
67C2×Dic181—21no—
68C2×Dic181—21no—
69C2×Dic181—41no—
70C2×Dic181—41no—
71C2×Dic181—41no—
72C2×Dic181—41no—
73C4×D181—81no—
74#13089678—41no—
75C23×C181C2×D411yescharacteristic
76Q8×C181—41no—
77C2×C4×C181—22no—
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—41no—
81C2×C4×C181—41no—
82C2×C4×C181—41no—
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C8×C181—41no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181#12011yescharacteristic
89C22×C181#13511yescharacteristic
90C22×C181#13711yescharacteristic
91C2×D181—42no—
92C2×D181—41no—
93C2×D181—41no—
94C2×D181—41no—
95C2×D181—41no—
96C2×D181—41no—
97C2×D181—41no—
98C2×D181—81no—
99C4×C181—21no—
100C4×C181—21no—
101C4×C181—42no—
102C4×C181—41no—
103C4×C181—41no—
104Dic181—41no—
105Dic181—41no—
106Dic181—81no—
107C2×C181—21no—
108C2×C181—21no—
109C2×C181#45211yescharacteristic
110C2×C181#45311yescharacteristic
111C2×C181#45811yescharacteristic
112D181—81no—
113D181—81no—
114C181#161811yescharacteristic, complemented, Sylow
115#1618—1811nomaximal, Sylow
116#538—1811no—
117#484—1811no—
118#481—1811no—
119#399—1811no—
120#328D18111yescharacteristic, complemented
121#327—1811no—
122#323—1811no—
123#126—3621no—
124#124—3621no—
125#123—3621no—
126C22⋊Q8C2×D18111yescharacteristic
127#121—3621no—
128#121—1811no—
129#120—3621no—
130C4×D4—3621no—
131C2×C4⋊C4—3621no—
132C2×C4⋊C4—1811no—
133C2×C22⋊C4—3621no—
134C2×C22⋊C4—1811no—
135#107—3621no—
136#106—3621no—
137#103—3621no—
138#102—3621no—
139C22⋊C8—1811no—
140C22⋊C8C2×D18111yescharacteristic
141#95C2×D18111yescharacteristic
142C24—1811no—
143C2×Q8D72411yescharacteristic
144C2×D4—3621no—
145C2×D4—3621no—
146C2×D4—3621no—
147C2×D4—1811no—
148C22×C4—21no—
149C22×C4—3621no—
150C22×C4—3621no—
151C22×C4—1811no—
152C22×C4C22×D18111yescharacteristic
153C2×C8—21no—
154C2×C8—3621no—
155C4⋊C4—21no—
156C4⋊C4—3621no—
157C4⋊C4—3621no—
158C4⋊C4—3621no—
159C4⋊C4—1811no—
160C4⋊C4—7242no—
161C4⋊C4D72411yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—3621no—
164C22⋊C4—3621no—
165C22⋊C4—3621no—
166C22⋊C4—3621no—
167C22⋊C4—3621no—
168C22⋊C4—3621no—
169C22⋊C4—7242no—
170C42—3621no—
171C23—3621no—
172C23—3621no—
173C23—3621no—
174C23—3621no—
175C23—7241no—
176C23—7241no—
177C23D4×D18111yescharacteristic
178Q8—41no—
179D4—14481no—
180D4—14481no—
181D4—7241no—
182D4—7241no—
183D4—7241no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3621no—
191C2×C4—3621no—
192C2×C4—14481no—
193C2×C4—7241no—
194C2×C4—7241no—
195C2×C4—7241no—
196C2×C4—7241no—
197C2×C4D4×D18111yescharacteristic
198C2×C4C2×D72411yescharacteristic
199C8—41no—
200C8—7241no—
201C22—21no—
202C22—21no—
203C22—14481no—
204C22—7242no—
205C22—7241no—
206C22—7241no—
207C22—7241no—
208C22—7241no—
209C22—7241no—
210C22—7241no—
211C22#1915378611yescharacteristic
212C22#1915383711yescharacteristic
213C22#1915384811yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—14481no—
220C4—7241no—
221C4—7241no—
222C2—21no—
223C2—21no—
224C2—14481no—
225C2—14481no—
226C2#2604144411yescharacteristic
227C2#2604144811yescharacteristic
228C2#2604147011yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#328×C181C2
socleC22×C181#120
prime radical#26041869C2
hypercenter#328D181
hypocenterC181#1618
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26041612C2semidirect
3. #26041598C2semidirect
4. #26041216C2semidirect
5. #26041012C2semidirect
6. #26041002C2semidirect
7. #328×C181C2semidirect
114. C181#1618semidirect
120. #328D181semidirect

Maximal subgroups

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

subgroupindex
#260418692
#260416122
#260415982
#260412162
#260410122
#260410022
#328×C1812
#1618181