Finite Group Databasebeta

← #37532679

Subgroups — #37532679

Order 23168.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#26041904C211yesmaximal, characteristic
2#26041437C211yesmaximal, characteristic, complemented
3#26041434C211yesmaximal, characteristic, complemented
4#26041192C211yesmaximal, characteristic
5#26041015C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic
7#328×C181C211yesmaximal, characteristic, complemented
8C22×Dic362C2211yescharacteristic
9C2×Dic724—21no—
10C2×#15668439—21no—
11#19153796—21no—
12#19153795C2211yescharacteristic
13#19153794—21no—
14#19153794—21no—
15#19153793—21no—
16#19153792—21no—
17C2×#15668426—21no—
18C2×#15668426C2211yescharacteristic
19C22⋊C4×D181—21no—
20#19153741—21no—
21#19153739—21no—
22C22⋊#13089678C2211yescharacteristic
23#19153723—21no—
24C22⋊Q8×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×C181C2211yescharacteristic
27C23×D181D411yescharacteristic
28C2×Dic362—21no—
29C2×Dic362—21no—
30C2×Dic362—21no—
31C2×Dic362—42no—
32C2×Dic362D411yescharacteristic
33C22×Dic181—21no—
34C22×Dic181D411yescharacteristic
35C2×C4×D181—21no—
36C2×C4×D181—21no—
37Dic724—42no—
38#15668439—42no—
39C2×#13089678—21no—
40C4⋊Dic181—21no—
41C4⋊Dic181D411yescharacteristic
42#15668428—21no—
43#15668428—21no—
44#15668428—21no—
45C22⋊Dic181—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—21no—
49#15668426—21no—
50#15668426—42no—
51C2×Q8×C181D411yescharacteristic
52C22×C4×C181—21no—
53C22×C4×C181C2311yescharacteristic
54C2×C8×C181—21no—
55C4⋊C4×C181—21no—
56C4⋊C4×C181D411yescharacteristic
57C22⋊C4×C181—21no—
58C22×D181—21no—
59C22×D181—21no—
60C22×D181—41no—
61C22×D181—41no—
62C22×D181Q1612yescomplemented
63Dic362—22no—
64Dic362—22no—
65Dic362—41no—
66Dic362—41no—
67Dic362—41no—
68Dic362—81no—
69C2×Dic181—22no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—41no—
74C2×Dic181—41no—
75C2×Dic181—41no—
76C4×D181—81no—
77C4×D181—81no—
78#13089678—41no—
79C23×C181C2×D411yescharacteristic
80Q8×C181—41no—
81C2×C4×C181—21no—
82C2×C4×C181—21no—
83C2×C4×C181—41no—
84C2×C4×C181—41no—
85C2×C4×C181—41no—
86C2×C4×C181—41no—
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2×D411yescharacteristic
89C8×C181—41no—
90C22×C181—21no—
91C22×C181—21no—
92C22×C181#12011yescharacteristic
93C22×C181C2×Q1611yescharacteristic
94C22×C181#13711yescharacteristic
95C2×D181—22no—
96C2×D181—41no—
97C2×D181—41no—
98C2×D181—42no—
99C2×D181—81no—
100C4×C181—21no—
101C4×C181—21no—
102C4×C181—41no—
103C4×C181—41no—
104C4×C181—81no—
105Dic181—41no—
106Dic181—41no—
107Dic181—41no—
108Dic181—42no—
109C2×C181—21no—
110C2×C181—21no—
111C2×C181#45111yescharacteristic
112C2×C181#45511yescharacteristic
113C2×C181B(4,𝔽2)11yescharacteristic
114D181—42no—
115C181#160211yescharacteristic, complemented, Sylow
116#1602—1811nomaximal, Sylow
117#543—1811no—
118#451—1811no—
119#451—1811no—
120#393—1811no—
121#328—1811no—
122#328D18111yescharacteristic, complemented
123#323—1811no—
124C22×Q8—1811no—
125C2×Q16—3621no—
126C2×Q16—3621no—
127C22⋊Q8—3621no—
128C22⋊Q8—3621no—
129C22⋊Q8—3621no—
130C22⋊Q8—3621no—
131C22⋊Q8—3621no—
132C22⋊Q8—1811no—
133C22⋊Q8C2×D18111yescharacteristic
134C2×C22⋊C4—3621no—
135C2×C22⋊C4—3621no—
136C2×C22⋊C4—1811no—
137#103—3621no—
138#103—3621no—
139C22⋊C8—1811no—
140C22⋊C8C2×D18111yescharacteristic
141#95—3621no—
142#95C2×D18111yescharacteristic
143C24—1811no—
144C2×Q8—3621no—
145C2×Q8—3621no—
146C2×Q8—3621no—
147C2×Q8—1811no—
148C2×Q8—7242no—
149C2×Q8D72411yescharacteristic
150C22×C4—21no—
151C22×C4—3621no—
152C22×C4—3621no—
153C22×C4—3621no—
154C22×C4—1811no—
155C22×C4C22×D18111yescharacteristic
156Q16—7242no—
157Q16—7242no—
158C2×C8—21no—
159C2×C8—3621no—
160C4⋊C4—21no—
161C4⋊C4—3621no—
162C4⋊C4—3621no—
163C4⋊C4—3621no—
164C4⋊C4—3621no—
165C4⋊C4—1811no—
166C4⋊C4D72411yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3621no—
169C22⋊C4—3621no—
170C22⋊C4—3621no—
171C22⋊C4—3621no—
172C22⋊C4—3621no—
173C22⋊C4—7242no—
174C23—3621no—
175C23—3621no—
176C23—1812no—
177C23—7241no—
178C23—7241no—
179C23D4×D18111yescharacteristic
180Q8—41no—
181Q8—3622no—
182Q8—3622no—
183Q8—14481no—
184Q8—7241no—
185Q8—7241no—
186Q8—7241no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3622no—
194C2×C4—3621no—
195C2×C4—3621no—
196C2×C4—3621no—
197C2×C4—14481no—
198C2×C4—14481no—
199C2×C4—7241no—
200C2×C4—7241no—
201C2×C4—7241no—
202C2×C4D4×D18111yescharacteristic
203C2×C4C2×D72411yescharacteristic
204C8—41no—
205C8—7241no—
206C22—21no—
207C22—21no—
208C22—3622no—
209C22—14481no—
210C22—7242no—
211C22—7241no—
212C22—7241no—
213C22#1915378611yescharacteristic
214C22Q16×D18111yescharacteristic
215C22#1915384511yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—7241no—
222C4—7241no—
223C4—7241no—
224C4—7242no—
225C2—21no—
226C2—21no—
227C2—7242no—
228C2#2604143611yescharacteristic
229C2#2604145811yescharacteristic
230C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#328×C181C2
socleC22×C181#120
prime radicalC23×D181D4
hypercenter#328D181
hypocenterC181#1602
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. #26041437C2semidirect
3. #26041434C2semidirect
5. #26041015C2semidirect
7. #328×C181C2semidirect
62. C22×D181Q16semidirect
115. C181#1602semidirect
122. #328D181semidirect

Maximal subgroups

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

subgroupindex
#260419042
#260414372
#260414342
#260411922
#260410152
#260410022
#328×C1812
#1602181