Finite Group Databasebeta

← #37532243

Subgroups — #37532243

Order 23168.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#26041810C211yesmaximal, characteristic
2#26041333C211yesmaximal, characteristic, complemented
3#26041130C211yesmaximal, characteristic, complemented
4#26041017C211yesmaximal, characteristic, complemented
5#26041015C211yesmaximal, characteristic, complemented
6#26041013C211yesmaximal, characteristic
7#327×C181C211yesmaximal, characteristic
8#19153809—21no—
9#19153807C411yescharacteristic, complemented
10#19153803C411yescharacteristic, complemented
11#19153797—21no—
12#19153795C2211yescharacteristic
13#19153781C411yescharacteristic, complemented
14D4×Dic181—21no—
15D4×Dic181C411yescharacteristic, complemented
16#19153772—21no—
17C2×#15668428—21no—
18C2×C4×Dic181—21no—
19C2×C4×Dic181C2211yescharacteristic
20#19153745—21no—
21C22⋊#13089678C2211yescharacteristic
22#19153726—21no—
23#19153723C2211yescharacteristic
24#121×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×C181C2211yescharacteristic
27C2×Dic362C2×C411yescharacteristic
28C22×Dic181—21no—
29C22×Dic181—21no—
30C22×Dic181D411yescharacteristic
31C22×Dic181D411yescharacteristic
32C2×#13089678—21no—
33C4⋊Dic181—21no—
34C4⋊Dic181C2×C411yescharacteristic
35#15668428—24no—
36#15668428—21no—
37#15668428—21no—
38#15668428—21no—
39C22⋊Dic181—21no—
40C22⋊Dic181—21no—
41C22⋊Dic181—21no—
42C22⋊Dic181—21no—
43C4×Dic181—21no—
44C4×Dic181—42no—
45C4×Dic181C2×C411yescharacteristic
46C4×Dic181C2×C411yescharacteristic
47C4×Dic181D411yescharacteristic
48C4×Dic181D411yescharacteristic
49C2×D4×C181—21no—
50C2×D4×C181C2×C411yescharacteristic
51C22×C4×C181—21no—
52C22×C4×C181C2311yescharacteristic
53C2×C8×C181—21no—
54C4⋊C4×C181C2×C411yescharacteristic
55C22⋊C4×C181—21no—
56Dic362—41no—
57C2×Dic181—21no—
58C2×Dic181—21no—
59C2×Dic181—22no—
60C2×Dic181—22no—
61C2×Dic181—21no—
62C2×Dic181—21no—
63C2×Dic181—41no—
64C2×Dic181—41no—
65C2×Dic181—42no—
66C2×Dic181C22⋊C411yescharacteristic
67C2×Dic181C22⋊C411yescharacteristic
68C2×Dic181C22⋊C411yescharacteristic
69C2×Dic181C22⋊C411yescharacteristic
70#13089678—41no—
71C23×C181—21no—
72C23×C181C2×D411yescharacteristic
73D4×C181—41no—
74D4×C181—41no—
75D4×C181—41no—
76C2×C4×C181—22no—
77C2×C4×C181—21no—
78C2×C4×C181—41no—
79C2×C4×C181—41no—
80C2×C4×C181—41no—
81C2×C4×C181C22×C411yescharacteristic
82C2×C4×C181C2×D411yescharacteristic
83C8×C181—41no—
84C22×C181—21no—
85C22×C181—21no—
86C22×C181—41no—
87C22×C181—41no—
88C22×C181—41no—
89C22×C181C2×C22⋊C411yescharacteristic
90C22×C181AGL(1,ℤ/8)11yescharacteristic
91C22×C181#13711yescharacteristic
92C4×C181—21no—
93C4×C181—21no—
94C4×C181—42no—
95C4×C181—41no—
96Dic181—21no—
97Dic181—21no—
98Dic181—21no—
99Dic181—42no—
100Dic181—41no—
101Dic181—41no—
102Dic181#10412yes—
103C2×C181—21no—
104C2×C181—21no—
105C2×C181—81no—
106C2×C181#41011yescharacteristic
107C2×C181#41711yescharacteristic
108C2×C181C2×#10411yescharacteristic
109C181#150911yescharacteristic, complemented, Sylow
110#1509—1811nomaximal, Sylow
111#529—1811no—
112#432—1811no—
113C4×C4⋊C4—1811no—
114#328—1811no—
115#328—1811no—
116#327—1811no—
117#327D18111yescharacteristic
118#126—3621no—
119#125—1811no—
120#124—1811no—
121#123—3621no—
122C22⋊Q8—1811no—
123#121C2×D18111yescharacteristic
124C4×Q8—1811no—
125C4×D4—3621no—
126C4×D4—1811no—
127#117—3621no—
128C2×C4⋊C4—3621no—
129C2×C42—3621no—
130C2×C42—1811no—
131C4⋊C8—3621no—
132C22⋊C8—1811no—
133C22⋊C8C2×D18111yescharacteristic
134#97—3621no—
135#95—1811no—
136#95C2×D18111yescharacteristic
137C2×Q8—1811no—
138C2×D4—21no—
139C2×D4C4×D18111yescharacteristic
140C22×C4—21no—
141C22×C4—3621no—
142C22×C4—3621no—
143C22×C4—1811no—
144C22×C4—1811no—
145C22×C4C22×D18111yescharacteristic
146C2×C8—21no—
147C2×C8—3621no—
148C4⋊C4—3624no—
149C4⋊C4—3621no—
150C4⋊C4—3621no—
151C4⋊C4—3621no—
152C4⋊C4—3621no—
153C4⋊C4—1811no—
154C4⋊C4C4×D18111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3621no—
157C22⋊C4—3621no—
158C22⋊C4—3621no—
159C22⋊C4—3621no—
160C42—3621no—
161C42—1811no—
162C42—1811no—
163C42—1811no—
164C42—1811no—
165C42—7242no—
166C23—21no—
167C23D4×D18111yescharacteristic
168Q8—7241no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—3621no—
178C2×C4—3621no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—3621no—
182C2×C4—3621no—
183C2×C4—1811no—
184C2×C4—1811no—
185C2×C4—1811no—
186C2×C4—1811no—
187C2×C4—7242no—
188C2×C4—7241no—
189C2×C4—7241no—
190C2×C4C2×C4×D18111yescharacteristic
191C2×C4D4×D18111yescharacteristic
192C8—41no—
193C8—7241no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D18111yescharacteristic
200C22#1915384011yescharacteristic
201C22#1915384511yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—3621no—
207C4—3621no—
208C4—3621no—
209C4—1812no—
210C4—7242no—
211C4—7241no—
212C4—7241no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2604125111yescharacteristic
217C2#2604127711yescharacteristic
218C2#104×D18111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D181
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#327×C181C2
socleC22×C181C2×C22⋊C4
prime radical#121×C181C22
cosocleC22×C4×C181C23
hypercenter#327D181
hypocenterC181#1509
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26041333C2semidirect
3. #26041130C2semidirect
4. #26041017C2semidirect
5. #26041015C2semidirect
9. #19153807C4semidirect
10. #19153803C4semidirect
13. #19153781C4semidirect
15. D4×Dic181C4semidirect
109. C181#1509semidirect

Maximal subgroups

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

subgroupindex
#260418102
#260413332
#260411302
#260410172
#260410152
#260410132
#327×C1812
#1509181