Finite Group Databasebeta

← #37549579

Subgroups — #37549579

Order 23168.

312 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×#19153883C211yesmaximal, characteristic, complemented
2C2×C4⋊#13089685C211yesmaximal, characteristic, complemented
3C2×C22⋊#13089685C212yesmaximal
4#26042145C212yesmaximal, complemented
5C2×#19153772C211yesmaximal, characteristic
6C22×#13089685C2212yes—
7C22×#13089685C2212yes—
8#19153883—22no—
9C4⋊#13089685—22no—
10C22⋊#13089685—24no—
11C23×Dic181C2211yescharacteristic
12#19153772—22no—
13#19153772—22no—
14C2×#15668428C411yescharacteristic
15C2×#15668428C411yescharacteristic
16C2×C22⋊Dic181C411yescharacteristic
17C2×C4×Dic181C2211yescharacteristic
18C2×C4×Dic181C2211yescharacteristic
19C2×C22⋊C4×C181C411yescharacteristic
20C2×#13089685—22no—
21C2×#13089685—22no—
22C2×#13089685—22no—
23C2×#13089685—22no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C22×Dic181—21no—
27C22×Dic181—21no—
28C22×Dic181—21no—
29C22×Dic181—21no—
30C22×Dic181C2×C411yescharacteristic
31C22×Dic181C2×C411yescharacteristic
32C22×Dic181C2×C411yescharacteristic
33C22×Dic181D412yes—
34C22×Dic181Q812yes—
35C22×Dic181C2311yescharacteristic
36#15668428—22no—
37#15668428—22no—
38C22⋊Dic181—22no—
39C4×Dic181—21no—
40C4×Dic181—21no—
41C4×Dic181D412yes—
42C4×Dic181D412yes—
43C24×C181C2×C411yescharacteristic
44C22×C4×C181C2×C411yescharacteristic
45C22×C4×C181C2×C411yescharacteristic
46C22⋊C4×C181—22no—
47#13089685—42no—
48#13089685—42no—
49C2×Dic181—21no—
50C2×Dic181—21no—
51C2×Dic181—22no—
52C2×Dic181—22no—
53C2×Dic181—22no—
54C2×Dic181—22no—
55C2×Dic181—22no—
56C2×Dic181—21no—
57C2×Dic181—22no—
58C2×Dic181C22⋊C412yes—
59C2×Dic181C4⋊C412yes—
60C2×Dic181C2×D411yescharacteristic
61C2×Dic181C2×D411yescharacteristic
62C2×Dic181C2×D411yescharacteristic
63C2×Dic181C2×Q811yescharacteristic
64C2×Dic181#4111yescharacteristic
65C2×Dic181#4111yescharacteristic
66C23×C181—21no—
67C23×C181—21no—
68C23×C181—22no—
69C23×C181—22no—
70C23×C181C4⋊C412yes—
71C23×C181C22×C411yescharacteristic
72C2×C4×C181—22no—
73C2×C4×C181—21no—
74C2×C4×C181—22no—
75C2×C4×C181C22⋊C412yes—
76C22×C181—22no—
77C22×C181—24no—
78C22×C181—41no—
79C22×C181—41no—
80C22×C181—42no—
81C22×C181C2×C22⋊C411yescharacteristic
82C22×C181C2×C4⋊C411yescharacteristic
83C22×C181#11711yescharacteristic
84C22×C181#13112yes—
85C22×C181#13112yes—
86C4×C181—22no—
87C4×C181—41no—
88Dic181—22no—
89Dic181—22no—
90Dic181—41no—
91Dic181#12111yescharacteristic
92Dic181#12111yescharacteristic
93Dic181C22⋊Q811yescharacteristic
94Dic181C22⋊Q811yescharacteristic
95C2×C181—42no—
96C2×C181#38011yescharacteristic
97C2×C181#40811yescharacteristic
98C2×C181#40811yescharacteristic
99C2×C181#42412yes—
100C2×C181#43312yes—
101C181#179811yescharacteristic, complemented, Sylow
102#1798—1811nomaximal, Sylow
103C2×#117—1811no—
104C2×C4⋊C8—1811no—
105C2×C4⋊C8—1811no—
106C2×C22⋊C8—1812no—
107#336—1812no—
108C23×C4—1811no—
109C22×C8—1812no—
110C22×C8—1812no—
111#117—3622no—
112#117—3622no—
113C2×C4⋊C4—1811no—
114C2×C4⋊C4—1811no—
115C2×C22⋊C4—1811no—
116C2×C22⋊C4#1062438211yescharacteristic
117C2×C42—1811no—
118C2×C42—1811no—
119C4⋊C8—3622no—
120C4⋊C8—3622no—
121C22⋊C8—3624no—
122C24C2×#1062438211yescharacteristic
123C22×C4—3621no—
124C22×C4—3621no—
125C22×C4—3621no—
126C22×C4—3621no—
127C22×C4—1811no—
128C22×C4—1812no—
129C22×C4—1812no—
130C22×C4—1811no—
131C22×C4—1811no—
132C22×C4—1811no—
133C22×C4C2×#1062438211yescharacteristic
134C22×C4C2×#1062438211yescharacteristic
135C2×C8—3622no—
136C2×C8—3622no—
137C2×C8—3622no—
138C2×C8—3622no—
139C2×C8—3622no—
140C2×C8—3622no—
141C4⋊C4—3622no—
142C4⋊C4—3622no—
143C22⋊C4—22no—
144C22⋊C4—3622no—
145C42—3621no—
146C42—3621no—
147C42—1812no—
148C42—1812no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23C4⋊#1062438212yes—
154C23C22×#1062438211yescharacteristic
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—3621no—
159C2×C4—3621no—
160C2×C4—3622no—
161C2×C4—3622no—
162C2×C4—3622no—
163C2×C4—3622no—
164C2×C4—3622no—
165C2×C4—3621no—
166C2×C4—3622no—
167C2×C4—1812no—
168C2×C4—1812no—
169C2×C4—1811no—
170C2×C4—1811no—
171C2×C4—1811no—
172C2×C4—1811no—
173C2×C4—1811no—
174C2×C4—1811no—
175C2×C4C22⋊#1062438212yes—
176C8—7242no—
177C8—7242no—
178C22—22no—
179C22—24no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22C2×C22⋊#1062438211yescharacteristic
184C22C2×C4⋊#1062438211yescharacteristic
185C22#1915389511yescharacteristic
186C22#1915390312yes—
187C22#1915390312yes—
188C4—22no—
189C4—41no—
190C4—3622no—
191C4—3622no—
192C4—1811no—
193C4—1811no—
194C4—1811no—
195C4—1811no—
196C4—7241no—
197C2—42no—
198C2#2604222111yescharacteristic
199C2#2604225211yescharacteristic
200C2#2604225311yescharacteristic
201C2#2604228512yes—
202C2#2604230312yes—
subgroupG/H
center, FrattiniC23C22×#10624382
commutator, socleC23×C181C22×C4
FittingC2×C22⋊C4×C181C4
prime radicalC24×C181C2×C4
cosocleC22×Dic181C23
hypercenterC2×C22⋊C4#10624382
hypocenterC181#1798
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19153883C2semidirect
2. C2×C4⋊#13089685C2semidirect
4. #26042145C2semidirect
101. C181#1798semidirect

Maximal subgroups

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

subgroupindex
C2×#191538832
C2×C4⋊#130896852
C2×C22⋊#130896852
#260421452
C2×#191537722
#1798181