Finite Group Databasebeta

← #37549077

Subgroups — #37549077

Order 23168.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#26042297C211yesmaximal, characteristic, complemented
2#26042283C211yesmaximal, characteristic, complemented
3Q8⋊#13089685C212yesmaximal, complemented
4D4⋊#13089685C212yesmaximal, complemented
5#26041808C211yesmaximal, characteristic
6C2×#15668459—21no—
7#19153883—21no—
8C4⋊#13089685—21no—
9C4⋊#13089685C2212yes—
10#19153873—21no—
11C4×#13089685C2212yes—
12#19153809—21no—
13#19153808C411yescharacteristic
14#19153798—21no—
15#19153792C411yescharacteristic
16#19153787C411yescharacteristic
17Q8×Dic181C2211yescharacteristic
18D4×Dic181C2211yescharacteristic
19#19153776—21no—
20#19153772—21no—
21C42×D181C2211yescharacteristic
22#124×C181C411yescharacteristic
23#15668459—42no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C2×C22⋊D181—21no—
27C22×Dic181—21no—
28C2×C4×D181C2×C411yescharacteristic
29C2×C4×D181D411yescharacteristic
30C2×C4×D181D411yescharacteristic
31C4⋊Dic181—21no—
32C4⋊Dic181C2×C411yescharacteristic
33C4⋊Dic181C2×C411yescharacteristic
34#15668428—21no—
35#15668428—21no—
36#15668428—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39#15668426—21no—
40C4×Dic181—21no—
41C4×Dic181D411yescharacteristic
42C4×Dic181D411yescharacteristic
43C4×Dic181C2311yescharacteristic
44C2×Q8×C181C2×C411yescharacteristic
45C2×D4×C181C2×C411yescharacteristic
46C22⋊C4×C181—21no—
47C22⋊C4×C181—21no—
48C42×C181C2×C411yescharacteristic
49#13089685—42no—
50#13089685—42no—
51C22×D181C22⋊C411yescharacteristic
52C22⋊D181—42no—
53C2×Dic181—21no—
54C2×Dic181—21no—
55C2×Dic181—41no—
56C2×Dic181—41no—
57C2×Dic181C22⋊C411yescharacteristic
58C2×Dic181C2×D411yescharacteristic
59C2×Dic181C2×D411yescharacteristic
60C4×D181—22no—
61C4×D181—21no—
62C4×D181—21no—
63C4×D181—41no—
64C23×C181—21no—
65Q8×C181—22no—
66D4×C181—22no—
67C2×C4×C181—21no—
68C2×C4×C181C22⋊C411yescharacteristic
69C2×C4×C181C22⋊C411yescharacteristic
70C2×C4×C181C22×C411yescharacteristic
71C22×C181—41no—
72C22×C181—41no—
73C22×C181—41no—
74C22×C181C2×C22⋊C411yescharacteristic
75C2×D181—21no—
76C2×D181—21no—
77C2×D181—21no—
78C4×C181—21no—
79C4×C181—21no—
80C4×C181—41no—
81C4×C181#13112yes—
82Dic181—21no—
83Dic181—21no—
84Dic181—41no—
85Dic181—41no—
86Dic181AGL(1,ℤ/8)11yescharacteristic
87Dic181#13711yescharacteristic
88C2×C181—81no—
89C2×C181#40811yescharacteristic
90C2×C181#41711yescharacteristic
91C2×C181#42111yescharacteristic
92D181—41no—
93C181#149511yescharacteristic, complemented, Sylow
94#1495—1811nomaximal, Sylow
95#529—1811no—
96#431—1811no—
97#423—1811no—
98Q8⋊C8—1812no—
99D4⋊C8—1812no—
100C2×M16—3621no—
101#126—3621no—
102#125—1811no—
103#124#1062438211yescharacteristic
104#123—3621no—
105C22⋊Q8—1811no—
106#121—1811no—
107C4×Q8—1811no—
108C4×D4—3621no—
109C4×D4—1811no—
110#117—3621no—
111C2×C42—1811no—
112C4⋊C8—3621no—
113C4⋊C8—3621no—
114C4⋊C8—1812no—
115C22⋊C8—3621no—
116C4×C8—1812no—
117C2×Q8C2×#1062438211yescharacteristic
118C2×D4—3621no—
119C2×D4C2×#1062438211yescharacteristic
120C22×C4—3621no—
121C22×C4—1811no—
122C22×C4—1811no—
123C22×C4—1811no—
124M16—7242no—
125C2×C8—3622no—
126C2×C8—3622no—
127C4⋊C4—3621no—
128C4⋊C4—3621no—
129C4⋊C4—3621no—
130C4⋊C4—3621no—
131C4⋊C4—1811no—
132C4⋊C4—1811no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—3621no—
136C22⋊C4—3621no—
137C22⋊C4—3621no—
138C42—3621no—
139C42—1811no—
140C42—1811no—
141C42—1811no—
142C42C2×#1062438211yescharacteristic
143C23—21no—
144C23—1811no—
145Q8—22no—
146D4—22no—
147D4—7242no—
148C2×C4—21no—
149C2×C4—3621no—
150C2×C4—3621no—
151C2×C4—3622no—
152C2×C4—3621no—
153C2×C4—3621no—
154C2×C4—1811no—
155C2×C4—1811no—
156C2×C4—1811no—
157C2×C4—7241no—
158C2×C4—7241no—
159C2×C4—7241no—
160C2×C4C22⋊#1062438211yescharacteristic
161C2×C4C22⋊#1062438211yescharacteristic
162C2×C4C22×#1062438211yescharacteristic
163C8—7242no—
164C8—7242no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—3621no—
169C22—3621no—
170C22—3621no—
171C22C2×C22⋊#1062438211yescharacteristic
172C4—21no—
173C4—21no—
174C4—41no—
175C4—3621no—
176C4—3621no—
177C4—1811no—
178C4—1811no—
179C4—7241no—
180C4—7241no—
181C4#1915390312yes—
182C2—81no—
183C2—7241no—
184C2#2604225311yescharacteristic
185C2#2604227011yescharacteristic
186C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#124×C181C4
socleC22×C181C2×C22⋊C4
prime radical#19153787C4
cosocleC4×Dic181C23
hypercenter#124#10624382
hypocenterC181#1495
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26042297C2semidirect
2. #26042283C2semidirect
3. Q8⋊#13089685C2semidirect
4. D4⋊#13089685C2semidirect
93. C181#1495semidirect

Maximal subgroups

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

subgroupindex
#260422972
#260422832
Q8⋊#130896852
D4⋊#130896852
#260418082
#1495181