Finite Group Databasebeta

← C2×#26041884 #37547081

Subgroups — C2×#26041884 #37547081

Order 23168.

1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 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#26041884C2116yesmaximal, direct product factor
2C2×#19153813C211yesmaximal, characteristic, complemented
3C2×#19153808C211yesmaximal, characteristic, complemented
4C2×#19153801C214yesmaximal, complemented
5C2×#19153790C212yesmaximal, complemented
6C2×#19153789C214yesmaximal, complemented
7C2×C4×D724C212yesmaximal, complemented
8C2×#117×C181C211yesmaximal, characteristic, complemented
9C22×D724C2212yes—
10C22×D724C2214yes—
11C22×C22⋊D181C2214yes—
12C22×C4×D181C2214yes—
13#19153813C2218yescomplemented
14#19153808C2218yescomplemented
15#19153801C22132yescomplemented
16#19153790C22116yescomplemented
17#19153789C22132yescomplemented
18C4×D724C22116yescomplemented
19C2×C4⋊Dic181C2212yes—
20C2×#15668428C2214yes—
21C2×#15668426C2214yes—
22C2×#15668426C2214yes—
23C23×C4×C181C2211yescharacteristic
24#117×C181C2218yescomplemented
25C2×C4⋊C4×C181C2212yes—
26C2×C22⋊C4×C181C2212yes—
27C2×C42×C181C2211yescharacteristic
28C2×C42×C181C2211yescharacteristic
29C23×D181C2314yes—
30C2×D724—28no—
31C2×D724—48no—
32C2×D724C2318yes—
33C2×D724C23116yes—
34C2×C22⋊D181—48no—
35C2×C22⋊D181C23116yes—
36C22×Dic181C2314yes—
37C2×C4×D181—48no—
38C2×C4×D181C23116yes—
39C4⋊Dic181C2318yes—
40#15668428C23116yes—
41#15668426C23116yes—
42#15668426C23116yes—
43C24×C181C2311yescharacteristic
44C22×C4×C181—24no—
45C22×C4×C181C2312yes—
46C22×C4×C181C2312yes—
47C22×C4×C181C2311yescharacteristic
48C22×C4×C181C2314yes—
49C22×C4×C181C2311yescharacteristic
50C4⋊C4×C181C2318yes—
51C22⋊C4×C181C2318yes—
52C42×C181C2314yes—
53C42×C181C2314yes—
54C22×D181—28no—
55C22×D181—28no—
56C22×D181—28no—
57C22×D181C2418yes—
58D724—216no—
59D724—416no—
60C22⋊D181—416no—
61C2×Dic181—28no—
62C2×Dic181C2418yes—
63C4×D181—416no—
64C23×C181—22no—
65C23×C181—24no—
66C23×C181C2412yes—
67C23×C181C2411yescharacteristic
68C2×C4×C181—22no—
69C2×C4×C181—24no—
70C2×C4×C181—24no—
71C2×C4×C181—28no—
72C2×C4×C181#4114yes—
73C2×C4×C181C2414yes—
74C2×C4×C181C2414yes—
75C2×C4×C181C2412yes—
76C2×C4×C181C2412yes—
77C22×C181—22no—
78C22×C181—24no—
79C22×C181—44no—
80C22×C181C2×#4112yes—
81C22×C181H(5,𝔽2)14yes—
82C22×C181C2511yescharacteristic
83C2×D181—28no—
84C2×D181—28no—
85C2×D181—28no—
86C2×D181—416no—
87C4×C181—22no—
88C4×C181—24no—
89C4×C181—24no—
90C4×C181C2×#4114yes—
91Dic181—28no—
92C2×C181—42no—
93C2×C181#54014yescomplemented
94C2×C181C22×#4111yescharacteristic
95C2×C181C2×H(5,𝔽2)12yes—
96D181—48no—
97C181C2×#54011yescharacteristic, complemented, Sylow
98C2×#540—1811nomaximal, Sylow
99#540—18116no—
100C2×#127—1811no—
101C2×#125—1811no—
102C2×#123—1814no—
103C2×#121—1812no—
104C2×#121—1814no—
105C2×C4×D4—1812no—
106C2×#117D18111yescharacteristic, complemented
107C22×D4—1814no—
108C22×D4—1812no—
109C22×D4—1814no—
110C23×C4—1814no—
111C23×C4C2×D18111yescharacteristic
112#127—1818no—
113#125—1818no—
114#123—18132no—
115#121—18116no—
116#121—18132no—
117C4×D4—18116no—
118#117C2×D18118yescomplemented
119C2×C4⋊C4—1812no—
120C2×C4⋊C4—1814no—
121C2×C4⋊C4C2×D18112yes—
122C2×C22⋊C4—1814no—
123C2×C22⋊C4—1814no—
124C2×C22⋊C4C2×D18112yes—
125C2×C42C2×D18111yescharacteristic
126C2×C42C2×D18111yescharacteristic
127C24—1814no—
128C24C22×D18111yescharacteristic
129C2×D4—3628no—
130C2×D4—18116no—
131C2×D4—1818no—
132C2×D4—18116no—
133C2×D4—7248no—
134C2×D4—7248no—
135C22×C4—24no—
136C22×C4—1814no—
137C22×C4—18116no—
138C22×C4—7248no—
139C22×C4C22×D18114yes—
140C22×C4C22×D18112yes—
141C22×C4C22×D18112yes—
142C22×C4C22×D18111yescharacteristic
143C22×C4C22×D18111yescharacteristic
144C4⋊C4—1818no—
145C4⋊C4—18116no—
146C4⋊C4C22×D18118yes—
147C22⋊C4—18116no—
148C22⋊C4—18116no—
149C22⋊C4C22×D18118yes—
150C42C22×D18114yes—
151C42C22×D18114yes—
152C23—22no—
153C23—24no—
154C23—3628no—
155C23—3628no—
156C23—3628no—
157C23—1818no—
158C23C23×D18112yes—
159C23C23×D18111yescharacteristic
160D4—36216no—
161D4—72416no—
162D4—72416no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—28no—
167C2×C4—3628no—
168C2×C4—1818no—
169C2×C4—72416no—
170C2×C4#1566844914yes—
171C2×C4C23×D18114yes—
172C2×C4C23×D18114yes—
173C2×C4C23×D18112yes—
174C2×C4C23×D18112yes—
175C22—22no—
176C22—24no—
177C22—44no—
178C22—3628no—
179C22—3628no—
180C22—3628no—
181C22—72416no—
182C22C2×#1566844912yes—
183C22#1915386214yes—
184C22C24×D18111yescharacteristic
185C4—22no—
186C4—24no—
187C4—24no—
188C4—3628no—
189C4C2×#1566844914yes—
190C2—42no—
191C2—7248no—
192C2#2604188414yesdirect product factor
193C2C22×#1566844911yescharacteristic
194C2C2×#1915386212yes—
subgroupG/H
centerC23C23×D181
commutator, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC2×#117×C181C2
socleC23×C181C24
hypercenterC2×#117D181
hypocenterC181C2×#540
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #26041884C2direct (N × H)
2. C2×#19153813C2semidirect
3. C2×#19153808C2semidirect
4. C2×#19153801C2semidirect
5. C2×#19153790C2semidirect
6. C2×#19153789C2semidirect
7. C2×C4×D724C2semidirect
8. C2×#117×C181C2semidirect
13. #19153813C22semidirect
14. #19153808C22semidirect
15. #19153801C22semidirect
16. #19153790C22semidirect
17. #19153789C22semidirect
18. C4×D724C22semidirect
24. #117×C181C22semidirect
93. C2×C181#540semidirect
97. C181C2×#540semidirect
106. C2×#117D181semidirect
118. #117C2×D181semidirect
192. C2#26041884direct (N × H)

Maximal subgroups

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

subgroupindex
#260418842
C2×#191538132
C2×#191538082
C2×#191538012
C2×#191537902
C2×#191537892
C2×C4×D7242
C2×#117×C1812
C2×#540181