Finite Group Databasebeta

← #37547133

Subgroups — #37547133

Order 23168.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#26041878C216yesmaximal, complemented
2#26041869C216yesmaximal, complemented
3#26041841C213yesmaximal, complemented
4#26041815C216yesmaximal, complemented
5#26041792C216yesmaximal, complemented
6C2×#19153776C213yesmaximal, complemented
7#543×C181C211yesmaximal, characteristic, complemented
8C2×#15668450C2212yescomplemented
9C22×C22⋊D181C2213yes—
10C23×Dic181C2211yescharacteristic
11C22×C4×D181C2213yescomplemented
12#19153811C22112yescomplemented
13#19153807C2216yescomplemented
14#19153805C2216yescomplemented
15#19153801C22112yescomplemented
16#19153800C22112yescomplemented
17#19153792C2216yescomplemented
18#19153791C2216yescomplemented
19#19153790C2216yes—
20#19153790C2216yes—
21#19153786C22112yescomplemented
22Q8×Dic181C2212yescomplemented
23#19153778C2216yescomplemented
24#19153776C22112yescomplemented
25#19153776C22112yescomplemented
26#19153774C2216yescomplemented
27C2×#15668428C2213yescomplemented
28C2×#15668426C2213yescomplemented
29C2×C4×Dic181C2213yescomplemented
30C22⋊Q8×C181C2216yescomplemented
31C22⋊Q8×C181C2216yescomplemented
32C2×C22⋊C4×C181C2213yes—
33C23×D181C2313yes—
34#15668450—44no—
35C2×D724C2316yes—
36C2×D724C2316yes—
37C2×C22⋊D181—212no—
38C2×C22⋊D181C23112yes—
39C22×Dic181—42no—
40C22×Dic181C2313yes—
41C22×Dic181C2313yes—
42C22×Dic181C2313yes—
43C2×C4×D181—46no—
44C2×C4×D181C2316yes—
45C2×C4×D181C2316yes—
46C4⋊Dic181C2316yes—
47C4⋊Dic181C2316yes—
48#15668428C23112yes—
49#15668426C23112yes—
50#15668426C23112yes—
51#15668426C23112yes—
52C4×Dic181C2316yes—
53C4×Dic181C2316yes—
54C24×C181C2311yescharacteristic
55C2×Q8×C181C2312yes—
56C2×Q8×C181C2312yes—
57C22×C4×C181C2313yes—
58C22×C4×C181C2313yes—
59C4⋊C4×C181C2316yes—
60C4⋊C4×C181C2316yes—
61C22⋊C4×C181C23112yes—
62C22×D181—26no—
63C22×D181—43no—
64C22×D181—43no—
65C22×D181C2416yes—
66D724—46no—
67D724—46no—
68C22⋊D181—224no—
69C2×Dic181—26no—
70C2×Dic181—26no—
71C2×Dic181—43no—
72C2×Dic181C2413yes—
73C2×Dic181C2416yes—
74C2×Dic181C2411yescharacteristic
75C4×D181—212no—
76C4×D181—46no—
77C23×C181—41no—
78C23×C181—41no—
79C23×C181#4114yescomplemented
80C23×C181C2413yes—
81Q8×C181—42no—
82Q8×C181—42no—
83C2×C4×C181—43no—
84C2×C4×C181—43no—
85C2×C4×C181C2416yes—
86C2×C4×C181C2416yes—
87C22×C181—23no—
88C22×C181—23no—
89C22×C181—44no—
90C22×C181C2×#4116yes—
91C22×C181C2511yescharacteristic
92C2×D181—26no—
93C2×D181—26no—
94C2×D181—26no—
95C2×D181—412no—
96C4×C181—26no—
97C4×C181—26no—
98Dic181—26no—
99Dic181—23no—
100Dic181H(5,𝔽2)12yes—
101C2×C181—26no—
102C2×C181C22×#4111yescharacteristic
103C2×C181C2×H(5,𝔽2)11yescharacteristic
104C2×C181#58511yescharacteristic
105D181—46no—
106C181#345911yescharacteristic, complemented, Sylow
107#3459—1811nomaximal, Sylow
108#543D18111yescharacteristic, complemented
109#539—1816no—
110#538—1816no—
111#534—1813no—
112#529—1816no—
113#525—1816no—
114C2×C4×D4—1813no—
115C2×#41—1812no—
116C22×D4—1813no—
117C23×C4—1811no—
118C23×C4—1813no—
119#126—18112no—
120#125—1816no—
121#124—1816no—
122#123—18112no—
123#123—18112no—
124C22⋊Q8—1816no—
125C22⋊Q8C2×D18116yescomplemented
126C22⋊Q8C2×D18116yescomplemented
127#121—1816no—
128#121—1816no—
129#121—1816no—
130#120—18112no—
131C4×Q8—1812no—
132C4×D4—1816no—
133C4×D4—18112no—
134C4×D4—18112no—
135#117—1816no—
136C2×C4⋊C4—1813no—
137C2×C22⋊C4—1813no—
138C2×C22⋊C4C2×D18113yes—
139C2×C42—1813no—
140C24—1813no—
141C24C22×D18111yescharacteristic
142#41—7244no—
143C2×Q8C22×D18112yes—
144C2×Q8C22×D18112yes—
145C2×D4—36212no—
146C2×D4—18112no—
147C2×D4—1816no—
148C2×D4—1816no—
149C22×C4—1813no—
150C22×C4—1813no—
151C22×C4—1816no—
152C22×C4—1813no—
153C22×C4—1816no—
154C22×C4—7246no—
155C22×C4—7242no—
156C22×C4C22×D18113yes—
157C22×C4C22×D18113yes—
158C4⋊C4—18112no—
159C4⋊C4—1816no—
160C4⋊C4—1816no—
161C4⋊C4C22×D18116yes—
162C4⋊C4C22×D18116yes—
163C22⋊C4—18112no—
164C22⋊C4—18112no—
165C22⋊C4—18112no—
166C22⋊C4C22×D181112yes—
167C42—1816no—
168C42—1816no—
169C23—41no—
170C23—41no—
171C23—3626no—
172C23—1816no—
173C23—7243no—
174C23—7243no—
175C23#1566845014yescomplemented
176C23C23×D18113yes—
177Q8—42no—
178Q8—42no—
179D4—36224no—
180D4—7246no—
181D4—7246no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—36212no—
185C2×C4—3626no—
186C2×C4—3626no—
187C2×C4—1816no—
188C2×C4—1813no—
189C2×C4—1811no—
190C2×C4—7246no—
191C2×C4—7243no—
192C2×C4C23×D18116yes—
193C2×C4C23×D18116yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—3626no—
198C22—3626no—
199C22—3626no—
200C22—72412no—
201C22C2×#1566845016yes—
202C22C24×D18111yescharacteristic
203C4—26no—
204C4—26no—
205C4—3626no—
206C4—3623no—
207C4—1812no—
208C2—26no—
209C2—7246no—
210C2C22×#1566845011yescharacteristic
211C2H(5,𝔽2)×D18111yescharacteristic
212C2#2604211411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#543×C181C2
hypercenter#543D181
hypocenterC181#3459
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #26041878C2semidirect
2. #26041869C2semidirect
3. #26041841C2semidirect
4. #26041815C2semidirect
5. #26041792C2semidirect
6. C2×#19153776C2semidirect
7. #543×C181C2semidirect
8. C2×#15668450C22semidirect
11. C22×C4×D181C22semidirect
12. #19153811C22semidirect
13. #19153807C22semidirect
14. #19153805C22semidirect
15. #19153801C22semidirect
16. #19153800C22semidirect
17. #19153792C22semidirect
18. #19153791C22semidirect
21. #19153786C22semidirect
22. Q8×Dic181C22semidirect
23. #19153778C22semidirect
24. #19153776C22semidirect
25. #19153776C22semidirect
26. #19153774C22semidirect
27. C2×#15668428C22semidirect
28. C2×#15668426C22semidirect
29. C2×C4×Dic181C22semidirect
30. C22⋊Q8×C181C22semidirect
31. C22⋊Q8×C181C22semidirect
79. C23×C181#41semidirect
106. C181#3459semidirect
108. #543D181semidirect
125. C22⋊Q8C2×D181semidirect
126. C22⋊Q8C2×D181semidirect
175. C23#15668450semidirect

Maximal subgroups

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

subgroupindex
#260418782
#260418692
#260418412
#260418152
#260417922
C2×#191537762
#543×C1812
#3459181