Finite Group Databasebeta

← #15686809

Subgroups — #15686809

Order 2944.

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#13093016C216yesmaximal, complemented
2#13093007C216yesmaximal, complemented
3#13092979C213yesmaximal, complemented
4#13092953C216yesmaximal, complemented
5#13092930C216yesmaximal, complemented
6C2×(C22⋊D23)⋊C4C213yesmaximal, complemented
7C23×#543C211yesmaximal, characteristic, complemented
8C22×C22⋊D23C2213yes—
9C23×Dic23C2211yescharacteristic
10C2×#67292C2212yescomplemented
11C22×C4×D23C2213yescomplemented
12C23×C22⋊Q8C2216yescomplemented
13C23×C22⋊Q8C2216yescomplemented
14C2×C22⋊C4×C23C2213yes—
15#10625118C2216yescomplemented
16#10625117C2216yescomplemented
17Q8×Dic23C2212yescomplemented
18#10625101C2216yes—
19#10625101C2216yes—
20C2×(C2×D23)⋊C4C2213yescomplemented
21C2×Dic23⋊C4C2213yescomplemented
22C2×C4×Dic23C2213yescomplemented
23#10625069C22112yescomplemented
24#10625066C2216yescomplemented
25#10625065C22112yescomplemented
26#10625064C2216yescomplemented
27#10625063C2216yescomplemented
28#10625061C2216yescomplemented
29#10625057C22112yescomplemented
30#10625053C22112yescomplemented
31(C22⋊D23)⋊C4C22112yescomplemented
32(C22⋊D23)⋊C4C22112yescomplemented
33C24×C23C2311yescharacteristic
34C23×D23C2313yes—
35C2×Q8×C23C2312yes—
36C2×Q8×C23C2312yes—
37C22×C4×C23C2313yes—
38C22×C4×C23C2313yes—
39C2×C22⋊D23—212no—
40C2×C22⋊D23C23112yes—
41C22×Dic23—42no—
42C22×Dic23C2313yes—
43C22×Dic23C2313yes—
44C22×Dic23C2313yes—
45#67292—44no—
46C2×D92C2316yes—
47C2×D92C2316yes—
48C2×C4×D23—46no—
49C2×C4×D23C2316yes—
50C2×C4×D23C2316yes—
51C4⋊C4×C23C2316yes—
52C4⋊C4×C23C2316yes—
53C22⋊C4×C23C23112yes—
54(C2×D23)⋊C4C23112yes—
55(C2×D23)⋊C4C23112yes—
56(C2×D23)⋊C4C23112yes—
57C4⋊Dic23C2316yes—
58C4⋊Dic23C2316yes—
59Dic23⋊C4C23112yes—
60C4×Dic23C2316yes—
61C4×Dic23C2316yes—
62C23×C23—41no—
63C23×C23—41no—
64C23×C23#4114yescomplemented
65C23×C23C2413yes—
66C22×D23—26no—
67C22×D23—43no—
68C22×D23—43no—
69C22×D23C2416yes—
70Q8×C23—42no—
71Q8×C23—42no—
72C2×C4×C23—43no—
73C2×C4×C23—43no—
74C2×C4×C23C2416yes—
75C2×C4×C23C2416yes—
76C22⋊D23—224no—
77C2×Dic23—26no—
78C2×Dic23—26no—
79C2×Dic23—43no—
80C2×Dic23C2413yes—
81C2×Dic23C2416yes—
82C2×Dic23C2411yescharacteristic
83D92—46no—
84D92—46no—
85C4×D23—212no—
86C4×D23—46no—
87#3459—231nomaximal, Sylow
88C22×C23—23no—
89C22×C23—23no—
90C22×C23—44no—
91C22×C23C2×#4116yes—
92C22×C23C2511yescharacteristic
93C2×D23—26no—
94C2×D23—26no—
95C2×D23—26no—
96C2×D23—412no—
97C4×C23—26no—
98C4×C23—26no—
99Dic23—26no—
100Dic23—23no—
101Dic23H(5,𝔽2)12yes—
102#543D2311yescharacteristic, complemented
103#539—236no—
104#538—236no—
105#534—233no—
106#529—236no—
107#525—236no—
108C2×C4×D4—233no—
109C2×C23—26no—
110C2×C23C22×#4111yescharacteristic
111C2×C23C2×H(5,𝔽2)11yescharacteristic
112C2×C23#58511yescharacteristic
113D23—46no—
114C2×#41—232no—
115C22×D4—233no—
116C23×C4—231no—
117C23×C4—233no—
118#126—2312no—
119#125—236no—
120#124—236no—
121#123—2312no—
122#123—2312no—
123C22⋊Q8—236no—
124C22⋊Q8C2×D2316yescomplemented
125C22⋊Q8C2×D2316yescomplemented
126#121—236no—
127#121—236no—
128#121—236no—
129#120—2312no—
130C4×Q8—232no—
131C4×D4—236no—
132C4×D4—2312no—
133C4×D4—2312no—
134#117—236no—
135C2×C4⋊C4—233no—
136C2×C22⋊C4—233no—
137C2×C22⋊C4C2×D2313yes—
138C2×C42—233no—
139C23#345911yescharacteristic, complemented, Sylow
140C24—233no—
141C24C22×D2311yescharacteristic
142#41—924no—
143C2×Q8C22×D2312yes—
144C2×Q8C22×D2312yes—
145C2×D4—2312no—
146C2×D4—236no—
147C2×D4—236no—
148C2×D4—4612no—
149C22×C4—233no—
150C22×C4—233no—
151C22×C4—233no—
152C22×C4—236no—
153C22×C4—236no—
154C22×C4—926no—
155C22×C4—922no—
156C22×C4C22×D2313yes—
157C22×C4C22×D2313yes—
158C4⋊C4—2312no—
159C4⋊C4—236no—
160C4⋊C4—236no—
161C4⋊C4C22×D2316yes—
162C4⋊C4C22×D2316yes—
163C22⋊C4—2312no—
164C22⋊C4—2312no—
165C22⋊C4—2312no—
166C22⋊C4C22×D23112yes—
167C42—236no—
168C42—236no—
169C23—41no—
170C23—41no—
171C23—236no—
172C23—466no—
173C23—923no—
174C23—923no—
175C23#6729214yescomplemented
176C23C23×D2313yes—
177Q8—42no—
178Q8—42no—
179D4—4624no—
180D4—926no—
181D4—926no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—231no—
185C2×C4—236no—
186C2×C4—233no—
187C2×C4—4612no—
188C2×C4—466no—
189C2×C4—466no—
190C2×C4—926no—
191C2×C4—923no—
192C2×C4C23×D2316yes—
193C2×C4C23×D2316yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—466no—
198C22—466no—
199C22—466no—
200C22—9212no—
201C22C2×#6729216yes—
202C22C24×D2311yescharacteristic
203C4—26no—
204C4—26no—
205C4—232no—
206C4—466no—
207C4—463no—
208C2—26no—
209C2—926no—
210C2C22×#6729211yescharacteristic
211C2H(5,𝔽2)×D2311yescharacteristic
212C2#1309325211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D23
commutator, socle, cosocleC22×C23C25
FittingC23×#543C2
hypercenter#543D23
hypocenterC23#3459
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #13093016C2semidirect
2. #13093007C2semidirect
3. #13092979C2semidirect
4. #13092953C2semidirect
5. #13092930C2semidirect
6. C2×(C22⋊D23)⋊C4C2semidirect
7. C23×#543C2semidirect
10. C2×#67292C22semidirect
11. C22×C4×D23C22semidirect
12. C23×C22⋊Q8C22semidirect
13. C23×C22⋊Q8C22semidirect
15. #10625118C22semidirect
16. #10625117C22semidirect
17. Q8×Dic23C22semidirect
20. C2×(C2×D23)⋊C4C22semidirect
21. C2×Dic23⋊C4C22semidirect
22. C2×C4×Dic23C22semidirect
23. #10625069C22semidirect
24. #10625066C22semidirect
25. #10625065C22semidirect
26. #10625064C22semidirect
27. #10625063C22semidirect
28. #10625061C22semidirect
29. #10625057C22semidirect
30. #10625053C22semidirect
31. (C22⋊D23)⋊C4C22semidirect
32. (C22⋊D23)⋊C4C22semidirect
64. C23×C23#41semidirect
102. #543D23semidirect
124. C22⋊Q8C2×D23semidirect
125. C22⋊Q8C2×D23semidirect
139. C23#3459semidirect
175. C23#67292semidirect

Maximal subgroups

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

subgroupindex
#130930162
#130930072
#130929792
#130929532
#130929302
C2×(C22⋊D23)⋊C42
C23×#5432
#345923