Finite Group Databasebeta

← #27155148

Subgroups — #27155148

Order 12416.

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#20261334C216yesmaximal, complemented
2#20261325C216yesmaximal, complemented
3#20261297C213yesmaximal, complemented
4#20261271C216yesmaximal, complemented
5#20261248C216yesmaximal, complemented
6C2×#15690770C213yesmaximal, complemented
7#543×C97C211yesmaximal, characteristic, complemented
8C2×#13095512C2212yescomplemented
9C22×C22⋊D97C2213yes—
10C23×Dic97C2211yescharacteristic
11C22×C4×D97C2213yescomplemented
12#15690805C22112yescomplemented
13#15690801C2216yescomplemented
14#15690799C2216yescomplemented
15#15690795C22112yescomplemented
16#15690794C22112yescomplemented
17#15690786C2216yescomplemented
18#15690785C2216yescomplemented
19#15690784C2216yes—
20#15690784C2216yes—
21#15690780C22112yescomplemented
22Q8×Dic97C2212yescomplemented
23#15690772C2216yescomplemented
24#15690770C22112yescomplemented
25#15690770C22112yescomplemented
26#15690768C2216yescomplemented
27C2×#13095490C2213yescomplemented
28C2×#13095488C2213yescomplemented
29C2×C4×Dic97C2213yescomplemented
30C22⋊Q8×C97C2216yescomplemented
31C22⋊Q8×C97C2216yescomplemented
32C2×C22⋊C4×C97C2213yes—
33C23×D97C2313yes—
34#13095512—44no—
35C2×D388C2316yes—
36C2×D388C2316yes—
37C2×C22⋊D97—212no—
38C2×C22⋊D97C23112yes—
39C22×Dic97—42no—
40C22×Dic97C2313yes—
41C22×Dic97C2313yes—
42C22×Dic97C2313yes—
43C2×C4×D97—46no—
44C2×C4×D97C2316yes—
45C2×C4×D97C2316yes—
46C4⋊Dic97C2316yes—
47C4⋊Dic97C2316yes—
48#13095490C23112yes—
49#13095488C23112yes—
50#13095488C23112yes—
51#13095488C23112yes—
52C4×Dic97C2316yes—
53C4×Dic97C2316yes—
54C24×C97C2311yescharacteristic
55C2×Q8×C97C2312yes—
56C2×Q8×C97C2312yes—
57C22×C4×C97C2313yes—
58C22×C4×C97C2313yes—
59C4⋊C4×C97C2316yes—
60C4⋊C4×C97C2316yes—
61C22⋊C4×C97C23112yes—
62C23×C97—41no—
63C23×C97—41no—
64C23×C97#4114yescomplemented
65C23×C97C2413yes—
66C22×D97—26no—
67C22×D97—43no—
68C22×D97—43no—
69C22×D97C2416yes—
70Q8×C97—42no—
71Q8×C97—42no—
72C2×C4×C97—43no—
73C2×C4×C97—43no—
74C2×C4×C97C2416yes—
75C2×C4×C97C2416yes—
76C22⋊D97—224no—
77C2×Dic97—26no—
78C2×Dic97—26no—
79C2×Dic97—43no—
80C2×Dic97C2413yes—
81C2×Dic97C2416yes—
82C2×Dic97C2411yescharacteristic
83D388—46no—
84D388—46no—
85C4×D97—212no—
86C4×D97—46no—
87C22×C97—23no—
88C22×C97—23no—
89C22×C97—44no—
90C22×C97C2×#4116yes—
91C22×C97C2511yescharacteristic
92C2×D97—26no—
93C2×D97—26no—
94C2×D97—26no—
95C2×D97—412no—
96C4×C97—26no—
97C4×C97—26no—
98Dic97—26no—
99Dic97—23no—
100Dic97H(5,𝔽2)12yes—
101C2×C97—26no—
102C2×C97C22×#4111yescharacteristic
103C2×C97C2×H(5,𝔽2)11yescharacteristic
104C2×C97#58511yescharacteristic
105D97—46no—
106#3459—971nomaximal, Sylow
107C97#345911yescharacteristic, complemented, Sylow
108#543D9711yescharacteristic, complemented
109#539—976no—
110#538—976no—
111#534—973no—
112#529—976no—
113#525—976no—
114C2×C4×D4—973no—
115C2×#41—972no—
116C22×D4—973no—
117C23×C4—971no—
118C23×C4—973no—
119#126—9712no—
120#125—976no—
121#124—976no—
122#123—9712no—
123#123—9712no—
124C22⋊Q8—976no—
125C22⋊Q8C2×D9716yescomplemented
126C22⋊Q8C2×D9716yescomplemented
127#121—976no—
128#121—976no—
129#121—976no—
130#120—9712no—
131C4×Q8—972no—
132C4×D4—976no—
133C4×D4—9712no—
134C4×D4—9712no—
135#117—976no—
136C2×C4⋊C4—973no—
137C2×C22⋊C4—973no—
138C2×C22⋊C4C2×D9713yes—
139C2×C42—973no—
140C24—973no—
141C24C22×D9711yescharacteristic
142#41—3884no—
143C2×Q8C22×D9712yes—
144C2×Q8C22×D9712yes—
145C2×D4—9712no—
146C2×D4—976no—
147C2×D4—976no—
148C2×D4—19412no—
149C22×C4—973no—
150C22×C4—973no—
151C22×C4—973no—
152C22×C4—976no—
153C22×C4—976no—
154C22×C4—3886no—
155C22×C4—3882no—
156C22×C4C22×D9713yes—
157C22×C4C22×D9713yes—
158C4⋊C4—9712no—
159C4⋊C4—976no—
160C4⋊C4—976no—
161C4⋊C4C22×D9716yes—
162C4⋊C4C22×D9716yes—
163C22⋊C4—9712no—
164C22⋊C4—9712no—
165C22⋊C4—9712no—
166C22⋊C4C22×D97112yes—
167C42—976no—
168C42—976no—
169C23—41no—
170C23—41no—
171C23—976no—
172C23—3883no—
173C23—3883no—
174C23—1946no—
175C23#1309551214yescomplemented
176C23C23×D9713yes—
177Q8—42no—
178Q8—42no—
179D4—3886no—
180D4—3886no—
181D4—19424no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—971no—
185C2×C4—976no—
186C2×C4—973no—
187C2×C4—3886no—
188C2×C4—3883no—
189C2×C4—19412no—
190C2×C4—1946no—
191C2×C4—1946no—
192C2×C4C23×D9716yes—
193C2×C4C23×D9716yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—38812no—
198C22—1946no—
199C22—1946no—
200C22—1946no—
201C22C2×#1309551216yes—
202C22C24×D9711yescharacteristic
203C4—26no—
204C4—26no—
205C4—972no—
206C4—1946no—
207C4—1943no—
208C2—26no—
209C2—3886no—
210C2C22×#1309551211yescharacteristic
211C2H(5,𝔽2)×D9711yescharacteristic
212C2#2026157011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#543×C97C2
hypercenter#543D97
hypocenterC97#3459
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20261334C2semidirect
2. #20261325C2semidirect
3. #20261297C2semidirect
4. #20261271C2semidirect
5. #20261248C2semidirect
6. C2×#15690770C2semidirect
7. #543×C97C2semidirect
8. C2×#13095512C22semidirect
11. C22×C4×D97C22semidirect
12. #15690805C22semidirect
13. #15690801C22semidirect
14. #15690799C22semidirect
15. #15690795C22semidirect
16. #15690794C22semidirect
17. #15690786C22semidirect
18. #15690785C22semidirect
21. #15690780C22semidirect
22. Q8×Dic97C22semidirect
23. #15690772C22semidirect
24. #15690770C22semidirect
25. #15690770C22semidirect
26. #15690768C22semidirect
27. C2×#13095490C22semidirect
28. C2×#13095488C22semidirect
29. C2×C4×Dic97C22semidirect
30. C22⋊Q8×C97C22semidirect
31. C22⋊Q8×C97C22semidirect
64. C23×C97#41semidirect
107. C97#3459semidirect
108. #543D97semidirect
125. C22⋊Q8C2×D97semidirect
126. C22⋊Q8C2×D97semidirect
175. C23#13095512semidirect

Maximal subgroups

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

subgroupindex
#202613342
#202613252
#202612972
#202612712
#202612482
C2×#156907702
#543×C972
#345997