Finite Group Databasebeta

← #399×D97 #27144982

Subgroups — #399×D97 #27144982

Order 12416.

908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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
1C2×C4⋊C4×D97C213yesmaximal, complemented
2C2×C22⋊C4×D97C213yesmaximal
3#20260672C211yesmaximal, characteristic, complemented
4#20260671C213yesmaximal, complemented
5#20260670C213yesmaximal, complemented
6#95×D97C211yesmaximal, characteristic, complemented
7#399×C97C211yesmaximal, characteristic, complemented
8C24×D97C2211yescharacteristic
9C22×C4×D97C2213yes—
10C22×C4×D97C2213yes—
11C2×C4⋊Dic97C2213yescomplemented
12C2×#13095490C2216yescomplemented
13C4⋊C4×D97—212no—
14C2×C22⋊Dic97C2213yes—
15C2×#13095488C2213yes—
16C2×#13095488C2213yes—
17C22⋊C4×D97—212no—
18#15690717C2213yescomplemented
19C2×C4⋊C4×C97C2213yescomplemented
20C2×C22⋊C4×C97C2213yes—
21#95×C97C2211yescharacteristic, complemented
22C23×D97—26no—
23C23×D97—26no—
24C23×D97Q814yes—
25C23×D97C2311yescharacteristic
26C23×D97C2311yescharacteristic
27C22×Dic97C2313yes—
28C22×Dic97C2313yes—
29C2×C4×D97—212no—
30C2×C4×D97—26no—
31C2×C4×D97—212no—
32C2×C4×D97D4112yes—
33C4⋊Dic97—26no—
34#13095490—212no—
35C22⋊Dic97—26no—
36#13095488—26no—
37#13095488—26no—
38C24×C97C2311yescharacteristic
39C22×C4×C97C2313yes—
40C22×C4×C97C2313yes—
41C4⋊C4×C97—26no—
42C22⋊C4×C97—26no—
43C23×C97—23no—
44C23×C97—23no—
45C23×C97C2×Q812yes—
46C23×C97C2411yescharacteristic
47C22×D97—23no—
48C22×D97—23no—
49C22×D97—224no—
50C22×D97—412no—
51C22×D97—44no—
52C22×D97C2×D416yes—
53C22×D97C2×Q812yes—
54C22×D97C2×Q812yes—
55C22×D97#4116yes—
56C2×C4×C97—26no—
57C2×C4×C97—23no—
58C2×C4×C97—26no—
59C2×C4×C97C2×D416yes—
60C2×Dic97—26no—
61C2×Dic97—23no—
62C2×Dic97—26no—
63C2×Dic97C2×D416yes—
64C4×D97—224no—
65C4×D97—412no—
66C22×C97—26no—
67C22×C97—43no—
68C22×C97—41no—
69C22×C97C22×D413yes—
70C22×C97C22×Q811yescharacteristic
71C22×C97C2×#4113yes—
72C2×D97—26no—
73C2×D97—43no—
74C2×D97—41no—
75C2×D97—416no—
76C2×D97C22⋊Q8112yes—
77C2×D97#123112yes—
78C2×D97#12714yes—
79C4×C97—26no—
80C4×C97—43no—
81Dic97—26no—
82Dic97—43no—
83C2×C97—42no—
84C2×C97C2×C22⋊Q813yes—
85C2×C97C2×#12313yes—
86C2×C97C2×#12711yescharacteristic
87D97—42no—
88D97#39918yesdirect product factor
89C2×#399—971nomaximal, Sylow
90C97C2×#39911yescharacteristic, complemented, Sylow
91C22×C4⋊C4—973no—
92C22×C22⋊C4—973no—
93#399—973no—
94#399—973no—
95#399—971no—
96#399D9711yescharacteristic, direct product factor
97C2×#95—971no—
98C25—971no—
99C23×C4—973no—
100C23×C4—973no—
101C2×C4⋊C4—976no—
102C2×C4⋊C4—973no—
103C2×C4⋊C4—19412no—
104C2×C4⋊C4C2×D9713yescomplemented
105C2×C22⋊C4—973no—
106C2×C22⋊C4—973no—
107C2×C22⋊C4—973no—
108C2×C22⋊C4—19412no—
109C2×C22⋊C4C2×D9713yes—
110#95—973no—
111#95C2×D9711yescharacteristic, complemented
112C24—971no—
113C24—974no—
114C24—971no—
115C24—1946no—
116C24—1946no—
117C24C22×D9711yescharacteristic
118C22×C4—973no—
119C22×C4—9712no—
120C22×C4—973no—
121C22×C4—1946no—
122C22×C4—19412no—
123C22×C4—19412no—
124C22×C4C22×D9713yes—
125C22×C4C22×D9713yes—
126C4⋊C4—26no—
127C4⋊C4—19412no—
128C4⋊C4—1946no—
129C22⋊C4—26no—
130C22⋊C4—1946no—
131C22⋊C4—1946no—
132C22⋊C4—1946no—
133C23—23no—
134C23—23no—
135C23—976no—
136C23—972no—
137C23—976no—
138C23—972no—
139C23—38812no—
140C23—3884no—
141C23—19424no—
142C23—1943no—
143C23—1943no—
144C23Q8×D9712yes—
145C23C23×D9711yescharacteristic
146C2×C4—26no—
147C2×C4—23no—
148C2×C4—26no—
149C2×C4—976no—
150C2×C4—38812no—
151C2×C4—19424no—
152C2×C4—1943no—
153C2×C4—1946no—
154C2×C4—1946no—
155C2×C4D4×D9716yes—
156C22—26no—
157C22—43no—
158C22—41no—
159C22—9712no—
160C22—9712no—
161C22—974no—
162C22—38816no—
163C22—3883no—
164C22—3881no—
165C22—1946no—
166C22C2×D4×D9713yes—
167C22C2×Q8×D9711yescharacteristic
168C22#41×D9713yes—
169C4—26no—
170C4—43no—
171C4—3883no—
172C4—1946no—
173C2—42no—
174C2—978no—
175C2—3882no—
176C2C22⋊Q8×D9713yes—
177C2#123×D9713yes—
178C2#127×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#399×C97C2
prime radicalC24×D97C22
hypercenter#399D97
hypocenterC97C2×#399
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D97C2semidirect
3. #20260672C2semidirect
4. #20260671C2semidirect
5. #20260670C2semidirect
6. #95×D97C2semidirect
7. #399×C97C2semidirect
11. C2×C4⋊Dic97C22semidirect
12. C2×#13095490C22semidirect
18. #15690717C22semidirect
19. C2×C4⋊C4×C97C22semidirect
21. #95×C97C22semidirect
88. D97#399direct (N × H)
90. C97C2×#399semidirect
96. #399D97direct (N × H)
104. C2×C4⋊C4C2×D97semidirect
111. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
C2×C22⋊C4×D972
#202606722
#202606712
#202606702
#95×D972
#399×C972
C2×#39997