Finite Group Databasebeta

← #27145014

Subgroups — #27145014

Order 12416.

840 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 167 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×D4×Dic97C213yesmaximal, complemented
2C2×#15690770C213yesmaximal, complemented
3#20260672C211yesmaximal, characteristic, complemented
4#20260671C213yesmaximal, complemented
5#20260643C213yesmaximal, complemented
6#20260580C211yesmaximal, characteristic, complemented
7#392×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2213yes—
9C23×Dic97C2213yes—
10C22×C4×D97C2211yescharacteristic
11D4×Dic97—212no—
12#15690770—212no—
13C2×C4⋊Dic97C2213yes—
14C2×#13095490C2213yes—
15C2×C22⋊Dic97C2216yes—
16C2×#13095488C2213yes—
17C2×C4×Dic97C2213yes—
18#15690717C2213yes—
19C22×D4×C97C2213yes—
20C2×C22⋊C4×C97C2213yes—
21#95×C97C2211yescharacteristic
22C23×D97C2311yescharacteristic
23C2×C22⋊D97—26no—
24C2×C22⋊D97—26no—
25C2×C22⋊D97—26no—
26C22×Dic97—26no—
27C22×Dic97—212no—
28C22×Dic97C2313yes—
29C22×Dic97C2313yes—
30C22×Dic97C2311yescharacteristic
31C2×C4×D97—26no—
32C4⋊Dic97—26no—
33#13095490—26no—
34C22⋊Dic97—212no—
35#13095488—26no—
36C4×Dic97D4112yescomplemented
37C24×C97C2313yes—
38C2×D4×C97—212no—
39C2×D4×C97—26no—
40C22×C4×C97C2311yescharacteristic
41C22×C4×C97C2313yes—
42C22⋊C4×C97—26no—
43C23×C97—23no—
44C23×C97—26no—
45C23×C97—26no—
46C23×C97—23no—
47C23×C97#4116yes—
48C23×C97C2411yescharacteristic
49C22×D97—23no—
50C22×D97—23no—
51C22×D97#4112yes—
52D4×C97—224no—
53C2×C4×C97—26no—
54C2×C4×C97—23no—
55C2×C4×C97C2×D416yes—
56C22⋊D97—224no—
57C2×Dic97—23no—
58C2×Dic97—23no—
59C2×Dic97—23no—
60C2×Dic97—26no—
61C2×Dic97—412no—
62C2×Dic97C2×D416yes—
63C2×Dic97C2×D416yes—
64C4×D97—44no—
65C22×C97—26no—
66C22×C97—212no—
67C22×C97—43no—
68C22×C97—46no—
69C22×C97—43no—
70C22×C97C22×D413yes—
71C22×C97C2×#4111yescharacteristic
72C22×C97C2×#4113yes—
73C2×D97—26no—
74C2×D97—43no—
75C2×D97—41no—
76C4×C97—26no—
77C4×C97—41no—
78Dic97—26no—
79Dic97—43no—
80Dic97#12714yes—
81C2×C97—46no—
82C2×C97#52513yes—
83C2×C97#52513yes—
84C2×C97C2×#12711yescharacteristic
85D97—42no—
86#2397—971nomaximal, Sylow
87C97#239711yescharacteristic, complemented, Sylow
88C2×C4×D4—973no—
89C2×C4×D4—973no—
90#399—973no—
91#399—971no—
92#392—973no—
93#392D9711yescharacteristic, complemented
94#376—971no—
95C22×D4—973no—
96C22×D4C2×D9713yes—
97C23×C4—973no—
98C23×C4—971no—
99C4×D4—19412no—
100C4×D4—19412no—
101C2×C4⋊C4—973no—
102C2×C4⋊C4—973no—
103C2×C22⋊C4—976no—
104C2×C22⋊C4—973no—
105C2×C22⋊C4C2×D9713yes—
106C2×C42—973no—
107#95—973no—
108#95C2×D9711yescharacteristic
109C24—971no—
110C24C22×D9713yes—
111C2×D4—212no—
112C2×D4—26no—
113C2×D4—1946no—
114C2×D4—1946no—
115C2×D4—1946no—
116C22×C4—971no—
117C22×C4—973no—
118C22×C4—973no—
119C22×C4—1946no—
120C22×C4—1946no—
121C22×C4—19412no—
122C22×C4C22×D9711yescharacteristic
123C22×C4C22×D9713yes—
124C4⋊C4—1946no—
125C4⋊C4—1946no—
126C22⋊C4—26no—
127C22⋊C4—19412no—
128C22⋊C4—1946no—
129C42—9712no—
130C23—23no—
131C23—26no—
132C23—23no—
133C23—26no—
134C23—972no—
135C23—1943no—
136C23—1943no—
137C23#1309551016yes—
138C23C23×D9711yescharacteristic
139D4—224no—
140D4—19424no—
141C2×C4—26no—
142C2×C4—23no—
143C2×C4—976no—
144C2×C4—976no—
145C2×C4—3884no—
146C2×C4—38812no—
147C2×C4—1946no—
148C2×C4—1943no—
149C2×C4—1943no—
150C2×C4—1943no—
151C2×C4D4×D9716yes—
152C22—212no—
153C22—26no—
154C22—43no—
155C22—46no—
156C22—43no—
157C22—3883no—
158C22—3881no—
159C22—1946no—
160C22C2×D4×D9713yes—
161C22#41×D9711yescharacteristic
162C22C2×#1309551013yes—
163C4—26no—
164C4—41no—
165C4—974no—
166C4—3883no—
167C4—1946no—
168C2—46no—
169C2—3882no—
170C2#2026124113yes—
171C2#2026124513yes—
172C2#127×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#392×C97C2
hypercenter#392D97
hypocenterC97#2397
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic97C2semidirect
2. C2×#15690770C2semidirect
3. #20260672C2semidirect
4. #20260671C2semidirect
5. #20260643C2semidirect
6. #20260580C2semidirect
7. #392×C97C2semidirect
36. C4×Dic97D4semidirect
87. C97#2397semidirect
93. #392D97semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic972
C2×#156907702
#202606722
#202606712
#202606432
#202605802
#392×C972
#239797