Finite Group Databasebeta

← #27145016

Subgroups — #27145016

Order 12416.

840 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 163 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×#15690772C213yesmaximal, complemented
2C2×#15690770C213yesmaximal, complemented
3#20260670C213yesmaximal, complemented
4#20260645C211yesmaximal, characteristic, complemented
5#20260644C213yesmaximal, complemented
6#20260580C211yesmaximal, characteristic, complemented
7#399×C97C211yesmaximal, characteristic, complemented
8C22×D388C2213yes—
9C22×C22⋊D97C2213yes—
10C23×Dic97C2211yescharacteristic
11C22×C4×D97C2213yes—
12#15690772—212no—
13#15690770—212no—
14C2×#13095490C2213yes—
15C2×#13095488C2216yes—
16C2×#13095488C2213yes—
17C2×C4×Dic97C2213yes—
18#15690717C2213yes—
19C2×C4⋊C4×C97C2213yes—
20C2×C22⋊C4×C97C2213yes—
21#95×C97C2211yescharacteristic
22C23×D97C2313yes—
23C2×D388—212no—
24C2×D388—26no—
25C2×C22⋊D97—26no—
26C2×C22⋊D97—26no—
27C2×C22⋊D97—26no—
28C22×Dic97—26no—
29C22×Dic97C2311yescharacteristic
30C22×Dic97C2313yes—
31C22×Dic97C2311yescharacteristic
32C2×C4×D97—26no—
33C2×C4×D97—212no—
34#13095490—26no—
35#13095488—26no—
36#13095488—212no—
37C4×Dic97D4112yescomplemented
38C24×C97C2311yescharacteristic
39C22×C4×C97C2313yes—
40C22×C4×C97C2313yes—
41C4⋊C4×C97—26no—
42C22⋊C4×C97—26no—
43C23×C97—23no—
44C23×C97—23no—
45C23×C97#4112yes—
46C23×C97C2411yescharacteristic
47C22×D97—23no—
48C22×D97—26no—
49C22×D97—23no—
50C22×D97—26no—
51C22×D97#4116yes—
52C2×C4×C97—26no—
53C2×C4×C97—23no—
54C2×C4×C97—26no—
55C2×C4×C97C2×D416yes—
56C22⋊D97—224no—
57C2×Dic97—23no—
58C2×Dic97—23no—
59C2×Dic97—23no—
60C2×Dic97—44no—
61C2×Dic97C2×D416yes—
62C2×Dic97C2×D416yes—
63D388—224no—
64C4×D97—412no—
65C22×C97—26no—
66C22×C97—43no—
67C22×C97—41no—
68C22×C97C22×D413yes—
69C22×C97C2×#4113yes—
70C22×C97C2×#4111yescharacteristic
71C2×D97—212no—
72C2×D97—26no—
73C2×D97—43no—
74C2×D97—46no—
75C2×D97—43no—
76C4×C97—26no—
77C4×C97—43no—
78Dic97—26no—
79Dic97—41no—
80Dic97#12714yes—
81C2×C97—42no—
82C2×C97#52513yes—
83C2×C97#52513yes—
84C2×C97C2×#12711yescharacteristic
85D97—46no—
86#2397—971nomaximal, Sylow
87C97#239711yescharacteristic, complemented, Sylow
88C2×C4×D4—973no—
89C2×C4×D4—973no—
90#399—973no—
91#399D9711yescharacteristic, complemented
92#392—973no—
93#392—971no—
94#376—971no—
95C22×D4—973no—
96C22×D4—973no—
97C23×C4—971no—
98C23×C4—973no—
99C4×D4—19412no—
100C4×D4—19412no—
101C2×C4⋊C4—973no—
102C2×C4⋊C4C2×D9713yes—
103C2×C22⋊C4—976no—
104C2×C22⋊C4—973no—
105C2×C22⋊C4C2×D9713yes—
106C2×C42—973no—
107#95—973no—
108#95C2×D9711yescharacteristic
109C24—973no—
110C24C22×D9711yescharacteristic
111C2×D4—19412no—
112C2×D4—1946no—
113C2×D4—1946no—
114C2×D4—1946no—
115C2×D4—1946no—
116C22×C4—971no—
117C22×C4—973no—
118C22×C4—971no—
119C22×C4—1946no—
120C22×C4—19412no—
121C22×C4—1946no—
122C22×C4C22×D9713yes—
123C22×C4C22×D9713yes—
124C4⋊C4—26no—
125C4⋊C4—1946no—
126C22⋊C4—26no—
127C22⋊C4—19412no—
128C22⋊C4—1946no—
129C42—9712no—
130C23—23no—
131C23—23no—
132C23—976no—
133C23—1943no—
134C23—1946no—
135C23—1946no—
136C23—1943no—
137C23#1309551212yes—
138C23C23×D9711yescharacteristic
139D4—19424no—
140D4—19424no—
141C2×C4—26no—
142C2×C4—23no—
143C2×C4—26no—
144C2×C4—976no—
145C2×C4—976no—
146C2×C4—38812no—
147C2×C4—3884no—
148C2×C4—1943no—
149C2×C4—1943no—
150C2×C4—1943no—
151C2×C4D4×D9716yes—
152C22—26no—
153C22—43no—
154C22—41no—
155C22—3883no—
156C22—3886no—
157C22—3883no—
158C22—19412no—
159C22—1946no—
160C22C2×D4×D9713yes—
161C22#41×D9713yes—
162C22C2×#1309551211yescharacteristic
163C4—26no—
164C4—43no—
165C4—974no—
166C4—3881no—
167C4—1946no—
168C2—42no—
169C2—3886no—
170C2#2026124313yes—
171C2#2026124813yes—
172C2#127×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#399×C97C2
hypercenter#399D97
hypocenterC97#2397
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15690772C2semidirect
2. C2×#15690770C2semidirect
3. #20260670C2semidirect
4. #20260645C2semidirect
5. #20260644C2semidirect
6. #20260580C2semidirect
7. #399×C97C2semidirect
37. C4×Dic97D4semidirect
87. C97#2397semidirect
91. #399D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907722
C2×#156907702
#202606702
#202606452
#202606442
#202605802
#399×C972
#239797