Finite Group Databasebeta

← #27150059

Subgroups — #27150059

Order 12416.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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×#15690793C211yesmaximal, characteristic
2C22×C4×Dic97C211yesmaximal, characteristic, complemented
3#20260627C214yesmaximal, complemented
4#20260620C214yesmaximal, complemented
5#20260612C212yesmaximal
6#20260610C212yesmaximal, complemented
7#381×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2211yescharacteristic
9C23×Dic97C2212yes—
10#15690793—24no—
11C2×C4⋊Dic97C2214yes—
12C2×#13095490C2212yes—
13C2×C22⋊Dic97C2211yescharacteristic
14C2×#13095488C2214yes—
15C2×#13095488C2212yes—
16C2×C4×Dic97—24no—
17C2×C4×Dic97C2214yes—
18#15690717C2214yes—
19#15690717C2214yes—
20C23×C4×C97C2211yescharacteristic
21C2×C22⋊C4×C97C2212yes—
22#95×C97C2214yes—
23C23×D97C2311yescharacteristic
24C2×C22⋊D97—22no—
25C2×C22⋊D97—24no—
26C22×Dic97—24no—
27C22×Dic97—28no—
28C22×Dic97C2312yes—
29C22×Dic97C2311yescharacteristic
30C22×Dic97C2314yes—
31C4⋊Dic97—28no—
32#13095490—24no—
33C22⋊Dic97—22no—
34#13095488—24no—
35#13095488—28no—
36C4×Dic97—28no—
37C24×C97C2311yescharacteristic
38C22×C4×C97—24no—
39C22×C4×C97D414yes—
40C22×C4×C97C2314yes—
41C22×C4×C97C2312yes—
42C22⋊C4×C97—24no—
43C23×C97—24no—
44C23×C97C2×D412yes—
45C23×C97#4114yes—
46C23×C97C2411yescharacteristic
47C22×D97—21no—
48C22×D97—24no—
49C22×D97—22no—
50C2×C4×C97—28no—
51C2×C4×C97—24no—
52C2×C4×C97—24no—
53C2×C4×C97—24no—
54C2×C4×C97—24no—
55C2×C4×C97C2×D414yes—
56C22⋊D97—44no—
57C2×Dic97—216no—
58C2×Dic97—24no—
59C2×Dic97—24no—
60C2×Dic97—22no—
61C2×Dic97—21no—
62C2×Dic97—24no—
63C2×Dic97—22no—
64C2×Dic97#4118yes—
65C22×C97—24no—
66C22×C97—28no—
67C22×C97#12314yes—
68C22×C97C22×D411yescharacteristic
69C22×C97C2×#4114yes—
70C22×C97C2×#4112yes—
71C2×D97—41no—
72C2×D97—42no—
73C2×D97—44no—
74C4×C97—24no—
75C4×C97—44no—
76Dic97—28no—
77Dic97—41no—
78Dic97—42no—
79C2×C97—24no—
80C2×C97C2×#12311yescharacteristic
81C2×C97#52512yes—
82C2×C97#52914yes—
83D97—81no—
84#2853—971nomaximal, Sylow
85C97#285311yescharacteristic, complemented, Sylow
86C2×#123—971no—
87C22×C42—971no—
88#388—974no—
89#387—974no—
90#386—972no—
91#385—972no—
92#381D9711yescharacteristic, complemented
93C22×D4—971no—
94C23×C4—972no—
95C23×C4C2×D9711yescharacteristic
96#123—1944no—
97C2×C4⋊C4—972no—
98C2×C4⋊C4—974no—
99C2×C22⋊C4—971no—
100C2×C22⋊C4—972no—
101C2×C22⋊C4—974no—
102C2×C22⋊C4C2×D9712yes—
103C2×C42—974no—
104C2×C42—1944no—
105#95—974no—
106#95—974no—
107#95C2×D9714yes—
108C24—971no—
109C24C22×D9711yescharacteristic
110C2×D4—1942no—
111C2×D4—1944no—
112C22×C4—24no—
113C22×C4—974no—
114C22×C4—971no—
115C22×C4—972no—
116C22×C4—1948no—
117C22×C4—1944no—
118C22×C4D38814yes—
119C22×C4C22×D9714yes—
120C22×C4C22×D9712yes—
121C4⋊C4—1948no—
122C4⋊C4—1944no—
123C22⋊C4—24no—
124C22⋊C4—1948no—
125C22⋊C4—1942no—
126C22⋊C4—1944no—
127C42—1948no—
128C23—24no—
129C23—1941no—
130C23—1944no—
131C23—1942no—
132C23C2×D38812yes—
133C23#1309551014yes—
134C23C23×D9711yescharacteristic
135D4—3884no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—978no—
142C2×C4—1942no—
143C2×C4—1941no—
144C2×C4—1944no—
145C2×C4—1942no—
146C2×C4—19416no—
147C2×C4—1944no—
148C2×C4—1944no—
149C2×C4C2×D38814yes—
150C22—24no—
151C22—28no—
152C22—3881no—
153C22—3882no—
154C22—3884no—
155C22#1569079414yes—
156C22C22×D38811yescharacteristic
157C22#41×D9714yes—
158C22C2×#1309551012yes—
159C4—24no—
160C4—44no—
161C4—3881no—
162C4—3882no—
163C4—1948no—
164C2—24no—
165C2—7761no—
166C2C2×#1569079411yescharacteristic
167C2#2026124712yes—
168C2#2026126214yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#381×C97C2
prime radicalC22×C22⋊D97C22
hypercenter#381D97
hypocenterC97#2853
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic97C2semidirect
3. #20260627C2semidirect
4. #20260620C2semidirect
6. #20260610C2semidirect
7. #381×C97C2semidirect
85. C97#2853semidirect
92. #381D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
C22×C4×Dic972
#202606272
#202606202
#202606122
#202606102
#381×C972
#285397