Finite Group Databasebeta

← #27142134

Subgroups — #27142134

Order 12416.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15690806C211yesmaximal, characteristic
2#20260739C212yesmaximal, complemented
3C2×#15690727C211yesmaximal, characteristic, complemented
4#20260494C212yesmaximal, complemented
5#518×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C22×C22⋊D97—22no—
8C2×#13095495C2212yes—
9#15690806—22no—
10#15690806C414yescomplemented
11C2×C4×Dic97C2211yescharacteristic
12#15690730—24no—
13#15690727—22no—
14C22×D4×C97C2211yescharacteristic
15C4×D4×C97—24no—
16#117×C97C2212yes—
17C2×C22⋊C4×C97—22no—
18C23×D97—21no—
19C2×D388—22no—
20C2×D388—22no—
21C2×D388—21no—
22C2×D388C2×C412yescomplemented
23C2×C22⋊D97—28no—
24C2×C22⋊D97—48no—
25C22×Dic97—21no—
26#13095495—22no—
27#13095495—22no—
28C2×#11715895—22no—
29C4×Dic97—21no—
30C4×Dic97C2×C412yescomplemented
31C24×C97D412yes—
32C2×D4×C97—22no—
33C2×D4×C97—22no—
34C2×D4×C97C2×C412yes—
35C2×D4×C97D412yes—
36C22×C4×C97—24no—
37C22×C4×C97C2311yescharacteristic
38C4⋊C4×C97D414yescomplemented
39C22⋊C4×C97—24no—
40C22⋊C4×C97—22no—
41C42×C97—22no—
42C23×C97—22no—
43C23×C97—22no—
44C23×C97—44no—
45C23×C97C22⋊C414yes—
46C23×C97C2×D411yescharacteristic
47C22×D97—22no—
48C22×D97—44no—
49C22×D97—41no—
50C22×D97—41no—
51D4×C97—24no—
52D4×C97—24no—
53C2×C4×C97—24no—
54C2×C4×C97—84no—
55C2×C4×C97C22×C411yescharacteristic
56C2×C4×C97C2×D412yes—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97#4112yes—
59C22⋊D97—416no—
60C2×Dic97—24no—
61C2×Dic97—22no—
62D388—24no—
63D388—42no—
64#11715895—42no—
65C22×C97—22no—
66C22×C97—24no—
67C22×C97—44no—
68C22×C97—44no—
69C22×C97—82no—
70C22×C97C2×C22⋊C411yescharacteristic
71C22×C97#12011yescharacteristic
72C22×C97#12311yescharacteristic
73C2×D97—44no—
74C2×D97—42no—
75C2×D97—82no—
76C2×D97—82no—
77C4×C97—44no—
78C4×C97C4×D412yes—
79C4×C97#12112yes—
80Dic97—24no—
81C2×C97—21no—
82C2×C97—21no—
83C2×C97—44no—
84C2×C97#38611yescharacteristic
85C2×C97#45312yes—
86D97—82no—
87#1887—971nomaximal, Sylow
88C97#188711yescharacteristic, complemented, Sylow
89C2×#127—971no—
90#518D9711yescharacteristic, complemented
91#418—972no—
92C2×#100—971no—
93#339—972no—
94C22×D4—971no—
95C22×D4—1942no—
96C22×D4C2×D9711yescharacteristic
97C2×M16—972no—
98#127—974no—
99#127—1942no—
100C4×D4—24no—
101#117C2×D9712yes—
102C2×C22⋊C4—22no—
103C2×C42—971no—
104#102—1944no—
105#100—1942no—
106C24—1941no—
107C24C22⋊D9712yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—972no—
111C2×D4—3888no—
112C2×D4—1948no—
113C2×D4—1942no—
114C2×D4—1941no—
115C2×D4—1942no—
116C2×D4C4×D9712yes—
117C2×D4D38812yes—
118C22×C4—24no—
119C22×C4—1941no—
120C22×C4C22×D9711yescharacteristic
121M16—1942no—
122M16—1942no—
123C2×C8—1942no—
124C4⋊C4C22⋊D9714yescomplemented
125C22⋊C4—24no—
126C22⋊C4—22no—
127C42—22no—
128C42—972no—
129C42—1941no—
130C23—22no—
131C23—22no—
132C23—44no—
133C23—3884no—
134C23—3881no—
135C23—3881no—
136C23—1942no—
137C23#1309548814yes—
138C23C2×C22⋊D9711yescharacteristic
139D4—24no—
140D4—24no—
141D4—38816no—
142D4—3882no—
143D4—1944no—
144C2×C4—24no—
145C2×C4—84no—
146C2×C4—1944no—
147C2×C4—1942no—
148C2×C4C2×C4×D9711yescharacteristic
149C2×C4C2×C22⋊D9712yes—
150C2×C4C2×D38811yescharacteristic
151C2×C4#1309551112yes—
152C8—3882no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—82no—
158C22—7762no—
159C22—7762no—
160C22—3884no—
161C22—3882no—
162C22C2×#1309548811yescharacteristic
163C22#1569077911yescharacteristic
164C22#1569079311yescharacteristic
165C4—44no—
166C4—1944no—
167C4C4×C22⋊D9712yes—
168C4#1569078412yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—7762no—
173C2#2026061411yescharacteristic
174C2#2026090312yes—
subgroupG/H
centerC22#15690779
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#518×C97C2
socleC22×C97#120
prime radicalC2×#15690806C2
cosocleC22×C4×C97C23
hypercenter#518D97
hypocenterC97#1887
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20260739C2semidirect
3. C2×#15690727C2semidirect
4. #20260494C2semidirect
5. #518×C97C2semidirect
10. #15690806C4semidirect
22. C2×D388C2×C4semidirect
30. C4×Dic97C2×C4semidirect
38. C4⋊C4×C97D4semidirect
88. C97#1887semidirect
90. #518D97semidirect
124. C4⋊C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908062
#202607392
C2×#156907272
#202604942
#518×C972
#188797