Finite Group Databasebeta

← #27158014

Subgroups — #27158014

Order 12416.

286 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D97⋊#99C212yesmaximal, complemented
2#20261641C212yesmaximal, complemented
3#20261640C212yesmaximal, complemented
4#20261303C211yesmaximal, characteristic
5C2×C22⋊#87604—22no—
6#15690869C2214yes—
7#15690868—22no—
8C2×#13095512C411yescharacteristic, complemented
9C2×D4×D97C2211yescharacteristic
10#15690807C2211yescharacteristic
11#15690806C411yescharacteristic, complemented
12#15690799C2211yescharacteristic
13#15690783—22no—
14#15690780—22no—
15#15690767C411yescharacteristic, complemented
16#124×C97C411yescharacteristic, complemented
17C22×#87604—42no—
18C22⋊#87604—24no—
19C22⋊#87604—42no—
20C23×D97D412yes—
21#13095512—42no—
22C2×D388—21no—
23C2×D388—22no—
24C2×D388C2×C411yescharacteristic, complemented
25C2×D388C2311yescharacteristic
26C2×C22⋊D97—22no—
27C2×C22⋊D97D412yes—
28D4×D97—42no—
29C2×C4×D97—21no—
30C2×C4×D97C2×C411yescharacteristic
31#13095490—21no—
32#13095488—21no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic, complemented
35C2×Q8×C97C2×C411yescharacteristic, complemented
36C2×D4×C97C2×C411yescharacteristic
37C22⋊C4×C97—22no—
38C42×C97C2×C411yescharacteristic, complemented
39C23×C97C22⋊C412yes—
40C22×D97—22no—
41C22×D97—22no—
42C22×D97—21no—
43C22×D97—82no—
44C22×D97C22⋊C411yescharacteristic
45C22×D97C2×D412yes—
46C2×#87604—44no—
47C2×#87604—84no—
48Q8×C97—41no—
49D4×C97—41no—
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97C22×C411yescharacteristic
53C22⋊D97—42no—
54C22⋊D97—82no—
55C2×Dic97—21no—
56C2×Dic97C22⋊C411yescharacteristic
57D388—41no—
58D388—41no—
59D388—42no—
60D388—82no—
61C4×D97—22no—
62C4×D97—42no—
63C22×C97—22no—
64C22×C97—42no—
65C22×C97C2×C22⋊C411yescharacteristic
66C2×D97—22no—
67C2×D97—41no—
68C2×D97—44no—
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—41no—
72C2×D97—82no—
73C2×D97#9912yes—
74#87604—84no—
75C4×C97—21no—
76C4×C97—41no—
77C4×C97—41no—
78Dic97—21no—
79Dic97—41no—
80C2×C97—21no—
81C2×C97—42no—
82C2×C97C2×#9911yescharacteristic
83D97—22no—
84D97—42no—
85D97—81no—
86#2127—971nomaximal, Sylow
87C97#212711yescharacteristic, complemented, Sylow
88#535—971no—
89C2×#99—972no—
90#354—972no—
91#353—972no—
92C2×#41—971no—
93C22×D4—971no—
94#127—971no—
95#127—971no—
96#124—971no—
97#124#8760411yescharacteristic, complemented
98#121—1942no—
99#120—1942no—
100#117—971no—
101C2×C22⋊C4—1942no—
102#99—974no—
103#99—1942no—
104C24—972no—
105#41—3882no—
106C2×Q8C2×#8760411yescharacteristic, complemented
107C2×D4—972no—
108C2×D4—971no—
109C2×D4—971no—
110C2×D4—3882no—
111C2×D4—1942no—
112C2×D4—1941no—
113C2×D4—1942no—
114C2×D4C2×#8760411yescharacteristic
115C22×C4—971no—
116C22×C4—3882no—
117C22×C4—1941no—
118C4⋊C4—1941no—
119C22⋊C4—22no—
120C22⋊C4—3882no—
121C22⋊C4—1941no—
122C22⋊C4—1942no—
123C22⋊C4—1944no—
124C42—971no—
125C42C2×#8760411yescharacteristic, complemented
126C23—7762no—
127C23—971no—
128C23—972no—
129C23—1942no—
130C23—1942no—
131C23—1941no—
132C23C22⋊#8760412yes—
133Q8—41no—
134D4—41no—
135D4—7762no—
136D4—7762no—
137D4—3881no—
138D4—3881no—
139D4—3882no—
140D4—3882no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—7764no—
144C2×C4—971no—
145C2×C4—3882no—
146C2×C4—3884no—
147C2×C4—1942no—
148C2×C4—1941no—
149C2×C4C22×#8760411yescharacteristic
150C22—22no—
151C22—42no—
152C22—7762no—
153C22—972no—
154C22—3882no—
155C22—3882no—
156C22—3881no—
157C22—3884no—
158C22—3881no—
159C22—1942no—
160C22C2×C22⋊#8760411yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—7764no—
165C4—3881no—
166C4—1941no—
167C2—21no—
168C2—42no—
169C2—7761no—
170C2—3882no—
171C2—1942no—
172C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC2×C97C2×#99
prime radical#20261303C2
cosocleC2×D388C23
hypercenter#124#87604
hypocenterC97#2127
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D97⋊#99C2semidirect
2. #20261641C2semidirect
3. #20261640C2semidirect
8. C2×#13095512C4semidirect
11. #15690806C4semidirect
15. #15690767C4semidirect
16. #124×C97C4semidirect
24. C2×D388C2×C4semidirect
34. C4×Dic97C2×C4semidirect
35. C2×Q8×C97C2×C4semidirect
38. C42×C97C2×C4semidirect
87. C97#2127semidirect
97. #124#87604semidirect
106. C2×Q8C2×#87604semidirect
125. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
D97⋊#992
#202616412
#202616402
#202613032
#212797