Finite Group Databasebeta

← #27143406

Subgroups — #27143406

Order 12416.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#20261298C211yesmaximal, characteristic
2#20260707C212yesmaximal, complemented
3#20260528C211yesmaximal, characteristic, complemented
4#20260527C212yesmaximal, complemented
5#353×C97C211yesmaximal, characteristic, complemented
6C2×#13095510C2211yescharacteristic
7C2×D4×D97C411yescharacteristic, complemented
8#15690806C2211yescharacteristic
9#15690798C411yescharacteristic, complemented
10#15690797C411yescharacteristic, complemented
11#15690782—22no—
12#15690778—22no—
13#15690767C411yescharacteristic, complemented
14#15690766—22no—
15#15690725C2212yescomplemented
16#15690724—22no—
17#127×C97C2211yescharacteristic
18#99×C97C2212yescomplemented
19C23×D97—21no—
20#13095510—42no—
21C2×Dic194C2×C411yescharacteristic
22C2×D388C2×C411yescharacteristic
23C2×C22⋊D97—22no—
24C2×C22⋊D97—21no—
25C2×C22⋊D97D412yes—
26D4×D97—42no—
27C22×Dic97D412yes—
28C2×C4×D97C2×C411yescharacteristic
29#13095490—21no—
30#13095490—42no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33#13095488—21no—
34#13095488—22no—
35C4×Dic97—42no—
36C4×Dic97C2×C411yescharacteristic
37C2×D4×C97—22no—
38C2×D4×C97C2×C411yescharacteristic
39C2×D4×C97C2311yescharacteristic
40C22⋊C4×C97—22no—
41C42×C97C2×C411yescharacteristic
42C23×C97—21no—
43C23×C97C2×D412yes—
44C22×D97—21no—
45C22×D97—42no—
46C22×D97—82no—
47C22×D97C22⋊C411yescharacteristic
48D4×C97—41no—
49D4×C97—41no—
50D4×C97—82no—
51C2×C4×C97—21no—
52C2×C4×C97—42no—
53C2×C4×C97C22×C411yescharacteristic
54C22⋊D97—42no—
55C22⋊D97—42no—
56C22⋊D97—82no—
57C2×Dic97—22no—
58C2×Dic97—22no—
59C2×Dic97—21no—
60C2×Dic97—42no—
61C2×Dic97C22⋊C412yes—
62C2×Dic97C22⋊C411yescharacteristic
63D388—41no—
64C4×D97—22no—
65Dic194—41no—
66C22×C97—22no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97—41no—
70C22×C97C2×C22⋊C411yescharacteristic
71C2×D97—21no—
72C2×D97—41no—
73C2×D97—42no—
74C2×D97—41no—
75C2×D97—84no—
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—82no—
79Dic97—22no—
80Dic97—41no—
81Dic97—82no—
82Dic97#9912yes—
83C2×C97—21no—
84C2×C97—42no—
85C2×C97—81no—
86C2×C97C2×#9911yescharacteristic
87D97—41no—
88D97—81no—
89#2128—971nomaximal, Sylow
90C97#212811yescharacteristic, complemented, Sylow
91#535—971no—
92#410—972no—
93#353—971no—
94#353—972no—
95#353D9711yescharacteristic, complemented
96C2×#41—971no—
97C22×D4—971no—
98#127—971no—
99#127C2×D9711yescharacteristic
100#124—971no—
101#124—971no—
102#121—1942no—
103#120—1942no—
104#117—971no—
105#117—1942no—
106#99—972no—
107#99—1942no—
108#99C2×D9712yescomplemented
109C24—1941no—
110#41—3882no—
111C2×Q8—971no—
112C2×D4—22no—
113C2×D4—972no—
114C2×D4—971no—
115C2×D4—3882no—
116C2×D4—1941no—
117C2×D4—1942no—
118C2×D4C4×D9711yescharacteristic
119C2×D4C22×D9711yescharacteristic
120C22×C4—972no—
121C22×C4—971no—
122C4⋊C4—3882no—
123C4⋊C4—1941no—
124C22⋊C4—22no—
125C22⋊C4—1942no—
126C22⋊C4—1941no—
127C22⋊C4—1942no—
128C22⋊C4—1942no—
129C42—971no—
130C42—3882no—
131C42C4×D9711yescharacteristic
132C23—21no—
133C23—7762no—
134C23—971no—
135C23—3882no—
136C23—1941no—
137C23D4×D9712yes—
138Q8—3881no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—7762no—
143D4—3881no—
144D4—3882no—
145D4—3882no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—971no—
149C2×C4—972no—
150C2×C4—3882no—
151C2×C4—1942no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4—1941no—
155C2×C4C2×C4×D9711yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—7764no—
161C22—3882no—
162C22—3881no—
163C22—3881no—
164C22—1941no—
165C22C22⋊C4×D9711yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—7762no—
170C4—972no—
171C4—3881no—
172C4—1942no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—7761no—
177C2—3881no—
178C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#353×C97C2
socleC2×C97C2×#99
prime radical#20261298C2
cosocleC2×D4×C97C23
hypercenter#353D97
hypocenterC97#2128
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #20260707C2semidirect
3. #20260528C2semidirect
4. #20260527C2semidirect
5. #353×C97C2semidirect
7. C2×D4×D97C4semidirect
9. #15690798C4semidirect
10. #15690797C4semidirect
13. #15690767C4semidirect
15. #15690725C22semidirect
18. #99×C97C22semidirect
90. C97#2128semidirect
95. #353D97semidirect
108. #99C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612982
#202607072
#202605282
#202605272
#353×C972
#212897