Finite Group Databasebeta

← #27143402

Subgroups — #27143402

Order 12416.

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

normal subgroups only

This group has 286 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#99×D97C212yesmaximal, complemented
3#20260529C212yesmaximal, complemented
4#20260528C211yesmaximal, characteristic, complemented
5#353×C97C211yesmaximal, characteristic, complemented
6C2×#13095510C411yescharacteristic, complemented
7C2×D4×D97C2211yescharacteristic
8#15690806C411yescharacteristic, complemented
9#15690798C411yescharacteristic, complemented
10#15690797C2211yescharacteristic
11#15690782—22no—
12#15690778—22no—
13#15690767C411yescharacteristic, complemented
14C22⋊C4×D97—22no—
15#15690725C2212yes—
16#15690724—22no—
17#127×C97C2211yescharacteristic
18#99×C97C2212yes—
19C23×D97D412yes—
20#13095510—42no—
21C2×Dic194C2×C411yescharacteristic, complemented
22C2×D388C2×C411yescharacteristic
23C2×C22⋊D97—21no—
24C2×C22⋊D97—22no—
25C2×C22⋊D97D412yes—
26D4×D97—42no—
27C22×Dic97—21no—
28C2×C4×D97—42no—
29C2×C4×D97C2×C411yescharacteristic
30#13095490—21no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33#13095488—21no—
34#13095488—22no—
35#13095488—42no—
36C4×Dic97C2×C411yescharacteristic, complemented
37C2×D4×C97—22no—
38C2×D4×C97C2×C411yescharacteristic, complemented
39C2×D4×C97C2311yescharacteristic
40C22⋊C4×C97—22no—
41C42×C97C2×C411yescharacteristic, complemented
42C23×C97—21no—
43C23×C97C2×D412yes—
44C22×D97—22no—
45C22×D97—22no—
46C22×D97—82no—
47C22×D97C22⋊C412yes—
48C22×D97C22⋊C411yescharacteristic
49D4×C97—41no—
50D4×C97—41no—
51D4×C97—82no—
52C2×C4×C97—21no—
53C2×C4×C97—42no—
54C2×C4×C97C22×C411yescharacteristic
55C22⋊D97—42no—
56C22⋊D97—42no—
57C22⋊D97—82no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—42no—
61C2×Dic97—42no—
62C2×Dic97C22⋊C411yescharacteristic
63D388—41no—
64C4×D97—22no—
65C4×D97—84no—
66Dic194—41no—
67C22×C97—22no—
68C22×C97—42no—
69C22×C97—42no—
70C22×C97—41no—
71C22×C97C2×C22⋊C411yescharacteristic
72C2×D97—22no—
73C2×D97—41no—
74C2×D97—44no—
75C2×D97—42no—
76C2×D97—82no—
77C2×D97#9912yes—
78C4×C97—21no—
79C4×C97—41no—
80C4×C97—82no—
81Dic97—21no—
82Dic97—41no—
83Dic97—41no—
84Dic97—82no—
85C2×C97—21no—
86C2×C97—42no—
87C2×C97—81no—
88C2×C97C2×#9911yescharacteristic
89D97—22no—
90D97—42no—
91#2127—971nomaximal, Sylow
92C97#212711yescharacteristic, complemented, Sylow
93#535—971no—
94C2×#99—972no—
95#354—972no—
96#353—971no—
97#353D9711yescharacteristic, complemented
98C2×#41—971no—
99C22×D4—971no—
100#127—971no—
101#127C2×D9711yescharacteristic
102#124—971no—
103#124—971no—
104#121—1942no—
105#120—1942no—
106#117—971no—
107C2×C22⋊C4—1942no—
108#99—972no—
109#99—1942no—
110#99C2×D9712yes—
111C24—972no—
112#41—3882no—
113C2×Q8—971no—
114C2×D4—22no—
115C2×D4—972no—
116C2×D4—971no—
117C2×D4—3882no—
118C2×D4—1942no—
119C2×D4—1941no—
120C2×D4C4×D9711yescharacteristic, complemented
121C2×D4C22×D9711yescharacteristic
122C22×C4—971no—
123C22×C4—3882no—
124C22×C4—1941no—
125C4⋊C4—1941no—
126C22⋊C4—22no—
127C22⋊C4—3882no—
128C22⋊C4—1941no—
129C22⋊C4—1942no—
130C22⋊C4—1942no—
131C22⋊C4—1942no—
132C42—971no—
133C42C4×D9711yescharacteristic, complemented
134C23—21no—
135C23—7762no—
136C23—971no—
137C23—972no—
138C23—1942no—
139C23—1942no—
140C23D4×D9712yes—
141Q8—3881no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—7762no—
146D4—3881no—
147D4—3882no—
148D4—3882no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—7764no—
152C2×C4—971no—
153C2×C4—3882no—
154C2×C4—3882no—
155C2×C4—1942no—
156C2×C4—1941no—
157C2×C4—1941no—
158C2×C4C2×C4×D9711yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—7762no—
164C22—972no—
165C22—3882no—
166C22—3881no—
167C22—3884no—
168C22—1942no—
169C22C22⋊C4×D9711yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—7762no—
174C4—3881no—
175C4—3881no—
176C4—1941no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—3882no—
181C2—1942no—
182C2#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#2127
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. #99×D97C2semidirect
3. #20260529C2semidirect
4. #20260528C2semidirect
5. #353×C97C2semidirect
6. C2×#13095510C4semidirect
8. #15690806C4semidirect
9. #15690798C4semidirect
13. #15690767C4semidirect
21. C2×Dic194C2×C4semidirect
36. C4×Dic97C2×C4semidirect
38. C2×D4×C97C2×C4semidirect
41. C42×C97C2×C4semidirect
92. C97#2127semidirect
97. #353D97semidirect
120. C2×D4C4×D97semidirect
133. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202612982
#99×D972
#202605292
#202605282
#353×C972
#212797