Finite Group Databasebeta

← #27143426

Subgroups — #27143426

Order 12416.

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

normal subgroups only

This group has 262 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#20261303C211yesmaximal, characteristic
2#20260718C212yesmaximal, complemented
3#20260532C211yesmaximal, characteristic, complemented
4#20260531C212yesmaximal, complemented
5#355×C97C211yesmaximal, characteristic, complemented
6C2×#13095512C2211yescharacteristic
7C2×D4×D97C411yescharacteristic
8M16×D97—22no—
9#15690807C411yescharacteristic
10#15690806C411yescharacteristic
11#15690799C2211yescharacteristic
12#15690783—22no—
13#15690780—22no—
14#15690767C411yescharacteristic
15#15690729C2212yescomplemented
16#15690726—22no—
17#124×C97C2211yescharacteristic
18#101×C97C2212yescomplemented
19C23×D97—21no—
20#13095512—42no—
21C2×D388—22no—
22C2×D388C2×C411yescharacteristic
23C2×D388C2×C411yescharacteristic
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C2×C22⋊D97—21no—
27D4×D97—42no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D412yes—
30#13095495—22no—
31#13095494—42no—
32C8×D97—42no—
33#13095490—21no—
34#13095488—21no—
35#13095488—22no—
36C4×Dic97C2×C411yescharacteristic
37C2×Q8×C97C2311yescharacteristic
38C2×D4×C97C2×C411yescharacteristic
39M16×C97—22no—
40C22⋊C4×C97—22no—
41C42×C97C2×C411yescharacteristic
42C23×C97—21no—
43C22×D97—21no—
44C22×D97—42no—
45C22×D97—82no—
46C22×D97C22⋊C411yescharacteristic
47C22×D97C22⋊C412yes—
48Q8×C97—41no—
49D4×C97—41no—
50C2×C4×C97—21no—
51C2×C4×C97C22×C411yescharacteristic
52C2×C4×C97C2×D412yes—
53C22⋊D97—42no—
54C22⋊D97—82no—
55C2×Dic97—21no—
56C2×Dic97C22⋊C411yescharacteristic
57D388—41no—
58D388—42no—
59D388—41no—
60D388—82no—
61C4×D97—22no—
62C4×D97—22no—
63C4×D97—22no—
64C8×C97—42no—
65#11715895—42no—
66C22×C97—42no—
67C22×C97—41no—
68C22×C97C2×C22⋊C411yescharacteristic
69C2×D97—21no—
70C2×D97—22no—
71C2×D97—42no—
72C2×D97—41no—
73C2×D97—42no—
74C2×D97—41no—
75C2×D97—84no—
76C4×C97—22no—
77C4×C97—21no—
78C4×C97—41no—
79Dic97—41no—
80Dic97#9912yes—
81C2×C97—21no—
82C2×C97—81no—
83C2×C97C2×#9911yescharacteristic
84D97—42no—
85D97—41no—
86D97—81no—
87#2134—971nomaximal, Sylow
88C97#213411yescharacteristic, complemented, Sylow
89#535—971no—
90#413—972no—
91#355—971no—
92#355—972no—
93#355D9711yescharacteristic, complemented
94C2×#41—971no—
95C22×D4—971no—
96C2×M16—1942no—
97#127—971no—
98#127—971no—
99#124—971no—
100#124C2×D9711yescharacteristic
101#121—1942no—
102#120—1942no—
103#117—971no—
104#101—972no—
105#101C2×D9712yescomplemented
106#100—1942no—
107C24—1941no—
108#41—3882no—
109C2×Q8C22×D9711yescharacteristic
110C2×D4—971no—
111C2×D4—971no—
112C2×D4—972no—
113C2×D4—3882no—
114C2×D4—1942no—
115C2×D4—1941no—
116C2×D4—1942no—
117C2×D4C4×D9711yescharacteristic
118C22×C4—971no—
119C22×C4—972no—
120M16—22no—
121M16—3882no—
122M16—1942no—
123C2×C8—3882no—
124C4⋊C4—1941no—
125C22⋊C4—22no—
126C22⋊C4—1941no—
127C22⋊C4—1942no—
128C42—971no—
129C42C4×D9711yescharacteristic
130C23—21no—
131C23—7762no—
132C23—971no—
133C23—972no—
134C23—3882no—
135C23—1941no—
136Q8—41no—
137D4—41no—
138D4—7762no—
139D4—7762no—
140D4—3881no—
141D4—3882no—
142D4—3881no—
143D4—3882no—
144C2×C4—21no—
145C2×C4—971no—
146C2×C4—1942no—
147C2×C4—1942no—
148C2×C4—1942no—
149C2×C4—1941no—
150C2×C4C2×C4×D9711yescharacteristic
151C2×C4D4×D9712yes—
152C8—42no—
153C8—3882no—
154C22—42no—
155C22—41no—
156C22—7764no—
157C22—3882no—
158C22—3882no—
159C22—3881no—
160C22—3881no—
161C22—1941no—
162C22—1942no—
163C22C22⋊C4×D9711yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—972no—
168C4—3881no—
169C2—21no—
170C2—81no—
171C2—7761no—
172C2—3881no—
173C2—3882no—
174C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#355×C97C2
socleC2×C97C2×#99
prime radical#20261303C2
cosocleC2×Q8×C97C23
hypercenter#355D97
hypocenterC97#2134
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #20260718C2semidirect
3. #20260532C2semidirect
4. #20260531C2semidirect
5. #355×C97C2semidirect
15. #15690729C22semidirect
18. #101×C97C22semidirect
88. C97#2134semidirect
93. #355D97semidirect
105. #101C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613032
#202607182
#202605322
#202605312
#355×C972
#213497