Finite Group Databasebeta

← #27140705

Subgroups — #27140705

Order 12416.

282 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 57 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#20260873C212yesmaximal, complemented
2#20260655C211yesmaximal, characteristic, complemented
3#20260469C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×#13095499—22no—
8C2×#13095490—21no—
9#15690733—22no—
10C22⋊#11715895C2212yes—
11#15690717—21no—
12#15690717C2211yescharacteristic
13C2×#41×C97—21no—
14C22⋊Q8×C97—22no—
15#121×C97—22no—
16#121×C97—21no—
17#121×C97C2212yes—
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×Dic194—22no—
20C2×Dic194—42no—
21C22×Dic97—21no—
22C22×Dic97—21no—
23#13095499—44no—
24C2×#11715895—22no—
25#13095490—42no—
26#41×C97—42no—
27#41×C97—42no—
28C2×Q8×C97—21no—
29C2×D4×C97—21no—
30C2×D4×C97—22no—
31C2×D4×C97—21no—
32C2×D4×C97D412yes—
33C22×C4×C97—22no—
34C22×C4×C97D411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C4⋊C4×C97—21no—
38C4⋊C4×C97—22no—
39C4⋊C4×C97D412yes—
40C22⋊C4×C97—22no—
41C22⋊C4×C97—22no—
42C23×C97—22no—
43C23×C97C2×D411yescharacteristic
44Q8×C97—42no—
45D4×C97—42no—
46D4×C97—42no—
47D4×C97—42no—
48D4×C97—42no—
49D4×C97—81no—
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—22no—
54C2×C4×C97—41no—
55C2×C4×C97—44no—
56C2×C4×C97—41no—
57C2×C4×C97SD1612yescomplemented
58C2×C4×C97C2×D412yes—
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63C2×Dic97—41no—
64C2×Dic97—41no—
65Dic194—24no—
66Dic194—42no—
67#11715895—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—42no—
71C22×C97—42no—
72C22×C97—42no—
73C22×C97#12011yescharacteristic
74C22×C97C2×SD1611yescharacteristic
75C22×C97AGL(1,ℤ/8)11yescharacteristic
76C4×C97—22no—
77C4×C97—22no—
78C4×C97—42no—
79C4×C97—41no—
80Dic97—42no—
81Dic97—42no—
82C2×C97—21no—
83C2×C97—21no—
84C2×C97—82no—
85C2×C97#45011yescharacteristic
86C2×C97#45611yescharacteristic
87C2×C97B(4,𝔽2)11yescharacteristic
88#1604—971nomaximal, Sylow
89C97#160411yescharacteristic, complemented, Sylow
90#537D9711yescharacteristic
91#448—972no—
92#395—971no—
93#327—972no—
94#324—971no—
95C2×#41—21no—
96C22×Q8—971no—
97C2×SD16—1942no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C2×D9712yes—
102C2×C4⋊C4—1941no—
103C2×C4⋊C4C2×D9711yescharacteristic
104#103—1942no—
105C22⋊C8—972no—
106#95—971no—
107#95—1941no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—3882no—
112C2×Q8—1942no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4C22⋊D9712yes—
117C22×C4—22no—
118C22×C4—1941no—
119C22×C4—1941no—
120C22×C4C22⋊D9711yescharacteristic
121C22×C4C22⋊D9711yescharacteristic
122C22×C4C22×D9711yescharacteristic
123SD16—3884no—
124C2×C8—1942no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—3882no—
128C4⋊C4C22⋊D9712yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×C22⋊D9711yescharacteristic
133Q8—42no—
134Q8—3882no—
135Q8—1944no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—3881no—
149C2×C4—3881no—
150C2×C4—3881no—
151C2×C4—3881no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4#1309549912yescomplemented
155C2×C4C2×C22⋊D9712yes—
156C8—3882no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#1569077911yescharacteristic
163C22C2×#1309549911yescharacteristic
164C22#1569083611yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—41no—
169C4—3882no—
170C4—3882no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2026088611yescharacteristic
175C2#2026091811yescharacteristic
176C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#1604
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20260873C2semidirect
2. #20260655C2semidirect
3. #20260469C2semidirect
4. #20260461C2semidirect
57. C2×C4×C97SD16semidirect
89. C97#1604semidirect
154. C2×C4#13095499semidirect

Maximal subgroups

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

subgroupindex
#202608732
#202606552
#202604692
#202604612
#537×C972
#160497