Finite Group Databasebeta

← #34099309

Subgroups — #34099309

Order 20096.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757997C211yesmaximal, characteristic
2#23757625C211yesmaximal, characteristic, complemented
3#23757569C211yesmaximal, characteristic, complemented
4#23757564C211yesmaximal, characteristic
5#23757003C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic
7#332×C157C211yesmaximal, characteristic, complemented
8#19097771C2211yescharacteristic
9#19097766—22no—
10#19097760—21no—
11#19097751—21no—
12Q8×Dic157—21no—
13C4×Dic314C2211yescharacteristic
14C4×D628C2211yescharacteristic
15#19097729—21no—
16#19097705—21no—
17#19097703—21no—
18#19097696—21no—
19#19097694—22no—
20#19097693—21no—
21C4×#11929434C2211yescharacteristic
22C4⋊Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4⋊C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×D628D411yescharacteristic
28C2×C4×D157—21no—
29C2×#11929434—21no—
30C4⋊Dic157—22no—
31C4⋊Dic157—21no—
32C4⋊Dic157—21no—
33C4⋊Dic157D411yescharacteristic
34C4⋊Dic157D411yescharacteristic
35#14578933—21no—
36#14578931—22no—
37#14578931—22no—
38#14578931—21no—
39C4×Dic157—22no—
40C4×Dic157—21no—
41C2×Q8×C157—21no—
42C2×C8×C157—21no—
43C2×C8×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157D412yes—
46C42×C157C2311yescharacteristic
47C22×D157—21no—
48Dic314—21no—
49Dic314—21no—
50Dic314—42no—
51D628—21no—
52D628—21no—
53C2×Dic157—21no—
54C2×Dic157—21no—
55C2×Dic157—21no—
56C2×Dic157—41no—
57C4×D157—41no—
58C4×D157—41no—
59#11929434—41no—
60Q8×C157—41no—
61Q8×C157—41no—
62C2×C4×C157—22no—
63C2×C4×C157C2×D411yescharacteristic
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157C2×D411yescharacteristic
66C8×C157—41no—
67C8×C157—41no—
68C22×C157#12011yescharacteristic
69C2×D157—41no—
70C2×D157—41no—
71C2×D157—41no—
72C4×C157—21no—
73C4×C157—42no—
74C4×C157#13511yescharacteristic
75C4×C157#13511yescharacteristic
76C4×C157#13711yescharacteristic
77C4×C157#13711yescharacteristic
78Dic157—41no—
79Dic157—41no—
80Dic157—41no—
81Dic157—81no—
82C2×C157#45211yescharacteristic
83C2×C157#45211yescharacteristic
84C2×C157#45411yescharacteristic
85D157—81no—
86C157#163811yescharacteristic, complemented, Sylow
87#1638—1571nomaximal, Sylow
88#556—1571no—
89#487—1571no—
90#477—1571no—
91#476—1571no—
92#332D15711yescharacteristic, complemented
93Q8⋊C8—1571no—
94D4⋊C8—1571no—
95C4⋊Q8—1571no—
96C4⋊Q8C2×D15711yescharacteristic
97#126—3142no—
98#124—3141no—
99C22⋊Q8—3141no—
100C4×Q8—3141no—
101C4×Q8—1571no—
102C4×D4—1571no—
103#117—3141no—
104#107—3141no—
105#106—3141no—
106C4⋊C8C2×D15711yescharacteristic
107C4⋊C8C2×D15711yescharacteristic
108#103—3141no—
109#103—3142no—
110#102—3141no—
111C4×C8—1571no—
112C2×Q8—21no—
113C2×Q8—3141no—
114C2×Q8—1571no—
115C2×D4—1571no—
116C22×C4—3141no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3141no—
120C4⋊C4—21no—
121C4⋊C4—3142no—
122C4⋊C4—3141no—
123C4⋊C4—3141no—
124C4⋊C4—3141no—
125C4⋊C4—1571no—
126C4⋊C4—1571no—
127C4⋊C4D62812yes—
128C22⋊C4—3142no—
129C22⋊C4—3142no—
130C22⋊C4—3141no—
131C42—3142no—
132C42—3141no—
133C42C22×D15711yescharacteristic
134C23—3141no—
135Q8—41no—
136Q8—41no—
137Q8—3141no—
138Q8—3141no—
139Q8—6282no—
140D4—3141no—
141D4—3141no—
142C2×C4—22no—
143C2×C4—3141no—
144C2×C4—3141no—
145C2×C4—3141no—
146C2×C4—6281no—
147C2×C4—6281no—
148C2×C4—6281no—
149C2×C4D4×D15711yescharacteristic
150C2×C4D4×D15711yescharacteristic
151C2×C4C2×D62811yescharacteristic
152C8—41no—
153C8—41no—
154C8—6281no—
155C22—6281no—
156C22—6281no—
157C22—6281no—
158C22#1909774111yescharacteristic
159C4—21no—
160C4—42no—
161C4—6281no—
162C4—6281no—
163C4—6281no—
164C4—12561no—
165C4#1909779011yescharacteristic
166C4#1909779211yescharacteristic
167C4#1909780011yescharacteristic
168C4#1909780311yescharacteristic
169C2—12561no—
170C2#2375743611yescharacteristic
171C2#2375743811yescharacteristic
172C2#2375744811yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#332×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#332D157
hypocenterC157#1638
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #23757625C2semidirect
3. #23757569C2semidirect
5. #23757003C2semidirect
7. #332×C157C2semidirect
86. C157#1638semidirect
92. #332D157semidirect

Maximal subgroups

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

subgroupindex
#237579972
#237576252
#237575692
#237575642
#237570032
#237570002
#332×C1572
#1638157