Finite Group Databasebeta

← #34099271

Subgroups — #34099271

Order 20096.

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

normal subgroups only

This group has 214 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#23757984C211yesmaximal, characteristic
2#23757652C211yesmaximal, characteristic, complemented
3#23757569C211yesmaximal, characteristic, complemented
4#23757564C211yesmaximal, characteristic
5#23757003C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic
7#333×C157C211yesmaximal, characteristic, complemented
8C2×#14578944—22no—
9#19097771C2211yescharacteristic
10#19097762—21no—
11#19097751—21no—
12#19097747—22no—
13C4×Dic314C2211yescharacteristic
14C4×D628C2211yescharacteristic
15#19097733—21no—
16C4⋊C4×D157—21no—
17#19097705—21no—
18#19097703—21no—
19#19097696—21no—
20#19097693—21no—
21C4×#11929434C2211yescharacteristic
22C4⋊Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4⋊C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×D628D411yescharacteristic
28C2×C4×D157—22no—
29C2×C4×D157—21no—
30#14578944—44no—
31C2×#11929434—21no—
32C4⋊Dic157—22no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157D411yescharacteristic
36C4⋊Dic157D411yescharacteristic
37#14578933—22no—
38#14578933—21no—
39#14578931—22no—
40#14578931—21no—
41C4×Dic157—21no—
42C2×Q8×C157D412yes—
43C2×C8×C157—21no—
44C2×C8×C157—21no—
45C4⋊C4×C157—21no—
46C4⋊C4×C157—21no—
47C42×C157C2311yescharacteristic
48C22×D157—21no—
49Dic314—21no—
50Dic314—21no—
51Dic314—42no—
52D628—21no—
53D628Q1612yescomplemented
54C2×Dic157—21no—
55C2×Dic157—21no—
56C2×Dic157—21no—
57C2×Dic157—41no—
58C4×D157—41no—
59C4×D157—41no—
60C4×D157—82no—
61#11929434—22no—
62Q8×C157—42no—
63C2×C4×C157—22no—
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157C2×D411yescharacteristic
66C2×C4×C157C2×D411yescharacteristic
67C8×C157—41no—
68C8×C157—41no—
69C22×C157#12011yescharacteristic
70C2×D157—22no—
71C2×D157—41no—
72C2×D157—41no—
73C4×C157—21no—
74C4×C157—42no—
75C4×C157C2×Q1611yescharacteristic
76C4×C157#13511yescharacteristic
77C4×C157#13711yescharacteristic
78C4×C157#13711yescharacteristic
79Dic157—41no—
80Dic157—41no—
81Dic157—41no—
82Dic157—81no—
83C2×C157#45111yescharacteristic
84C2×C157#45211yescharacteristic
85C2×C157#45311yescharacteristic
86D157—42no—
87C157#163211yescharacteristic, complemented, Sylow
88#1632—1571nomaximal, Sylow
89#554—1571no—
90#494—1571no—
91#477—1571no—
92#476—1571no—
93#333D15711yescharacteristic, complemented
94Q8⋊C8—1571no—
95D4⋊C8—1571no—
96C2×Q16—3142no—
97C4⋊Q8—1571no—
98C4⋊Q8C2×D15711yescharacteristic
99#125—3141no—
100C22⋊Q8—3141no—
101C22⋊Q8—3142no—
102C4×Q8—1571no—
103C4×D4—3141no—
104C4×D4—1571no—
105C2×C4⋊C4—3141no—
106#107—3141no—
107#106—3141no—
108C4⋊C8C2×D15711yescharacteristic
109C4⋊C8C2×D15711yescharacteristic
110#103—3141no—
111#102—3141no—
112C4×C8—1571no—
113C2×Q8—3141no—
114C2×Q8—1571no—
115C2×Q8D62812yes—
116C2×D4—1571no—
117C22×C4—3142no—
118C22×C4—3141no—
119Q16—6284no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3141no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—3142no—
126C4⋊C4—3142no—
127C4⋊C4—3141no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4—1571no—
131C4⋊C4—1571no—
132C22⋊C4—3142no—
133C22⋊C4—3141no—
134C42—3141no—
135C42C22×D15711yescharacteristic
136C23—3141no—
137Q8—42no—
138Q8—3141no—
139Q8—3141no—
140Q8—6282no—
141D4—3141no—
142D4—1572no—
143C2×C4—22no—
144C2×C4—3141no—
145C2×C4—3141no—
146C2×C4—3141no—
147C2×C4—6281no—
148C2×C4—6281no—
149C2×C4—6281no—
150C2×C4—12562no—
151C2×C4D4×D15711yescharacteristic
152C2×C4D4×D15711yescharacteristic
153C2×C4C2×D62811yescharacteristic
154C8—41no—
155C8—41no—
156C8—3142no—
157C22—3142no—
158C22—6281no—
159C22—6281no—
160C22#1909774111yescharacteristic
161C4—21no—
162C4—42no—
163C4—6281no—
164C4—6281no—
165C4—6281no—
166C4—12561no—
167C4Q16×D15711yescharacteristic
168C4#1909779211yescharacteristic
169C4#1909780011yescharacteristic
170C4#1909780311yescharacteristic
171C2—6282no—
172C2#2375743011yescharacteristic
173C2#2375743811yescharacteristic
174C2#2375744311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#333×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#333D157
hypocenterC157#1632
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #23757652C2semidirect
3. #23757569C2semidirect
5. #23757003C2semidirect
7. #333×C157C2semidirect
53. D628Q16semidirect
87. C157#1632semidirect
93. #333D157semidirect

Maximal subgroups

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

subgroupindex
#237579842
#237576522
#237575692
#237575642
#237570032
#237570002
#333×C1572
#1632157