Finite Group Databasebeta

← #34099260

Subgroups — #34099260

Order 20096.

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

normal subgroups only

This group has 256 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#23757943C211yesmaximal, characteristic
2#23757565C211yesmaximal, characteristic, complemented
3#23757560C211yesmaximal, characteristic, complemented
4#23757017C211yesmaximal, characteristic, complemented
5#23757005C211yesmaximal, characteristic, complemented
6#23757002C211yesmaximal, characteristic, complemented
7#492×C157C211yesmaximal, characteristic, complemented
8C2×#14578953—21no—
9#19097767—21no—
10#19097743—22no—
11#19097742—21no—
12C4×Dic314C2211yescharacteristic
13C4×D628C2211yescharacteristic
14D4×Dic157—21no—
15#19097704—21no—
16#19097702—21no—
17C4⋊#11929434C2211yescharacteristic
18C4⋊#11929434C2211yescharacteristic
19#19097694—21no—
20#19097691—21no—
21C2×SD16×C157—22no—
22C4⋊Q8×C157C2211yescharacteristic
23#127×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25#14578953—42no—
26#14578953—42no—
27C2×Dic314D411yescharacteristic
28C2×D628D411yescharacteristic
29C2×C22⋊D157—22no—
30C2×C22⋊D157—22no—
31C22×Dic157—22no—
32C2×C4×D157—21no—
33C2×#11929434—21no—
34C2×#11929434—21no—
35C4⋊Dic157D411yescharacteristic
36C4⋊Dic157D411yescharacteristic
37#14578933—21no—
38C22⋊Dic157—22no—
39#14578931—21no—
40C4×Dic157—21no—
41C2×Q8×C157—21no—
42C2×D4×C157—21no—
43C2×D4×C157—21no—
44C2×D4×C157D412yes—
45SD16×C157—44no—
46C2×C8×C157—21no—
47C4⋊C4×C157—21no—
48C4⋊C4×C157—21no—
49C42×C157C2311yescharacteristic
50C22×D157—21no—
51Dic314—21no—
52Dic314SD1612yescomplemented
53D628—21no—
54D628—21no—
55C22⋊D157—44no—
56C22⋊D157—82no—
57C2×Dic157—21no—
58C2×Dic157—21no—
59C2×Dic157—21no—
60C2×Dic157—44no—
61C4×D157—41no—
62C4×D157—41no—
63#11929434—41no—
64#11929434—41no—
65C23×C157—22no—
66Q8×C157—42no—
67D4×C157—42no—
68D4×C157—41no—
69D4×C157—41no—
70D4×C157—41no—
71D4×C157—41no—
72C2×C4×C157—41no—
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C8×C157—22no—
77C22×C157—42no—
78C22×C157—42no—
79C22×C157—42no—
80C22×C157#12011yescharacteristic
81C2×D157—41no—
82C2×D157—41no—
83C2×D157—41no—
84C4×C157—21no—
85C4×C157—81no—
86C4×C157C2×SD1611yescharacteristic
87C4×C157#13511yescharacteristic
88C4×C157AGL(1,ℤ/8)11yescharacteristic
89C4×C157#13711yescharacteristic
90Dic157—22no—
91Dic157—41no—
92Dic157—41no—
93C2×C157—82no—
94C2×C157#45011yescharacteristic
95C2×C157#45211yescharacteristic
96C2×C157#45311yescharacteristic
97D157—81no—
98C157#163111yescharacteristic, complemented, Sylow
99#1631—1571nomaximal, Sylow
100#550—1571no—
101#492D15711yescharacteristic, complemented
102#477—1571no—
103#476—1571no—
104#331—1571no—
105Q8⋊C8—1571no—
106D4⋊C8—1571no—
107C2×#41—3141no—
108C2×SD16—22no—
109C4⋊Q8C2×D15711yescharacteristic
110#127—3141no—
111#127C2×D15711yescharacteristic
112#121—3142no—
113#121—3141no—
114C4×Q8—1571no—
115C4×D4—3141no—
116C4×D4—1571no—
117#107—3141no—
118#106—3141no—
119C4⋊C8—1571no—
120C4⋊C8—1571no—
121#103—3141no—
122#102—3141no—
123C4×C8C2×D15711yescharacteristic
124#41—6282no—
125#41—6282no—
126C2×Q8—21no—
127C2×Q8—1571no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—3142no—
131C2×D4—3142no—
132C2×D4—1571no—
133C2×D4C22⋊D15712yes—
134C22×C4—3142no—
135C22×C4—3141no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—3141no—
139C2×C8—3141no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—3141no—
143C4⋊C4—1571no—
144C4⋊C4—1571no—
145C22⋊C4—3142no—
146C22⋊C4—3141no—
147C42—3141no—
148C42C22×D15711yescharacteristic
149C23—22no—
150C23—3141no—
151Q8—42no—
152Q8—3141no—
153Q8—1572no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—3141no—
160D4—3141no—
161D4—6284no—
162D4—12562no—
163C2×C4—41no—
164C2×C4—3141no—
165C2×C4—3141no—
166C2×C4—3141no—
167C2×C4—6284no—
168C2×C4—6281no—
169C2×C4—6281no—
170C2×C4D4×D15711yescharacteristic
171C2×C4D4×D15711yescharacteristic
172C2×C4C2×C22⋊D15711yescharacteristic
173C8—22no—
174C8—6281no—
175C8—6281no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—6281no—
180C22—6281no—
181C22—6281no—
182C22#1909773911yescharacteristic
183C4—21no—
184C4—81no—
185C4—3142no—
186C4—6281no—
187C4—6281no—
188C4SD16×D15711yescharacteristic
189C4#1909779211yescharacteristic
190C4#1909779811yescharacteristic
191C4#1909780011yescharacteristic
192C2—82no—
193C2—12561no—
194C2#2375742211yescharacteristic
195C2#2375743211yescharacteristic
196C2#2375744011yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#492×C157C2
socleC22×C157#120
prime radical#23757943C2
hypercenter#492D157
hypocenterC157#1631
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #23757565C2semidirect
3. #23757560C2semidirect
4. #23757017C2semidirect
5. #23757005C2semidirect
6. #23757002C2semidirect
7. #492×C157C2semidirect
52. Dic314SD16semidirect
98. C157#1631semidirect
101. #492D157semidirect

Maximal subgroups

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

subgroupindex
#237579432
#237575652
#237575602
#237570172
#237570052
#237570022
#492×C1572
#1631157