Finite Group Databasebeta

← #34099298

Subgroups — #34099298

Order 20096.

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

normal subgroups only

This group has 228 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#23757938C211yesmaximal, characteristic
2#23757565C211yesmaximal, characteristic, complemented
3#23757560C211yesmaximal, characteristic
4#23757019C211yesmaximal, characteristic, complemented
5#23757005C211yesmaximal, characteristic, complemented
6#23757002C211yesmaximal, characteristic
7#487×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157—21no—
9#19097770—21no—
10#19097749—22no—
11#19097747—21no—
12C4×Dic314C2211yescharacteristic
13C4×D628C2211yescharacteristic
14#19097733—21no—
15#19097704—21no—
16#19097702—21no—
17C4⋊#11929434C2211yescharacteristic
18C4⋊#11929434C2211yescharacteristic
19#19097694—21no—
20#19097691—21no—
21C4⋊Q8×C157C2211yescharacteristic
22C4⋊Q8×C157C2211yescharacteristic
23#103×C157—22no—
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—22no—
26C2×Dic314D411yescharacteristic
27Q8×D157—42no—
28Q8×D157—42no—
29C2×D628D411yescharacteristic
30C2×C4×D157—22no—
31C2×C4×D157—21no—
32C2×#11929434—21no—
33C2×#11929434—21no—
34C4⋊Dic157D411yescharacteristic
35C4⋊Dic157D411yescharacteristic
36#14578933—22no—
37#14578933—21no—
38#14578931—22no—
39#14578931—21no—
40C4×Dic157—21no—
41C2×Q8×C157—21no—
42C2×Q8×C157—21no—
43C2×C8×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157—21no—
46C4⋊C4×C157—21no—
47C4⋊C4×C157D412yes—
48C42×C157C2311yescharacteristic
49C22×D157—21no—
50Dic314—21no—
51Dic314—44no—
52Dic314SD1612yes—
53D628—21no—
54D628Q1612yes—
55C2×Dic157—21no—
56C2×Dic157—21no—
57C2×Dic157—21no—
58C4×D157—44no—
59C4×D157—41no—
60C4×D157—41no—
61#11929434—41no—
62#11929434—41no—
63Q8×C157—42no—
64Q8×C157—41no—
65Q8×C157—41no—
66C2×C4×C157—22no—
67C2×C4×C157—41no—
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C8×C157—41no—
72C22×C157#12011yescharacteristic
73C2×D157—22no—
74C2×D157—41no—
75C2×D157—41no—
76C4×C157—21no—
77C4×C157—42no—
78C4×C157—81no—
79C4×C157C2×SD1611yescharacteristic
80C4×C157C2×Q1611yescharacteristic
81C4×C157AGL(1,ℤ/8)11yescharacteristic
82C4×C157#13711yescharacteristic
83Dic157—22no—
84Dic157—41no—
85Dic157—41no—
86C2×C157#45011yescharacteristic
87C2×C157#45111yescharacteristic
88C2×C157#45411yescharacteristic
89D157—42no—
90C157#163711yescharacteristic, complemented, Sylow
91#1637—1571nomaximal, Sylow
92D4×Q8—1571no—
93#487D15711yescharacteristic, complemented
94#477—1571no—
95#476—1571no—
96#332—1571no—
97Q8⋊C8—1571no—
98D4⋊C8—1571no—
99C22×Q8—3141no—
100C4⋊Q8—3141no—
101C4⋊Q8C2×D15711yescharacteristic
102C4⋊Q8C2×D15711yescharacteristic
103C22⋊Q8—3141no—
104C22⋊Q8—3142no—
105C4×Q8—1571no—
106C4×D4—3141no—
107C4×D4—1571no—
108#107—3141no—
109#106—3141no—
110C4⋊C8—1571no—
111C4⋊C8—1571no—
112#103—22no—
113#103—3141no—
114#102—3141no—
115C4×C8C2×D15711yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—3142no—
119C2×Q8—6282no—
120C2×Q8—6282no—
121C2×Q8—1571no—
122C2×D4—1571no—
123C22×C4—3142no—
124C22×C4—3141no—
125C2×C8—21no—
126C2×C8—3141no—
127C2×C8—3141no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3142no—
132C4⋊C4—3141no—
133C4⋊C4—1571no—
134C4⋊C4—1571no—
135C4⋊C4C22⋊D15712yes—
136C22⋊C4—3142no—
137C22⋊C4—3141no—
138C42—3141no—
139C42C22×D15711yescharacteristic
140C23—3141no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—3141no—
145Q8—6284no—
146Q8—1572no—
147D4—3141no—
148D4—1572no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—3141no—
152C2×C4—3141no—
153C2×C4—3141no—
154C2×C4—6284no—
155C2×C4—6281no—
156C2×C4—6281no—
157C2×C4D4×D15711yescharacteristic
158C2×C4D4×D15711yescharacteristic
159C2×C4C2×C22⋊D15711yescharacteristic
160C8—41no—
161C8—6281no—
162C8—6281no—
163C22—3142no—
164C22—6281no—
165C22—6281no—
166C22#1909773911yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—3142no—
171C4—6281no—
172C4—6281no—
173C4SD16×D15711yescharacteristic
174C4Q16×D15711yescharacteristic
175C4#1909779811yescharacteristic
176C4#1909780311yescharacteristic
177C2—6282no—
178C2#2375742211yescharacteristic
179C2#2375742711yescharacteristic
180C2#2375744511yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#487×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#487D157
hypocenterC157#1637
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #23757565C2semidirect
4. #23757019C2semidirect
5. #23757005C2semidirect
7. #487×C157C2semidirect
90. C157#1637semidirect
93. #487D157semidirect

Maximal subgroups

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

subgroupindex
#237579382
#237575652
#237575602
#237570192
#237570052
#237570022
#487×C1572
#1637157