Finite Group Databasebeta

← #34099073

Subgroups — #34099073

Order 20096.

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

normal subgroups only

This group has 310 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#23757854C211yesmaximal, characteristic, complemented
2#23757409C212yesmaximal, complemented
3#23757190C211yesmaximal, characteristic, complemented
4#23757008C212yesmaximal, complemented
5#324×C157C211yesmaximal, characteristic, complemented
6C2×#14578954—21no—
7C22×D628C2211yescharacteristic
8C2×D1256—22no—
9#19097751—22no—
10#19097746—22no—
11#19097745—21no—
12#19097745C2212yes—
13C2×#14578931—21no—
14C2×#14578931—21no—
15#19097693—22no—
16#19097678C2211yescharacteristic
17C2×C4⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19C23×D157—21no—
20#14578954—42no—
21#14578954—42no—
22C2×Dic314—21no—
23C2×D628—21no—
24C2×D628—21no—
25C2×D628—22no—
26C2×D628—42no—
27C2×D628D412yes—
28C2×C22⋊D157—22no—
29C22×Dic157—21no—
30C2×C4×D157—22no—
31D1256—44no—
32C4⋊Dic157—21no—
33C4⋊Dic157D412yes—
34#14578931—22no—
35#14578931—22no—
36#14578931—42no—
37#14578931—42no—
38C22×C4×C157D411yescharacteristic
39C22×C4×C157D411yescharacteristic
40C22×C4×C157C2311yescharacteristic
41C2×C8×C157—22no—
42C4⋊C4×C157—22no—
43C22×D157—22no—
44C22×D157—22no—
45C22×D157—41no—
46C22×D157—41no—
47C22×D157—41no—
48C22×D157—41no—
49C22×D157—41no—
50C22×D157—41no—
51Dic314—42no—
52D628—24no—
53D628—42no—
54D628—42no—
55D628—42no—
56D628—81no—
57C22⋊D157—42no—
58C22⋊D157—42no—
59C2×Dic157—22no—
60C2×Dic157—41no—
61C2×Dic157—41no—
62C2×Dic157—41no—
63C4×D157—44no—
64C23×C157C2×D411yescharacteristic
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—21no—
68C2×C4×C157—41no—
69C2×C4×C157—41no—
70C2×C4×C157D812yescomplemented
71C2×C4×C157C2×D412yes—
72C8×C157—42no—
73C22×C157—21no—
74C22×C157—21no—
75C22×C157#12011yescharacteristic
76C22×C157C2×D811yescharacteristic
77C22×C157AGL(1,ℤ/8)11yescharacteristic
78C2×D157—42no—
79C2×D157—42no—
80C2×D157—42no—
81C2×D157—42no—
82C2×D157—42no—
83C2×D157—42no—
84C2×D157—81no—
85C2×D157—81no—
86C2×D157—81no—
87C2×D157—81no—
88C4×C157—22no—
89C4×C157—22no—
90C4×C157—41no—
91Dic157—42no—
92Dic157—81no—
93C2×C157—21no—
94C2×C157—21no—
95C2×C157#44711yescharacteristic
96C2×C157#45411yescharacteristic
97C2×C157B(4,𝔽2)11yescharacteristic
98D157—82no—
99D157—82no—
100C157#159811yescharacteristic, complemented, Sylow
101#1598—1571nomaximal, Sylow
102#537—1571no—
103#447—1572no—
104#394—1571no—
105#327—1572no—
106#324D15711yescharacteristic, complemented
107C2×#41—3141no—
108C22×D4—1571no—
109C2×D8—3142no—
110C22⋊Q8—3142no—
111#121—3142no—
112#121—3141no—
113#121—1572no—
114C2×C4⋊C4C2×D15711yescharacteristic
115C2×C22⋊C4—3141no—
116C2×C22⋊C4—3141no—
117#102—3142no—
118C22⋊C8C2×D15712yes—
119#95—1571no—
120C24—3141no—
121#41—6282no—
122#41—6282no—
123C2×Q8—3141no—
124C2×D4—3141no—
125C2×D4—3141no—
126C2×D4—3142no—
127C2×D4—3142no—
128C2×D4—6282no—
129C2×D4—1572no—
130C22×C4—3142no—
131C22×C4—3141no—
132C22×C4D62811yescharacteristic
133C22×C4D62811yescharacteristic
134C22×C4C22×D15711yescharacteristic
135D8—6284no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—3141no—
139C4⋊C4—1572no—
140C22⋊C4—3142no—
141C22⋊C4—3142no—
142C22⋊C4—6282no—
143C22⋊C4—6282no—
144C23—3142no—
145C23—3142no—
146C23—6281no—
147C23—6281no—
148C23—6281no—
149C23—6281no—
150C23—6281no—
151C23—6281no—
152C23C2×D62811yescharacteristic
153Q8—6282no—
154D4—3144no—
155D4—6282no—
156D4—6282no—
157D4—6282no—
158D4—6282no—
159D4—6282no—
160D4—12561no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—3142no—
167C2×C4—6284no—
168C2×C4—6281no—
169C2×C4—6281no—
170C2×C4—6281no—
171C2×C4D125612yescomplemented
172C2×C4D4×D15712yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—6282no—
177C22—6282no—
178C22—6282no—
179C22—6282no—
180C22—6282no—
181C22—6282no—
182C22—12561no—
183C22—12561no—
184C22—12561no—
185C22—12561no—
186C22#1909774111yescharacteristic
187C22C2×D125611yescharacteristic
188C22#1909779911yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—6282no—
193C4—12561no—
194C2—21no—
195C2—21no—
196C2—12562no—
197C2—12562no—
198C2#2375740911yescharacteristic
199C2#2375744811yescharacteristic
200C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#324×C157C2
socleC22×C157#120
hypercenter#324D157
hypocenterC157#1598
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #23757854C2semidirect
2. #23757409C2semidirect
3. #23757190C2semidirect
4. #23757008C2semidirect
5. #324×C157C2semidirect
70. C2×C4×C157D8semidirect
100. C157#1598semidirect
106. #324D157semidirect
171. C2×C4D1256semidirect

Maximal subgroups

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

subgroupindex
#237578542
#237574092
#237571902
#237570082
#324×C1572
#1598157