Finite Group Databasebeta

← #34115634

Subgroups — #34115634

Order 20096.

256 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 68 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#23758290C211yesmaximal, characteristic, complemented
2C2×#19097838C211yesmaximal, characteristic, complemented
3#23758132C212yesmaximal, complemented
4#23758131C212yesmaximal, complemented
5#23757818C211yesmaximal, characteristic
6C22×#11929441—21no—
7#19097838—21no—
8#19097838C2212yes—
9#19097838C2212yes—
10C22⋊#11929441—21no—
11C4×#11929441—21no—
12C2×#14578953—21no—
13#19097770C2211yescharacteristic
14#19097767C2211yescharacteristic
15#19097757—21no—
16#19097745C411yescharacteristic
17#19097743—21no—
18D4×Dic157C411yescharacteristic
19#19097733C411yescharacteristic
20#19097731—21no—
21C2×C4×Dic157C2211yescharacteristic
22#121×C157C411yescharacteristic
23C2×#11929441—22no—
24C2×#11929441—22no—
25C2×#11929441—42no—
26#14578953—42no—
27#14578953—42no—
28C2×Dic314—21no—
29C2×D628C2×C411yescharacteristic
30C2×C22⋊D157—21no—
31C2×C22⋊D157—21no—
32C2×C22⋊D157—21no—
33C22×Dic157—21no—
34C22×Dic157D411yescharacteristic
35C22×Dic157D411yescharacteristic
36C2×C4×D157—21no—
37C4⋊Dic157C2×C411yescharacteristic
38#14578933—21no—
39C22⋊Dic157—21no—
40#14578931—21no—
41#14578931—21no—
42C4×Dic157C2×C411yescharacteristic
43C4×Dic157D411yescharacteristic
44C4×Dic157D411yescharacteristic
45C4×Dic157C2311yescharacteristic
46C2×D4×C157—21no—
47C2×D4×C157C2×C411yescharacteristic
48C22×C4×C157C2×C411yescharacteristic
49C4⋊C4×C157C2×C411yescharacteristic
50C22⋊C4×C157—21no—
51#11929441—42no—
52#11929441—42no—
53C22×D157—21no—
54Dic314—41no—
55Dic314—41no—
56D628—41no—
57C22⋊D157—42no—
58C22⋊D157—41no—
59C22⋊D157—41no—
60C22⋊D157—42no—
61C2×Dic157—22no—
62C2×Dic157—21no—
63C2×Dic157—42no—
64C2×Dic157C22⋊C411yescharacteristic
65C2×Dic157C22⋊C411yescharacteristic
66C2×Dic157D812yes—
67C2×Dic157SD1612yes—
68C2×Dic157C2×D411yescharacteristic
69C2×Dic157C2×D411yescharacteristic
70C4×D157—42no—
71C23×C157—21no—
72C23×C157C22⋊C411yescharacteristic
73D4×C157—41no—
74D4×C157—42no—
75C2×C4×C157—21no—
76C2×C4×C157—41no—
77C2×C4×C157C22⋊C411yescharacteristic
78C2×C4×C157C22×C411yescharacteristic
79C22×C157—21no—
80C22×C157—21no—
81C22×C157—41no—
82C22×C157—41no—
83C22×C157—41no—
84C22×C157#10212yes—
85C22×C157C2×C22⋊C411yescharacteristic
86C2×D157—41no—
87C2×D157—41no—
88C2×D157—41no—
89C4×C157—21no—
90C4×C157—21no—
91C4×C157—41no—
92Dic157—21no—
93Dic157—41no—
94Dic157#10212yes—
95Dic157C2×D811yescharacteristic
96Dic157C2×SD1611yescharacteristic
97Dic157#13512yes—
98C2×C157—22no—
99C2×C157—81no—
100C2×C157#41311yescharacteristic
101C2×C157C2×#10211yescharacteristic
102C2×C157#41611yescharacteristic
103D157—81no—
104C157#154011yescharacteristic, complemented, Sylow
105#1540—1571nomaximal, Sylow
106#532—1571no—
107#431—1571no—
108C2×C4⋊C8—1571no—
109#332—1572no—
110#331—1572no—
111C2×#41—3141no—
112C22×C8—3141no—
113C4⋊Q8—1571no—
114#127—1571no—
115#124—3141no—
116#121—3141no—
117#121—1571no—
118#121#1059784511yescharacteristic
119C4×D4—3141no—
120C4×D4—1571no—
121C4×D4—1571no—
122C2×C42—1571no—
123C4⋊C8—3141no—
124C4⋊C8—1572no—
125C4⋊C8—1572no—
126C22⋊C8—3141no—
127C4×C8—3141no—
128#41—6282no—
129#41—6282no—
130C2×Q8—3141no—
131C2×D4—21no—
132C2×D4—3141no—
133C2×D4—3141no—
134C2×D4—3141no—
135C2×D4—1571no—
136C2×D4C2×#1059784511yescharacteristic
137C22×C4—3141no—
138C22×C4—3141no—
139C22×C4—1571no—
140C22×C4—1571no—
141C22×C4C2×#1059784511yescharacteristic
142C2×C8—3142no—
143C2×C8—3142no—
144C2×C8—6282no—
145C4⋊C4—3141no—
146C4⋊C4—1571no—
147C4⋊C4C2×#1059784511yescharacteristic
148C22⋊C4—21no—
149C22⋊C4—3141no—
150C22⋊C4—3141no—
151C22⋊C4—3141no—
152C42—1571no—
153C42—1571no—
154C42—1571no—
155C42—1571no—
156C23—21no—
157C23—3141no—
158C23C22⋊#1059784511yescharacteristic
159Q8—6281no—
160Q8—6281no—
161D4—41no—
162D4—42no—
163D4—6281no—
164D4—6282no—
165D4—6281no—
166D4—6281no—
167D4—6282no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—3142no—
171C2×C4—3141no—
172C2×C4—6282no—
173C2×C4—6282no—
174C2×C4—1571no—
175C2×C4—1572no—
176C2×C4—1572no—
177C2×C4—1571no—
178C2×C4—1571no—
179C2×C4—1571no—
180C2×C4C22⋊#1059784511yescharacteristic
181C2×C4C22×#1059784511yescharacteristic
182C8—6282no—
183C8—6282no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—6281no—
190C22—6281no—
191C22—6281no—
192C22#1909783312yes—
193C22C2×C22⋊#1059784511yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—3141no—
198C4—6281no—
199C4—1571no—
200C4—1572no—
201C4—1571no—
202C4—1572no—
203C2—22no—
204C2—81no—
205C2—12561no—
206C2#2375826011yescharacteristic
207C2C2×#1909783311yescharacteristic
208C2#2375826311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597845
commutatorC2×C4×C157C22×C4
FrattiniC2×C4C22×#10597845
Fitting#121×C157C4
socleC22×C157C2×C22⋊C4
prime radical#23757818C2
cosocleC4×Dic157C23
hypercenter#121#10597845
hypocenterC157#1540
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #23758290C2semidirect
2. C2×#19097838C2semidirect
3. #23758132C2semidirect
4. #23758131C2semidirect
104. C157#1540semidirect

Maximal subgroups

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

subgroupindex
#237582902
C2×#190978382
#237581322
#237581312
#237578182
#1540157