Finite Group Databasebeta

← #34100655

Subgroups — #34100655

Order 20096.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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#23758039C211yesmaximal, characteristic
2#23757847C211yesmaximal, characteristic
3#23757282C211yesmaximal, characteristic, complemented
4#23757246C211yesmaximal, characteristic
5#23757030C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7#419×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097779—22no—
10#19097777—22no—
11C2×#14578938C2211yescharacteristic
12M16×D157—22no—
13#19097769C412yescomplemented
14#19097760C412yescomplemented
15#19097754—22no—
16#19097747—22no—
17#19097730C2211yescharacteristic
18#19097730C2211yescharacteristic
19#19097695—22no—
20#19097685—22no—
21C22×Q8×C157C2211yescharacteristic
22C2×M16×C157C2211yescharacteristic
23#117×C157C2211yescharacteristic
24#103×C157—22no—
25#14578954—21no—
26#14578954—21no—
27#14578954—21no—
28#14578954—21no—
29C2×Dic314C2×C411yescharacteristic
30C2×D628C2×C411yescharacteristic
31C2×C22⋊D157D412yes—
32C2×C4×D157D412yes—
33#14578938—21no—
34#14578938—21no—
35#14578937—24no—
36C2×#11929434D412yes—
37C8×D157—24no—
38C4⋊Dic157—21no—
39C4⋊Dic157—21no—
40#14578933—24no—
41C22⋊Dic157—21no—
42C22⋊Dic157—21no—
43#14578931—24no—
44C4×Dic157—21no—
45C4×Dic157C2×C412yes—
46C2×Q8×C157—21no—
47C2×Q8×C157—21no—
48C2×Q8×C157—21no—
49C2×Q8×C157—21no—
50C2×Q8×C157—21no—
51C2×Q8×C157C2×C412yes—
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—21no—
55M16×C157—21no—
56C2×C8×C157D412yes—
57C4⋊C4×C157—21no—
58C22⋊C4×C157—21no—
59C42×C157—21no—
60C22×D157C22⋊C412yes—
61Dic314—21no—
62Dic314—21no—
63D628—21no—
64D628—21no—
65C22⋊D157—22no—
66C22⋊D157—22no—
67C2×Dic157—22no—
68C2×Dic157—41no—
69C2×Dic157C22⋊C412yes—
70C4×D157—22no—
71C4×D157—22no—
72#11929434—22no—
73C23×C157C2×D411yescharacteristic
74Q8×C157—22no—
75Q8×C157—22no—
76Q8×C157—41no—
77Q8×C157—41no—
78C2×C4×C157—22no—
79C2×C4×C157—41no—
80C2×C4×C157—41no—
81C2×C4×C157—41no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157#4111yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C8×C157—22no—
89C22×C157—41no—
90C22×C157C2×C22⋊C411yescharacteristic
91C22×C157#12111yescharacteristic
92C22×C157#12111yescharacteristic
93C2×D157—22no—
94C2×D157—42no—
95C4×C157—42no—
96C4×C157—81no—
97C4×C157C4×D411yescharacteristic
98C4×C157C4×D411yescharacteristic
99C4×C157#12011yescharacteristic
100C4×C157#12311yescharacteristic
101Dic157—22no—
102Dic157—42no—
103Dic157—81no—
104C2×C157—21no—
105C2×C157—21no—
106C2×C157—21no—
107C2×C157#38611yescharacteristic
108D157—42no—
109C157#190911yescharacteristic, complemented, Sylow
110#1909—1571nomaximal, Sylow
111#568—1571no—
112#536—1571no—
113#419—1571no—
114#419D15711yescharacteristic, complemented
115#410—1571no—
116H(3,ℤ/4)—1571no—
117H(3,ℤ/4)—1571no—
118C2×#41—1571no—
119C22×Q8C2×D15711yescharacteristic
120#131—3142no—
121#131—3142no—
122C2×M16—3142no—
123C2×M16—1571no—
124C2×M16C2×D15711yescharacteristic
125C4⋊Q8—1572no—
126#124—1572no—
127#123—3142no—
128C22⋊Q8—3142no—
129#117—1571no—
130#117—1571no—
131#117C2×D15711yescharacteristic
132#103—22no—
133#103—3142no—
134#99—3142no—
135#41—3141no—
136#41—3141no—
137#41—3141no—
138#41—3141no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—1571no—
145C2×Q8C4×D15712yes—
146C2×D4—1572no—
147C2×D4—1571no—
148C22×C4—21no—
149C22×C4—1572no—
150C22×C4C22×D15711yescharacteristic
151M16—21no—
152M16—21no—
153M16—3144no—
154M16—3141no—
155M16—3141no—
156C2×C8—3144no—
157C2×C8—1572no—
158C2×C8C22⋊D15712yes—
159C4⋊C4—21no—
160C4⋊C4—3144no—
161C4⋊C4—3141no—
162C4⋊C4—3141no—
163C22⋊C4—21no—
164C22⋊C4—3144no—
165C22⋊C4—3141no—
166C22⋊C4—3141no—
167C42—21no—
168C42—3141no—
169C42—1572no—
170C23—1572no—
171C23D4×D15711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—3141no—
177Q8—3141no—
178D4—3142no—
179D4—3142no—
180D4—3141no—
181D4—3141no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3142no—
187C2×C4—3142no—
188C2×C4—3142no—
189C2×C4—6281no—
190C2×C4—1572no—
191C2×C4C2×C4×D15711yescharacteristic
192C2×C4D4×D15711yescharacteristic
193C2×C4D4×D15711yescharacteristic
194C2×C4C2×C22⋊D15711yescharacteristic
195C2×C4#1457895311yescharacteristic
196C2×C4#1457895411yescharacteristic
197C8—22no—
198C8—3142no—
199C22—41no—
200C22—3142no—
201C22—6282no—
202C22C22⋊C4×D15711yescharacteristic
203C22#1909774211yescharacteristic
204C22#1909774411yescharacteristic
205C4—42no—
206C4—81no—
207C4—3142no—
208C4—6282no—
209C4—12561no—
210C4#1909773111yescharacteristic
211C4C4×C22⋊D15711yescharacteristic
212C4#1909773911yescharacteristic
213C4#1909775311yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—6282no—
218C2#2375714911yescharacteristic
subgroupG/H
centerC2#23757149
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#419×C157C2
socleC2×C157#386
prime radicalC2×#14578954C22
cosocleC22×C4×C157C23
hypercenter#419D157
hypocenterC157#1909
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #23757282C2semidirect
5. #23757030C2semidirect
6. #23757029C2semidirect
7. #419×C157C2semidirect
13. #19097769C4semidirect
14. #19097760C4semidirect
109. C157#1909semidirect
114. #419D157semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237578472
#237572822
#237572462
#237570302
#237570292
#419×C1572
#1909157