Finite Group Databasebeta

← #34111602

Subgroups — #34111602

Order 20096.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#23757896C212yesmaximal, complemented
2#23757562C212yesmaximal, complemented
3#23757556C212yesmaximal, complemented
4#23757526C211yesmaximal, characteristic
5#23757504C212yesmaximal, complemented
6#23757497C212yesmaximal, complemented
7#23757491C212yesmaximal, complemented
8#23757301C211yesmaximal, characteristic, complemented
9#470×C157C211yesmaximal, characteristic, complemented
10C2×#14578941C2212yes—
11C2×D1256C2211yescharacteristic
12C22×#11929434C2211yescharacteristic
13#19097770C2212yes—
14#19097766—22no—
15#19097760C2212yes—
16#19097757—22no—
17#19097753—22no—
18#19097751—22no—
19#19097745C2212yes—
20Q8×Dic157C2212yes—
21#19097733C2212yes—
22#19097730C2211yescharacteristic
23#19097703C2211yescharacteristic
24#19097702C2212yes—
25#19097700C2212yes—
26#19097700C2212yes—
27#19097695C2212yes—
28#19097693C2212yes—
29#19097691C2212yes—
30C2×Q16×C157C2211yescharacteristic
31C2×M16×C157C2211yescharacteristic
32C22⋊Q8×C157C2212yes—
33#106×C157C2211yescharacteristic
34#103×C157C2212yes—
35C2×Dic314—22no—
36C2×D628C2312yes—
37C2×C22⋊D157—22no—
38C2×C4×D157—22no—
39#14578941—42no—
40D1256—41no—
41C2×#11929434—41no—
42C2×#11929434C2312yes—
43C4⋊Dic157—22no—
44C4⋊Dic157D412yes—
45C4⋊Dic157C2312yes—
46#14578933—22no—
47C22⋊Dic157D411yescharacteristic
48C22⋊Dic157D411yescharacteristic
49#14578931—22no—
50#14578931—22no—
51#14578931—22no—
52#14578931—22no—
53C4×Dic157—22no—
54C4×Dic157C2312yes—
55C2×Q8×C157C2312yes—
56C22×C4×C157C2311yescharacteristic
57Q16×C157—41no—
58M16×C157—41no—
59C2×C8×C157C2311yescharacteristic
60C2×C8×C157C2311yescharacteristic
61C4⋊C4×C157—22no—
62C4⋊C4×C157C2312yes—
63C22⋊C4×C157—22no—
64C22×D157—22no—
65Dic314—82no—
66D628—22no—
67D628—22no—
68C22⋊D157—82no—
69C2×Dic157—22no—
70C2×Dic157C2×D412yes—
71C2×Dic157C2×D412yes—
72C4×D157—82no—
73#11929434—22no—
74C23×C157C2×D411yescharacteristic
75Q8×C157—22no—
76Q8×C157—22no—
77C2×C4×C157—21no—
78C2×C4×C157—21no—
79C2×C4×C157—22no—
80C2×C4×C157—22no—
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2411yescharacteristic
83C8×C157—21no—
84C8×C157—21no—
85C22×C157—21no—
86C22×C157—22no—
87C22×C157C22×D411yescharacteristic
88C2×D157—42no—
89C2×D157—42no—
90C2×D157—42no—
91C4×C157—21no—
92C4×C157—42no—
93C4×C157—42no—
94C4×C157H(5,𝔽2)11yescharacteristic
95C4×C1572−1+411yescharacteristic
96Dic157—22no—
97Dic157—22no—
98Dic157—42no—
99C2×C157—41no—
100C2×C157#53711yescharacteristic
101C2×C157#57712yes—
102D157—82no—
103C157#326311yescharacteristic, complemented, Sylow
104#3263—1571nomaximal, Sylow
105#542—1572no—
106#476—1572no—
107#475—1572no—
108#470D15711yescharacteristic, complemented
109#469—1571no—
110#465—1572no—
111#464—1572no—
112#463—1572no—
113#424—1571no—
114C2×Q16C2×D15711yescharacteristic
115C2×SD16—1572no—
116C2×D8—1571no—
117C2×M16C2×D15711yescharacteristic
118C22×C8—1571no—
119C4⋊Q8—1572no—
120#126—3142no—
121#124—3142no—
122#124—1572no—
123#123—3142no—
124C22⋊Q8—3142no—
125C22⋊Q8C2×D15712yes—
126#121—1572no—
127C4×Q8—1572no—
128C4×D4—1572no—
129#117—1571no—
130#106—1571no—
131#106—1572no—
132#106C2×D15711yescharacteristic
133C4⋊C8—1572no—
134C4⋊C8—1572no—
135#103—1572no—
136#103C2×D15712yes—
137#102—1572no—
138#102—1572no—
139C2×Q8—3142no—
140C2×Q8C22×D15712yes—
141C2×D4—3142no—
142C2×D4—1572no—
143C22×C4—3142no—
144C22×C4C22×D15711yescharacteristic
145Q16—41no—
146SD16—6282no—
147D8—6281no—
148M16—41no—
149C2×C8—6281no—
150C2×C8—1572no—
151C2×C8C22×D15711yescharacteristic
152C2×C8C22×D15711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—3142no—
155C4⋊C4—3142no—
156C4⋊C4—1572no—
157C4⋊C4—1572no—
158C4⋊C4C22×D15712yes—
159C22⋊C4—22no—
160C22⋊C4—3142no—
161C22⋊C4—3142no—
162C22⋊C4—3142no—
163C22⋊C4—3142no—
164C22⋊C4—1571no—
165C22⋊C4—1571no—
166C42—3142no—
167C42—1572no—
168C23—3142no—
169C23D4×D15711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—12562no—
173D4—3142no—
174D4—3142no—
175D4—12562no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—3142no—
181C2×C4—1572no—
182C2×C4—1572no—
183C2×C4—12562no—
184C2×C4D4×D15711yescharacteristic
185C2×C4C23×D15711yescharacteristic
186C8—21no—
187C8—21no—
188C8—3142no—
189C22—22no—
190C22—21no—
191C22—6282no—
192C22—6282no—
193C22—6282no—
194C22C2×D4×D15711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—3142no—
199C4—3142no—
200C4—6282no—
201C4#1909781611yescharacteristic
202C4#1909781811yescharacteristic
203C2—41no—
204C2—12562no—
205C2#2375785011yescharacteristic
206C2#2375808212yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#470×C157C2
socleC22×C157C22×D4
prime radical#23757526C2
hypercenter#470D157
hypocenterC157#3263
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #23757896C2semidirect
2. #23757562C2semidirect
3. #23757556C2semidirect
5. #23757504C2semidirect
6. #23757497C2semidirect
7. #23757491C2semidirect
8. #23757301C2semidirect
9. #470×C157C2semidirect
103. C157#3263semidirect
108. #470D157semidirect

Maximal subgroups

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

subgroupindex
#237578962
#237575622
#237575562
#237575262
#237575042
#237574972
#237574912
#237573012
#470×C1572
#3263157