Finite Group Databasebeta

← #34111607

Subgroups — #34111607

Order 20096.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#23757894C212yesmaximal, complemented
2#23757564C212yesmaximal, complemented
3#23757559C212yesmaximal, complemented
4#23757531C211yesmaximal, characteristic, complemented
5#23757526C211yesmaximal, characteristic
6#23757502C212yesmaximal, complemented
7#23757496C212yesmaximal, complemented
8#23757489C212yesmaximal, complemented
9#424×C157C211yesmaximal, characteristic, complemented
10C2×Dic628C2211yescharacteristic
11C2×#14578943C2212yes—
12C2×D1256C2211yescharacteristic
13#19097771C2212yes—
14#19097766—22no—
15#19097759C2212yes—
16#19097757—22no—
17#19097753—22no—
18#19097750C2212yes—
19#19097749—22no—
20#19097745C2212yes—
21C4×Dic314C2212yes—
22C4×D628C2212yes—
23#19097703C2212yes—
24#19097703C2211yescharacteristic
25#19097703C2211yescharacteristic
26#19097696C2212yes—
27#19097696C2212yes—
28#19097693C2212yes—
29#19097693C2212yes—
30C2×M16×C157C2211yescharacteristic
31C22×C8×C157C2211yescharacteristic
32#117×C157C2211yescharacteristic
33C4⋊C8×C157C2212yes—
34C4⋊C8×C157C2212yes—
35C2×Dic314—22no—
36C2×Dic314C2312yes—
37C2×D628C2312yes—
38C2×C22⋊D157—22no—
39C2×C4×D157—22no—
40Dic628—41no—
41#14578943—42no—
42D1256—41no—
43C4⋊Dic157—22no—
44C4⋊Dic157C2312yes—
45C4⋊Dic157C2312yes—
46#14578933—22no—
47#14578933—22no—
48C22⋊Dic157—22no—
49#14578931—22no—
50#14578931—22no—
51#14578931—22no—
52#14578931—22no—
53C4×Dic157—22no—
54C22×C4×C157C2311yescharacteristic
55M16×C157—41no—
56C2×C8×C157—41no—
57C2×C8×C157C2312yes—
58C2×C8×C157C2311yescharacteristic
59C2×C8×C157C2311yescharacteristic
60C4⋊C4×C157D412yes—
61C22⋊C4×C157D411yescharacteristic
62C22⋊C4×C157D411yescharacteristic
63C42×C157C2312yes—
64C22×D157—22no—
65Dic314—22no—
66Dic314—22no—
67Dic314—82no—
68D628—22no—
69D628—22no—
70C22⋊D157—82no—
71C2×Dic157—22no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C4×D157—82no—
75C23×C157C2×D411yescharacteristic
76C2×C4×C157—21no—
77C2×C4×C157—21no—
78C2×C4×C157C2×D412yes—
79C2×C4×C157C2×D412yes—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2411yescharacteristic
82C8×C157—21no—
83C8×C157—22no—
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—22no—
93C4×C157—22no—
94C4×C157H(5,𝔽2)11yescharacteristic
95C4×C1572−1+411yescharacteristic
96Dic157—42no—
97Dic157—42no—
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#470—1571no—
109#469—1571no—
110#465—1572no—
111#464—1572no—
112#463—1572no—
113#424D15711yescharacteristic, complemented
114C2×Q16—1571no—
115C2×SD16—1572no—
116C2×D8—1571no—
117C2×M16C2×D15711yescharacteristic
118C22×C8C2×D15711yescharacteristic
119C4⋊Q8—1572no—
120#126—3142no—
121#124—3142no—
122#124—1572no—
123#123—3142no—
124C22⋊Q8—3142no—
125C22⋊Q8—1572no—
126#121—1572no—
127C4×Q8—1572no—
128C4×D4—1572no—
129#117C2×D15711yescharacteristic
130#106—1572no—
131#106—1571no—
132#106—1571no—
133C4⋊C8C2×D15712yes—
134C4⋊C8C2×D15712yes—
135#103—1572no—
136#103—1572no—
137#102—1572no—
138#102—1572no—
139C2×Q8—3142no—
140C2×Q8—1572no—
141C2×D4—3142no—
142C2×D4—1572no—
143C22×C4—3142no—
144C22×C4C22×D15711yescharacteristic
145Q16—6281no—
146SD16—6282no—
147D8—6281no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D15711yescharacteristic
151C2×C8C22×D15712yes—
152C2×C8C22×D15711yescharacteristic
153C4⋊C4—3142no—
154C4⋊C4—3142no—
155C4⋊C4—3142no—
156C4⋊C4—1572no—
157C4⋊C4—1572no—
158C4⋊C4D62812yes—
159C22⋊C4—3142no—
160C22⋊C4—3142no—
161C22⋊C4—3142no—
162C22⋊C4—3142no—
163C22⋊C4—3142no—
164C22⋊C4D62811yescharacteristic
165C22⋊C4D62811yescharacteristic
166C42—3142no—
167C42C22×D15712yes—
168C23—3142no—
169C23C2×D62811yescharacteristic
170Q8—3142no—
171Q8—3142no—
172Q8—12562no—
173D4—3142no—
174D4—3142no—
175D4—12562no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—3142no—
179C2×C4—3142no—
180C2×C4—3142no—
181C2×C4—12562no—
182C2×C4C2×D62812yes—
183C2×C4C2×D62812yes—
184C2×C4C2×D62811yescharacteristic
185C2×C4C23×D15711yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—6282no—
192C22—6282no—
193C22—6282no—
194C22C22×D62811yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—6282no—
199C4—6282no—
200C4—6282no—
201C4#1909781611yescharacteristic
202C4#1909781911yescharacteristic
203C2—41no—
204C2—12562no—
205C2#2375785411yescharacteristic
206C2#2375808512yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#424×C157C2
socleC22×C157C22×D4
prime radical#23757526C2
hypercenter#424D157
hypocenterC157#3263
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #23757894C2semidirect
2. #23757564C2semidirect
3. #23757559C2semidirect
4. #23757531C2semidirect
6. #23757502C2semidirect
7. #23757496C2semidirect
8. #23757489C2semidirect
9. #424×C157C2semidirect
103. C157#3263semidirect
113. #424D157semidirect

Maximal subgroups

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

subgroupindex
#237578942
#237575642
#237575592
#237575312
#237575262
#237575022
#237574962
#237574892
#424×C1572
#3263157