Finite Group Databasebeta

← #34108398

Subgroups — #34108398

Order 20096.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757214C212yesmaximal, complemented
2#23757206C212yesmaximal, complemented
3#23757203C212yesmaximal, complemented
4#23757196C212yesmaximal
5#23757193C212yesmaximal, complemented
6#23757185C211yesmaximal, characteristic
7#23757134C211yesmaximal, characteristic
8#23757131C212yesmaximal
9#399×C157C211yesmaximal, characteristic
10C22×Dic314C2211yescharacteristic
11C23×Dic157C2211yescharacteristic
12C2×C4⋊Dic157C2212yes—
13C2×#14578933C2212yes—
14C2×#14578933C2212yes—
15C2×#14578933C2212yes—
16C2×#14578933C2212yes—
17C2×C22⋊Dic157C2212yes—
18C2×C22⋊Dic157C2211yescharacteristic
19#19097678C2212yes—
20#19097678C2212yes—
21#19097678C2212yes—
22#19097678C2212yes—
23#19097678C2211yescharacteristic
24#19097678C2212yes—
25#19097678C2212yes—
26C2×C4⋊C4×C157C2212yes—
27C2×C4⋊C4×C157C2211yescharacteristic
28C2×C22⋊C4×C157C2212yes—
29C2×C22⋊C4×C157C2211yescharacteristic
30#95×C157C2211yescharacteristic
31C2×Dic314—41no—
32C2×Dic314—42no—
33C22×Dic157—41no—
34C22×Dic157—41no—
35C22×Dic157Q814yescomplemented
36C22×Dic157C2312yes—
37C22×Dic157C2312yes—
38C22×Dic157C2312yes—
39C22×Dic157C2312yes—
40C4⋊Dic157—42no—
41#14578933—42no—
42#14578933—42no—
43#14578933—42no—
44#14578933—42no—
45C22⋊Dic157—41no—
46C22⋊Dic157—42no—
47C24×C157C2311yescharacteristic
48C22×C4×C157C2311yescharacteristic
49C22×C4×C157C2312yes—
50C22×C4×C157C2311yescharacteristic
51C22×C4×C157C2312yes—
52C4⋊C4×C157—42no—
53C4⋊C4×C157—41no—
54C22⋊C4×C157—41no—
55C22⋊C4×C157—42no—
56Dic314—44no—
57C2×Dic157—22no—
58C2×Dic157—22no—
59C2×Dic157—22no—
60C2×Dic157—22no—
61C2×Dic157—22no—
62C2×Dic157—22no—
63C2×Dic157—22no—
64C2×Dic157—22no—
65C2×Dic157—22no—
66C2×Dic157—22no—
67C2×Dic157—22no—
68C2×Dic157—44no—
69C2×Dic157C2×Q814yes—
70C23×C157—21no—
71C23×C157—21no—
72C23×C157—22no—
73C23×C157—22no—
74C23×C157C2×Q812yes—
75C23×C157C2411yescharacteristic
76C2×C4×C157—21no—
77C2×C4×C157—22no—
78C2×C4×C157—22no—
79C2×C4×C157—21no—
80C2×C4×C157—22no—
81C2×C4×C157—22no—
82C2×C4×C157—22no—
83C2×C4×C157—22no—
84C2×C4×C157—22no—
85C2×C4×C157—22no—
86C22×C157—22no—
87C22×C157—24no—
88C22×C157—41no—
89C22×C157—42no—
90C22×C157—41no—
91C22×C157C22×Q811yescharacteristic
92C22×C157H(5,𝔽2)11yescharacteristic
93C22×C157H(5,𝔽2)12yes—
94C22×C157H(5,𝔽2)11yescharacteristic
95C22×C1572−1+412yes—
96C4×C157—42no—
97C4×C157—41no—
98C4×C157—42no—
99C4×C157—41no—
100Dic157—24no—
101Dic157—42no—
102Dic157—42no—
103Dic157—42no—
104C2×C157—42no—
105C2×C157#54311yescharacteristic
106C2×C157#54412yes—
107C2×C157#56011yescharacteristic
108C2×C157#56212yes—
109C2×C157#56311yescharacteristic
110C157#284111yescharacteristic, complemented, Sylow
111#2841—1571nomaximal, Sylow
112#400—1572no—
113#399D15711yescharacteristic
114#398—1572no—
115#398—1572no—
116#396—1572no—
117#395—1572no—
118#393—1571no—
119#381—1571no—
120#380—1572no—
121C22×Q8—1571no—
122C23×C4—1571no—
123C2×C4⋊C4—1572no—
124C2×C4⋊C4—1572no—
125C2×C4⋊C4—1572no—
126C2×C4⋊C4—1572no—
127C2×C4⋊C4—1572no—
128C2×C4⋊C4C2×D15712yes—
129C2×C4⋊C4C2×D15711yescharacteristic
130C2×C22⋊C4—1572no—
131C2×C22⋊C4—1571no—
132C2×C22⋊C4C2×D15712yes—
133C2×C22⋊C4C2×D15711yescharacteristic
134#95—1572no—
135#95—1572no—
136#95—1572no—
137#95—1572no—
138#95—1571no—
139#95—1572no—
140#95—1572no—
141#95C2×D15711yescharacteristic
142C24C22×D15711yescharacteristic
143C2×Q8—6281no—
144C2×Q8—6282no—
145C22×C4—6281no—
146C22×C4—6281no—
147C22×C4—1572no—
148C22×C4—1572no—
149C22×C4—1572no—
150C22×C4—1572no—
151C22×C4—1574no—
152C22×C4C22×D15711yescharacteristic
153C22×C4C22×D15711yescharacteristic
154C22×C4C22×D15712yes—
155C22×C4C22×D15712yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—6282no—
159C4⋊C4—6282no—
160C4⋊C4—6282no—
161C4⋊C4—6282no—
162C4⋊C4—6282no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—6282no—
166C22⋊C4—6281no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D15712yes—
172C23C23×D15711yescharacteristic
173Q8—6284no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—3142no—
185C2×C4—3142no—
186C2×C4—3142no—
187C2×C4—3142no—
188C2×C4—3142no—
189C2×C4—3142no—
190C2×C4—3142no—
191C2×C4—3142no—
192C2×C4—3142no—
193C2×C4—3142no—
194C2×C4—3142no—
195C2×C4—6284no—
196C2×C4—1574no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D15711yescharacteristic
203C22#1909781611yescharacteristic
204C22#1909781612yes—
205C22#1909781711yescharacteristic
206C22#1909781812yes—
207C4—41no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—3144no—
212C4—6282no—
213C4—6282no—
214C4—6282no—
215C2—42no—
216C2#2375789811yescharacteristic
217C2#2375790212yes—
218C2#2375801011yescharacteristic
219C2#2375802012yes—
220C2#2375802611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#399×C157C2
prime radicalC24×C157C23
hypercenter#399D157
hypocenterC157#2841
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #23757214C2semidirect
2. #23757206C2semidirect
3. #23757203C2semidirect
5. #23757193C2semidirect
35. C22×Dic157Q8semidirect
110. C157#2841semidirect

Maximal subgroups

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

subgroupindex
#237572142
#237572062
#237572032
#237571962
#237571932
#237571852
#237571342
#237571312
#399×C1572
#2841157