Finite Group Databasebeta

← #34099098

Subgroups — #34099098

Order 20096.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#23757872C211yesmaximal, characteristic
2#23757412C211yesmaximal, characteristic
3#23757192C211yesmaximal, characteristic, complemented
4#23757010C211yesmaximal, characteristic, complemented
5#23757008C211yesmaximal, characteristic
6#23756998C211yesmaximal, characteristic, complemented
7#451×C157C211yesmaximal, characteristic, complemented
8C2×#14578941—21no—
9#19097766—21no—
10#19097762—21no—
11#19097756—21no—
12#19097755—21no—
13#19097747—21no—
14#19097745C2211yescharacteristic
15#19097731—21no—
16C2×#14578933—21no—
17C2×#14578933C2211yescharacteristic
18#19097695—21no—
19C22⋊#11929434C2211yescharacteristic
20#19097678—21no—
21#19097678C2211yescharacteristic
22C22×Q8×C157C2211yescharacteristic
23C2×Q16×C157—21no—
24C22⋊Q8×C157C2211yescharacteristic
25#103×C157—21no—
26C22⋊C8×C157C2211yescharacteristic
27C2×D628D411yescharacteristic
28C2×C22⋊D157—21no—
29C22×Dic157—21no—
30C22×Dic157D411yescharacteristic
31C22×Dic157D411yescharacteristic
32C2×C4×D157—21no—
33#14578941—42no—
34C2×#11929434—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38#14578933—21no—
39#14578933—21no—
40#14578933—21no—
41#14578933—42no—
42#14578931—21no—
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46C4×Dic157—21no—
47C2×Q8×C157—21no—
48C2×Q8×C157—21no—
49C2×Q8×C157—42no—
50C2×Q8×C157D411yescharacteristic
51C22×C4×C157—21no—
52C22×C4×C157C2311yescharacteristic
53Q16×C157—42no—
54C2×C8×C157—21no—
55C4⋊C4×C157—21no—
56C4⋊C4×C157D411yescharacteristic
57C22⋊C4×C157—21no—
58C22×D157—21no—
59D628—41no—
60C22⋊D157—41no—
61C22⋊D157—41no—
62C2×Dic157—22no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C4×D157—81no—
72#11929434—41no—
73C23×C157C2×D411yescharacteristic
74Q8×C157—22no—
75Q8×C157—22no—
76Q8×C157—41no—
77Q8×C157—41no—
78Q8×C157—41no—
79C2×C4×C157—22no—
80C2×C4×C157—21no—
81C2×C4×C157—21no—
82C2×C4×C157—41no—
83C2×C4×C157—41no—
84C2×C4×C157—41no—
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C8×C157—41no—
88C22×C157—21no—
89C22×C157—21no—
90C22×C157#12011yescharacteristic
91C22×C157#13511yescharacteristic
92C22×C157#13711yescharacteristic
93C2×D157—41no—
94C2×D157—41no—
95C2×D157—41no—
96C4×C157—21no—
97C4×C157—21no—
98C4×C157—41no—
99C4×C157—42no—
100C4×C157—41no—
101Dic157—42no—
102Dic157—41no—
103Dic157—41no—
104Dic157—41no—
105C2×C157—21no—
106C2×C157—21no—
107C2×C157#45211yescharacteristic
108C2×C157#45611yescharacteristic
109C2×C157B(4,𝔽2)11yescharacteristic
110D157—81no—
111C157#160311yescharacteristic, complemented, Sylow
112#1603—1571nomaximal, Sylow
113#539—1571no—
114#451D15711yescharacteristic, complemented
115#448—1571no—
116#395—1571no—
117#328—1571no—
118#327—1571no—
119#324—1571no—
120C22×Q8C2×D15711yescharacteristic
121C2×Q16—21no—
122C2×SD16—3141no—
123#126—3141no—
124#125—3141no—
125#123—3141no—
126#123—3141no—
127C22⋊Q8—3141no—
128C22⋊Q8C2×D15711yescharacteristic
129#121—1571no—
130C4×D4—3141no—
131C2×C4⋊C4—3141no—
132C2×C4⋊C4—1571no—
133#103—21no—
134#103—3141no—
135C22⋊C8—1571no—
136C22⋊C8C2×D15711yescharacteristic
137#95—3141no—
138#95—1571no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D15711yescharacteristic
143C2×D4—3141no—
144C2×D4—1571no—
145C22×C4—21no—
146C22×C4—3141no—
147C22×C4—3141no—
148C22×C4—1571no—
149C22×C4—1571no—
150C22×C4C22×D15711yescharacteristic
151Q16—42no—
152SD16—6282no—
153C2×C8—21no—
154C2×C8—3141no—
155C4⋊C4—21no—
156C4⋊C4—3141no—
157C4⋊C4—3141no—
158C4⋊C4—3141no—
159C4⋊C4—3141no—
160C4⋊C4—3141no—
161C4⋊C4—6282no—
162C4⋊C4—1571no—
163C4⋊C4C22⋊D15711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3141no—
166C22⋊C4—3141no—
167C22⋊C4—3141no—
168C22⋊C4—3141no—
169C42—3141no—
170C23—3141no—
171C23D4×D15711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—6281no—
178D4—6281no—
179D4—6281no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3142no—
187C2×C4—3141no—
188C2×C4—3141no—
189C2×C4—3141no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—6281no—
193C2×C4—6281no—
194C2×C4—6281no—
195C2×C4—12561no—
196C2×C4D4×D15711yescharacteristic
197C2×C4C2×C22⋊D15711yescharacteristic
198C8—41no—
199C8—6281no—
200C22—21no—
201C22—21no—
202C22—6281no—
203C22—6281no—
204C22—6281no—
205C22#1909773911yescharacteristic
206C22#1909779411yescharacteristic
207C22#1909780311yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—6282no—
214C4—6281no—
215C4—6281no—
216C4—6281no—
217C2—21no—
218C2—21no—
219C2—12561no—
220C2#2375743411yescharacteristic
221C2#2375745411yescharacteristic
222C2#2375745911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#451×C157C2
socleC22×C157#120
prime radical#19097745C22
hypercenter#451D157
hypocenterC157#1603
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #23757192C2semidirect
4. #23757010C2semidirect
6. #23756998C2semidirect
7. #451×C157C2semidirect
111. C157#1603semidirect
114. #451D157semidirect

Maximal subgroups

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

subgroupindex
#237578722
#237574122
#237571922
#237570102
#237570082
#237569982
#451×C1572
#1603157