Finite Group Databasebeta

← #34102127

Subgroups — #34102127

Order 20096.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#23757951C211yesmaximal, characteristic, complemented
2#23757462C211yesmaximal, characteristic, complemented
3#23757461C211yesmaximal, characteristic
4#23757453C211yesmaximal, characteristic, complemented
5#23757067C211yesmaximal, characteristic, complemented
6#23757061C211yesmaximal, characteristic, complemented
7#352×C157C211yesmaximal, characteristic, complemented
8#19097816C2211yescharacteristic
9#19097804—21no—
10#19097764—21no—
11#19097758C2211yescharacteristic
12#19097754—21no—
13#19097753—21no—
14#19097749—21no—
15#19097741—21no—
16#19097739C2211yescharacteristic
17C4×C22⋊D157—21no—
18#19097729—21no—
19C22⋊C4×D157—21no—
20#19097697—21no—
21#19097686C2211yescharacteristic
22#19097685—21no—
23#19097685—21no—
24#123×C157C2211yescharacteristic
25#100×C157C2211yescharacteristic
26#99×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28#14578954—21no—
29#14578953—41no—
30C2×Dic314D411yescharacteristic, complemented
31C2×D628—41no—
32C2×C22⋊D157—21no—
33C2×C22⋊D157—21no—
34C2×C22⋊D157—21no—
35D4×D157—41no—
36C2×C4×D157—21no—
37C2×C4×D157—21no—
38Dic628—41no—
39#14578943—41no—
40C4⋊Dic157—21no—
41#14578933—21no—
42#14578933—21no—
43#14578933—41no—
44C22⋊Dic157—21no—
45C22⋊Dic157—21no—
46C22⋊Dic157—21no—
47C22⋊Dic157D411yescharacteristic
48#14578931—21no—
49#14578931—21no—
50#14578931—21no—
51#14578931—41no—
52#14578931—41no—
53C4×Dic157—21no—
54C4×Dic157D411yescharacteristic, complemented
55C2×D4×C157C2311yescharacteristic
56C22×C4×C157D411yescharacteristic, complemented
57M16×C157—21no—
58C4⋊C4×C157—21no—
59C22⋊C4×C157—21no—
60C22⋊C4×C157—21no—
61C22⋊C4×C157D411yescharacteristic, complemented
62C22×D157—21no—
63C22×D157—21no—
64C22×D157—21no—
65C22×D157—42no—
66C22×D157—41no—
67Dic314—21no—
68Dic314—41no—
69D628—21no—
70D628—82no—
71C22⋊D157—42no—
72C22⋊D157—41no—
73C22⋊D157—41no—
74C22⋊D157—41no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—21no—
78C2×Dic157—21no—
79C2×Dic157—21no—
80C2×Dic157—41no—
81C4×D157—41no—
82C4×D157—42no—
83C4×D157—81no—
84C23×C157C2×D411yescharacteristic
85C23×C157C2×D411yescharacteristic
86D4×C157—41no—
87C2×C4×C157—21no—
88C2×C4×C157—21no—
89C2×C4×C157—41no—
90C2×C4×C157—41no—
91C2×C4×C157C2×D411yescharacteristic
92C8×C157—41no—
93C22×C157—21no—
94C22×C157—21no—
95C22×C157—41no—
96C22×C157—41no—
97C22×C157#12011yescharacteristic
98C2×D157—22no—
99C2×D157—41no—
100C2×D157—41no—
101C2×D157—42no—
102C2×D157—41no—
103C2×D157—81no—
104C2×D157—81no—
105C2×D157—81no—
106C4×C157—21no—
107C4×C157—41no—
108C4×C157—41no—
109C4×C157—81no—
110Dic157—41no—
111Dic157—41no—
112Dic157—41no—
113Dic157—41no—
114Dic157—81no—
115C2×C157—21no—
116C2×C157—41no—
117C2×C157—41no—
118C2×C157B(4,𝔽2)11yescharacteristic
119D157—42no—
120D157—81no—
121C157#219711yescharacteristic, complemented, Sylow
122#2197—1571nomaximal, Sylow
123#551—1571no—
124#458—1571no—
125B(4,𝔽2)—1571no—
126#455—1571no—
127#354—1571no—
128#352D15711yescharacteristic, complemented
129AΣL(1,𝔽16)—1571no—
130H(5,𝔽2)—1571no—
131#137—3141no—
132#126—3141no—
133#124—1571no—
134#123—3141no—
135#123—3141no—
136#123C2×D15711yescharacteristic
137C22⋊Q8—3141no—
138#120—3141no—
139#120—1571no—
140C4×D4—3141no—
141#117—3141no—
142C2×C22⋊C4—3141no—
143#104—3141no—
144#100C2×D15711yescharacteristic
145#99—3141no—
146#99—3141no—
147#99—1571no—
148#99C2×D15711yescharacteristic
149C24—1571no—
150#41—3141no—
151#41—6281no—
152C2×Q8—1571no—
153C2×D4—3141no—
154C2×D4—3141no—
155C2×D4—3141no—
156C2×D4—6281no—
157C2×D4—6281no—
158C2×D4C22×D15711yescharacteristic
159C22×C4—3141no—
160C22×C4—3141no—
161C22×C4D62811yescharacteristic, complemented
162Q16—6281no—
163SD16—6281no—
164M16—21no—
165C4⋊C4—21no—
166C4⋊C4—3141no—
167C4⋊C4—3141no—
168C4⋊C4—3141no—
169C4⋊C4—6281no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3141no—
173C22⋊C4—3141no—
174C22⋊C4—3141no—
175C22⋊C4—3141no—
176C22⋊C4—3141no—
177C22⋊C4—3141no—
178C22⋊C4—6281no—
179C22⋊C4—6281no—
180C22⋊C4—1571no—
181C22⋊C4D62811yescharacteristic, complemented
182C42—3141no—
183C42—1571no—
184C23—3141no—
185C23—3141no—
186C23—3141no—
187C23—6281no—
188C23—6282no—
189C23D4×D15711yescharacteristic
190C23C2×D62811yescharacteristic
191Q8—3141no—
192Q8—6281no—
193D4—41no—
194D4—3141no—
195D4—6281no—
196D4—6281no—
197D4—6282no—
198D4—6281no—
199D4—12562no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—3141no—
205C2×C4—3141no—
206C2×C4—3141no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—6281no—
210C2×C4—6282no—
211C2×C4—6281no—
212C2×C4—12561no—
213C2×C4D4×D15711yescharacteristic
214C8—41no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—3142no—
220C22—6281no—
221C22—6282no—
222C22—6281no—
223C22—6281no—
224C22—12561no—
225C22—12561no—
226C22—12561no—
227C22#1909774111yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—6281no—
233C4—6281no—
234C4—6281no—
235C4—6281no—
236C4—12561no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—6282no—
241C2—12561no—
242C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×D4×C157C23
FrattiniC2×D4C22×D157
Fitting#352×C157C2
socleC2×C157B(4,𝔽2)
prime radical#23757461C2
hypercenter#352D157
hypocenterC157#2197
LCSG ▷ C2×D4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #23757951C2semidirect
2. #23757462C2semidirect
4. #23757453C2semidirect
5. #23757067C2semidirect
6. #23757061C2semidirect
7. #352×C157C2semidirect
30. C2×Dic314D4semidirect
54. C4×Dic157D4semidirect
56. C22×C4×C157D4semidirect
61. C22⋊C4×C157D4semidirect
121. C157#2197semidirect
128. #352D157semidirect
161. C22×C4D628semidirect
181. C22⋊C4D628semidirect

Maximal subgroups

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

subgroupindex
#237579512
#237574622
#237574612
#237574532
#237570672
#237570612
#352×C1572
#2197157