Finite Group Databasebeta

← #34101774

Subgroups — #34101774

Order 20096.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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#23757779C211yesmaximal, characteristic
2#23757254C211yesmaximal, characteristic, complemented
3#23757245C211yesmaximal, characteristic, complemented
4#23757064C211yesmaximal, characteristic, complemented
5#23757062C211yesmaximal, characteristic, complemented
6#23757060C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C157C211yesmaximal, characteristic, complemented
8C2×#14578953C2211yescharacteristic
9C23×Dic157C411yescharacteristic, complemented
10M16×D157—21no—
11#19097755C411yescharacteristic, complemented
12#19097752C2211yescharacteristic
13#19097748—21no—
14#19097743—21no—
15#19097740C411yescharacteristic, complemented
16D4×Dic157—21no—
17#19097731—21no—
18#19097727—21no—
19#19097727C411yescharacteristic, complemented
20#19097689—21no—
21#19097688C2211yescharacteristic
22#19097686C2211yescharacteristic
23#19097685—21no—
24#120×C157C2211yescharacteristic
25#100×C157C2211yescharacteristic
26#99×C157C2211yescharacteristic
27#14578953—41no—
28#14578953—41no—
29C2×Dic314D411yescharacteristic
30C2×C22⋊D157—21no—
31C2×C22⋊D157C2×C411yescharacteristic
32C2×C22⋊D157D411yescharacteristic
33C22×Dic157—21no—
34C22×Dic157—21no—
35C22×Dic157—41no—
36C22×Dic157—41no—
37C22×Dic157C2×C411yescharacteristic
38C22×Dic157C2×C411yescharacteristic
39C22×Dic157D411yescharacteristic
40C2×C4×D157D411yescharacteristic
41#14578938—21no—
42#14578937—41no—
43C8×D157—41no—
44C4⋊Dic157—21no—
45#14578933—21no—
46#14578933—21no—
47#14578933—41no—
48C22⋊Dic157—21no—
49C22⋊Dic157—21no—
50C22⋊Dic157—21no—
51C22⋊Dic157C2×C411yescharacteristic
52#14578931—21no—
53C4×Dic157—21no—
54C4×Dic157—41no—
55C24×C157C2×C411yescharacteristic
56C2×D4×C157—21no—
57C2×D4×C157C2311yescharacteristic
58M16×C157—21no—
59C22⋊C4×C157—21no—
60C22⋊C4×C157—21no—
61C22⋊C4×C157C2×C411yescharacteristic
62C22×D157C22⋊C411yescharacteristic
63Dic314—41no—
64C22⋊D157—41no—
65C22⋊D157—41no—
66C22⋊D157—42no—
67C2×Dic157—21no—
68C2×Dic157—21no—
69C2×Dic157—21no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—42no—
74C2×Dic157—41no—
75C2×Dic157—41no—
76C2×Dic157—41no—
77C2×Dic157—41no—
78C2×Dic157C22⋊C411yescharacteristic
79C2×Dic157C22⋊C411yescharacteristic
80C2×Dic157C22⋊C411yescharacteristic
81C4×D157—21no—
82C4×D157—21no—
83#11929434—41no—
84C23×C157—21no—
85C23×C157—41no—
86C23×C157—42no—
87C23×C157C22×C411yescharacteristic
88C23×C157C2×D411yescharacteristic
89D4×C157—41no—
90D4×C157—42no—
91C2×C4×C157—21no—
92C2×C4×C157—41no—
93C2×C4×C157C2×D411yescharacteristic
94C8×C157—41no—
95C22×C157—22no—
96C22×C157—21no—
97C22×C157—21no—
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157—42no—
102C22×C157—42no—
103C22×C157—41no—
104C22×C157C2×C22⋊C411yescharacteristic
105C2×D157—21no—
106C2×D157—41no—
107C4×C157—21no—
108C4×C157—41no—
109C4×C157—81no—
110Dic157—21no—
111Dic157—21no—
112Dic157—41no—
113Dic157—41no—
114Dic157—81no—
115Dic157#9911yescharacteristic
116Dic157#9911yescharacteristic
117C2×C157—21no—
118C2×C157—41no—
119C2×C157—42no—
120C2×C157—41no—
121C2×C157C2×#9911yescharacteristic
122D157—41no—
123C157#212011yescharacteristic, complemented, Sylow
124#2120—1571nomaximal, Sylow
125#525—1571no—
126#413—1571no—
127#410—1571no—
128#352—1571no—
129#352—1571no—
130AΣL(1,𝔽16)—1571no—
131AΣL(1,𝔽16)D15711yescharacteristic, complemented
132C2×#41—1571no—
133C23×C4—1571no—
134C2×M16—3141no—
135#123—1571no—
136#123—1571no—
137C22⋊Q8—3141no—
138#121—3141no—
139#120—1571no—
140#120C2×D15711yescharacteristic
141C4×D4—3141no—
142C4×D4—3141no—
143#117—3141no—
144#117—1571no—
145#101—3141no—
146#100—1571no—
147#100C2×D15711yescharacteristic
148#99—3141no—
149#99—1571no—
150#99C2×D15711yescharacteristic
151C24C4×D15711yescharacteristic
152#41—6281no—
153#41—6281no—
154C2×Q8—1571no—
155C2×D4—21no—
156C2×D4—3141no—
157C2×D4—1571no—
158C2×D4—1571no—
159C2×D4C22×D15711yescharacteristic
160C22×C4—3141no—
161C22×C4—3141no—
162C22×C4—6281no—
163C22×C4—6281no—
164C22×C4—1571no—
165C22×C4—1571no—
166C22×C4—1571no—
167C22×C4—1571no—
168M16—21no—
169M16—3141no—
170M16—6281no—
171C2×C8—6281no—
172C4⋊C4—3141no—
173C4⋊C4—3141no—
174C4⋊C4—3141no—
175C4⋊C4—6281no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—3141no—
179C22⋊C4—3141no—
180C22⋊C4—3141no—
181C22⋊C4—3141no—
182C22⋊C4—1571no—
183C22⋊C4C4×D15711yescharacteristic
184C42—3141no—
185C42—6281no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—1571no—
190C23C2×C4×D15711yescharacteristic
191C23D4×D15711yescharacteristic
192Q8—6281no—
193D4—41no—
194D4—42no—
195D4—6281no—
196D4—6281no—
197D4—6282no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—3141no—
201C2×C4—3141no—
202C2×C4—3141no—
203C2×C4—3141no—
204C2×C4—3141no—
205C2×C4—3141no—
206C2×C4—3141no—
207C2×C4—3141no—
208C2×C4—6282no—
209C2×C4—6281no—
210C2×C4—6281no—
211C2×C4—6281no—
212C2×C4—6281no—
213C2×C4—1571no—
214C2×C4—1571no—
215C2×C4—1571no—
216C2×C4D4×D15711yescharacteristic
217C8—41no—
218C8—6281no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—3141no—
229C22—6281no—
230C22C22⋊C4×D15711yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—3141no—
235C4—3141no—
236C4—6281no—
237C4—6281no—
238C4—1571no—
239C4—1571no—
240C4—12561no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—6281no—
246C2#99×D15711yescharacteristic
subgroupG/H
centerC2#99×D157
commutatorC23×C157C22×C4
FrattiniC2×D4C22×D157
FittingAΣL(1,𝔽16)×C157C2
socleC2×C157C2×#99
prime radical#23757779C2
cosocleC2×D4×C157C23
hypercenterAΣL(1,𝔽16)D157
hypocenterC157#2120
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #23757254C2semidirect
3. #23757245C2semidirect
4. #23757064C2semidirect
5. #23757062C2semidirect
6. #23757060C2semidirect
7. AΣL(1,𝔽16)×C157C2semidirect
9. C23×Dic157C4semidirect
11. #19097755C4semidirect
15. #19097740C4semidirect
19. #19097727C4semidirect
123. C157#2120semidirect
131. AΣL(1,𝔽16)D157semidirect

Maximal subgroups

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

subgroupindex
#237577792
#237572542
#237572452
#237570642
#237570622
#237570602
AΣL(1,𝔽16)×C1572
#2120157