Finite Group Databasebeta

← #34099116

Subgroups — #34099116

Order 20096.

260 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757428C211yesmaximal, characteristic, complemented
2#23757411C211yesmaximal, characteristic, complemented
3#23757185C211yesmaximal, characteristic, complemented
4#23757010C211yesmaximal, characteristic, complemented
5#23757007C211yesmaximal, characteristic, complemented
6#23756997C211yesmaximal, characteristic, complemented
7#538×C157C211yesmaximal, characteristic
8C22×Dic314C2211yescharacteristic
9C2×#14578944—21no—
10C2×#14578942—21no—
11C2×C22⋊Dic157—21no—
12#19097694—21no—
13#19097694—21no—
14C22⋊#11929434C2211yescharacteristic
15C22⋊#11929434C2211yescharacteristic
16#19097678—21no—
17#19097678C2211yescharacteristic
18#126×C157—21no—
19#124×C157—21no—
20#123×C157—21no—
21C22⋊Q8×C157C2211yescharacteristic
22#121×C157—21no—
23#121×C157C2211yescharacteristic
24#120×C157—21no—
25C4×D4×C157—21no—
26C2×C22⋊C4×C157C2211yescharacteristic
27C2×Dic314—21no—
28C2×Dic314—21no—
29C2×Dic314—42no—
30C22×Dic157—21no—
31C22×Dic157—21no—
32#14578944—42no—
33#14578942—42no—
34C2×#11929434—21no—
35C2×#11929434—21no—
36C22⋊Dic157—42no—
37C24×C157D411yescharacteristic
38C2×Q8×C157D411yescharacteristic
39C2×D4×C157—21no—
40C2×D4×C157—21no—
41C2×D4×C157—21no—
42C2×D4×C157D411yescharacteristic
43C22×C4×C157—21no—
44C22×C4×C157D411yescharacteristic
45C22×C4×C157C2311yescharacteristic
46C4⋊C4×C157—21no—
47C4⋊C4×C157—21no—
48C4⋊C4×C157D411yescharacteristic
49C4⋊C4×C157D411yescharacteristic
50C22⋊C4×C157—21no—
51C22⋊C4×C157—21no—
52C22⋊C4×C157—21no—
53C22⋊C4×C157—21no—
54C22⋊C4×C157—21no—
55C22⋊C4×C157—21no—
56C22⋊C4×C157—21no—
57C42×C157—21no—
58Dic314—22no—
59Dic314—22no—
60Dic314—41no—
61Dic314—41no—
62C2×Dic157—22no—
63C2×Dic157—41no—
64C2×Dic157—41no—
65C2×Dic157—41no—
66C2×Dic157—41no—
67C2×Dic157—41no—
68#11929434—41no—
69#11929434—41no—
70C23×C157—21no—
71C23×C157—21no—
72C23×C157—21no—
73C23×C157—21no—
74C23×C157—41no—
75C23×C157—41no—
76C23×C157C2×D411yescharacteristic
77Q8×C157—41no—
78D4×C157—41no—
79D4×C157—41no—
80D4×C157—41no—
81D4×C157—81no—
82D4×C157—81no—
83C2×C4×C157—21no—
84C2×C4×C157—21no—
85C2×C4×C157—21no—
86C2×C4×C157—21no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157—81no—
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D411yescharacteristic
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—42no—
95C22×C157—41no—
96C22×C157—41no—
97C22×C157—41no—
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157—81no—
102C22×C157#12011yescharacteristic
103C22×C157#13511yescharacteristic
104C22×C157#13711yescharacteristic
105C4×C157—21no—
106C4×C157—21no—
107C4×C157—41no—
108C4×C157—41no—
109C4×C157—41no—
110C4×C157—41no—
111Dic157—42no—
112Dic157—81no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157—81no—
116C2×C157—81no—
117C2×C157#45211yescharacteristic
118C2×C157#45511yescharacteristic
119C2×C157B(4,𝔽2)11yescharacteristic
120C157#160611yescharacteristic, complemented, Sylow
121#1606—1571nomaximal, Sylow
122#538D15711yescharacteristic
123#451—1571no—
124#448—1571no—
125#393—1571no—
126#328—1571no—
127#327—1571no—
128#323—1571no—
129C22×Q8—1571no—
130C2×Q16—3141no—
131C2×SD16—3141no—
132#126—21no—
133#124—21no—
134#123—21no—
135C22⋊Q8C2×D15711yescharacteristic
136#121—21no—
137#121C2×D15711yescharacteristic
138#120—21no—
139C4×D4—21no—
140C2×C22⋊C4—3141no—
141C2×C22⋊C4C2×D15711yescharacteristic
142#103—3141no—
143#103—3141no—
144C22⋊C8—1571no—
145C22⋊C8—1571no—
146#95—3141no—
147#95—1571no—
148C24C22⋊D15711yescharacteristic
149C2×Q8—3141no—
150C2×Q8—3141no—
151C2×Q8—6282no—
152C2×Q8C22⋊D15711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C22⋊D15711yescharacteristic
157C22×C4—21no—
158C22×C4—3141no—
159C22×C4—3141no—
160C22×C4C22⋊D15711yescharacteristic
161C22×C4C22×D15711yescharacteristic
162Q16—6282no—
163SD16—6282no—
164C2×C8—3141no—
165C2×C8—3141no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4C22⋊D15711yescharacteristic
169C4⋊C4C22⋊D15711yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—6282no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23C2×C22⋊D15711yescharacteristic
186Q8—41no—
187Q8—3142no—
188Q8—3142no—
189Q8—6281no—
190Q8—6281no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—3142no—
204C2×C4—6281no—
205C2×C4—6281no—
206C2×C4—6281no—
207C2×C4—6281no—
208C2×C4—6281no—
209C2×C4C2×C22⋊D15711yescharacteristic
210C2×C4C2×C22⋊D15711yescharacteristic
211C8—6281no—
212C8—6281no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22#1909774011yescharacteristic
224C22#1909779111yescharacteristic
225C22#1909780111yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—6282no—
233C4—12561no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2#2375743511yescharacteristic
239C2#2375745111yescharacteristic
240C2#2375746011yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting, prime radical#538×C157C2
socleC22×C157#120
hypercenter#538D157
hypocenterC157#1606
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #23757428C2semidirect
2. #23757411C2semidirect
3. #23757185C2semidirect
4. #23757010C2semidirect
5. #23757007C2semidirect
6. #23756997C2semidirect
120. C157#1606semidirect

Maximal subgroups

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

subgroupindex
#237574282
#237574112
#237571852
#237570102
#237570072
#237569972
#538×C1572
#1606157