Finite Group Databasebeta

← #34099106

Subgroups — #34099106

Order 20096.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#23757850C211yesmaximal, characteristic
2#23757414C211yesmaximal, characteristic, complemented
3#23757411C211yesmaximal, characteristic, complemented
4#23757194C211yesmaximal, characteristic, complemented
5#23757006C211yesmaximal, characteristic, complemented
6#23756998C211yesmaximal, characteristic, complemented
7#327×C157C211yesmaximal, characteristic, complemented
8C2×#14578953—21no—
9C22×Dic314C2211yescharacteristic
10C2×#14578943—21no—
11C2×#14578942—21no—
12#19097749—21no—
13#19097748—21no—
14#19097745C2211yescharacteristic
15#19097744—21no—
16#19097743—21no—
17#19097743—21no—
18C2×C4⋊Dic157—21no—
19C2×#14578933C2211yescharacteristic
20#19097696—21no—
21#19097694—21no—
22C22⋊#11929434C2211yescharacteristic
23#19097678—21no—
24#121×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27#14578953—42no—
28#14578953—42no—
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314—42no—
33C2×D628D411yescharacteristic
34C2×C22⋊D157—21no—
35C2×C22⋊D157—21no—
36C2×C22⋊D157—21no—
37C22×Dic157—21no—
38C22×Dic157—21no—
39C22×Dic157D411yescharacteristic
40C22×Dic157D411yescharacteristic
41C2×C4×D157—21no—
42#14578943—42no—
43#14578942—42no—
44C2×#11929434—21no—
45C4⋊Dic157—42no—
46C4⋊Dic157D411yescharacteristic
47#14578933—21no—
48#14578933—21no—
49#14578933—21no—
50C22⋊Dic157—21no—
51#14578931—21no—
52#14578931—21no—
53C2×D4×C157—21no—
54C2×D4×C157D411yescharacteristic
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57C2×C8×C157—21no—
58C4⋊C4×C157D411yescharacteristic
59C22⋊C4×C157—21no—
60C22×D157—21no—
61Dic314—22no—
62Dic314—22no—
63Dic314—42no—
64Dic314—41no—
65Dic314—41no—
66D628—41no—
67C22⋊D157—42no—
68C22⋊D157—41no—
69C22⋊D157—41no—
70C22⋊D157—81no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—22no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—42no—
77C2×Dic157—41no—
78C2×Dic157—41no—
79C2×Dic157—41no—
80C2×Dic157SD1612yescomplemented
81C4×D157—42no—
82#11929434—41no—
83C23×C157—21no—
84C23×C157C2×D411yescharacteristic
85D4×C157—41no—
86D4×C157—41no—
87D4×C157—41no—
88C2×C4×C157—22no—
89C2×C4×C157—21no—
90C2×C4×C157—41no—
91C2×C4×C157—41no—
92C2×C4×C157—41no—
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157C2×D411yescharacteristic
95C8×C157—41no—
96C22×C157—21no—
97C22×C157—21no—
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157#12011yescharacteristic
102C22×C157C2×SD1611yescharacteristic
103C22×C157AGL(1,ℤ/8)11yescharacteristic
104C2×D157—41no—
105C2×D157—41no—
106C2×D157—41no—
107C4×C157—21no—
108C4×C157—21no—
109C4×C157—42no—
110C4×C157—41no—
111Dic157—22no—
112Dic157—41no—
113Dic157—42no—
114Dic157—41no—
115C2×C157—21no—
116C2×C157—21no—
117C2×C157—81no—
118C2×C157#45011yescharacteristic
119C2×C157#45611yescharacteristic
120C2×C157B(4,𝔽2)11yescharacteristic
121D157—81no—
122C157#160411yescharacteristic, complemented, Sylow
123#1604—1571nomaximal, Sylow
124#537—1571no—
125#448—1571no—
126#448—1571no—
127#395—1571no—
128#327—1571no—
129#327D15711yescharacteristic, complemented
130#324—1571no—
131C2×#41—3141no—
132C22×Q8—1571no—
133C2×SD16—3141no—
134C2×SD16—3141no—
135C22⋊Q8—3141no—
136C22⋊Q8—3141no—
137#121—3141no—
138#121—3141no—
139#121—3141no—
140#121—1571no—
141#121C2×D15711yescharacteristic
142C2×C4⋊C4—3141no—
143C2×C4⋊C4—1571no—
144#103—3141no—
145#103—3141no—
146C22⋊C8—1571no—
147C22⋊C8C2×D15711yescharacteristic
148#95—3141no—
149#95C2×D15711yescharacteristic
150#41—6282no—
151#41—6282no—
152C2×Q8—3141no—
153C2×Q8—3141no—
154C2×Q8—3141no—
155C2×Q8—6282no—
156C2×D4—21no—
157C2×D4—3141no—
158C2×D4—3141no—
159C2×D4—3141no—
160C2×D4—1571no—
161C2×D4D62811yescharacteristic
162C22×C4—21no—
163C22×C4—3141no—
164C22×C4—3141no—
165C22×C4—3141no—
166C22×C4—1571no—
167C22×C4—1571no—
168C22×C4C22×D15711yescharacteristic
169SD16—6282no—
170SD16—6282no—
171C2×C8—21no—
172C2×C8—3141no—
173C4⋊C4—3141no—
174C4⋊C4—3141no—
175C4⋊C4—3141no—
176C4⋊C4—6282no—
177C4⋊C4—1571no—
178C4⋊C4D62811yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—3141no—
181C22⋊C4—3141no—
182C22⋊C4—3141no—
183C23—21no—
184C23—3141no—
185C23D4×D15711yescharacteristic
186Q8—3142no—
187Q8—3142no—
188Q8—6281no—
189Q8—6281no—
190Q8—6282no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—6281no—
195D4—6282no—
196D4—6281no—
197D4—6281no—
198D4—12561no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—3141no—
205C2×C4—3141no—
206C2×C4—3142no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—6282no—
210C2×C4—6282no—
211C2×C4—6281no—
212C2×C4—6281no—
213C2×C4—6281no—
214C2×C4—1572no—
215C2×C4D4×D15711yescharacteristic
216C2×C4C2×D62811yescharacteristic
217C8—41no—
218C8—6281no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—6281no—
225C22—6281no—
226C22—6281no—
227C22#1909774111yescharacteristic
228C22SD16×D15711yescharacteristic
229C22#1909779511yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—3142no—
235C4—6281no—
236C4—6282no—
237C4—6281no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—12561no—
242C2#2375742511yescharacteristic
243C2#2375745711yescharacteristic
244C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#327×C157C2
socleC22×C157#120
prime radical#23757850C2
hypercenter#327D157
hypocenterC157#1604
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #23757414C2semidirect
3. #23757411C2semidirect
4. #23757194C2semidirect
5. #23757006C2semidirect
6. #23756998C2semidirect
7. #327×C157C2semidirect
80. C2×Dic157SD16semidirect
122. C157#1604semidirect
129. #327D157semidirect

Maximal subgroups

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

subgroupindex
#237578502
#237574142
#237574112
#237571942
#237570062
#237569982
#327×C1572
#1604157