Finite Group Databasebeta

← #34099187

Subgroups — #34099187

Order 20096.

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

normal subgroups only

This group has 240 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#23757863C211yesmaximal, characteristic
2#23757606C211yesmaximal, characteristic, complemented
3#23757592C211yesmaximal, characteristic, complemented
4#23757210C211yesmaximal, characteristic, complemented
5#23757006C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic, complemented
7#328×C157C211yesmaximal, characteristic, complemented
8#19097766—21no—
9#19097760—21no—
10#19097756—21no—
11#19097746—21no—
12#19097745C2211yescharacteristic
13#19097741—21no—
14#19097731—21no—
15C2×C4⋊Dic157—21no—
16C2×C4⋊Dic157C2211yescharacteristic
17C2×#14578931—21no—
18C2×#14578931C2211yescharacteristic
19#19097704—21no—
20#19097703—21no—
21#19097695—21no—
22#19097693—21no—
23C22⋊#11929434C2211yescharacteristic
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28C2×D628—21no—
29C2×D628—21no—
30C2×D628D411yescharacteristic
31C2×C22⋊D157—21no—
32C22×Dic157—21no—
33C22×Dic157D411yescharacteristic
34C2×C4×D157—21no—
35C2×#11929434—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157—21no—
38C4⋊Dic157—42no—
39C4⋊Dic157D411yescharacteristic
40#14578933—21no—
41#14578931—21no—
42#14578931—21no—
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46#14578931—21no—
47#14578931—42no—
48C4×Dic157—21no—
49C2×Q8×C157D411yescharacteristic
50C22×C4×C157—21no—
51C22×C4×C157C2311yescharacteristic
52C2×C8×C157—21no—
53C4⋊C4×C157—21no—
54C4⋊C4×C157D411yescharacteristic
55C22⋊C4×C157—21no—
56C22×D157—21no—
57C22×D157—21no—
58C22×D157—21no—
59C22×D157—21no—
60C22×D157—41no—
61C22×D157—41no—
62D628—41no—
63D628—81no—
64D628—81no—
65C22⋊D157—41no—
66C22⋊D157—41no—
67C2×Dic157—21no—
68C2×Dic157—21no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C2×Dic157—41no—
73C4×D157—81no—
74#11929434—41no—
75C23×C157C2×D411yescharacteristic
76Q8×C157—41no—
77C2×C4×C157—22no—
78C2×C4×C157—21no—
79C2×C4×C157—21no—
80C2×C4×C157—41no—
81C2×C4×C157—41no—
82C2×C4×C157—41no—
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C8×C157—41no—
86C22×C157—21no—
87C22×C157—21no—
88C22×C157#12011yescharacteristic
89C22×C157#13511yescharacteristic
90C22×C157#13711yescharacteristic
91C2×D157—42no—
92C2×D157—41no—
93C2×D157—41no—
94C2×D157—41no—
95C2×D157—41no—
96C2×D157—41no—
97C2×D157—41no—
98C2×D157—81no—
99C4×C157—21no—
100C4×C157—21no—
101C4×C157—42no—
102C4×C157—41no—
103C4×C157—41no—
104Dic157—41no—
105Dic157—41no—
106Dic157—81no—
107C2×C157—21no—
108C2×C157—21no—
109C2×C157#45211yescharacteristic
110C2×C157#45311yescharacteristic
111C2×C157#45811yescharacteristic
112D157—81no—
113D157—81no—
114C157#161811yescharacteristic, complemented, Sylow
115#1618—1571nomaximal, Sylow
116#538—1571no—
117#484—1571no—
118#481—1571no—
119#399—1571no—
120#328D15711yescharacteristic, complemented
121#327—1571no—
122#323—1571no—
123#126—3141no—
124#124—3141no—
125#123—3141no—
126C22⋊Q8C2×D15711yescharacteristic
127#121—3141no—
128#121—1571no—
129#120—3141no—
130C4×D4—3141no—
131C2×C4⋊C4—3141no—
132C2×C4⋊C4—1571no—
133C2×C22⋊C4—3141no—
134C2×C22⋊C4—1571no—
135#107—3141no—
136#106—3141no—
137#103—3141no—
138#102—3141no—
139C22⋊C8—1571no—
140C22⋊C8C2×D15711yescharacteristic
141#95C2×D15711yescharacteristic
142C24—1571no—
143C2×Q8D62811yescharacteristic
144C2×D4—3141no—
145C2×D4—3141no—
146C2×D4—3141no—
147C2×D4—1571no—
148C22×C4—21no—
149C22×C4—3141no—
150C22×C4—3141no—
151C22×C4—1571no—
152C22×C4C22×D15711yescharacteristic
153C2×C8—21no—
154C2×C8—3141no—
155C4⋊C4—21no—
156C4⋊C4—3141no—
157C4⋊C4—3141no—
158C4⋊C4—3141no—
159C4⋊C4—6282no—
160C4⋊C4—1571no—
161C4⋊C4D62811yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—3141no—
166C22⋊C4—3141no—
167C22⋊C4—3141no—
168C22⋊C4—3141no—
169C22⋊C4—6282no—
170C42—3141no—
171C23—3141no—
172C23—3141no—
173C23—3141no—
174C23—3141no—
175C23—6281no—
176C23—6281no—
177C23D4×D15711yescharacteristic
178Q8—41no—
179D4—6281no—
180D4—6281no—
181D4—6281no—
182D4—12561no—
183D4—12561no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—6281no—
193C2×C4—6281no—
194C2×C4—6281no—
195C2×C4—6281no—
196C2×C4—12561no—
197C2×C4D4×D15711yescharacteristic
198C2×C4C2×D62811yescharacteristic
199C8—41no—
200C8—6281no—
201C22—21no—
202C22—21no—
203C22—6282no—
204C22—6281no—
205C22—6281no—
206C22—6281no—
207C22—6281no—
208C22—6281no—
209C22—6281no—
210C22—12561no—
211C22#1909774111yescharacteristic
212C22#1909779211yescharacteristic
213C22#1909780311yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—6281no—
220C4—6281no—
221C4—12561no—
222C2—21no—
223C2—21no—
224C2—12561no—
225C2—12561no—
226C2#2375743811yescharacteristic
227C2#2375744211yescharacteristic
228C2#2375746411yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157#120
prime radical#23757863C2
hypercenter#328D157
hypocenterC157#1618
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #23757606C2semidirect
3. #23757592C2semidirect
4. #23757210C2semidirect
5. #23757006C2semidirect
6. #23756996C2semidirect
7. #328×C157C2semidirect
114. C157#1618semidirect
120. #328D157semidirect

Maximal subgroups

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

subgroupindex
#237578632
#237576062
#237575922
#237572102
#237570062
#237569962
#328×C1572
#1618157