Finite Group Databasebeta

← #34098694

Subgroups — #34098694

Order 20096.

264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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#23757818C211yesmaximal, characteristic
2#23757323C211yesmaximal, characteristic, complemented
3#23757137C211yesmaximal, characteristic, complemented
4#23757008C211yesmaximal, characteristic, complemented
5#23757007C211yesmaximal, characteristic, complemented
6#23757006C211yesmaximal, characteristic, complemented
7#327×C157C211yesmaximal, characteristic, complemented
8C2×#14578953—21no—
9#19097770C411yescharacteristic, complemented
10#19097767C411yescharacteristic, complemented
11#19097757—21no—
12#19097745C2211yescharacteristic
13#19097743—21no—
14D4×Dic157C411yescharacteristic, complemented
15#19097733C411yescharacteristic, complemented
16#19097731—21no—
17C2×#14578933—21no—
18C2×C4×Dic157C2211yescharacteristic
19#19097700—21no—
20C22⋊#11929434C2211yescharacteristic
21C8×Dic157—21no—
22#19097678—21no—
23#19097678C2211yescharacteristic
24#121×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27#14578953—42no—
28#14578953—42no—
29C2×Dic314—21no—
30C2×D628C2×C411yescharacteristic
31C2×C22⋊D157—21no—
32C2×C22⋊D157—21no—
33C2×C22⋊D157—21no—
34C22×Dic157—21no—
35C22×Dic157—21no—
36C22×Dic157D411yescharacteristic
37C22×Dic157D411yescharacteristic
38C2×C4×D157—21no—
39C2×#11929434—21no—
40C4⋊Dic157C2×C411yescharacteristic
41#14578933—21no—
42#14578933—42no—
43C22⋊Dic157—21no—
44#14578931—21no—
45#14578931—21no—
46C4×Dic157C2×C411yescharacteristic
47C4×Dic157C2×C411yescharacteristic
48C4×Dic157D411yescharacteristic
49C4×Dic157D411yescharacteristic
50C2×D4×C157—21no—
51C2×D4×C157C2×C411yescharacteristic
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54C2×C8×C157—21no—
55C4⋊C4×C157C2×C411yescharacteristic
56C22⋊C4×C157—21no—
57C22×D157—21no—
58Dic314—42no—
59D628—41no—
60C22⋊D157—42no—
61C22⋊D157—41no—
62C22⋊D157—41no—
63C22⋊D157—42no—
64C2×Dic157—21no—
65C2×Dic157—41no—
66C2×Dic157—42no—
67C2×Dic157—41no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157C22⋊C411yescharacteristic
71C2×Dic157C22⋊C411yescharacteristic
72C2×Dic157C22⋊C411yescharacteristic
73C2×Dic157C22⋊C411yescharacteristic
74C2×Dic157D812yes—
75C2×Dic157SD1612yes—
76C4×D157—42no—
77#11929434—41no—
78C23×C157—21no—
79C23×C157C2×D411yescharacteristic
80D4×C157—41no—
81D4×C157—41no—
82D4×C157—41no—
83C2×C4×C157—21no—
84C2×C4×C157—41no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157C22×C411yescharacteristic
89C2×C4×C157C2×D411yescharacteristic
90C8×C157—41no—
91C22×C157—21no—
92C22×C157—21no—
93C22×C157—41no—
94C22×C157—41no—
95C22×C157—41no—
96C22×C157C2×C22⋊C411yescharacteristic
97C22×C157C2×D811yescharacteristic
98C22×C157C2×SD1611yescharacteristic
99C2×D157—41no—
100C2×D157—41no—
101C2×D157—41no—
102C4×C157—21no—
103C4×C157—21no—
104C4×C157—41no—
105C4×C157—81no—
106Dic157—21no—
107Dic157—21no—
108Dic157—41no—
109Dic157—81no—
110Dic157#10212yes—
111Dic157#10212yes—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—81no—
115C2×C157#41011yescharacteristic
116C2×C157C2×#10211yescharacteristic
117C2×C157#42111yescharacteristic
118D157—81no—
119C157#151711yescharacteristic, complemented, Sylow
120#1517—1571nomaximal, Sylow
121#532—1571no—
122#431—1571no—
123#382—1571no—
124#327—1571no—
125#327—1571no—
126#327—1571no—
127#327D15711yescharacteristic, complemented
128C2×#41—3141no—
129C4⋊Q8—1571no—
130#127—1571no—
131#124—3141no—
132#121—3141no—
133#121—1571no—
134#121C2×D15711yescharacteristic
135C4×D4—3141no—
136C4×D4—1571no—
137C4×D4—1571no—
138C2×C4⋊C4—3141no—
139C2×C42—1571no—
140C4⋊C8—3141no—
141C22⋊C8—1571no—
142C22⋊C8C2×D15711yescharacteristic
143C4×C8—3141no—
144#95—3141no—
145#95—1571no—
146#95C2×D15711yescharacteristic
147#41—6282no—
148#41—6282no—
149C2×Q8—3141no—
150C2×D4—21no—
151C2×D4—3141no—
152C2×D4—3141no—
153C2×D4—3141no—
154C2×D4—1571no—
155C2×D4C4×D15711yescharacteristic
156C22×C4—21no—
157C22×C4—3141no—
158C22×C4—3141no—
159C22×C4—3141no—
160C22×C4—1571no—
161C22×C4—1571no—
162C22×C4C22×D15711yescharacteristic
163C2×C8—21no—
164C2×C8—3141no—
165C4⋊C4—3141no—
166C4⋊C4—6282no—
167C4⋊C4—1571no—
168C4⋊C4C4×D15711yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—3141no—
171C22⋊C4—3141no—
172C22⋊C4—3141no—
173C42—1571no—
174C42—1571no—
175C42—1571no—
176C42—1571no—
177C23—21no—
178C23—3141no—
179C23D4×D15711yescharacteristic
180Q8—6282no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—6281no—
185D4—6282no—
186D4—6281no—
187D4—6281no—
188D4—6282no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—3141no—
195C2×C4—6282no—
196C2×C4—6281no—
197C2×C4—6282no—
198C2×C4—6281no—
199C2×C4—6281no—
200C2×C4—6281no—
201C2×C4—1571no—
202C2×C4—1571no—
203C2×C4—1571no—
204C2×C4—1571no—
205C2×C4—1572no—
206C2×C4—1572no—
207C2×C4C2×C4×D15711yescharacteristic
208C2×C4D4×D15711yescharacteristic
209C8—41no—
210C8—6281no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—6281no—
217C22—6281no—
218C22—6281no—
219C22C22⋊C4×D15711yescharacteristic
220C22D8×D15711yescharacteristic
221C22SD16×D15711yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—3141no—
227C4—3141no—
228C4—6281no—
229C4—1572no—
230C4—1572no—
231C4—12561no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—12561no—
236C2#2375724511yescharacteristic
237C2#102×D15711yescharacteristic
238C2#2375728811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#327×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757818C2
cosocleC22×C4×C157C23
hypercenter#327D157
hypocenterC157#1517
LCSG ▷ C2×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. #23757323C2semidirect
3. #23757137C2semidirect
4. #23757008C2semidirect
5. #23757007C2semidirect
6. #23757006C2semidirect
7. #327×C157C2semidirect
9. #19097770C4semidirect
10. #19097767C4semidirect
14. D4×Dic157C4semidirect
15. #19097733C4semidirect
119. C157#1517semidirect
127. #327D157semidirect

Maximal subgroups

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

subgroupindex
#237578182
#237573232
#237571372
#237570082
#237570072
#237570062
#327×C1572
#1517157