Finite Group Databasebeta

← #34100155

Subgroups — #34100155

Order 20096.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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#23757741C211yesmaximal, characteristic
2#23757296C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic
4C2×#19097697C211yesmaximal, characteristic
5C4×#14578933C211yesmaximal, characteristic, complemented
6#23757032C211yesmaximal, characteristic, complemented
7#339×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097736C411yescharacteristic, complemented
11#19097736C411yescharacteristic, complemented
12#19097733C411yescharacteristic, complemented
13#19097733C411yescharacteristic, complemented
14#19097731—21no—
15#19097731—21no—
16#19097728—21no—
17C2×#14578933—21no—
18C2×C4×Dic157C2211yescharacteristic
19C2×C4×Dic157C2211yescharacteristic
20C42×D157—21no—
21#19097700—21no—
22#19097700—21no—
23#19097699—22no—
24#19097697—22no—
25#19097678—21no—
26C2×M16×C157C2211yescharacteristic
27#117×C157C2211yescharacteristic
28C2×C42×C157C2211yescharacteristic
29#14578954—21no—
30#14578954—21no—
31#14578954D412yes—
32#14578954Q812yes—
33C2×Dic314C2×C411yescharacteristic
34C2×D628C2×C411yescharacteristic
35C2×C22⋊D157—21no—
36C22×Dic157—21no—
37C22×Dic157D411yescharacteristic
38C22×Dic157D411yescharacteristic
39C2×C4×D157—21no—
40C2×C4×D157—21no—
41C2×C4×D157—21no—
42#14578938—21no—
43#14578938—21no—
44C2×#11929434—21no—
45#14578933—24no—
46#14578933—21no—
47#14578933—21no—
48#14578931—21no—
49#14578931—21no—
50C4×Dic157—21no—
51C4×Dic157—21no—
52C4×Dic157—21no—
53C4×Dic157—21no—
54C4×Dic157C2×C411yescharacteristic
55C4×Dic157C2×C411yescharacteristic
56C4×Dic157D411yescharacteristic
57C4×Dic157D411yescharacteristic
58C22×C4×C157—21no—
59C22×C4×C157C2311yescharacteristic
60M16×C157—21no—
61M16×C157—21no—
62C2×C8×C157—21no—
63C4⋊C4×C157C2×C411yescharacteristic
64C4⋊C4×C157C2×C411yescharacteristic
65C22⋊C4×C157—21no—
66C42×C157—21no—
67C42×C157—21no—
68C42×C157—21no—
69C22×D157—21no—
70Dic314—21no—
71Dic314C4⋊C412yescomplemented
72D628—21no—
73D628C4⋊C412yescomplemented
74C22⋊D157—22no—
75C22⋊D157—22no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C2×Dic157—22no—
79C2×Dic157—41no—
80C2×Dic157—41no—
81C2×Dic157—41no—
82C2×Dic157C22⋊C411yescharacteristic
83C2×Dic157C22⋊C411yescharacteristic
84C2×Dic157C22⋊C411yescharacteristic
85C2×Dic157C22⋊C411yescharacteristic
86C4×D157—22no—
87C4×D157—22no—
88C4×D157—42no—
89C4×D157—42no—
90#11929434—41no—
91C23×C157C2×D411yescharacteristic
92C2×C4×C157—22no—
93C2×C4×C157—21no—
94C2×C4×C157—21no—
95C2×C4×C157—41no—
96C2×C4×C157—41no—
97C2×C4×C157C22×C411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×Q811yescharacteristic
101C2×C4×C157#4111yescharacteristic
102C2×C4×C157#4111yescharacteristic
103C8×C157—41no—
104C22×C157—21no—
105C22×C157—21no—
106C22×C157C2×C22⋊C411yescharacteristic
107C22×C157#12111yescharacteristic
108C22×C157C22⋊Q811yescharacteristic
109C2×D157—22no—
110C2×D157—41no—
111C2×D157—41no—
112C4×C157—42no—
113C4×C157—41no—
114C4×C157—41no—
115C4×C157C2×C4⋊C411yescharacteristic
116C4×C157#11711yescharacteristic
117C4×C157#12111yescharacteristic
118C4×C157C22⋊Q811yescharacteristic
119Dic157—22no—
120Dic157—21no—
121Dic157—21no—
122Dic157—21no—
123Dic157—42no—
124Dic157#10412yescomplemented
125C2×C157—21no—
126C2×C157—21no—
127C2×C157#38011yescharacteristic
128C2×C157C2×#10411yescharacteristic
129C2×C157#42111yescharacteristic
130D157—42no—
131C157#180611yescharacteristic, complemented, Sylow
132#1806—1571nomaximal, Sylow
133C4×#41—1571no—
134#423—1571no—
135C2×#104—1571no—
136C2×#104—1571no—
137C4×C4⋊C4—1571no—
138#339—1571no—
139#339D15711yescharacteristic, complemented
140C2×#41—1571no—
141C2×M16—1571no—
142C2×M16C2×D15711yescharacteristic
143C4×Q8—1571no—
144C4×Q8—1571no—
145C4×D4—3141no—
146C4×D4—3141no—
147C4×D4—1571no—
148C4×D4—1571no—
149#117—3141no—
150#117C2×D15711yescharacteristic
151C2×C4⋊C4—3141no—
152C2×C42—3141no—
153C2×C42—1571no—
154C2×C42—1571no—
155C2×C42C2×D15711yescharacteristic
156C4⋊C8—3141no—
157C4⋊C8—3141no—
158#104—3142no—
159#104—3142no—
160#95—3141no—
161#41—3141no—
162#41—3141no—
163#41—1572no—
164#41—1572no—
165C2×Q8—1571no—
166C2×D4—3141no—
167C2×D4—1571no—
168C22×C4—21no—
169C22×C4—3141no—
170C22×C4—3141no—
171C22×C4—3141no—
172C22×C4—3141no—
173C22×C4—1571no—
174C22×C4—1571no—
175C22×C4C22×D15711yescharacteristic
176M16—21no—
177M16—21no—
178M16—3141no—
179M16—3141no—
180C2×C8—21no—
181C2×C8—3141no—
182C4⋊C4—3144no—
183C4⋊C4—3141no—
184C4⋊C4—3141no—
185C4⋊C4C4×D15711yescharacteristic
186C4⋊C4C4×D15711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—3141no—
189C22⋊C4—3141no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—3141no—
194C42—3141no—
195C42—3141no—
196C42—3141no—
197C42—1571no—
198C42—1571no—
199C42—1571no—
200C42—1571no—
201C23—3141no—
202C23D4×D15711yescharacteristic
203Q8—3141no—
204Q8—1572no—
205D4—3142no—
206D4—3142no—
207D4—3141no—
208D4—1572no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3141no—
215C2×C4—3142no—
216C2×C4—3142no—
217C2×C4—3142no—
218C2×C4—3142no—
219C2×C4—6282no—
220C2×C4—6282no—
221C2×C4—6281no—
222C2×C4—6281no—
223C2×C4—6281no—
224C2×C4—1571no—
225C2×C4—1571no—
226C2×C4—1571no—
227C2×C4—1571no—
228C2×C4C2×C4×D15711yescharacteristic
229C2×C4D4×D15711yescharacteristic
230C2×C4D4×D15711yescharacteristic
231C2×C4Q8×D15711yescharacteristic
232C2×C4#1457895411yescharacteristic
233C2×C4#1457895411yescharacteristic
234C8—41no—
235C8—6281no—
236C22—21no—
237C22—21no—
238C22—3142no—
239C22—6281no—
240C22—6281no—
241C22C22⋊C4×D15711yescharacteristic
242C22#1909774411yescharacteristic
243C22#1909774911yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—3142no—
248C4—3141no—
249C4—3141no—
250C4—3141no—
251C4—6282no—
252C4—1572no—
253C4C4⋊C4×D15711yescharacteristic
254C4#1909772811yescharacteristic
255C4#1909774411yescharacteristic
256C4#1909774911yescharacteristic
257C2—21no—
258C2—21no—
259C2—6282no—
260C2#2375713211yescharacteristic
261C2#104×D15711yescharacteristic
262C2#2375728811yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC2×#14578954C22
cosocleC22×C4×C157C23
hypercenter#339D157
hypocenterC157#1806
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #23757296C2semidirect
5. C4×#14578933C2semidirect
6. #23757032C2semidirect
7. #339×C157C2semidirect
10. #19097736C4semidirect
11. #19097736C4semidirect
12. #19097733C4semidirect
13. #19097733C4semidirect
71. Dic314C4⋊C4semidirect
73. D628C4⋊C4semidirect
124. Dic157#104semidirect
131. C157#1806semidirect
139. #339D157semidirect

Maximal subgroups

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

subgroupindex
#237577412
#237572962
C2×#190976992
C2×#190976972
C4×#145789332
#237570322
#339×C1572
#1806157