Finite Group Databasebeta

← #34109963

Subgroups — #34109963

Order 20096.

398 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 141 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#23757904C211yesmaximal, characteristic, complemented
2#23757748C211yesmaximal, characteristic
3#23757575C212yesmaximal, complemented
4#23757571C212yesmaximal, complemented
5#23757561C212yesmaximal, complemented
6#23757551C212yesmaximal, complemented
7#23757313C211yesmaximal, characteristic, complemented
8#23757311C211yesmaximal, characteristic, complemented
9#23757299C211yesmaximal, characteristic, complemented
10#23757278C211yesmaximal, characteristic
11C2×#102×C157C211yesmaximal, characteristic
12C2×#14578938C2211yescharacteristic
13#19097770—21no—
14#19097770C2212yescomplemented
15#19097762—21no—
16#19097762C2212yescomplemented
17#19097750—22no—
18D4×Dic157—21no—
19D4×Dic157—21no—
20D4×Dic157C2212yes—
21D4×Dic157C2212yes—
22#19097730C2211yescharacteristic
23#19097730C2211yescharacteristic
24#19097730C2211yescharacteristic
25C2×C22⋊Dic157—22no—
26#19097705C2211yescharacteristic, complemented
27#19097705C2211yescharacteristic, complemented
28#19097704C2212yescomplemented
29#19097703C2211yescharacteristic, complemented
30#19097703C2211yescharacteristic, complemented
31#19097702C2212yescomplemented
32#19097700C2212yescomplemented
33#19097700C2212yescomplemented
34#19097692C2212yes—
35#19097692C2212yes—
36C22×D4×C157C2211yescharacteristic
37C22×C8×C157C2211yescharacteristic
38C2×C4⋊C4×C157C2211yescharacteristic
39#102×C157C2212yes—
40#102×C157C2212yes—
41C2×Dic314—22no—
42C22×Dic157—22no—
43C22×Dic157—22no—
44#14578938—22no—
45C2×#11929434C2312yes—
46C4⋊Dic157—22no—
47C4⋊Dic157D411yescharacteristic
48C4⋊Dic157D411yescharacteristic
49C4⋊Dic157C2311yescharacteristic
50C4⋊Dic157C2311yescharacteristic
51C4⋊Dic157C2311yescharacteristic
52C4⋊Dic157C2311yescharacteristic
53#14578933—22no—
54#14578933—22no—
55C22⋊Dic157—22no—
56C22⋊Dic157—22no—
57C22⋊Dic157—21no—
58C22⋊Dic157—21no—
59C22⋊Dic157D412yes—
60C4×Dic157—21no—
61C4×Dic157—21no—
62C4×Dic157C2312yes—
63C24×C157—21no—
64C2×D4×C157—21no—
65C2×D4×C157—22no—
66C2×D4×C157Q814yescomplemented
67C2×D4×C157C2312yes—
68C22×C4×C157—21no—
69C22×C4×C157C2311yescharacteristic
70C2×C8×C157—22no—
71C2×C8×C157C2311yescharacteristic
72C2×C8×C157C2311yescharacteristic
73C4⋊C4×C157—21no—
74C4⋊C4×C157C2312yes—
75Dic314—82no—
76C2×Dic157—21no—
77C2×Dic157—21no—
78C2×Dic157—21no—
79C2×Dic157—21no—
80C2×Dic157—82no—
81C2×Dic157—82no—
82C2×Dic157C2×D412yes—
83C2×Dic157C2×D412yes—
84#11929434—22no—
85C23×C157—22no—
86C23×C157—24no—
87C23×C157—81no—
88C23×C157—81no—
89C23×C157C2×D411yescharacteristic
90D4×C157—24no—
91D4×C157—22no—
92D4×C157C2×Q814yes—
93C2×C4×C157—22no—
94C2×C4×C157—81no—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×Q812yes—
97C2×C4×C157#4112yes—
98C2×C4×C157C2411yescharacteristic
99C8×C157—21no—
100C8×C157—21no—
101C22×C157—21no—
102C22×C157—21no—
103C22×C157—24no—
104C22×C157—42no—
105C22×C157—42no—
106C22×C157—84no—
107C22×C157C22⋊Q812yes—
108C22×C157C22×D411yescharacteristic
109C4×C157—42no—
110C4×C157C22⋊Q812yes—
111C4×C157C22×Q811yescharacteristic
112C4×C157C2×#4111yescharacteristic
113Dic157—22no—
114Dic157—22no—
115Dic157—41no—
116Dic157—41no—
117Dic157—41no—
118Dic157—41no—
119C2×C157—22no—
120C2×C157—44no—
121C2×C157C2×C22⋊Q811yescharacteristic
122C2×C157#57611yescharacteristic
123C2×C157#57711yescharacteristic
124C157#308111yescharacteristic, complemented, Sylow
125#3081—1571nomaximal, Sylow
126#544—1571no—
127#518—1571no—
128#478—1572no—
129#478—1572no—
130#476—1572no—
131#474—1572no—
132#428—1571no—
133#427—1571no—
134#424—1571no—
135#418—1571no—
136C2×#102D15711yescharacteristic
137C22×D4C2×D15711yescharacteristic
138C2×M16—1571no—
139C22×C8C2×D15711yescharacteristic
140C4⋊Q8—3141no—
141C4⋊Q8—1572no—
142#125—3141no—
143#125—1572no—
144C22⋊Q8—3142no—
145C4×D4—3141no—
146C4×D4—3141no—
147C4×D4—1572no—
148C4×D4—1572no—
149#117—1571no—
150#117—1571no—
151#117—1571no—
152C2×C4⋊C4C2×D15711yescharacteristic
153C2×C22⋊C4—3142no—
154#107—1571no—
155#107—1571no—
156#107—1572no—
157#106—1571no—
158#106—1571no—
159#106—1572no—
160C4⋊C8—1572no—
161C4⋊C8—1572no—
162#102—1572no—
163#102—1572no—
164#102C2×D15712yes—
165#102C2×D15712yes—
166C24—21no—
167C2×Q8—3142no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic31414yescomplemented
171C2×D4C22×D15712yes—
172C22×C4—21no—
173C22×C4—3142no—
174C22×C4—3142no—
175C22×C4C22×D15711yescharacteristic
176M16—3142no—
177C2×C8—22no—
178C2×C8—1572no—
179C2×C8C22×D15711yescharacteristic
180C2×C8C22×D15711yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—3142no—
183C4⋊C4—3142no—
184C4⋊C4—3142no—
185C4⋊C4—1571no—
186C4⋊C4—1571no—
187C4⋊C4—1571no—
188C4⋊C4—1571no—
189C4⋊C4—1571no—
190C4⋊C4—1571no—
191C4⋊C4C22×D15712yes—
192C22⋊C4—3141no—
193C22⋊C4—3141no—
194C22⋊C4—3142no—
195C22⋊C4—3142no—
196C22⋊C4—1572no—
197C42—3141no—
198C42—3141no—
199C42—1572no—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23D4×D15711yescharacteristic
205Q8—12562no—
206D4—24no—
207D4—22no—
208D4C2×Dic31414yes—
209C2×C4—22no—
210C2×C4—81no—
211C2×C4—3141no—
212C2×C4—3141no—
213C2×C4—3141no—
214C2×C4—3141no—
215C2×C4—1572no—
216C2×C4—1572no—
217C2×C4—12562no—
218C2×C4—12562no—
219C2×C4D4×D15711yescharacteristic
220C2×C4C2×Dic31412yes—
221C2×C4#1457895312yes—
222C2×C4C23×D15711yescharacteristic
223C8—21no—
224C8—21no—
225C8—3142no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1909774812yes—
233C22C2×D4×D15711yescharacteristic
234C4—42no—
235C4—3142no—
236C4—3142no—
237C4—6281no—
238C4—6281no—
239C4—6281no—
240C4—6281no—
241C4#1909774812yes—
242C4C22×Dic31411yescharacteristic
243C4C2×#1457895311yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1909774811yescharacteristic
247C2#2375807411yescharacteristic
248C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#102×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenterC2×#102D157
hypocenterC157#3081
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #23757904C2semidirect
3. #23757575C2semidirect
4. #23757571C2semidirect
5. #23757561C2semidirect
6. #23757551C2semidirect
7. #23757313C2semidirect
8. #23757311C2semidirect
9. #23757299C2semidirect
14. #19097770C22semidirect
16. #19097762C22semidirect
26. #19097705C22semidirect
27. #19097705C22semidirect
28. #19097704C22semidirect
29. #19097703C22semidirect
30. #19097703C22semidirect
31. #19097702C22semidirect
32. #19097700C22semidirect
33. #19097700C22semidirect
66. C2×D4×C157Q8semidirect
124. C157#3081semidirect
170. C2×D4Dic314semidirect

Maximal subgroups

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

subgroupindex
#237579042
#237577482
#237575752
#237575712
#237575612
#237575512
#237573132
#237573112
#237572992
#237572782
C2×#102×C1572
#3081157