Finite Group Databasebeta

← #497×D157 #34110572

Subgroups — #497×D157 #34110572

Order 20096.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×Q16×D157C212yesmaximal, complemented
2C2×SD16×D157C212yesmaximal
3C4⋊Q8×D157C211yesmaximal, characteristic, complemented
4#124×D157C211yesmaximal, characteristic
5#23757668C211yesmaximal, characteristic, complemented
6#23757667C211yesmaximal, characteristic, complemented
7#23757666C212yesmaximal, complemented
8#23757665C211yesmaximal, characteristic, complemented
9#23757664C212yesmaximal, complemented
10#97×D157C211yesmaximal, characteristic, complemented
11#497×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C2×Q8×D157C2212yes—
14C2×D4×D157C2211yescharacteristic
15C2×Dic628C2212yescomplemented
16C2×#14578944C2212yescomplemented
17C2×#14578944C2212yescomplemented
18Q16×D157—28no—
19C2×#14578943C2212yes—
20C2×#14578942C2212yes—
21C2×#14578941C2212yes—
22SD16×D157—28no—
23C2×C8×D157C2212yes—
24#19097771C2211yescharacteristic, complemented
25#19097770—22no—
26#19097769C2212yescomplemented
27#19097760C2211yescharacteristic
28#19097759C2211yescharacteristic
29#19097758C2211yescharacteristic
30#19097757—22no—
31C4⋊C4×D157—22no—
32C22⋊C4×D157—22no—
33C42×D157C2211yescharacteristic
34#19097682C2211yescharacteristic, complemented
35#19097681C2212yescomplemented
36C2×Q16×C157C2212yescomplemented
37C2×SD16×C157C2212yes—
38C4⋊Q8×C157C2211yescharacteristic, complemented
39#124×C157C2211yescharacteristic
40#97×C157C2211yescharacteristic, complemented
41C23×D157—21no—
42C2×Dic314—21no—
43C2×Dic314—22no—
44C2×Dic314C2311yescharacteristic
45C2×Dic314C2312yes—
46Q8×D157—24no—
47Q8×D157—24no—
48Q8×D157—24no—
49C2×D628C2311yescharacteristic
50C2×C22⋊D157—21no—
51D4×D157—24no—
52C2×C4×D157—21no—
53C2×C4×D157—22no—
54C2×C4×D157D412yes—
55C2×C4×D157C2311yescharacteristic
56Dic628—24no—
57#14578944—24no—
58#14578944—24no—
59#14578943—24no—
60#14578942—24no—
61#14578941—24no—
62C2×#11929434C2312yes—
63C8×D157D418yes—
64C4⋊Dic157—22no—
65#14578933—24no—
66C22⋊Dic157—22no—
67#14578931—22no—
68#14578931—22no—
69C4×Dic157D412yes—
70C4×Dic157C2311yescharacteristic
71C2×Q8×C157C2311yescharacteristic
72C2×Q8×C157C2312yes—
73C2×D4×C157C2311yescharacteristic
74Q16×C157—24no—
75SD16×C157—24no—
76C2×C8×C157C2312yes—
77C4⋊C4×C157—22no—
78C22⋊C4×C157—22no—
79C42×C157C2311yescharacteristic
80C22×D157—21no—
81C22×D157—42no—
82C22×D157—44no—
83C22×D157C2×D411yescharacteristic
84Dic314—22no—
85Dic314—22no—
86Dic314—22no—
87Dic314—44no—
88Dic314—42no—
89D628—22no—
90C22⋊D157—42no—
91C2×Dic157—21no—
92C2×Dic157—22no—
93C2×Dic157C2×D412yes—
94C2×Dic157C2×D411yescharacteristic
95C4×D157—24no—
96C4×D157—42no—
97C4×D157—44no—
98C4×D157C2×D414yes—
99#11929434C2×D414yes—
100C23×C157—21no—
101Q8×C157—22no—
102Q8×C157—22no—
103Q8×C157—22no—
104D4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—22no—
107C2×C4×C157C2×D412yes—
108C2×C4×C157C2411yescharacteristic
109C8×C157C2×D414yes—
110C22×C157—41no—
111C22×C157—42no—
112C22×C157C22×D411yescharacteristic
113C2×D157—41no—
114C2×D157—42no—
115C2×D157—84no—
116C2×D157#12712yes—
117C2×D157#13714yes—
118C4×C157—22no—
119C4×C157—41no—
120C4×C157—42no—
121C4×C157C22×D412yes—
122Dic157—22no—
123Dic157—41no—
124Dic157—42no—
125Dic157#12712yes—
126C2×C157—81no—
127C2×C157C2×#12711yescharacteristic
128C2×C157C2×#13712yes—
129D157—81no—
130D157#49714yesdirect product factor
131C157C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—1571nomaximal, Sylow
133C22×Q16—1572no—
134C22×SD16—1572no—
135C2×C4⋊Q8—1571no—
136C2×#124—1571no—
137#497—1572no—
138#497—1572no—
139#497—1571no—
140#497—1571no—
141#497—1571no—
142#497D15711yescharacteristic, direct product factor
143C2×#97—1571no—
144C22×Q8—1571no—
145C22×Q8—1572no—
146C22×D4—1571no—
147C2×Q16—3148no—
148C2×Q16—1572no—
149C2×Q16—1572no—
150C2×Q16—1572no—
151C2×Q16C2×D15712yescomplemented
152C2×SD16—3148no—
153C2×SD16—1572no—
154C2×SD16—1572no—
155C2×SD16—1572no—
156C2×SD16C2×D15712yes—
157C22×C8—1572no—
158C4⋊Q8—3142no—
159C4⋊Q8—1572no—
160C4⋊Q8—1571no—
161C4⋊Q8C2×D15711yescharacteristic, complemented
162#124—3142no—
163#124—1571no—
164#124—1571no—
165#124—1571no—
166#124C2×D15711yescharacteristic
167C2×C4⋊C4—3142no—
168C2×C22⋊C4—3142no—
169C2×C42—1571no—
170#97—1572no—
171#97—1571no—
172#97C2×D15711yescharacteristic, complemented
173C24—3141no—
174C2×Q8—3141no—
175C2×Q8—3142no—
176C2×Q8—3144no—
177C2×Q8—3144no—
178C2×Q8—3144no—
179C2×Q8—1571no—
180C2×Q8—1572no—
181C2×Q8C22×D15711yescharacteristic
182C2×Q8C22×D15712yes—
183C2×D4—3144no—
184C2×D4—3141no—
185C2×D4—1571no—
186C2×D4C22×D15711yescharacteristic
187C22×C4—3141no—
188C22×C4—3142no—
189C22×C4—1572no—
190C22×C4—1571no—
191Q16—24no—
192Q16—3144no—
193Q16—3144no—
194Q16—3144no—
195SD16—24no—
196SD16—3144no—
197SD16—3144no—
198SD16—3144no—
199C2×C8—1572no—
200C2×C8—1578no—
201C2×C8C22×D15712yes—
202C4⋊C4—22no—
203C4⋊C4—3144no—
204C4⋊C4—3142no—
205C22⋊C4—22no—
206C22⋊C4—3142no—
207C22⋊C4—3142no—
208C22⋊C4—3142no—
209C42—1572no—
210C42—1571no—
211C42C22×D15711yescharacteristic
212C23—21no—
213C23—3141no—
214C23—6282no—
215C23—6284no—
216C23—1571no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—3142no—
221Q8—3142no—
222Q8—3142no—
223Q8—6282no—
224Q8—6284no—
225D4—22no—
226D4—3142no—
227D4—6282no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—3144no—
231C2×C4—3141no—
232C2×C4—3142no—
233C2×C4—6282no—
234C2×C4—6284no—
235C2×C4—1572no—
236C2×C4—1571no—
237C2×C4—1574no—
238C2×C4D4×D15712yes—
239C2×C4C23×D15711yescharacteristic
240C8—1574no—
241C8D4×D15714yes—
242C22—42no—
243C22—41no—
244C22—6282no—
245C22—6281no—
246C22—1574no—
247C22—1572no—
248C22—12564no—
249C22C2×D4×D15711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—3142no—
254C4—6281no—
255C4—6282no—
256C4—1572no—
257C4C2×D4×D15712yes—
258C2—81no—
259C2—1574no—
260C2—12561no—
261C2#127×D15711yescharacteristic
262C2#137×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#497×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#497D157
hypocenterC157C2×#497
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D157C2semidirect
3. C4⋊Q8×D157C2semidirect
5. #23757668C2semidirect
6. #23757667C2semidirect
7. #23757666C2semidirect
8. #23757665C2semidirect
9. #23757664C2semidirect
10. #97×D157C2semidirect
11. #497×C157C2semidirect
15. C2×Dic628C22semidirect
16. C2×#14578944C22semidirect
17. C2×#14578944C22semidirect
24. #19097771C22semidirect
26. #19097769C22semidirect
34. #19097682C22semidirect
35. #19097681C22semidirect
36. C2×Q16×C157C22semidirect
38. C4⋊Q8×C157C22semidirect
40. #97×C157C22semidirect
130. D157#497direct (N × H)
131. C157C2×#497semidirect
142. #497D157direct (N × H)
151. C2×Q16C2×D157semidirect
161. C4⋊Q8C2×D157semidirect
172. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1572
C2×SD16×D1572
C4⋊Q8×D1572
#124×D1572
#237576682
#237576672
#237576662
#237576652
#237576642
#97×D1572
#497×C1572
C2×#497157