Finite Group Databasebeta

← #34109614

Subgroups — #34109614

Order 20096.

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

normal subgroups only

This group has 584 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×#19097796C211yesmaximal, characteristic, complemented
2C2×#19097767C211yesmaximal, characteristic, complemented
3#41×Dic157C211yesmaximal, characteristic, complemented
4#23757471C214yesmaximal, complemented
5#23757466C214yesmaximal, complemented
6#23757296C211yesmaximal, characteristic, complemented
7#23757279C211yesmaximal, characteristic, complemented
8#23757278C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
10C22×D628C2211yescharacteristic
11C22×C22⋊D157—22no—
12#19097796—24no—
13C2×#14578941C2212yescomplemented
14C2×#14578939C2212yescomplemented
15C2×#14578938C2211yescharacteristic
16#19097767—21no—
17#19097767—21no—
18#19097767C2214yes—
19Q8×Dic157C2212yescomplemented
20D4×Dic157—22no—
21D4×Dic157C2212yescomplemented
22#19097730—21no—
23#19097730C2211yescharacteristic
24C2×C4×Dic157—21no—
25C2×C4×Dic157C2211yescharacteristic
26#19097700C2214yescomplemented
27#19097693C2214yescomplemented
28#19097692C2214yescomplemented
29C2×#41×C157C2211yescharacteristic
30C22×D4×C157C2211yescharacteristic
31AGL(1,ℤ/8)×C157—24no—
32C2×SD16×C157C2212yescomplemented
33C2×D8×C157C2212yescomplemented
34C2×M16×C157C2211yescharacteristic
35C23×D157—21no—
36C2×D628—22no—
37C2×D628—21no—
38C2×D628—22no—
39C2×D628C2312yes—
40C2×C22⋊D157—24no—
41C2×C22⋊D157—24no—
42C2×C22⋊D157—48no—
43C22×Dic157—22no—
44C22×Dic157D412yes—
45#14578941—24no—
46#14578939—24no—
47#14578938—22no—
48C2×#11929434C2312yes—
49C4⋊Dic157—22no—
50C4⋊Dic157C2312yes—
51C22⋊Dic157—22no—
52C22⋊Dic157—21no—
53C4×Dic157—22no—
54C4×Dic157—21no—
55C4×Dic157D411yescharacteristic
56C4×Dic157D411yescharacteristic
57C4×Dic157C2312yes—
58C24×C157—21no—
59#41×C157—22no—
60#41×C157D414yescomplemented
61C2×Q8×C157C2311yescharacteristic
62C2×D4×C157—21no—
63C2×D4×C157—22no—
64C2×D4×C157—21no—
65C2×D4×C157—22no—
66C2×D4×C157C2312yes—
67C2×D4×C157C2311yescharacteristic
68C22×C4×C157—21no—
69C22×C4×C157C2311yescharacteristic
70SD16×C157—24no—
71D8×C157—24no—
72M16×C157—22no—
73C2×C8×C157C2312yes—
74C22×D157—22no—
75C22×D157—44no—
76C22×D157—42no—
77D628—22no—
78D628—22no—
79D628—42no—
80C22⋊D157—48no—
81C22⋊D157—48no—
82C2×Dic157—24no—
83C2×Dic157—22no—
84C2×Dic157—44no—
85C2×Dic157C2×D412yes—
86C2×Dic157C2×D412yes—
87#11929434—22no—
88C23×C157—21no—
89C23×C157—22no—
90C23×C157—44no—
91C23×C157—42no—
92C23×C157C2×D411yescharacteristic
93Q8×C157—21no—
94Q8×C157C2×D412yescomplemented
95D4×C157—24no—
96D4×C157—22no—
97D4×C157—21no—
98D4×C157—22no—
99D4×C157—42no—
100D4×C157C2×D412yescomplemented
101C2×C4×C157—21no—
102C2×C4×C157—24no—
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D412yes—
105C2×C4×C157#4112yes—
106C2×C4×C157C2411yescharacteristic
107C8×C157—22no—
108C22×C157—22no—
109C22×C157—22no—
110C22×C157—42no—
111C22×C157—42no—
112C22×C157—44no—
113C22×C157—84no—
114C22×C157#12112yes—
115C22×C157C22×D411yescharacteristic
116C2×D157—42no—
117C2×D157—44no—
118C2×D157—84no—
119C4×C157—22no—
120C4×C157#12112yes—
121C4×C157C22×D411yescharacteristic
122C4×C157C2×#4111yescharacteristic
123Dic157—22no—
124Dic157—42no—
125Dic157AGL(1,ℤ/8)14yescomplemented
126C2×C157—22no—
127C2×C157—42no—
128C2×C157—82no—
129C2×C157C2×#12111yescharacteristic
130C2×C157C2×AGL(1,ℤ/8)12yes—
131D157—82no—
132C157#303811yescharacteristic, complemented, Sylow
133#3038—1571nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1571no—
135C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
136C2×#127—1571no—
137C4×#41—1571no—
138#460—1574no—
139#459—1574no—
140#423—1571no—
141#418—1571no—
142#418—1571no—
143C2×#41C2×D15711yescharacteristic
144C22×D4—3142no—
145C22×D4—1571no—
146C22×D4C2×D15711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—3144no—
149C2×SD16—1572no—
150C2×SD16C2×D15712yescomplemented
151C2×D8—1572no—
152C2×D8C2×D15712yescomplemented
153C2×M16—1571no—
154C2×M16C2×D15711yescharacteristic
155#127—3141no—
156#127—3141no—
157#127—1574no—
158C4×Q8—1572no—
159C4×D4—3142no—
160C4×D4—1572no—
161#117—3141no—
162#117—1571no—
163C2×C42—3141no—
164C2×C42—1571no—
165C4⋊C8—1574no—
166#102—1574no—
167#102—1574no—
168C24—21no—
169C24—3141no—
170#41—22no—
171#41C22⋊D15714yescomplemented
172C2×Q8C22×D15711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—3144no—
178C2×D4—3144no—
179C2×D4—3141no—
180C2×D4—3142no—
181C2×D4—3142no—
182C2×D4—6288no—
183C2×D4—1572no—
184C2×D4C22×D15712yes—
185C2×D4C22×D15711yescharacteristic
186C22×C4—21no—
187C22×C4—3142no—
188C22×C4—1572no—
189C22×C4C22×D15711yescharacteristic
190SD16—24no—
191SD16—3144no—
192D8—24no—
193D8—3144no—
194M16—22no—
195M16—3142no—
196C2×C8—1572no—
197C2×C8C22×D15712yes—
198C4⋊C4—3142no—
199C4⋊C4—1572no—
200C22⋊C4—3141no—
201C22⋊C4—3142no—
202C42—3142no—
203C42—3141no—
204C42—1571no—
205C42—1571no—
206C42—1572no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—3142no—
212C23—6284no—
213C23—6282no—
214C23D4×D15711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D15712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—3142no—
223D4—3142no—
224D4—6288no—
225D4—6288no—
226D4—6282no—
227D4C2×C22⋊D15712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—3144no—
231C2×C4—3142no—
232C2×C4—6284no—
233C2×C4—1572no—
234C2×C4—1572no—
235C2×C4D4×D15711yescharacteristic
236C2×C4C2×C22⋊D15712yes—
237C2×C4#1457895312yes—
238C2×C4C23×D15711yescharacteristic
239C8—22no—
240C8—3142no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—6282no—
248C22—6284no—
249C22—12564no—
250C22#1909774312yes—
251C22C2×D4×D15711yescharacteristic
252C4—22no—
253C4—3142no—
254C4—6282no—
255C4—1574no—
256C4#1909774312yes—
257C4C22×C22⋊D15711yescharacteristic
258C4C2×#1457895311yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—12562no—
263C2C2×#1909774311yescharacteristic
264C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3038
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097796C2semidirect
2. C2×#19097767C2semidirect
3. #41×Dic157C2semidirect
4. #23757471C2semidirect
5. #23757466C2semidirect
6. #23757296C2semidirect
7. #23757279C2semidirect
8. #23757278C2semidirect
9. C2×AGL(1,ℤ/8)×C157C2semidirect
13. C2×#14578941C22semidirect
14. C2×#14578939C22semidirect
19. Q8×Dic157C22semidirect
21. D4×Dic157C22semidirect
26. #19097700C22semidirect
27. #19097693C22semidirect
28. #19097692C22semidirect
32. C2×SD16×C157C22semidirect
33. C2×D8×C157C22semidirect
60. #41×C157D4semidirect
94. Q8×C157C2×D4semidirect
100. D4×C157C2×D4semidirect
125. Dic157AGL(1,ℤ/8)semidirect
132. C157#3038semidirect
135. C2×AGL(1,ℤ/8)D157semidirect
150. C2×SD16C2×D157semidirect
152. C2×D8C2×D157semidirect
171. #41C22⋊D157semidirect
216. Q8C2×C22⋊D157semidirect
227. D4C2×C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977962
C2×#190977672
#41×Dic1572
#237574712
#237574662
#237572962
#237572792
#237572782
C2×AGL(1,ℤ/8)×C1572
#3038157