Finite Group Databasebeta

← #34109852

Subgroups — #34109852

Order 20096.

452 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 137 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#23758071C212yesmaximal, complemented
2C2×#19097777C211yesmaximal, characteristic, complemented
3#23757545C212yesmaximal, complemented
4#23757540C212yesmaximal, complemented
5#23757533C214yesmaximal, complemented
6C2×#19097707C211yesmaximal, characteristic, complemented
7#23757257C212yesmaximal, complemented
8C2×#135×C157C211yesmaximal, characteristic, complemented
9C2×#14578954C2211yescharacteristic
10#19097802C2214yes—
11#19097801—22no—
12#19097797C2214yes—
13#19097796—22no—
14#19097791—24no—
15#19097777—21no—
16#19097777—21no—
17#19097777C2212yes—
18#19097777C2212yes—
19C2×#14578938C2212yes—
20C2×#14578938C2211yescharacteristic
21C2×#14578937—21no—
22C2×C8×D157—21no—
23#19097707C2212yescomplemented
24#19097707C2212yescomplemented
25#19097690C2214yescomplemented
26#19097688C2214yescomplemented
27C2×#41×C157C2212yes—
28#135×C157—21no—
29#135×C157—21no—
30#135×C157—21no—
31#135×C157—21no—
32C2×Q16×C157C2211yescharacteristic, complemented
33C2×SD16×C157C2212yescomplemented
34C2×D8×C157C2211yescharacteristic, complemented
35C22×C8×C157C2211yescharacteristic
36#14578954—21no—
37#14578954—21no—
38#14578954D412yes—
39#14578954C2312yes—
40C2×Dic314D411yescharacteristic
41C2×D628D411yescharacteristic
42C2×C22⋊D157—21no—
43C2×C4×D157—21no—
44#14578944—24no—
45#14578942—24no—
46#14578941—24no—
47#14578939—24no—
48#14578938—21no—
49#14578938—22no—
50#14578938C2312yes—
51#14578938C2314yes—
52#14578937—22no—
53#14578937—22no—
54C2×#11929434—21no—
55C2×#11929434—22no—
56C8×D157—22no—
57C8×D157—22no—
58#41×C157—22no—
59#41×C157—22no—
60#41×C157—22no—
61#41×C157—22no—
62C2×Q8×C157C2312yes—
63C2×D4×C157—22no—
64C2×D4×C157C2312yes—
65C22×C4×C157—22no—
66C22×C4×C157C2311yescharacteristic
67Q16×C157—21no—
68Q16×C157—21no—
69SD16×C157—22no—
70SD16×C157—22no—
71D8×C157—21no—
72D8×C157—21no—
73C2×C8×C157D411yescharacteristic
74C2×C8×C157D411yescharacteristic
75C2×C8×C157D411yescharacteristic
76C2×C8×C157D411yescharacteristic
77C2×C8×C157C2311yescharacteristic
78C2×C8×C157C2311yescharacteristic
79C22×D157—21no—
80Dic314—21no—
81Dic314C2×D412yes—
82D628—21no—
83D628C2×D412yes—
84C22⋊D157—22no—
85C22⋊D157—22no—
86C2×Dic157—21no—
87C4×D157—22no—
88C4×D157—22no—
89#11929434—24no—
90#11929434—22no—
91C23×C157—22no—
92C23×C157#4111yescharacteristic
93Q8×C157—22no—
94Q8×C157—22no—
95D4×C157—22no—
96D4×C157—22no—
97D4×C157—42no—
98D4×C157—42no—
99C2×C4×C157—22no—
100C2×C4×C157—42no—
101C2×C4×C157—42no—
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157#4111yescharacteristic
107C2×C4×C157C2411yescharacteristic
108C8×C157C2×D411yescharacteristic
109C8×C157C2×D411yescharacteristic
110C8×C157C2×D411yescharacteristic
111C8×C157C2×D411yescharacteristic
112C22×C157—41no—
113C22×C157—42no—
114C22×C157—82no—
115C22×C157#12111yescharacteristic
116C22×C157#12111yescharacteristic
117C22×C157C2×#4111yescharacteristic
118C2×D157—22no—
119C2×D157—42no—
120C4×C157—42no—
121C4×C157#12111yescharacteristic
122C4×C157#12111yescharacteristic
123C4×C157C22×D411yescharacteristic
124C4×C157C22×D411yescharacteristic
125Dic157—22no—
126C2×C157—21no—
127C2×C157—21no—
128C2×C157—21no—
129C2×C157—82no—
130C2×C157C2×#12111yescharacteristic
131D157—42no—
132C157#306711yescharacteristic, complemented, Sylow
133#3067—1571nomaximal, Sylow
134#575—1572no—
135C2×#135D15711yescharacteristic, complemented
136C2×#131—1571no—
137#473—1572no—
138#472—1572no—
139#471—1574no—
140C2×#108—1571no—
141#413—1572no—
142C2×#41—1571no—
143C2×#41C2×D15712yes—
144#137—3142no—
145#137—1574no—
146AGL(1,ℤ/8)—3142no—
147AGL(1,ℤ/8)—1574no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—3144no—
153C2×Q16C2×D15711yescharacteristic, complemented
154C2×SD16C2×D15712yescomplemented
155C2×D8C2×D15711yescharacteristic, complemented
156#131—3141no—
157#131—3141no—
158#131—1572no—
159#131—1572no—
160C2×M16—3141no—
161C2×M16—1571no—
162C2×M16—1572no—
163C22×C8—3141no—
164C22×C8C2×D15711yescharacteristic
165#108—1572no—
166#108—1572no—
167#101—1574no—
168#100—1574no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—3141no—
174#41—3141no—
175#41—1572no—
176#41—1572no—
177C2×Q8—1571no—
178C2×Q8C22×D15712yes—
179C2×D4—22no—
180C2×D4—3141no—
181C2×D4—1571no—
182C2×D4C22×D15712yes—
183C22×C4—22no—
184C22×C4—3141no—
185C22×C4C22×D15711yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—3144no—
189SD16—22no—
190SD16—22no—
191SD16—3144no—
192SD16—3144no—
193D8—21no—
194D8—21no—
195D8—3144no—
196M16—3142no—
197M16—3141no—
198M16—3142no—
199M16—3142no—
200M16—1572no—
201M16—1574no—
202C2×C8—3142no—
203C2×C8—3141no—
204C2×C8—3142no—
205C2×C8—3142no—
206C2×C8C22⋊D15711yescharacteristic
207C2×C8C22⋊D15711yescharacteristic
208C2×C8C22⋊D15711yescharacteristic
209C2×C8C22⋊D15711yescharacteristic
210C2×C8C22×D15711yescharacteristic
211C2×C8C22×D15711yescharacteristic
212C23—22no—
213C23—3141no—
214C23#1457895311yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—3141no—
218Q8—1572no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—3142no—
224D4—3142no—
225D4—3141no—
226D4—1572no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—3141no—
231C2×C4—3142no—
232C2×C4—3142no—
233C2×C4D4×D15711yescharacteristic
234C2×C4D4×D15711yescharacteristic
235C2×C4C2×C22⋊D15711yescharacteristic
236C2×C4C2×C22⋊D15711yescharacteristic
237C2×C4#1457895311yescharacteristic
238C2×C4C23×D15711yescharacteristic
239C8—3144no—
240C8—3142no—
241C8C2×C22⋊D15711yescharacteristic
242C8C2×C22⋊D15711yescharacteristic
243C8C2×C22⋊D15711yescharacteristic
244C8C2×C22⋊D15711yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—3142no—
249C22—6282no—
250C22#1909774211yescharacteristic
251C22#1909774211yescharacteristic
252C22C2×#1457895311yescharacteristic
253C4—42no—
254C4—3142no—
255C4#1909774211yescharacteristic
256C4#1909774211yescharacteristic
257C4C2×D4×D15711yescharacteristic
258C4C22×C22⋊D15711yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—6282no—
264C2C2×#1909774211yescharacteristic
subgroupG/H
centerC4#19097742
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#135×C157C2
socleC2×C157C2×#121
hypercenterC2×#135D157
hypocenterC157#3067
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #23758071C2semidirect
2. C2×#19097777C2semidirect
3. #23757545C2semidirect
4. #23757540C2semidirect
5. #23757533C2semidirect
6. C2×#19097707C2semidirect
7. #23757257C2semidirect
8. C2×#135×C157C2semidirect
23. #19097707C22semidirect
24. #19097707C22semidirect
25. #19097690C22semidirect
26. #19097688C22semidirect
32. C2×Q16×C157C22semidirect
33. C2×SD16×C157C22semidirect
34. C2×D8×C157C22semidirect
132. C157#3067semidirect
135. C2×#135D157semidirect
153. C2×Q16C2×D157semidirect
154. C2×SD16C2×D157semidirect
155. C2×D8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
C2×#190977772
#237575452
#237575402
#237575332
C2×#190977072
#237572572
C2×#135×C1572
#3067157