Finite Group Databasebeta

← #34101187

Subgroups — #34101187

Order 20096.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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#23758071C211yesmaximal, characteristic, complemented
2#23757840C211yesmaximal, characteristic, complemented
3#23757289C211yesmaximal, characteristic, complemented
4#23757278C211yesmaximal, characteristic, complemented
5#23757247C211yesmaximal, characteristic, complemented
6#23757045C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—21no—
10#19097801—21no—
11#19097797—21no—
12#19097796—21no—
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15#19097767—21no—
16#19097758—21no—
17#19097742—22no—
18#19097740—22no—
19#19097730C2211yescharacteristic
20#19097730C2211yescharacteristic
21#19097697—22no—
22#19097692—22no—
23#19097686—22no—
24C2×#41×C157C2211yescharacteristic
25C22×D4×C157C2211yescharacteristic
26AGL(1,ℤ/8)×C157—22no—
27AGL(1,ℤ/8)×C157—22no—
28C2×SD16×C157—21no—
29C2×D8×C157—21no—
30C2×M16×C157C2211yescharacteristic
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35C2×Dic314D411yescharacteristic
36C2×D628D411yescharacteristic
37C2×C22⋊D157—24no—
38C2×C22⋊D157—21no—
39C2×C4×D157—21no—
40#14578944—22no—
41#14578942—22no—
42#14578941—22no—
43#14578939—22no—
44#14578938—21no—
45#14578938—21no—
46C2×#11929434D412yescomplemented
47C4⋊Dic157—21no—
48C4⋊Dic157—21no—
49C22⋊Dic157—24no—
50C22⋊Dic157—21no—
51C22⋊Dic157D412yes—
52C4×Dic157—21no—
53C4×Dic157—21no—
54C24×C157—21no—
55#41×C157—21no—
56#41×C157—21no—
57#41×C157—21no—
58#41×C157—21no—
59C2×Q8×C157D411yescharacteristic, complemented
60C2×D4×C157—21no—
61C2×D4×C157—21no—
62C2×D4×C157—21no—
63C2×D4×C157—21no—
64C2×D4×C157—21no—
65C2×D4×C157—21no—
66C2×D4×C157D412yes—
67C2×D4×C157D411yescharacteristic, complemented
68C22×C4×C157—21no—
69C22×C4×C157C2311yescharacteristic
70SD16×C157—24no—
71D8×C157—24no—
72M16×C157—21no—
73M16×C157D412yescomplemented
74C2×C8×C157—21no—
75C22×D157—21no—
76Dic314—21no—
77Dic314—21no—
78D628—21no—
79D628—21no—
80C22⋊D157—41no—
81C22⋊D157—41no—
82C22⋊D157—84no—
83C2×Dic157—21no—
84C2×Dic157—22no—
85C2×Dic157—41no—
86C4×D157—41no—
87C4×D157—41no—
88#11929434—22no—
89C23×C157—22no—
90C23×C157—21no—
91C23×C157—41no—
92C23×C157—41no—
93C23×C157—82no—
94C23×C157C2×D411yescharacteristic
95Q8×C157—21no—
96Q8×C157—21no—
97D4×C157—21no—
98D4×C157—22no—
99D4×C157—21no—
100D4×C157—22no—
101D4×C157—41no—
102D4×C157—41no—
103D4×C157—41no—
104D4×C157—41no—
105C2×C4×C157—21no—
106C2×C4×C157—41no—
107C2×C4×C157—41no—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157#4111yescharacteristic
114C8×C157—22no—
115C22×C157—41no—
116C22×C157—44no—
117C22×C157—41no—
118C22×C157—42no—
119C22×C157—81no—
120C22×C157—82no—
121C22×C157—82no—
122C22×C157#12011yescharacteristic
123C22×C157#12111yescharacteristic
124C22×C157#12711yescharacteristic
125C2×D157—42no—
126C2×D157—41no—
127C4×C157—41no—
128C4×C157#12011yescharacteristic
129C4×C157#12011yescharacteristic
130C4×C157#12111yescharacteristic
131C4×C157#12111yescharacteristic
132Dic157—42no—
133Dic157—41no—
134Dic157—81no—
135C2×C157—21no—
136C2×C157—21no—
137C2×C157—21no—
138C2×C157—82no—
139C2×C157—81no—
140C2×C157#39211yescharacteristic
141D157—81no—
142C157#200811yescharacteristic, complemented, Sylow
143#2008—1571nomaximal, Sylow
144#575—1571no—
145C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
146#535—1571no—
147#421—1571no—
148#418—1571no—
149#410—1571no—
150#344—1571no—
151C2×#41—1571no—
152C2×#41C2×D15711yescharacteristic
153C22×D4C2×D15711yescharacteristic
154#137—3141no—
155#137—3141no—
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—22no—
158AGL(1,ℤ/8)—3141no—
159AGL(1,ℤ/8)—3141no—
160#135—3142no—
161C2×SD16—21no—
162C2×D8—21no—
163C2×M16—1571no—
164C2×M16C2×D15711yescharacteristic
165#127—3141no—
166#124—3141no—
167#121—3142no—
168#120—3142no—
169#117—1571no—
170#117—1571no—
171#104—3142no—
172#102—3142no—
173#99—3142no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—3141no—
180#41—3141no—
181#41—3141no—
182#41—3141no—
183C2×Q8—1571no—
184C2×Q8C22⋊D15711yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—3141no—
192C2×D4—3144no—
193C2×D4—1571no—
194C2×D4C22⋊D15712yes—
195C2×D4C22⋊D15711yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—3141no—
198C22×C4C22×D15711yescharacteristic
199Q16—3142no—
200SD16—24no—
201SD16—3142no—
202SD16—3142no—
203D8—24no—
204D8—3142no—
205M16—21no—
206M16—3141no—
207M16—3141no—
208M16C22⋊D15712yescomplemented
209C2×C8—21no—
210C2×C8—1572no—
211C4⋊C4—3141no—
212C4⋊C4—3141no—
213C22⋊C4—3144no—
214C22⋊C4—3141no—
215C22⋊C4—1572no—
216C42—3141no—
217C42—3141no—
218C23—22no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23—3141no—
224C23D4×D15711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—3141no—
228Q8—3141no—
229D4—21no—
230D4—21no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—3141no—
238D4—3141no—
239D4—6281no—
240D4—6281no—
241D4—12564no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—3142no—
246C2×C4—3141no—
247C2×C4—6281no—
248C2×C4—6281no—
249C2×C4—6281no—
250C2×C4D4×D15711yescharacteristic
251C2×C4D4×D15711yescharacteristic
252C2×C4C2×C22⋊D15711yescharacteristic
253C2×C4C2×C22⋊D15711yescharacteristic
254C2×C4C2×C22⋊D15711yescharacteristic
255C2×C4#1457895311yescharacteristic
256C8—22no—
257C8—3142no—
258C22—41no—
259C22—44no—
260C22—41no—
261C22—42no—
262C22—81no—
263C22—82no—
264C22—82no—
265C22—6282no—
266C22—6281no—
267C22#1909773911yescharacteristic
268C22#1909774311yescharacteristic
269C22#1909776711yescharacteristic
270C4—41no—
271C4—6282no—
272C4—6281no—
273C4—12561no—
274C4#1909773911yescharacteristic
275C4#1909774011yescharacteristic
276C4#1909774211yescharacteristic
277C4#1909774311yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—81no—
283C2—12561no—
284C2#2375718111yescharacteristic
subgroupG/H
centerC2#23757181
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC2×C157#392
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#2008
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23758071C2semidirect
2. #23757840C2semidirect
3. #23757289C2semidirect
4. #23757278C2semidirect
5. #23757247C2semidirect
6. #23757045C2semidirect
7. C2×AGL(1,ℤ/8)×C157C2semidirect
46. C2×#11929434D4semidirect
59. C2×Q8×C157D4semidirect
67. C2×D4×C157D4semidirect
73. M16×C157D4semidirect
142. C157#2008semidirect
145. C2×AGL(1,ℤ/8)D157semidirect
184. C2×Q8C22⋊D157semidirect
195. C2×D4C22⋊D157semidirect
208. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237578402
#237572892
#237572782
#237572472
#237570452
C2×AGL(1,ℤ/8)×C1572
#2008157