Finite Group Databasebeta

← #34101421

Subgroups — #34101421

Order 20096.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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
2#23757296C211yesmaximal, characteristic, complemented
3C2×#19097698C211yesmaximal, characteristic, complemented
4C2×#19097697C211yesmaximal, characteristic, complemented
5#23757255C211yesmaximal, characteristic, complemented
6#23757031C211yesmaximal, characteristic, complemented
7#413×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—21no—
10#19097801—21no—
11#19097797—21no—
12#19097796—21no—
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15C2×#14578938C2211yescharacteristic
16C2×C4×Dic157C2211yescharacteristic
17#19097700—22no—
18#19097698—21no—
19#19097698—21no—
20#19097697—21no—
21#19097697—21no—
22#19097689—21no—
23#19097687—21no—
24C2×#41×C157C2211yescharacteristic
25C2×M16×C157C2211yescharacteristic
26C2×M16×C157C2211yescharacteristic
27#101×C157—21no—
28#100×C157—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33C2×Dic314D411yescharacteristic
34C2×D628D411yescharacteristic
35C2×C22⋊D157—21no—
36C22×Dic157—21no—
37C2×C4×D157—21no—
38#14578944—22no—
39#14578942—22no—
40#14578941—22no—
41#14578939—22no—
42#14578938—21no—
43#14578938—21no—
44#14578938—21no—
45#14578938—21no—
46C2×#11929434—21no—
47C2×#11929434D412yes—
48C4×Dic157—21no—
49C4×Dic157D411yescharacteristic
50C4×Dic157D411yescharacteristic
51#41×C157—21no—
52#41×C157—21no—
53#41×C157—21no—
54#41×C157—21no—
55C2×Q8×C157D411yescharacteristic
56C2×D4×C157—21no—
57C2×D4×C157D411yescharacteristic
58C22×C4×C157—21no—
59C22×C4×C157C2311yescharacteristic
60M16×C157—21no—
61M16×C157—21no—
62M16×C157—21no—
63M16×C157—21no—
64C2×C8×C157—21no—
65C2×C8×C157—21no—
66C22×D157—21no—
67Dic314—21no—
68Dic314—21no—
69D628—21no—
70D628—21no—
71C22⋊D157—41no—
72C22⋊D157—41no—
73C2×Dic157—21no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—42no—
77C4×D157—41no—
78C4×D157—41no—
79#11929434—22no—
80#11929434—41no—
81C23×C157—21no—
82C23×C157#4111yescharacteristic
83Q8×C157—21no—
84Q8×C157—21no—
85D4×C157—21no—
86D4×C157—21no—
87D4×C157—41no—
88D4×C157—41no—
89C2×C4×C157—21no—
90C2×C4×C157—41no—
91C2×C4×C157—41no—
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157C2×D411yescharacteristic
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157#4111yescharacteristic
97C2×C4×C157#4111yescharacteristic
98C8×C157—41no—
99C8×C157—41no—
100C22×C157—41no—
101C22×C157—41no—
102C22×C157—81no—
103C22×C157#12111yescharacteristic
104C22×C157#12311yescharacteristic
105C22×C157#12311yescharacteristic
106C2×D157—41no—
107C2×D157—81no—
108C4×C157—41no—
109C4×C157#12011yescharacteristic
110C4×C157#12111yescharacteristic
111C4×C157#12111yescharacteristic
112C4×C157#12411yescharacteristic
113Dic157—22no—
114Dic157—41no—
115Dic157—41no—
116C2×C157—21no—
117C2×C157—21no—
118C2×C157—21no—
119C2×C157—81no—
120C2×C157#39411yescharacteristic
121D157—81no—
122C157#205011yescharacteristic, complemented, Sylow
123#2050—1571nomaximal, Sylow
124#575—1571no—
125#423—1571no—
126C2×#104—1571no—
127C2×#104—1571no—
128#413—1571no—
129#413D15711yescharacteristic, complemented
130#338—1571no—
131C2×#41—1571no—
132C2×#41C2×D15711yescharacteristic
133#137—3141no—
134#137—3141no—
135AGL(1,ℤ/8)—3141no—
136AGL(1,ℤ/8)—3141no—
137#135—3142no—
138C2×M16—1571no—
139C2×M16—1571no—
140C2×M16C2×D15711yescharacteristic
141C2×M16C2×D15711yescharacteristic
142C2×C42—1571no—
143C4⋊C8—3142no—
144#104—3141no—
145#104—3141no—
146#104—3141no—
147#104—3141no—
148#101—21no—
149#101—3141no—
150#100—21no—
151#100—3141no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—3141no—
157#41—3141no—
158#41—3141no—
159#41—3141no—
160C2×Q8—1571no—
161C2×Q8C22⋊D15711yescharacteristic
162C2×D4—21no—
163C2×D4—3141no—
164C2×D4—1571no—
165C2×D4C22⋊D15711yescharacteristic
166C22×C4—21no—
167C22×C4—3141no—
168C22×C4—3141no—
169C22×C4C22×D15711yescharacteristic
170Q16—3142no—
171SD16—3142no—
172SD16—3142no—
173D8—3142no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—21no—
178M16—3141no—
179M16—3141no—
180M16—3141no—
181M16—3141no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—3141no—
185C2×C8—1572no—
186C42—3141no—
187C42—1571no—
188C42—1571no—
189C23—21no—
190C23—3141no—
191C23#1457895411yescharacteristic
192Q8—21no—
193Q8—21no—
194Q8—3141no—
195Q8—3141no—
196D4—21no—
197D4—21no—
198D4—41no—
199D4—41no—
200D4—3141no—
201D4—3141no—
202D4—6281no—
203D4—6281no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—3141no—
210C2×C4—6282no—
211C2×C4—6281no—
212C2×C4—6281no—
213C2×C4D4×D15711yescharacteristic
214C2×C4D4×D15711yescharacteristic
215C2×C4D4×D15711yescharacteristic
216C2×C4C2×C22⋊D15711yescharacteristic
217C2×C4#1457895311yescharacteristic
218C2×C4#1457895411yescharacteristic
219C8—41no—
220C8—41no—
221C8—3142no—
222C8—6281no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—6281no—
227C22—12561no—
228C22#1909774411yescharacteristic
229C22#1909775311yescharacteristic
230C22#1909775411yescharacteristic
231C4—41no—
232C4—3142no—
233C4—6281no—
234C4—6281no—
235C4#1909773911yescharacteristic
236C4#1909774311yescharacteristic
237C4#1909774411yescharacteristic
238C4#1909775711yescharacteristic
239C2—21no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—12561no—
244C2#2375718811yescharacteristic
subgroupG/H
centerC4#19097757
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#413×C157C2
socleC2×C157#394
prime radical#23758071C2
hypercenter#413D157
hypocenterC157#2050
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #23757296C2semidirect
3. C2×#19097698C2semidirect
4. C2×#19097697C2semidirect
5. #23757255C2semidirect
6. #23757031C2semidirect
7. #413×C157C2semidirect
122. C157#2050semidirect
129. #413D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237572962
C2×#190976982
C2×#190976972
#237572552
#237570312
#413×C1572
#2050157