Finite Group Databasebeta

← #34101254

Subgroups — #34101254

Order 20096.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19097801C211yesmaximal, characteristic
2C22×#14578944C211yesmaximal, characteristic, complemented
3C2×#19097690C211yesmaximal, characteristic, complemented
4C2×#19097689C211yesmaximal, characteristic, complemented
5#23757240C211yesmaximal, characteristic
6#23757037C211yesmaximal, characteristic, complemented
7C2×#137×C157C211yesmaximal, characteristic
8C22×Dic314C2211yescharacteristic
9#19097801—22no—
10#19097801—22no—
11C2×#14578944—22no—
12C2×#14578944—22no—
13C2×#14578944—21no—
14C2×#14578944—22no—
15C2×#14578942—21no—
16C2×#14578938C2211yescharacteristic
17C2×#14578938C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19#19097690—22no—
20#19097689—22no—
21C22⋊#11929434—22no—
22C2×#41×C157C2211yescharacteristic
23C22×Q8×C157C2211yescharacteristic
24#137×C157—22no—
25#137×C157—22no—
26C2×Q16×C157—21no—
27C2×SD16×C157—21no—
28C2×M16×C157C2211yescharacteristic
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314—21no—
33C2×Dic314—21no—
34C2×Dic314D412yes—
35C22×Dic157—21no—
36#14578944—28no—
37#14578944—24no—
38#14578942—24no—
39#14578938—21no—
40#14578938—21no—
41#14578938—21no—
42#14578938D412yes—
43C2×#11929434—21no—
44C2×#11929434—21no—
45C2×#11929434—21no—
46C2×#11929434—21no—
47C2×#11929434D412yes—
48#41×C157—21no—
49#41×C157—21no—
50#41×C157—21no—
51#41×C157—21no—
52C2×Q8×C157—21no—
53C2×Q8×C157—21no—
54C2×Q8×C157—21no—
55C2×Q8×C157—21no—
56C2×Q8×C157—21no—
57C2×Q8×C157D411yescharacteristic
58C2×Q8×C157D412yes—
59C2×D4×C157—21no—
60C2×D4×C157D411yescharacteristic
61C22×C4×C157—21no—
62C22×C4×C157—21no—
63C22×C4×C157C2311yescharacteristic
64Q16×C157—24no—
65SD16×C157—24no—
66M16×C157—21no—
67M16×C157D412yes—
68C2×C8×C157—21no—
69Dic314—22no—
70Dic314—22no—
71Dic314—41no—
72Dic314—41no—
73C2×Dic157—22no—
74C2×Dic157—41no—
75C2×Dic157—41no—
76#11929434—22no—
77#11929434—22no—
78#11929434—41no—
79C23×C157—21no—
80C23×C157#4111yescharacteristic
81Q8×C157—22no—
82Q8×C157—21no—
83Q8×C157—22no—
84Q8×C157—21no—
85Q8×C157—41no—
86Q8×C157—41no—
87D4×C157—21no—
88D4×C157—21no—
89D4×C157—41no—
90D4×C157—41no—
91C2×C4×C157—21no—
92C2×C4×C157—22no—
93C2×C4×C157—41no—
94C2×C4×C157—41no—
95C2×C4×C157—41no—
96C2×C4×C157—41no—
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C8×C157—22no—
104C22×C157—22no—
105C22×C157—41no—
106C22×C157—42no—
107C22×C157#12111yescharacteristic
108C22×C157#12111yescharacteristic
109C22×C157#12111yescharacteristic
110C4×C157—41no—
111C4×C157—42no—
112C4×C157#12011yescharacteristic
113C4×C157#12011yescharacteristic
114C4×C157#12011yescharacteristic
115C4×C157#12711yescharacteristic
116Dic157—42no—
117C2×C157—21no—
118C2×C157—21no—
119C2×C157—81no—
120C2×C157#39211yescharacteristic
121C2×C157#47312yes—
122C157#202011yescharacteristic, complemented, Sylow
123#2020—1571nomaximal, Sylow
124C2×#137—1571no—
125C2×#137D15711yescharacteristic
126C22×Q16—1571no—
127C2×#101—1571no—
128C2×#101—1571no—
129#408—1571no—
130#341—1571no—
131C2×#41C2×D15711yescharacteristic
132C22×Q8—1571no—
133C22×Q8C2×D15711yescharacteristic
134#137—22no—
135#137—22no—
136#137—3142no—
137#137—3142no—
138C2×Q16—21no—
139C2×Q16—3142no—
140C2×Q16—3142no—
141C2×Q16—3141no—
142C2×Q16—3142no—
143C2×SD16—21no—
144C2×SD16—3141no—
145C2×M16—1571no—
146C2×M16—1571no—
147C2×M16C2×D15711yescharacteristic
148C22×C8—1571no—
149#101—3142no—
150#101—3142no—
151C22⋊C8—3142no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—3141no—
162C2×Q8—3141no—
163C2×Q8—3141no—
164C2×Q8—3141no—
165C2×Q8—3141no—
166C2×Q8—1572no—
167C2×Q8C22⋊D15711yescharacteristic
168C2×Q8C22⋊D15712yes—
169C2×D4—21no—
170C2×D4C22⋊D15711yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—3141no—
174C22×C4C22×D15711yescharacteristic
175Q16—24no—
176Q16—3148no—
177Q16—3144no—
178SD16—24no—
179SD16—3144no—
180M16—21no—
181M16—3141no—
182M16—3141no—
183M16—3141no—
184M16—1572no—
185M16C22⋊D15712yes—
186C2×C8—21no—
187C2×C8—3141no—
188C2×C8—3141no—
189C2×C8—3141no—
190C2×C8—3141no—
191C2×C8—1572no—
192C23—21no—
193C23#1457895311yescharacteristic
194Q8—22no—
195Q8—21no—
196Q8—22no—
197Q8—21no—
198Q8—41no—
199Q8—41no—
200Q8—3142no—
201Q8—3142no—
202Q8—6281no—
203Q8—6281no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3142no—
215C2×C4—6281no—
216C2×C4—6281no—
217C2×C4D4×D15711yescharacteristic
218C2×C4D4×D15711yescharacteristic
219C2×C4D4×D15711yescharacteristic
220C2×C4C2×C22⋊D15711yescharacteristic
221C2×C4C2×C22⋊D15711yescharacteristic
222C2×C4C2×C22⋊D15711yescharacteristic
223C8—22no—
224C8—3142no—
225C8—3142no—
226C8—6281no—
227C22—22no—
228C22—41no—
229C22—42no—
230C22#1909774211yescharacteristic
231C22#1909774311yescharacteristic
232C22#1909774311yescharacteristic
233C4—41no—
234C4—42no—
235C4—6282no—
236C4#1909773911yescharacteristic
237C4#1909773911yescharacteristic
238C4#1909774011yescharacteristic
239C4#1909776711yescharacteristic
240C2—21no—
241C2—21no—
242C2—81no—
243C2#2375718111yescharacteristic
244C2#2375754612yes—
subgroupG/H
centerC22#19097743
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#137×C157C2
socleC22×C157#121
prime radicalC2×#41×C157C22
hypercenterC2×#137D157
hypocenterC157#2020
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14578944C2semidirect
3. C2×#19097690C2semidirect
4. C2×#19097689C2semidirect
6. #23757037C2semidirect
122. C157#2020semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C22×#145789442
C2×#190976902
C2×#190976892
#237572402
#237570372
C2×#137×C1572
#2020157