Finite Group Databasebeta

← #34100501

Subgroups — #34100501

Order 20096.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#23758039C211yesmaximal, characteristic, complemented
3#23757290C211yesmaximal, characteristic, complemented
4#23757270C211yesmaximal, characteristic, complemented
5#23757044C211yesmaximal, characteristic, complemented
6#23757038C211yesmaximal, characteristic, complemented
7#408×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802C412yescomplemented
10#19097801—21no—
11#19097797C412yescomplemented
12#19097796—21no—
13#19097791—22no—
14#19097779—22no—
15#19097777—22no—
16C2×#14578938C2211yescharacteristic
17C2×#14578938C2211yescharacteristic
18M16×D157—22no—
19#19097730C2211yescharacteristic
20#19097698—22no—
21#19097696—21no—
22#19097693—21no—
23C2×#41×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25C22×C8×C157C2211yescharacteristic
26C22⋊C8×C157—22no—
27#14578954—21no—
28#14578954—21no—
29#14578954C2×C412yes—
30#14578954D412yes—
31C2×Dic314D411yescharacteristic
32C2×D628D411yescharacteristic
33C2×C22⋊D157—21no—
34C2×C4×D157—21no—
35#14578944—22no—
36#14578942—22no—
37#14578941—22no—
38#14578939—22no—
39#14578938—21no—
40#14578938—21no—
41#14578938C2×C412yes—
42#14578938D412yes—
43#14578937—24no—
44C2×#11929434—21no—
45C2×#11929434—21no—
46C8×D157—24no—
47C4⋊Dic157—21no—
48C22⋊Dic157—21no—
49C4×Dic157—21no—
50#41×C157—21no—
51#41×C157—21no—
52#41×C157—21no—
53#41×C157—21no—
54C2×Q8×C157C2×C411yescharacteristic
55C2×D4×C157—21no—
56C2×D4×C157C2×C411yescharacteristic
57C22×C4×C157—21no—
58C22×C4×C157C2311yescharacteristic
59M16×C157—21no—
60M16×C157—21no—
61C2×C8×C157—21no—
62C2×C8×C157—21no—
63C2×C8×C157—21no—
64C2×C8×C157D412yes—
65C22×D157—21no—
66Dic314—21no—
67Dic314C22⋊C412yes—
68D628—21no—
69D628C22⋊C412yes—
70C22⋊D157—22no—
71C22⋊D157—22no—
72C2×Dic157—21no—
73C2×Dic157—41no—
74C4×D157—22no—
75C4×D157—22no—
76#11929434—22no—
77#11929434—22no—
78C23×C157—21no—
79C23×C157#4111yescharacteristic
80Q8×C157—21no—
81Q8×C157—21no—
82D4×C157—21no—
83D4×C157—21no—
84D4×C157—41no—
85D4×C157—41no—
86C2×C4×C157—21no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157C22×C411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157#4111yescharacteristic
95C8×C157—22no—
96C8×C157—41no—
97C22×C157—41no—
98C22×C157—42no—
99C22×C157—41no—
100C22×C157C4×D411yescharacteristic
101C22×C157#12111yescharacteristic
102C22×C157#12311yescharacteristic
103C2×D157—22no—
104C2×D157—81no—
105C4×C157—41no—
106C4×C157C2×C22⋊C411yescharacteristic
107C4×C157C4×D411yescharacteristic
108C4×C157#12011yescharacteristic
109C4×C157#12111yescharacteristic
110Dic157—22no—
111Dic157—81no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157—81no—
116C2×C157#38611yescharacteristic
117D157—42no—
118C157#188111yescharacteristic, complemented, Sylow
119#1881—1571nomaximal, Sylow
120#575—1571no—
121#568—1571no—
122#421—1571no—
123#416—1571no—
124#408D15711yescharacteristic, complemented
125#344—1571no—
126#341—1571no—
127C2×#41—1571no—
128C2×#41C2×D15711yescharacteristic
129#137—3141no—
130#137—1572no—
131AGL(1,ℤ/8)—3141no—
132AGL(1,ℤ/8)—1572no—
133#135—3142no—
134#131—3142no—
135#131—3142no—
136C2×M16—3142no—
137C2×M16—1571no—
138C2×M16—1571no—
139C2×M16C2×D15711yescharacteristic
140C22×C8C2×D15711yescharacteristic
141#117—1571no—
142#104—3142no—
143#103—3141no—
144#102—3141no—
145C22⋊C8—22no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—3141no—
151#41—3141no—
152#41—1572no—
153#41—1572no—
154C2×Q8—1571no—
155C2×Q8C4×D15711yescharacteristic
156C2×D4—21no—
157C2×D4—3141no—
158C2×D4—1571no—
159C2×D4C4×D15711yescharacteristic
160C22×C4—21no—
161C22×C4—3141no—
162C22×C4C22×D15711yescharacteristic
163Q16—3142no—
164SD16—3142no—
165SD16—3142no—
166D8—3142no—
167M16—21no—
168M16—21no—
169M16—3144no—
170M16—3141no—
171M16—3141no—
172M16—1572no—
173M16—1572no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—3144no—
178C2×C8—3141no—
179C2×C8—3141no—
180C2×C8C22⋊D15712yes—
181C4⋊C4—3141no—
182C22⋊C4—3141no—
183C42—3141no—
184C23—21no—
185C23—3141no—
186C23#1457895411yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—3141no—
190Q8—1572no—
191D4—21no—
192D4—21no—
193D4—41no—
194D4—41no—
195D4—3142no—
196D4—3142no—
197D4—3141no—
198D4—1572no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3141no—
203C2×C4—3142no—
204C2×C4—3142no—
205C2×C4—6281no—
206C2×C4C2×C4×D15711yescharacteristic
207C2×C4D4×D15711yescharacteristic
208C2×C4D4×D15711yescharacteristic
209C2×C4D4×D15711yescharacteristic
210C2×C4C2×C22⋊D15711yescharacteristic
211C2×C4#1457895311yescharacteristic
212C8—22no—
213C8—41no—
214C8—3142no—
215C8—3142no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—3142no—
220C22—12561no—
221C22C4×C22⋊D15711yescharacteristic
222C22#1909774411yescharacteristic
223C22#1909775311yescharacteristic
224C4—41no—
225C4—3142no—
226C4—12561no—
227C4C22⋊C4×D15711yescharacteristic
228C4#1909773111yescharacteristic
229C4#1909773911yescharacteristic
230C4#1909774211yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—6282no—
236C2#2375714911yescharacteristic
subgroupG/H
centerC4#19097739
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#408×C157C2
socleC2×C157#386
prime radical#23758071C2
cosocleC22×C4×C157C23
hypercenter#408D157
hypocenterC157#1881
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #23758039C2semidirect
3. #23757290C2semidirect
4. #23757270C2semidirect
5. #23757044C2semidirect
6. #23757038C2semidirect
7. #408×C157C2semidirect
9. #19097802C4semidirect
11. #19097797C4semidirect
118. C157#1881semidirect
124. #408D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237580392
#237572902
#237572702
#237570442
#237570382
#408×C1572
#1881157