Finite Group Databasebeta

← #34100498

Subgroups — #34100498

Order 20096.

310 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 89 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#23757289C211yesmaximal, characteristic, complemented
3#23757269C211yesmaximal, characteristic, complemented
4#23757241C211yesmaximal, characteristic, complemented
5#23757044C211yesmaximal, characteristic, complemented
6#23757037C211yesmaximal, characteristic, complemented
7#568×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—21no—
10#19097801C412yescomplemented
11#19097797—21no—
12#19097796C412yescomplemented
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15C22×#11929434C2211yescharacteristic
16#19097730C2211yescharacteristic
17#19097697—22no—
18#19097695—21no—
19#19097692—21no—
20#19097683—22no—
21C2×#41×C157C2211yescharacteristic
22#131×C157—22no—
23#131×C157—22no—
24C2×M16×C157—22no—
25C2×M16×C157C2211yescharacteristic
26C2×M16×C157C2211yescharacteristic
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31C2×Dic314C2×C411yescharacteristic
32C2×D628C2×C411yescharacteristic
33C2×C22⋊D157—21no—
34C2×C4×D157—21no—
35#14578944—22no—
36#14578942—22no—
37#14578941—22no—
38#14578939—22no—
39#14578938—21no—
40#14578938C2×C412yes—
41C2×#11929434—21no—
42C2×#11929434—21no—
43C2×#11929434—21no—
44C2×#11929434—21no—
45C4⋊Dic157—21no—
46C22⋊Dic157—21no—
47C4×Dic157—21no—
48#41×C157—21no—
49#41×C157—21no—
50#41×C157C2×C412yes—
51#41×C157D412yes—
52C2×Q8×C157D411yescharacteristic
53C2×D4×C157—21no—
54C2×D4×C157D411yescharacteristic
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57M16×C157—24no—
58M16×C157—21no—
59M16×C157—21no—
60M16×C157—21no—
61M16×C157D412yes—
62C2×C8×C157—24no—
63C2×C8×C157—21no—
64C2×C8×C157D412yes—
65C22×D157—21no—
66Dic314—21no—
67Dic314—21no—
68D628—21no—
69D628—21no—
70C22⋊D157—41no—
71C22⋊D157—41no—
72C2×Dic157—21no—
73C2×Dic157—41no—
74C4×D157—41no—
75C4×D157—41no—
76#11929434—22no—
77#11929434—41no—
78C23×C157—21no—
79C23×C157#4111yescharacteristic
80Q8×C157—21no—
81Q8×C157C22⋊C412yes—
82D4×C157—22no—
83D4×C157—22no—
84D4×C157—21no—
85D4×C157C22⋊C412yes—
86C2×C4×C157—21no—
87C2×C4×C157—22no—
88C2×C4×C157—22no—
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—22no—
97C22×C157—22no—
98C22×C157—41no—
99C22×C157—81no—
100C22×C157C4×D411yescharacteristic
101C22×C157#12111yescharacteristic
102C22×C157#12311yescharacteristic
103C2×D157—42no—
104C2×D157—41no—
105C4×C157—22no—
106C4×C157C2×C22⋊C411yescharacteristic
107C4×C157C4×D411yescharacteristic
108C4×C157#12011yescharacteristic
109C4×C157#12111yescharacteristic
110Dic157—41no—
111Dic157—81no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157—42no—
116C2×C157#38611yescharacteristic
117D157—81no—
118C157#188111yescharacteristic, complemented, Sylow
119#1881—1571nomaximal, Sylow
120#575—1571no—
121#568D15711yescharacteristic, complemented
122#421—1571no—
123#416—1571no—
124#408—1571no—
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—22no—
135#131—22no—
136C2×M16—22no—
137C2×M16—1571no—
138C2×M16C2×D15711yescharacteristic
139C2×M16C2×D15711yescharacteristic
140C22×C8—1571no—
141#117—1571no—
142#104—3142no—
143#103—3141no—
144#102—3141no—
145C22⋊C8—3142no—
146#41—21no—
147#41—21no—
148#41—3141no—
149#41—3141no—
150#41—3141no—
151#41—3141no—
152#41C4×D15712yes—
153#41C22⋊D15712yes—
154C2×Q8—1571no—
155C2×Q8D62811yescharacteristic
156C2×D4—21no—
157C2×D4—3141no—
158C2×D4—1571no—
159C2×D4D62811yescharacteristic
160C22×C4—21no—
161C22×C4—3141no—
162C22×C4C22×D15711yescharacteristic
163Q16—3142no—
164SD16—3142no—
165SD16—3142no—
166D8—3142no—
167M16—24no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—3141no—
172M16—1572no—
173M16C22⋊D15712yes—
174C2×C8—24no—
175C2×C8—21no—
176C2×C8—3141no—
177C2×C8—3141no—
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—3141no—
189Q8—3141no—
190Q8#1457893112yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—3141no—
195D4—3141no—
196D4—6281no—
197D4—6281no—
198D4#1457893112yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—3141no—
203C2×C4—6281no—
204C2×C4—6281no—
205C2×C4—6281no—
206C2×C4C2×C4×D15711yescharacteristic
207C2×C4C2×C22⋊D15711yescharacteristic
208C2×C4C2×C22⋊D15711yescharacteristic
209C2×C4C2×C22⋊D15711yescharacteristic
210C2×C4C2×D62811yescharacteristic
211C2×C4#1457895411yescharacteristic
212C8—22no—
213C8—22no—
214C8—3142no—
215C8—6281no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—6282no—
220C22—6281no—
221C22C4×C22⋊D15711yescharacteristic
222C22#1909774511yescharacteristic
223C22#1909775411yescharacteristic
224C4—22no—
225C4—6281no—
226C4—12561no—
227C4C2×#1457893111yescharacteristic
228C4C4×C22⋊D15711yescharacteristic
229C4#1909774011yescharacteristic
230C4#1909774511yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—12561no—
236C2#2375715211yescharacteristic
subgroupG/H
centerC4#19097740
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#568×C157C2
socleC2×C157#386
prime radical#23758071C2
cosocleC22×C4×C157C23
hypercenter#568D157
hypocenterC157#1881
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23757289C2semidirect
3. #23757269C2semidirect
4. #23757241C2semidirect
5. #23757044C2semidirect
6. #23757037C2semidirect
7. #568×C157C2semidirect
10. #19097801C4semidirect
12. #19097796C4semidirect
118. C157#1881semidirect
121. #568D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237572892
#237572692
#237572412
#237570442
#237570372
#568×C1572
#1881157