Finite Group Databasebeta

← #34100553

Subgroups — #34100553

Order 20096.

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

normal subgroups only

This group has 342 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#23758039C211yesmaximal, characteristic, complemented
2#23757785C211yesmaximal, characteristic
3#23757255C211yesmaximal, characteristic, complemented
4#23757236C211yesmaximal, characteristic, complemented
5#23757236C211yesmaximal, characteristic, complemented
6#23757043C211yesmaximal, characteristic, complemented
7#344×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×C4×D157C2211yescharacteristic
10#19097779—22no—
11#19097777—22no—
12C2×#14578938C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14M16×D157—22no—
15#19097755C412yes—
16#19097751—21no—
17#19097746—21no—
18#19097741C412yes—
19C4×D628—22no—
20#19097689—21no—
21#19097687—21no—
22#19097683—22no—
23#19097683—22no—
24C2×M16×C157C2211yescharacteristic
25C2×M16×C157C2211yescharacteristic
26#117×C157C2211yescharacteristic
27C23×D157C2×C411yescharacteristic
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32C2×Dic314D411yescharacteristic
33C2×D628—22no—
34C2×D628D411yescharacteristic
35C2×C22⋊D157C2×C412yes—
36C22×Dic157C2×C411yescharacteristic
37C2×C4×D157—21no—
38C2×C4×D157—21no—
39C2×C4×D157—21no—
40C2×C4×D157—21no—
41C2×C4×D157—21no—
42C2×C4×D157—21no—
43C2×C4×D157D412yes—
44#14578938—21no—
45#14578938—21no—
46#14578938—21no—
47#14578938—21no—
48#14578937—24no—
49C2×#11929434—21no—
50C2×#11929434D412yescomplemented
51C8×D157—24no—
52C4⋊Dic157—22no—
53#14578931—22no—
54#14578931—22no—
55C22×C4×C157C2311yescharacteristic
56M16×C157—21no—
57M16×C157—21no—
58M16×C157—21no—
59M16×C157—21no—
60C2×C8×C157—21no—
61C2×C8×C157D412yescomplemented
62C4⋊C4×C157—21no—
63C22⋊C4×C157C2×C412yes—
64C42×C157—21no—
65C22×D157—21no—
66C22×D157—21no—
67C22×D157—21no—
68C22×D157—42no—
69C22×D157C22⋊C412yes—
70Dic314—21no—
71Dic314—21no—
72D628—21no—
73D628—21no—
74D628—44no—
75C22⋊D157—22no—
76C22⋊D157—22no—
77C2×Dic157—21no—
78C2×Dic157—21no—
79C2×Dic157—21no—
80C2×Dic157C22⋊C412yes—
81C4×D157—22no—
82C4×D157—22no—
83C4×D157—41no—
84C4×D157—41no—
85C4×D157—41no—
86C4×D157—41no—
87#11929434—22no—
88#11929434—41no—
89C23×C157C22×C411yescharacteristic
90C2×C4×C157—22no—
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157C2×D411yescharacteristic
95C2×C4×C157#4111yescharacteristic
96C2×C4×C157#4111yescharacteristic
97C8×C157—22no—
98C8×C157—41no—
99C22×C157—41no—
100C22×C157C2×C22⋊C411yescharacteristic
101C22×C157C4×D411yescharacteristic
102C22×C157C4×D411yescharacteristic
103C2×D157—22no—
104C2×D157—22no—
105C2×D157—22no—
106C2×D157—22no—
107C2×D157—41no—
108C2×D157—41no—
109C2×D157—41no—
110C2×D157—41no—
111C2×D157—42no—
112C4×C157—42no—
113C4×C157#12011yescharacteristic
114C4×C157#12111yescharacteristic
115C4×C157#12111yescharacteristic
116C4×C157#12311yescharacteristic
117Dic157—22no—
118Dic157—41no—
119C2×C157—21no—
120C2×C157—21no—
121C2×C157—21no—
122C2×C157#38611yescharacteristic
123D157—42no—
124D157—42no—
125C157#189111yescharacteristic, complemented, Sylow
126#1891—1571nomaximal, Sylow
127#568—1571no—
128#525—1571no—
129#413—1571no—
130#407—1571no—
131#407—1571no—
132#344—1571no—
133#344D15711yescharacteristic, complemented
134C2×#41—1571no—
135C23×C4—1571no—
136#131—3142no—
137#131—3142no—
138C2×M16—3142no—
139C2×M16—1571no—
140C2×M16—1571no—
141C2×M16C2×D15711yescharacteristic
142C2×M16C2×D15711yescharacteristic
143#123—1572no—
144C22⋊Q8—3141no—
145#121—3141no—
146#120—1572no—
147C4×D4—3142no—
148#117C2×D15711yescharacteristic
149#101—3141no—
150#100—3141no—
151C22⋊C8—3142no—
152C22⋊C8—3142no—
153C24—1571no—
154#41—3141no—
155#41—3141no—
156#41—3141no—
157#41—3141no—
158C2×Q8—1571no—
159C2×D4—3142no—
160C2×D4—1572no—
161C2×D4—1571no—
162C22×C4—3141no—
163C22×C4—3141no—
164C22×C4—3141no—
165C22×C4—3141no—
166C22×C4—3141no—
167C22×C4—3141no—
168C22×C4—1571no—
169C22×C4—1572no—
170C22×C4C22×D15711yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—3144no—
176M16—3141no—
177M16—3141no—
178M16—3141no—
179M16—3141no—
180C2×C8—21no—
181C2×C8—3144no—
182C2×C8—3141no—
183C2×C8—1572no—
184C2×C8D62812yescomplemented
185C4⋊C4—21no—
186C4⋊C4—3142no—
187C22⋊C4—3142no—
188C22⋊C4—3142no—
189C22⋊C4C4×D15712yes—
190C42—21no—
191C23—3141no—
192C23—3141no—
193C23—3141no—
194C23—6282no—
195C23—1572no—
196C23C2×C4×D15711yescharacteristic
197Q8—3141no—
198Q8—3141no—
199D4—3142no—
200D4—3142no—
201D4—3141no—
202D4—3141no—
203D4—6284no—
204C2×C4—22no—
205C2×C4—3142no—
206C2×C4—3142no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—3141no—
210C2×C4—6281no—
211C2×C4—6281no—
212C2×C4—6281no—
213C2×C4—6281no—
214C2×C4—1572no—
215C2×C4D4×D15711yescharacteristic
216C2×C4D4×D15711yescharacteristic
217C2×C4D4×D15711yescharacteristic
218C2×C4C2×D62811yescharacteristic
219C2×C4#1457895411yescharacteristic
220C2×C4#1457895511yescharacteristic
221C8—22no—
222C8—41no—
223C8—3142no—
224C8—6281no—
225C22—41no—
226C22—3142no—
227C22—3142no—
228C22—3142no—
229C22—3142no—
230C22—6282no—
231C22—6281no—
232C22—6281no—
233C22—6281no—
234C22—6281no—
235C22C22⋊C4×D15711yescharacteristic
236C22#1909773311yescharacteristic
237C22C4×D62811yescharacteristic
238C4—42no—
239C4—3142no—
240C4—6281no—
241C4#1909774111yescharacteristic
242C4#1909774411yescharacteristic
243C4#1909774611yescharacteristic
244C4#1909775611yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—6282no—
249C2—6282no—
250C2#2375715111yescharacteristic
subgroupG/H
centerC4#19097741
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#344×C157C2
socleC2×C157#386
prime radical#23757785C2
cosocleC22×C4×C157C23
hypercenter#344D157
hypocenterC157#1891
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #23758039C2semidirect
3. #23757255C2semidirect
4. #23757236C2semidirect
5. #23757236C2semidirect
6. #23757043C2semidirect
7. #344×C157C2semidirect
50. C2×#11929434D4semidirect
61. C2×C8×C157D4semidirect
125. C157#1891semidirect
133. #344D157semidirect
184. C2×C8D628semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237577852
#237572552
#237572362
#237570432
#344×C1572
#1891157