Finite Group Databasebeta

← #34100638

Subgroups — #34100638

Order 20096.

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

normal subgroups only

This group has 332 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#23758073C211yesmaximal, characteristic
2#23757741C211yesmaximal, characteristic, complemented
3#23757289C211yesmaximal, characteristic, complemented
4C2×#19097697C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757034C211yesmaximal, characteristic, complemented
7#344×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578954C2211yescharacteristic
10#19097804—21no—
11#19097804C412yescomplemented
12#19097799—21no—
13#19097799C412yescomplemented
14#19097793—22no—
15#19097736—21no—
16#19097733—21no—
17#19097731—22no—
18#19097730C2211yescharacteristic
19#19097728—21no—
20C2×C4×Dic157C2211yescharacteristic
21C42×D157—21no—
22#19097697—21no—
23#19097697—21no—
24#19097697—22no—
25#19097685—22no—
26C2×M16×C157C2211yescharacteristic
27C2×M16×C157C2211yescharacteristic
28#117×C157C2211yescharacteristic
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35#14578954C2×C412yes—
36#14578954D412yes—
37C2×Dic314C2×C411yescharacteristic, complemented
38C2×Dic314D411yescharacteristic
39C2×D628C2×C411yescharacteristic, complemented
40C2×D628D411yescharacteristic
41C2×C22⋊D157—21no—
42C2×C22⋊D157—21no—
43C22×Dic157—21no—
44C2×C4×D157—22no—
45C2×C4×D157—21no—
46C2×C4×D157—21no—
47Dic628—22no—
48#14578943—22no—
49#14578943—22no—
50D1256—22no—
51C4⋊Dic157—21no—
52#14578933—22no—
53C22⋊Dic157—21no—
54#14578931—22no—
55C4×Dic157—22no—
56C4×Dic157—21no—
57C4×Dic157—21no—
58C4×Dic157D411yescharacteristic
59C4×Dic157D411yescharacteristic
60C22×C4×C157C2311yescharacteristic
61M16×C157—21no—
62M16×C157—21no—
63M16×C157—21no—
64M16×C157C2×C412yescomplemented
65C2×C8×C157—21no—
66C2×C8×C157—21no—
67C4⋊C4×C157—21no—
68C22⋊C4×C157D412yes—
69C42×C157—21no—
70C22×D157—21no—
71C22×D157—21no—
72Dic314—21no—
73Dic314—21no—
74Dic314—21no—
75Dic314C22⋊C412yes—
76D628—21no—
77D628—21no—
78D628—21no—
79D628C22⋊C412yes—
80C22⋊D157—22no—
81C22⋊D157—22no—
82C22⋊D157—41no—
83C22⋊D157—41no—
84C2×Dic157—21no—
85C2×Dic157—21no—
86C2×Dic157—21no—
87C2×Dic157—21no—
88C2×Dic157—42no—
89C2×Dic157—41no—
90C4×D157—22no—
91C4×D157—22no—
92C4×D157—44no—
93C4×D157—41no—
94C4×D157—41no—
95C23×C157C2×D411yescharacteristic
96C2×C4×C157—22no—
97C2×C4×C157C22×C411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157#4111yescharacteristic
102C2×C4×C157#4111yescharacteristic
103C8×C157—22no—
104C8×C157—41no—
105C22×C157—41no—
106C22×C157C4×D411yescharacteristic
107C22×C157#12011yescharacteristic
108C22×C157#12111yescharacteristic
109C2×D157—22no—
110C2×D157—42no—
111C2×D157—41no—
112C2×D157—81no—
113C4×C157—42no—
114C4×C157C2×C22⋊C411yescharacteristic
115C4×C157C4×D411yescharacteristic
116C4×C157#12111yescharacteristic
117C4×C157#12311yescharacteristic
118Dic157—22no—
119Dic157—22no—
120Dic157—41no—
121Dic157—41no—
122Dic157—81no—
123C2×C157—21no—
124C2×C157—21no—
125C2×C157—21no—
126C2×C157#38611yescharacteristic
127D157—42no—
128D157—81no—
129C157#190611yescharacteristic, complemented, Sylow
130#1906—1571nomaximal, Sylow
131#575—1571no—
132C4×#41—1571no—
133#421—1571no—
134C2×#104—1571no—
135#410—1571no—
136#344D15711yescharacteristic, complemented
137#339—1571no—
138C2×#41—1571no—
139C2×#41—1571no—
140#137—3141no—
141#137—1572no—
142AGL(1,ℤ/8)—3141no—
143AGL(1,ℤ/8)—1572no—
144#135—3142no—
145C2×M16C2×D15711yescharacteristic
146C2×M16C2×D15711yescharacteristic
147C4×Q8—3141no—
148C4×D4—3141no—
149C4×D4—3142no—
150#117—3141no—
151#117—1571no—
152#117C2×D15711yescharacteristic
153C2×C42—3141no—
154C2×C42—1571no—
155#104—3142no—
156#104—3141no—
157#104—3141no—
158#99—3142no—
159#41—3141no—
160#41—3141no—
161#41—3141no—
162#41—3141no—
163#41—3141no—
164#41—3141no—
165#41—1572no—
166#41—1572no—
167C2×Q8—1571no—
168C2×Q8—1571no—
169C2×D4—3141no—
170C2×D4—3141no—
171C2×D4—1571no—
172C2×D4—1571no—
173C22×C4—3142no—
174C22×C4—3141no—
175C22×C4—3141no—
176C22×C4—3141no—
177C22×C4C22×D15711yescharacteristic
178Q16—3142no—
179SD16—3142no—
180SD16—3142no—
181D8—3142no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D15712yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—3142no—
190C4⋊C4—3141no—
191C22⋊C4—3142no—
192C22⋊C4—3141no—
193C22⋊C4D62812yes—
194C42—21no—
195C42—3142no—
196C42—3141no—
197C42—3141no—
198C42—1571no—
199C42—1571no—
200C23—3141no—
201C23—3141no—
202C23C2×D62811yescharacteristic
203Q8—3141no—
204Q8—3141no—
205Q8—3141no—
206Q8—1572no—
207D4—3142no—
208D4—3142no—
209D4—3141no—
210D4—3141no—
211D4—3141no—
212D4—6281no—
213D4—6281no—
214D4—1572no—
215C2×C4—22no—
216C2×C4—3141no—
217C2×C4—3141no—
218C2×C4—3142no—
219C2×C4—3142no—
220C2×C4—3141no—
221C2×C4—3141no—
222C2×C4—6284no—
223C2×C4—6282no—
224C2×C4—6281no—
225C2×C4—6281no—
226C2×C4—6281no—
227C2×C4C2×C4×D15711yescharacteristic
228C2×C4D4×D15711yescharacteristic
229C2×C4D4×D15711yescharacteristic
230C2×C4D4×D15711yescharacteristic
231C2×C4#1457895411yescharacteristic
232C2×C4#1457895511yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—3142no—
237C22—6282no—
238C22—6281no—
239C22—12561no—
240C22C4×D62811yescharacteristic
241C22#1909774111yescharacteristic
242C22#1909774611yescharacteristic
243C4—42no—
244C4—3142no—
245C4—3142no—
246C4—6281no—
247C4—6281no—
248C4—12561no—
249C4C22⋊C4×D15711yescharacteristic
250C4#1909773311yescharacteristic
251C4#1909774411yescharacteristic
252C4#1909775611yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—6282no—
257C2—12561no—
258C2#2375715111yescharacteristic
subgroupG/H
centerC4#19097744
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#344×C157C2
socleC2×C157#386
prime radical#23758073C2
cosocleC22×C4×C157C23
hypercenter#344D157
hypocenterC157#1906
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #23757741C2semidirect
3. #23757289C2semidirect
4. C2×#19097697C2semidirect
5. #23757246C2semidirect
6. #23757034C2semidirect
7. #344×C157C2semidirect
11. #19097804C4semidirect
13. #19097799C4semidirect
37. C2×Dic314C2×C4semidirect
39. C2×D628C2×C4semidirect
64. M16×C157C2×C4semidirect
129. C157#1906semidirect
136. #344D157semidirect
185. M16C4×D157semidirect

Maximal subgroups

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

subgroupindex
#237580732
#237577412
#237572892
C2×#190976972
#237572462
#237570342
#344×C1572
#1906157