Finite Group Databasebeta

← #34099316

Subgroups — #34099316

Order 20096.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757908C211yesmaximal, characteristic
2#23757619C211yesmaximal, characteristic, complemented
3#23757554C211yesmaximal, characteristic, complemented
4#23757479C211yesmaximal, characteristic, complemented
5#23757000C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7#331×C157C211yesmaximal, characteristic, complemented
8C2×D4×D157—21no—
9C2×D4×D157—21no—
10C2×#14578943—21no—
11#19097771C2211yescharacteristic, complemented
12#19097742—21no—
13#19097742—21no—
14#19097739—22no—
15C4×D628C2211yescharacteristic
16C4×D628C2211yescharacteristic
17#19097705—21no—
18#19097696—21no—
19#19097693—21no—
20#19097692—22no—
21C4×#11929434C2211yescharacteristic, complemented
22#127×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic, complemented
24C4⋊C8×C157C2211yescharacteristic, complemented
25C23×D157—22no—
26C2×Dic314—21no—
27C2×D628D411yescharacteristic
28C2×D628D411yescharacteristic
29C2×C22⋊D157—22no—
30C2×C22⋊D157—22no—
31D4×D157—42no—
32D4×D157—42no—
33D4×D157—42no—
34D4×D157—42no—
35C2×C4×D157—21no—
36C2×C4×D157—21no—
37#14578943—42no—
38C2×#11929434—21no—
39C4⋊Dic157—21no—
40C4⋊Dic157—21no—
41C4⋊Dic157D411yescharacteristic
42C4⋊Dic157D411yescharacteristic
43C22⋊Dic157—22no—
44#14578931—21no—
45#14578931—21no—
46C2×D4×C157—21no—
47C2×D4×C157—21no—
48C2×D4×C157D412yes—
49C2×C8×C157—21no—
50C2×C8×C157—21no—
51C42×C157C2311yescharacteristic
52C22×D157—21no—
53C22×D157—21no—
54C22×D157—44no—
55C22×D157—44no—
56C22×D157—82no—
57Dic314—41no—
58Dic314—41no—
59D628—21no—
60D628—21no—
61D628D812yes—
62D628SD1612yescomplemented
63C22⋊D157—44no—
64C22⋊D157—44no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—41no—
68C4×D157—41no—
69C4×D157—41no—
70C4×D157—41no—
71C4×D157—41no—
72#11929434—41no—
73C23×C157—22no—
74D4×C157—42no—
75D4×C157—41no—
76D4×C157—41no—
77D4×C157—41no—
78D4×C157—41no—
79C2×C4×C157C2×D411yescharacteristic
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C8×C157—41no—
83C8×C157—41no—
84C22×C157—42no—
85C22×C157—42no—
86C22×C157—42no—
87C22×C157#12011yescharacteristic
88C2×D157—22no—
89C2×D157—22no—
90C2×D157—41no—
91C2×D157—41no—
92C2×D157—41no—
93C2×D157—41no—
94C2×D157—88no—
95C4×C157—21no—
96C4×C157C2×D811yescharacteristic
97C4×C157C2×SD1611yescharacteristic
98C4×C157AGL(1,ℤ/8)11yescharacteristic
99C4×C157AGL(1,ℤ/8)11yescharacteristic
100Dic157—41no—
101Dic157—41no—
102Dic157—81no—
103C2×C157—82no—
104C2×C157#44711yescharacteristic
105C2×C157#45011yescharacteristic
106C2×C157#45511yescharacteristic
107D157—42no—
108D157—42no—
109C157#163911yescharacteristic, complemented, Sylow
110#1639—1571nomaximal, Sylow
111D42—1571no—
112#486—1571no—
113#474—1571no—
114#461—1571no—
115#331D15711yescharacteristic, complemented
116D4⋊C8—1571no—
117D4⋊C8—1571no—
118C22×D4—3141no—
119C22×D4—3141no—
120C2×SD16—3141no—
121C4⋊Q8—1571no—
122#127C2×D15711yescharacteristic
123#121—3141no—
124#121—3141no—
125#120—3142no—
126C4×D4—1571no—
127C4×D4—1571no—
128#107—3141no—
129C4⋊C8C2×D15711yescharacteristic, complemented
130C4⋊C8C2×D15711yescharacteristic, complemented
131#103—3141no—
132#102—3141no—
133#102—3142no—
134C4×C8—1571no—
135C24—3142no—
136C2×Q8—3141no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—3142no—
140C2×D4—3142no—
141C2×D4—6282no—
142C2×D4—6282no—
143C2×D4—6282no—
144C2×D4—6282no—
145C2×D4—1571no—
146C2×D4—1571no—
147C2×D4D62812yes—
148C22×C4—3141no—
149C22×C4—3141no—
150SD16—6282no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—3141no—
154C4⋊C4—3141no—
155C4⋊C4—3141no—
156C4⋊C4—1571no—
157C4⋊C4—1571no—
158C22⋊C4—3142no—
159C22⋊C4—3141no—
160C22⋊C4—3141no—
161C42C22×D15711yescharacteristic
162C23—22no—
163C23—3141no—
164C23—3141no—
165C23—6284no—
166C23—6284no—
167C23—12562no—
168Q8—6281no—
169Q8—6281no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—3141no—
176D4—3141no—
177D4—6284no—
178D4—6284no—
179D4—1572no—
180D4—1572no—
181C2×C4—3141no—
182C2×C4—3141no—
183C2×C4—6281no—
184C2×C4—6281no—
185C2×C4—6281no—
186C2×C4—6281no—
187C2×C4—6281no—
188C2×C4D4×D15711yescharacteristic
189C2×C4D4×D15711yescharacteristic
190C2×C4C2×D62811yescharacteristic
191C8—41no—
192C8—41no—
193C8—6281no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—3142no—
198C22—3142no—
199C22—6281no—
200C22—6281no—
201C22—6281no—
202C22—6281no—
203C22—12568no—
204C22#1909774111yescharacteristic
205C4—21no—
206C4—6281no—
207C4—6281no—
208C4—12561no—
209C4D8×D15711yescharacteristic
210C4SD16×D15711yescharacteristic
211C4#1909779511yescharacteristic
212C4#1909779811yescharacteristic
213C2—82no—
214C2—6282no—
215C2—6282no—
216C2#2375740811yescharacteristic
217C2#2375742511yescharacteristic
218C2#2375745311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157#120
prime radical#23757908C2
hypercenter#331D157
hypocenterC157#1639
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #23757619C2semidirect
3. #23757554C2semidirect
4. #23757479C2semidirect
5. #23757000C2semidirect
6. #23757000C2semidirect
7. #331×C157C2semidirect
11. #19097771C22semidirect
21. C4×#11929434C22semidirect
23. C4⋊C8×C157C22semidirect
24. C4⋊C8×C157C22semidirect
62. D628SD16semidirect
109. C157#1639semidirect
115. #331D157semidirect
129. C4⋊C8C2×D157semidirect
130. C4⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579082
#237576192
#237575542
#237574792
#237570002
#331×C1572
#1639157