Finite Group Databasebeta

← #34099313

Subgroups — #34099313

Order 20096.

308 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D628C211yesmaximal, characteristic
2#23757621C211yesmaximal, characteristic, complemented
3#23757550C211yesmaximal, characteristic, complemented
4#23757016C211yesmaximal, characteristic, complemented
5D4⋊#11929434C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7#461×C157C211yesmaximal, characteristic, complemented
8C22×D628—21no—
9C2×D4×D157—21no—
10#19097768C2211yescharacteristic
11#19097746—21no—
12#19097745—21no—
13#19097741—22no—
14C4×D628C2211yescharacteristic
15#19097704—21no—
16C4⋊#11929434C2211yescharacteristic, complemented
17#19097691—22no—
18#19097691—21no—
19C4×#11929434C2211yescharacteristic, complemented
20C2×SD16×C157—21no—
21C4⋊Q8×C157C2211yescharacteristic, complemented
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic, complemented
24#103×C157—21no—
25C23×D157—22no—
26C2×D628—22no—
27C2×D628—21no—
28C2×D628—21no—
29C2×D628—42no—
30C2×D628—42no—
31C2×D628D411yescharacteristic
32C2×D628D412yes—
33C2×C22⋊D157—22no—
34D4×D157—42no—
35D4×D157—42no—
36C2×C4×D157—21no—
37C2×#11929434—21no—
38C2×#11929434—21no—
39C4⋊Dic157D411yescharacteristic
40#14578931—22no—
41#14578931—21no—
42C2×Q8×C157—21no—
43C2×D4×C157D411yescharacteristic
44C22×C4×C157—21no—
45SD16×C157—42no—
46C2×C8×C157—21no—
47C4⋊C4×C157—21no—
48C4⋊C4×C157—21no—
49C4⋊C4×C157D411yescharacteristic
50C22⋊C4×C157—21no—
51C42×C157C2311yescharacteristic
52C22×D157—21no—
53C22×D157—22no—
54C22×D157—44no—
55C22×D157—44no—
56C22×D157—82no—
57D628—21no—
58D628—42no—
59D628—44no—
60D628—41no—
61D628—41no—
62D628—41no—
63D628—41no—
64D628SD1612yescomplemented
65C22⋊D157—44no—
66C2×Dic157—21no—
67C4×D157—41no—
68C4×D157—41no—
69#11929434—41no—
70#11929434—41no—
71C23×C157—21no—
72Q8×C157—41no—
73Q8×C157—41no—
74D4×C157—21no—
75D4×C157D812yes—
76C2×C4×C157—21no—
77C2×C4×C157—41no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C8×C157—41no—
84C22×C157—22no—
85C22×C157—41no—
86C22×C157—41no—
87C22×C157#12011yescharacteristic
88C2×D157—22no—
89C2×D157—41no—
90C2×D157—41no—
91C2×D157—42no—
92C2×D157—42no—
93C2×D157—42no—
94C2×D157—88no—
95C4×C157—21no—
96C4×C157—41no—
97C4×C157—81no—
98C4×C157C2×D811yescharacteristic
99C4×C157C2×SD1611yescharacteristic
100C4×C157AGL(1,ℤ/8)11yescharacteristic
101C4×C157AGL(1,ℤ/8)11yescharacteristic
102Dic157—41no—
103C2×C157—42no—
104C2×C157#44711yescharacteristic
105C2×C157#45011yescharacteristic
106C2×C157#45511yescharacteristic
107D157—42no—
108D157—82no—
109C157#163911yescharacteristic, complemented, Sylow
110#1639—1571nomaximal, Sylow
111D42—1571no—
112#486—1571no—
113#474—1571no—
114#461D15711yescharacteristic, complemented
115#331—1571no—
116D4⋊C8—1571no—
117D4⋊C8—1571no—
118C22×D4—3141no—
119C22×D4—3141no—
120C2×SD16—21no—
121C4⋊Q8C2×D15711yescharacteristic, complemented
122#127—1571no—
123#121—3141no—
124#121—3141no—
125#120—3142no—
126C4×D4—1571no—
127C4×D4C2×D15711yescharacteristic
128#107—3141no—
129C4⋊C8—1571no—
130C4⋊C8C2×D15711yescharacteristic, complemented
131#103—21no—
132#102—3141no—
133#102—3142no—
134C4×C8—1571no—
135C24—3142no—
136C2×Q8—21no—
137C2×D4—3142no—
138C2×D4—3142no—
139C2×D4—3141no—
140C2×D4—3141no—
141C2×D4—6282no—
142C2×D4—6282no—
143C2×D4—6282no—
144C2×D4—6282no—
145C2×D4—1571no—
146C2×D4—1572no—
147C2×D4C22⋊D15711yescharacteristic
148C22×C4—21no—
149C22×C4—3141no—
150SD16—42no—
151C2×C8—21no—
152C2×C8—3141no—
153C2×C8—3141no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1571no—
157C4⋊C4C22⋊D15711yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—3142no—
160C22⋊C4—3141no—
161C42C22×D15711yescharacteristic
162C23—21no—
163C23—3141no—
164C23—3142no—
165C23—6284no—
166C23—6284no—
167C23—12562no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—3141no—
172D4—6282no—
173D4—6284no—
174D4—6284no—
175D4—6281no—
176D4—6281no—
177D4—6281no—
178D4—6281no—
179D4—1572no—
180D4#1457893912yes—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—3141no—
186C2×C4—6281no—
187C2×C4—6281no—
188C2×C4D4×D15711yescharacteristic
189C2×C4D4×D15711yescharacteristic
190C2×C4C2×C22⋊D15711yescharacteristic
191C8—41no—
192C8—6281no—
193C8—6281no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—3142no—
198C22—6281no—
199C22—6281no—
200C22—6282no—
201C22—6282no—
202C22—6282no—
203C22—12568no—
204C22#1909773911yescharacteristic
205C4—21no—
206C4—41no—
207C4—81no—
208C4—6281no—
209C4C2×#1457893911yescharacteristic
210C4SD16×D15711yescharacteristic
211C4#1909779611yescharacteristic
212C4#1909779811yescharacteristic
213C2—42no—
214C2—6282no—
215C2—12562no—
216C2#2375740611yescharacteristic
217C2#2375742211yescharacteristic
218C2#2375744911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#461×C157C2
socleC22×C157#120
prime radicalD4×D628C2
hypercenter#461D157
hypocenterC157#1639
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #23757621C2semidirect
3. #23757550C2semidirect
4. #23757016C2semidirect
5. D4⋊#11929434C2semidirect
6. #23757000C2semidirect
7. #461×C157C2semidirect
16. C4⋊#11929434C22semidirect
19. C4×#11929434C22semidirect
21. C4⋊Q8×C157C22semidirect
23. C4⋊C8×C157C22semidirect
64. D628SD16semidirect
109. C157#1639semidirect
114. #461D157semidirect
121. C4⋊Q8C2×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
D4×D6282
#237576212
#237575502
#237570162
D4⋊#119294342
#237570002
#461×C1572
#1639157