Finite Group Databasebeta

← #34099327

Subgroups — #34099327

Order 20096.

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

normal subgroups only

This group has 250 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#23757945C211yesmaximal, characteristic
2#23757621C211yesmaximal, characteristic, complemented
3#23757550C211yesmaximal, characteristic, complemented
4#23757016C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7#462×C157C211yesmaximal, characteristic, complemented
8C2×#14578955—21no—
9#19097768—21no—
10#19097768C2211yescharacteristic
11#19097746—21no—
12#19097746—22no—
13C4×D628—21no—
14C4×D628C2211yescharacteristic
15#19097704—21no—
16C4⋊#11929434C2211yescharacteristic
17#19097691—22no—
18#19097691—21no—
19C4×#11929434C2211yescharacteristic
20C2×Q16×C157—21no—
21C4⋊Q8×C157C2211yescharacteristic
22C4×Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24#103×C157—21no—
25#14578955—42no—
26#14578955—42no—
27C2×D628—22no—
28C2×D628—22no—
29C2×D628—21no—
30C2×D628—21no—
31C2×D628D411yescharacteristic
32C2×D628D412yes—
33C2×C4×D157—22no—
34C2×C4×D157—21no—
35C2×#11929434—21no—
36C2×#11929434—21no—
37C4⋊Dic157D411yescharacteristic
38#14578931—22no—
39#14578931—21no—
40C2×Q8×C157—21no—
41C2×Q8×C157D411yescharacteristic
42Q16×C157—42no—
43C2×C8×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157—21no—
46C4⋊C4×C157—21no—
47C4⋊C4×C157D411yescharacteristic
48C42×C157—21no—
49C42×C157C2311yescharacteristic
50C22×D157—21no—
51C22×D157—22no—
52D628—21no—
53D628—21no—
54D628—42no—
55D628—44no—
56D628—41no—
57D628—41no—
58D628—41no—
59D628—41no—
60D628—82no—
61C2×Dic157—21no—
62C4×D157—44no—
63C4×D157—41no—
64C4×D157—41no—
65#11929434—41no—
66#11929434—41no—
67Q8×C157—21no—
68Q8×C157—41no—
69Q8×C157—41no—
70Q8×C157D812yes—
71C2×C4×C157—21no—
72C2×C4×C157—21no—
73C2×C4×C157—41no—
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C2×C4×C157C2×D411yescharacteristic
77C8×C157—41no—
78C22×C157#12011yescharacteristic
79C2×D157—42no—
80C2×D157—41no—
81C2×D157—41no—
82C2×D157—42no—
83C2×D157—42no—
84C2×D157—41no—
85C4×C157—22no—
86C4×C157—21no—
87C4×C157—41no—
88C4×C157—81no—
89C4×C157C2×D811yescharacteristic
90C4×C157#13511yescharacteristic
91C4×C157AGL(1,ℤ/8)11yescharacteristic
92C4×C157#13711yescharacteristic
93Dic157—41no—
94C2×C157#44711yescharacteristic
95C2×C157#45211yescharacteristic
96C2×C157#45511yescharacteristic
97D157—81no—
98D157—82no—
99C157#164111yescharacteristic, complemented, Sylow
100#1641—1571nomaximal, Sylow
101#550—1571no—
102#486—1571no—
103#474—1571no—
104#462D15711yescharacteristic, complemented
105#331—1571no—
106Q8⋊C8—1571no—
107D4⋊C8—1571no—
108C2×#41—3141no—
109C2×Q16—21no—
110C4⋊Q8C2×D15711yescharacteristic
111#127—3141no—
112#127—1571no—
113#121—3142no—
114#121—3141no—
115C4×Q8C2×D15711yescharacteristic
116C4×D4—3141no—
117C4×D4—1571no—
118#107—3141no—
119C4⋊C8—1571no—
120C4⋊C8C2×D15711yescharacteristic
121#103—21no—
122#102—3141no—
123#102—3142no—
124C4×C8—1571no—
125#41—6282no—
126#41—6282no—
127C2×Q8—21no—
128C2×Q8C22⋊D15711yescharacteristic
129C2×D4—3142no—
130C2×D4—3142no—
131C2×D4—3141no—
132C2×D4—3141no—
133C2×D4—1571no—
134C2×D4—1572no—
135C22×C4—3142no—
136C22×C4—3141no—
137Q16—42no—
138C2×C8—21no—
139C2×C8—3141no—
140C2×C8—3141no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1571no—
145C4⋊C4C22⋊D15711yescharacteristic
146C22⋊C4—3142no—
147C22⋊C4—3141no—
148C42—21no—
149C42C22×D15711yescharacteristic
150C23—3141no—
151C23—3142no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1457893912yes—
156D4—3141no—
157D4—3141no—
158D4—6282no—
159D4—6284no—
160D4—6281no—
161D4—6281no—
162D4—6281no—
163D4—6281no—
164D4—12562no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—3141no—
169C2×C4—6284no—
170C2×C4—6281no—
171C2×C4—6281no—
172C2×C4D4×D15711yescharacteristic
173C2×C4D4×D15711yescharacteristic
174C2×C4C2×C22⋊D15711yescharacteristic
175C8—41no—
176C8—6281no—
177C8—6281no—
178C22—6282no—
179C22—6281no—
180C22—6281no—
181C22—6282no—
182C22—6282no—
183C22—6281no—
184C22#1909773911yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—6281no—
190C4C2×#1457893911yescharacteristic
191C4#1909779411yescharacteristic
192C4#1909779611yescharacteristic
193C4#1909780311yescharacteristic
194C2—12561no—
195C2—12562no—
196C2#2375740611yescharacteristic
197C2#2375743411yescharacteristic
198C2#2375744911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#462×C157C2
socleC22×C157#120
prime radical#23757945C2
hypercenter#462D157
hypocenterC157#1641
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #23757621C2semidirect
3. #23757550C2semidirect
4. #23757016C2semidirect
5. Q8⋊#11929434C2semidirect
6. #23757000C2semidirect
7. #462×C157C2semidirect
99. C157#1641semidirect
104. #462D157semidirect

Maximal subgroups

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

subgroupindex
#237579452
#237576212
#237575502
#237570162
Q8⋊#119294342
#237570002
#462×C1572
#1641157