Finite Group Databasebeta

← #34099240

Subgroups — #34099240

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#23757938C211yesmaximal, characteristic, complemented
2#23757621C211yesmaximal, characteristic, complemented
3#23757474C211yesmaximal, characteristic, complemented
4#23757469C211yesmaximal, characteristic
5#23757003C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7#332×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157—21no—
9C2×#14578943—21no—
10C2×D1256—21no—
11#19097770—21no—
12#19097768C2211yescharacteristic
13#19097749—22no—
14#19097747—21no—
15C4×Dic314C2211yescharacteristic
16C4×D628C2211yescharacteristic
17#19097733—21no—
18#19097693—21no—
19#19097693—21no—
20#19097691—22no—
21C4×#11929434C2211yescharacteristic
22C4⋊Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4⋊C8×C157C2211yescharacteristic
25C2×Dic314—22no—
26C2×Dic314D411yescharacteristic
27Q8×D157—42no—
28Q8×D157—42no—
29C2×D628—21no—
30C2×D628—21no—
31C2×D628—21no—
32C2×D628D411yescharacteristic
33C2×C4×D157—22no—
34C2×C4×D157—21no—
35#14578943—42no—
36D1256—42no—
37C2×#11929434—21no—
38C4⋊Dic157D411yescharacteristic
39C4⋊Dic157D411yescharacteristic
40#14578933—22no—
41#14578933—21no—
42#14578931—22no—
43#14578931—21no—
44C4×Dic157—21no—
45C2×Q8×C157—21no—
46C2×C8×C157—21no—
47C2×C8×C157—21no—
48C4⋊C4×C157—21no—
49C4⋊C4×C157D412yes—
50C42×C157C2311yescharacteristic
51C22×D157—21no—
52C22×D157—41no—
53Dic314—21no—
54Dic314—44no—
55Dic314D812yescomplemented
56D628—21no—
57D628—42no—
58D628—41no—
59D628—41no—
60D628—41no—
61D628—41no—
62D628SD1612yescomplemented
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C4×D157—44no—
67C4×D157—41no—
68C4×D157—41no—
69#11929434—41no—
70Q8×C157—41no—
71Q8×C157—41no—
72C2×C4×C157—22no—
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C8×C157—41no—
77C8×C157—41no—
78C22×C157#12011yescharacteristic
79C2×D157—22no—
80C2×D157—41no—
81C2×D157—41no—
82C2×D157—81no—
83C2×D157—81no—
84C2×D157—81no—
85C4×C157—21no—
86C4×C157—42no—
87C4×C157C2×D811yescharacteristic
88C4×C157C2×SD1611yescharacteristic
89C4×C157AGL(1,ℤ/8)11yescharacteristic
90C4×C157#13711yescharacteristic
91Dic157—22no—
92Dic157—41no—
93Dic157—41no—
94C2×C157#44711yescharacteristic
95C2×C157#44811yescharacteristic
96C2×C157#45411yescharacteristic
97D157—42no—
98D157—161no—
99C157#162711yescharacteristic, complemented, Sylow
100#1627—1571nomaximal, Sylow
101D4×Q8—1571no—
102#486—1571no—
103#460—1571no—
104#459—1571no—
105#332D15711yescharacteristic, complemented
106Q8⋊C8—1571no—
107D4⋊C8—1571no—
108C22×Q8—3141no—
109C2×SD16—3141no—
110C2×D8—3141no—
111C4⋊Q8—3141no—
112C4⋊Q8C2×D15711yescharacteristic
113#127—1571no—
114C22⋊Q8—3141no—
115C22⋊Q8—3142no—
116C4×Q8—1571no—
117C4×D4—3141no—
118C4×D4—1571no—
119C4⋊C8C2×D15711yescharacteristic
120C4⋊C8C2×D15711yescharacteristic
121#102—3141no—
122#102—3141no—
123#102—3142no—
124C4×C8—1571no—
125C2×Q8—21no—
126C2×Q8—3142no—
127C2×Q8—6282no—
128C2×Q8—6282no—
129C2×Q8—1571no—
130C2×D4—3141no—
131C2×D4—3141no—
132C2×D4—3141no—
133C2×D4—1571no—
134C22×C4—3142no—
135C22×C4—3141no—
136SD16—6282no—
137D8—6282no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3141no—
141C4⋊C4—21no—
142C4⋊C4—3142no—
143C4⋊C4—3141no—
144C4⋊C4—1571no—
145C4⋊C4—1571no—
146C4⋊C4D62812yes—
147C22⋊C4—3142no—
148C22⋊C4—3141no—
149C42—3141no—
150C42C22×D15711yescharacteristic
151C23—3141no—
152C23—6281no—
153Q8—41no—
154Q8—41no—
155Q8—3141no—
156Q8—6284no—
157Q8—1572no—
158D4—3141no—
159D4—6282no—
160D4—6281no—
161D4—6281no—
162D4—6281no—
163D4—6281no—
164D4—1572no—
165C2×C4—22no—
166C2×C4—3141no—
167C2×C4—3141no—
168C2×C4—3141no—
169C2×C4—6284no—
170C2×C4—6281no—
171C2×C4—6281no—
172C2×C4D4×D15711yescharacteristic
173C2×C4D4×D15711yescharacteristic
174C2×C4C2×D62811yescharacteristic
175C8—41no—
176C8—41no—
177C8—6281no—
178C22—3142no—
179C22—6281no—
180C22—6281no—
181C22—12561no—
182C22—12561no—
183C22—12561no—
184C22#1909774111yescharacteristic
185C4—21no—
186C4—42no—
187C4—3142no—
188C4—6281no—
189C4—6281no—
190C4D8×D15711yescharacteristic
191C4SD16×D15711yescharacteristic
192C4#1909779811yescharacteristic
193C4#1909780311yescharacteristic
194C2—6282no—
195C2—25121no—
196C2#2375740811yescharacteristic
197C2#2375741311yescharacteristic
198C2#2375744811yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#332×C157C2
socleC22×C157#120
prime radical#23757469C2
hypercenter#332D157
hypocenterC157#1627
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #23757938C2semidirect
2. #23757621C2semidirect
3. #23757474C2semidirect
5. #23757003C2semidirect
6. #23757000C2semidirect
7. #332×C157C2semidirect
55. Dic314D8semidirect
62. D628SD16semidirect
99. C157#1627semidirect
105. #332D157semidirect

Maximal subgroups

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

subgroupindex
#237579382
#237576212
#237574742
#237574692
#237570032
#237570002
#332×C1572
#1627157