Finite Group Databasebeta

← #34098595

Subgroups — #34098595

Order 20096.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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#23757810C211yesmaximal, characteristic
2#23757329C211yesmaximal, characteristic, complemented
3C4×C8×D157C211yesmaximal, characteristic, complemented
4Q8⋊#11929434C211yesmaximal, characteristic, complemented
5#23757002C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×C8×D157—21no—
9#19097766—21no—
10#19097762C411yescharacteristic
11#19097760C411yescharacteristic
12#19097756—21no—
13#19097747C411yescharacteristic
14#19097746C411yescharacteristic
15C4×D628—21no—
16C4×D628C2211yescharacteristic
17#19097728—21no—
18C42×D157C2211yescharacteristic
19C4⋊#11929434C2211yescharacteristic
20#19097683—21no—
21#19097681—21no—
22C4×#11929434C2211yescharacteristic
23C8×Dic157—21no—
24C4×Q8×C157C2211yescharacteristic
25C4⋊C8×C157C2211yescharacteristic
26C4×C8×C157C2211yescharacteristic
27C2×D628—21no—
28C2×D628C2×C411yescharacteristic
29C2×C4×D157—21no—
30C2×C4×D157C2×C411yescharacteristic
31C2×C4×D157D411yescharacteristic
32C2×C4×D157D411yescharacteristic
33C2×#11929434—21no—
34C2×#11929434—21no—
35C8×D157—24no—
36C4⋊Dic157—21no—
37C4⋊Dic157C2×C411yescharacteristic
38#14578933—21no—
39#14578933—21no—
40#14578931—21no—
41#14578931—21no—
42#14578931—21no—
43#14578931—21no—
44#14578931—21no—
45C4×Dic157C2×C411yescharacteristic
46C4×Dic157D411yescharacteristic
47C4×Dic157D411yescharacteristic
48C2×Q8×C157C2×C411yescharacteristic
49C2×C8×C157—21no—
50C2×C8×C157—21no—
51C4⋊C4×C157—21no—
52C4⋊C4×C157C2×C411yescharacteristic
53C42×C157—21no—
54C42×C157C2311yescharacteristic
55C22×D157—21no—
56C22×D157C22⋊C411yescharacteristic
57D628—21no—
58D628—21no—
59D628—42no—
60C2×Dic157—21no—
61C2×Dic157C22⋊C411yescharacteristic
62C2×Dic157C22⋊C411yescharacteristic
63C2×Dic157C22⋊C411yescharacteristic
64C4×D157—22no—
65C4×D157—21no—
66C4×D157—21no—
67C4×D157—21no—
68C4×D157—21no—
69C4×D157—41no—
70C4×D157—41no—
71#11929434—22no—
72#11929434—41no—
73Q8×C157—21no—
74Q8×C157—21no—
75C2×C4×C157—21no—
76C2×C4×C157—21no—
77C2×C4×C157C22×C411yescharacteristic
78C2×C4×C157C2×D411yescharacteristic
79C2×C4×C157C2×D411yescharacteristic
80C8×C157—22no—
81C8×C157—41no—
82C22×C157C2×C22⋊C411yescharacteristic
83C2×D157—21no—
84C2×D157—21no—
85C2×D157—41no—
86C2×D157—41no—
87C2×D157—41no—
88C2×D157#10412yes—
89C4×C157—21no—
90C4×C157—41no—
91C4×C157—41no—
92C4×C157#13111yescharacteristic
93C4×C157#13111yescharacteristic
94C4×C157#13511yescharacteristic
95C4×C157#13511yescharacteristic
96Dic157—21no—
97Dic157—21no—
98Dic157—21no—
99Dic157—41no—
100C2×C157#40811yescharacteristic
101C2×C157#41611yescharacteristic
102C2×C157C2×#10411yescharacteristic
103D157—22no—
104D157—81no—
105C157#149411yescharacteristic, complemented, Sylow
106#1494—1571nomaximal, Sylow
107#529—1571no—
108#432—1571no—
109C2×C4×C8—1571no—
110Q8⋊C8—1571no—
111Q8⋊C8D15711yescharacteristic, complemented
112D4⋊C8—1571no—
113D4⋊C8—1571no—
114C22×C8—3141no—
115#126—3141no—
116#125—1571no—
117#124—1571no—
118#123—3141no—
119C22⋊Q8—1571no—
120#121—1571no—
121C4×Q8C2×D15711yescharacteristic
122C4×D4—3141no—
123C4×D4—1571no—
124#117—3141no—
125C2×C42—1571no—
126C4⋊C8—1571no—
127C4⋊C8C2×D15711yescharacteristic
128C22⋊C8—3141no—
129#97—3141no—
130C4×C8—3141no—
131C4×C8—1571no—
132C4×C8C2×D15711yescharacteristic
133C2×Q8C4×D15711yescharacteristic
134C2×D4—3141no—
135C2×D4—1571no—
136C22×C4—3141no—
137C22×C4—1571no—
138C22×C4—1571no—
139C22×C4—1571no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—3144no—
143C2×C8—3141no—
144C2×C8—3141no—
145C4⋊C4—21no—
146C4⋊C4—3141no—
147C4⋊C4—3141no—
148C4⋊C4—3141no—
149C4⋊C4—1571no—
150C4⋊C4C4×D15711yescharacteristic
151C22⋊C4—3141no—
152C22⋊C4—3141no—
153C22⋊C4—3141no—
154C22⋊C4—3141no—
155C22⋊C4—3141no—
156C42—21no—
157C42—1571no—
158C42—1571no—
159C42—1571no—
160C42C22×D15711yescharacteristic
161C23—3141no—
162C23—1571no—
163Q8—21no—
164Q8—21no—
165D4—3141no—
166D4—3141no—
167D4—6282no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—3142no—
171C2×C4—3141no—
172C2×C4—3141no—
173C2×C4—3141no—
174C2×C4—3141no—
175C2×C4—3141no—
176C2×C4—6281no—
177C2×C4—6281no—
178C2×C4—1571no—
179C2×C4—1571no—
180C2×C4—1571no—
181C2×C4C2×C4×D15711yescharacteristic
182C2×C4D4×D15711yescharacteristic
183C2×C4D4×D15711yescharacteristic
184C8—22no—
185C8—41no—
186C8—3142no—
187C8—6281no—
188C22—3141no—
189C22—3141no—
190C22—6281no—
191C22—6281no—
192C22—6281no—
193C22—1572no—
194C22C22⋊C4×D15711yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—3141no—
199C4—3141no—
200C4—3141no—
201C4—6281no—
202C4#1909777711yescharacteristic
203C4#1909777911yescharacteristic
204C4#1909779211yescharacteristic
205C4#1909779411yescharacteristic
206C2—3142no—
207C2—12561no—
208C2#2375723811yescharacteristic
209C2#2375726811yescharacteristic
210C2#104×D15711yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
FittingQ8⋊C8×C157C2
socleC22×C157C2×C22⋊C4
prime radical#19097746C4
cosocleC42×C157C23
hypercenterQ8⋊C8D157
hypocenterC157#1494
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757329C2semidirect
3. C4×C8×D157C2semidirect
4. Q8⋊#11929434C2semidirect
5. #23757002C2semidirect
6. #23757000C2semidirect
7. Q8⋊C8×C157C2semidirect
105. C157#1494semidirect
111. Q8⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237578102
#237573292
C4×C8×D1572
Q8⋊#119294342
#237570022
#237570002
Q8⋊C8×C1572
#1494157