Finite Group Databasebeta

← #34098543

Subgroups — #34098543

Order 20096.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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#23757325C211yesmaximal, characteristic, complemented
3#23757324C211yesmaximal, characteristic, complemented
4Q8⋊#11929434C211yesmaximal, characteristic, complemented
5#23757002C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8#19097766—21no—
9#19097762C411yescharacteristic
10#19097760C411yescharacteristic
11#19097756—21no—
12#19097747C411yescharacteristic
13#19097746C411yescharacteristic
14C4×D628—21no—
15C4×D628C2211yescharacteristic
16#19097728—21no—
17C42×D157C2211yescharacteristic
18C4⋊#11929434C2211yescharacteristic
19#19097700—21no—
20#19097683—21no—
21#19097683—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—
35C4⋊Dic157—21no—
36C4⋊Dic157C2×C411yescharacteristic
37#14578933—21no—
38#14578933—21no—
39#14578931—21no—
40#14578931—21no—
41#14578931—21no—
42#14578931—21no—
43#14578931—21no—
44C4×Dic157C2×C411yescharacteristic
45C4×Dic157D411yescharacteristic
46C4×Dic157D411yescharacteristic
47C2×Q8×C157C2×C411yescharacteristic
48C2×C8×C157—21no—
49C2×C8×C157—21no—
50C4⋊C4×C157—21no—
51C4⋊C4×C157C2×C411yescharacteristic
52C42×C157—21no—
53C42×C157C2311yescharacteristic
54C22×D157—21no—
55C22×D157C22⋊C411yescharacteristic
56D628—21no—
57D628—21no—
58D628—42no—
59C2×Dic157—21no—
60C2×Dic157C22⋊C411yescharacteristic
61C2×Dic157C22⋊C411yescharacteristic
62C2×Dic157C22⋊C411yescharacteristic
63C4×D157—21no—
64C4×D157—21no—
65C4×D157—21no—
66C4×D157—21no—
67C4×D157—41no—
68C4×D157—41no—
69C4×D157—41no—
70#11929434—41no—
71#11929434—41no—
72Q8×C157—21no—
73Q8×C157—21no—
74C2×C4×C157—21no—
75C2×C4×C157—21no—
76C2×C4×C157C22×C411yescharacteristic
77C2×C4×C157C2×D411yescharacteristic
78C2×C4×C157C2×D411yescharacteristic
79C8×C157—41no—
80C8×C157—41no—
81C22×C157C2×C22⋊C411yescharacteristic
82C2×D157—21no—
83C2×D157—21no—
84C2×D157—21no—
85C2×D157—41no—
86C2×D157—41no—
87C2×D157—41no—
88C4×C157—21no—
89C4×C157—41no—
90C4×C157—41no—
91C4×C157#13111yescharacteristic
92C4×C157#13111yescharacteristic
93C4×C157#13511yescharacteristic
94C4×C157#13511yescharacteristic
95Dic157—21no—
96Dic157—21no—
97Dic157—21no—
98Dic157—41no—
99C2×C157#40811yescharacteristic
100C2×C157#41611yescharacteristic
101C2×C157#42111yescharacteristic
102D157—41no—
103D157—81no—
104C157#148211yescharacteristic, complemented, Sylow
105#1482—1571nomaximal, Sylow
106#529—1571no—
107#431—1571no—
108#431—1571no—
109Q8⋊C8—1571no—
110Q8⋊C8D15711yescharacteristic, complemented
111D4⋊C8—1571no—
112D4⋊C8—1571no—
113#126—3141no—
114#125—1571no—
115#124—1571no—
116#123—3141no—
117C22⋊Q8—1571no—
118#121—1571no—
119C4×Q8C2×D15711yescharacteristic
120C4×D4—3141no—
121C4×D4—1571no—
122#117—3141no—
123C2×C42—1571no—
124C4⋊C8—3141no—
125C4⋊C8—1571no—
126C4⋊C8C2×D15711yescharacteristic
127C22⋊C8—3141no—
128C22⋊C8—3141no—
129C4×C8—3141no—
130C4×C8—1571no—
131C4×C8C2×D15711yescharacteristic
132C2×Q8C4×D15711yescharacteristic
133C2×D4—3141no—
134C2×D4—1571no—
135C22×C4—3141no—
136C22×C4—1571no—
137C22×C4—1571no—
138C22×C4—1571no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—3141no—
142C2×C8—3141no—
143C4⋊C4—21no—
144C4⋊C4—3141no—
145C4⋊C4—3141no—
146C4⋊C4—3141no—
147C4⋊C4—1571no—
148C4⋊C4C4×D15711yescharacteristic
149C22⋊C4—3141no—
150C22⋊C4—3141no—
151C22⋊C4—3141no—
152C22⋊C4—3141no—
153C22⋊C4—3141no—
154C42—21no—
155C42—1571no—
156C42—1571no—
157C42—1571no—
158C42C22×D15711yescharacteristic
159C23—3141no—
160C23—1571no—
161Q8—21no—
162Q8—21no—
163D4—3141no—
164D4—3141no—
165D4—6282no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—3141no—
169C2×C4—3141no—
170C2×C4—3141no—
171C2×C4—3141no—
172C2×C4—3141no—
173C2×C4—6281no—
174C2×C4—6281no—
175C2×C4—6281no—
176C2×C4—1571no—
177C2×C4—1571no—
178C2×C4—1571no—
179C2×C4C2×C4×D15711yescharacteristic
180C2×C4D4×D15711yescharacteristic
181C2×C4D4×D15711yescharacteristic
182C8—41no—
183C8—41no—
184C8—6281no—
185C8—6281no—
186C22—3141no—
187C22—3141no—
188C22—3141no—
189C22—6281no—
190C22—6281no—
191C22—6281no—
192C22C22⋊C4×D15711yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—3141no—
197C4—3141no—
198C4—3141no—
199C4—6281no—
200C4#1909777711yescharacteristic
201C4#1909777911yescharacteristic
202C4#1909779211yescharacteristic
203C4#1909779411yescharacteristic
204C2—6281no—
205C2—12561no—
206C2#2375723811yescharacteristic
207C2#2375726811yescharacteristic
208C2#2375728811yescharacteristic
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#1482
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. #23757325C2semidirect
3. #23757324C2semidirect
4. Q8⋊#11929434C2semidirect
5. #23757002C2semidirect
6. #23757000C2semidirect
7. Q8⋊C8×C157C2semidirect
104. C157#1482semidirect
110. Q8⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237578102
#237573252
#237573242
Q8⋊#119294342
#237570022
#237570002
Q8⋊C8×C1572
#1482157