Finite Group Databasebeta

← #360×D157 #34101866

Subgroups — #360×D157 #34101866

Order 20096.

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

normal subgroups only

This group has 232 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#135×D157C211yesmaximal, characteristic
2M32×D157C211yesmaximal, characteristic, complemented
3#106×D157C211yesmaximal, characteristic, complemented
4#23757084C211yesmaximal, characteristic, complemented
5#23757083C211yesmaximal, characteristic, complemented
6#23757082C211yesmaximal, characteristic, complemented
7#360×C157C211yesmaximal, characteristic, complemented
8#41×D157—21no—
9#19097794C411yescharacteristic, complemented
10#19097793C2211yescharacteristic
11#19097792—21no—
12#19097791—21no—
13#19097790C411yescharacteristic, complemented
14Q16×D157C411yescharacteristic, complemented
15SD16×D157—21no—
16D8×D157C411yescharacteristic, complemented
17C2×C8×D157C2211yescharacteristic
18C4⋊C4×D157—21no—
19#19097711C2211yescharacteristic, complemented
20#19097710—21no—
21C16×D157—21no—
22#19097703C2211yescharacteristic, complemented
23#19097702—21no—
24#135×C157C2211yescharacteristic
25M32×C157C2211yescharacteristic, complemented
26#106×C157C2211yescharacteristic, complemented
27#14578955—21no—
28#14578954—21no—
29#14578954—41no—
30#14578953—21no—
31#14578953—41no—
32Q8×D157—21no—
33D4×D157—21no—
34D4×D157—41no—
35C2×C4×D157—41no—
36C2×C4×D157—41no—
37C2×C4×D157D411yescharacteristic
38Dic628C2×C411yescharacteristic, complemented
39#14578944—21no—
40#14578943—21no—
41#14578942—21no—
42#14578941—21no—
43D1256C2×C411yescharacteristic, complemented
44#14578939—21no—
45C2×#11929434D411yescharacteristic
46C8×D157C2×C411yescharacteristic
47C8×D157C2×C411yescharacteristic
48C8×D157D411yescharacteristic
49C8×D157D411yescharacteristic
50C4⋊Dic157—21no—
51#14578933—41no—
52#14578929—21no—
53#41×C157—21no—
54Q16×C157C2×C411yescharacteristic, complemented
55SD16×C157—21no—
56D8×C157C2×C411yescharacteristic, complemented
57C2×C8×C157C2311yescharacteristic
58C4⋊C4×C157—21no—
59C16×C157—21no—
60C22×D157—41no—
61C22×D157D811yescharacteristic
62Dic314—21no—
63Dic314—41no—
64D628—21no—
65D628—41no—
66C22⋊D157—41no—
67C22⋊D157—41no—
68C22⋊D157—41no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157D811yescharacteristic
72C4×D157—41no—
73C4×D157—41no—
74C4×D157—41no—
75C4×D157—82no—
76C4×D157C22⋊C411yescharacteristic
77C4×D157C22⋊C411yescharacteristic
78C4×D157SD1611yescharacteristic
79C4×D157SD1611yescharacteristic
80#11929434C22⋊C411yescharacteristic
81#11929434C22⋊C411yescharacteristic
82Q8×C157—21no—
83D4×C157—21no—
84D4×C157—41no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
87C2×C4×C157C2×D411yescharacteristic
88C8×C157C22×C411yescharacteristic
89C8×C157C2×D411yescharacteristic
90C22×C157—41no—
91C22×C157C2×D811yescharacteristic
92C2×D157—22no—
93C2×D157—41no—
94C2×D157—82no—
95C2×D157#10211yescharacteristic
96C2×D157#10211yescharacteristic
97C4×C157—41no—
98C4×C157—81no—
99C4×C157C2×C22⋊C411yescharacteristic
100C4×C157C2×SD1611yescharacteristic
101Dic157—41no—
102Dic157—81no—
103Dic157#10211yescharacteristic
104Dic157#10211yescharacteristic
105C2×C157—21no—
106C2×C157—81no—
107C2×C157C2×#10211yescharacteristic
108D157—21no—
109D157—81no—
110D157#36012yesdirect product factor
111C157C2×#36011yescharacteristic, complemented, Sylow
112C2×#360—1571nomaximal, Sylow
113C2×#135—1571no—
114C2×M32—1571no—
115C2×#106—1571no—
116#360—1571no—
117#360—1571no—
118#360—1571no—
119#360D15711yescharacteristic, direct product factor
120C2×#41—3141no—
121#135—3141no—
122#135—3141no—
123#135—1571no—
124#135—1571no—
125#135—1571no—
126#135C2×D15711yescharacteristic
127C2×Q16—1571no—
128C2×SD16—3141no—
129C2×D8—1571no—
130C22×C8—1571no—
131C2×C4⋊C4—3141no—
132M32—3141no—
133M32—1571no—
134M32C2×D15711yescharacteristic, complemented
135C2×C16—3141no—
136#106—3141no—
137#106—1571no—
138#106C2×D15711yescharacteristic, complemented
139#41—21no—
140#41—3141no—
141#41—3141no—
142#41—3141no—
143#41—6281no—
144#41—6281no—
145C2×Q8—3141no—
146C2×D4—3141no—
147C2×D4—6281no—
148C22×C4—6281no—
149C22×C4—6281no—
150C22×C4—1571no—
151Q16—3141no—
152Q16—1571no—
153Q16C4×D15711yescharacteristic, complemented
154SD16—21no—
155SD16—3141no—
156SD16—3141no—
157SD16—3141no—
158D8—3141no—
159D8—1571no—
160D8C4×D15711yescharacteristic, complemented
161C2×C8—1571no—
162C2×C8—1571no—
163C2×C8—1571no—
164C2×C8—1571no—
165C2×C8—1571no—
166C2×C8C22×D15711yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—3141no—
169C4⋊C4—6281no—
170C16—21no—
171C16—3141no—
172C23—6281no—
173C23—1571no—
174Q8—21no—
175Q8—3141no—
176Q8—6281no—
177D4—21no—
178D4—41no—
179D4—3141no—
180D4—6281no—
181D4—6281no—
182D4—6281no—
183D4—6281no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—6281no—
187C2×C4—6281no—
188C2×C4—6281no—
189C2×C4—6281no—
190C2×C4—6281no—
191C2×C4—1571no—
192C2×C4—1571no—
193C2×C4—1571no—
194C2×C4—1571no—
195C2×C4—1571no—
196C2×C4—12562no—
197C2×C4D4×D15711yescharacteristic
198C8—1571no—
199C8—1571no—
200C8C2×C4×D15711yescharacteristic
201C8D4×D15711yescharacteristic
202C22—41no—
203C22—3142no—
204C22—6281no—
205C22—1571no—
206C22—1571no—
207C22—12562no—
208C22D8×D15711yescharacteristic
209C4—41no—
210C4—81no—
211C4—6281no—
212C4—1571no—
213C4—1571no—
214C4—12561no—
215C4C22⋊C4×D15711yescharacteristic
216C4SD16×D15711yescharacteristic
217C2—21no—
218C2—81no—
219C2—3141no—
220C2—1572no—
221C2—12561no—
222C2#102×D15711yescharacteristic
subgroupG/H
centerC2#102×D157
commutatorC8×C157C22×C4
FrattiniC2×C8C22×D157
Fitting#360×C157C2
socleC2×C157C2×#102
prime radical#135×D157C2
cosocleC2×C8×C157C23
hypercenter#360D157
hypocenterC157C2×#360
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. M32×D157C2semidirect
3. #106×D157C2semidirect
4. #23757084C2semidirect
5. #23757083C2semidirect
6. #23757082C2semidirect
7. #360×C157C2semidirect
9. #19097794C4semidirect
13. #19097790C4semidirect
14. Q16×D157C4semidirect
16. D8×D157C4semidirect
19. #19097711C22semidirect
22. #19097703C22semidirect
25. M32×C157C22semidirect
26. #106×C157C22semidirect
38. Dic628C2×C4semidirect
43. D1256C2×C4semidirect
54. Q16×C157C2×C4semidirect
56. D8×C157C2×C4semidirect
110. D157#360direct (N × H)
111. C157C2×#360semidirect
119. #360D157direct (N × H)
134. M32C2×D157semidirect
138. #106C2×D157semidirect
153. Q16C4×D157semidirect
160. D8C4×D157semidirect

Maximal subgroups

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

subgroupindex
#135×D1572
M32×D1572
#106×D1572
#237570842
#237570832
#237570822
#360×C1572
C2×#360157