Finite Group Databasebeta

← #360×D127 #30640460

Subgroups — #360×D127 #30640460

Order 16256.

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×D127C211yesmaximal, characteristic
2M32×D127C211yesmaximal, characteristic, complemented
3#106×D127C211yesmaximal, characteristic, complemented
4#22549687C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic, complemented
6#22549685C211yesmaximal, characteristic, complemented
7#360×C127C211yesmaximal, characteristic, complemented
8#41×D127—21no—
9#16861232C411yescharacteristic, complemented
10#16861231C2211yescharacteristic
11#16861230—21no—
12#16861229—21no—
13#16861228C411yescharacteristic, complemented
14Q16×D127C411yescharacteristic, complemented
15SD16×D127—21no—
16D8×D127C411yescharacteristic, complemented
17C2×C8×D127C2211yescharacteristic
18C4⋊C4×D127—21no—
19#16861149C2211yescharacteristic, complemented
20#16861148—21no—
21C16×D127—21no—
22#16861141C2211yescharacteristic, complemented
23#16861140—21no—
24#135×C127C2211yescharacteristic
25M32×C127C2211yescharacteristic, complemented
26#106×C127C2211yescharacteristic, complemented
27#14523249—21no—
28#14523248—21no—
29#14523248—41no—
30#14523247—21no—
31#14523247—41no—
32Q8×D127—21no—
33D4×D127—21no—
34D4×D127—41no—
35C2×C4×D127—41no—
36C2×C4×D127—41no—
37C2×C4×D127D411yescharacteristic
38Dic508C2×C411yescharacteristic, complemented
39#14523238—21no—
40#14523237—21no—
41#14523236—21no—
42#14523235—21no—
43D1016C2×C411yescharacteristic, complemented
44#14523233—21no—
45C2×C127⋊C8D411yescharacteristic
46C8×D127C2×C411yescharacteristic
47C8×D127C2×C411yescharacteristic
48C8×D127D411yescharacteristic
49C8×D127D411yescharacteristic
50C4⋊Dic127—21no—
51Dic127⋊C4—41no—
52C127⋊C16—21no—
53#41×C127—21no—
54Q16×C127C2×C411yescharacteristic, complemented
55SD16×C127—21no—
56D8×C127C2×C411yescharacteristic, complemented
57C2×C8×C127C2311yescharacteristic
58C4⋊C4×C127—21no—
59C16×C127—21no—
60C22×D127—41no—
61C22×D127D811yescharacteristic
62Dic254—21no—
63Dic254—41no—
64D508—21no—
65D508—41no—
66C22⋊D127—41no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127D811yescharacteristic
72C4×D127—41no—
73C4×D127—41no—
74C4×D127—41no—
75C4×D127—82no—
76C4×D127C22⋊C411yescharacteristic
77C4×D127C22⋊C411yescharacteristic
78C4×D127SD1611yescharacteristic
79C4×D127SD1611yescharacteristic
80C127⋊C8C22⋊C411yescharacteristic
81C127⋊C8C22⋊C411yescharacteristic
82Q8×C127—21no—
83D4×C127—21no—
84D4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127C2×D411yescharacteristic
88C8×C127C22×C411yescharacteristic
89C8×C127C2×D411yescharacteristic
90C22×C127—41no—
91C22×C127C2×D811yescharacteristic
92C2×D127—22no—
93C2×D127—41no—
94C2×D127—82no—
95C2×D127#10211yescharacteristic
96C2×D127#10211yescharacteristic
97C4×C127—41no—
98C4×C127—81no—
99C4×C127C2×C22⋊C411yescharacteristic
100C4×C127C2×SD1611yescharacteristic
101Dic127—41no—
102Dic127—81no—
103Dic127#10211yescharacteristic
104Dic127#10211yescharacteristic
105C2×C127—21no—
106C2×C127—81no—
107C2×C127C2×#10211yescharacteristic
108D127—21no—
109D127—81no—
110D127#36012yesdirect product factor
111C2×#360—1271nomaximal, Sylow
112C127C2×#36011yescharacteristic, complemented, Sylow
113C2×#135—1271no—
114C2×M32—1271no—
115C2×#106—1271no—
116#360—1271no—
117#360—1271no—
118#360—1271no—
119#360D12711yescharacteristic, direct product factor
120C2×#41—2541no—
121#135—1271no—
122#135—1271no—
123#135—1271no—
124#135—2541no—
125#135—2541no—
126#135C2×D12711yescharacteristic
127C2×Q16—1271no—
128C2×SD16—2541no—
129C2×D8—1271no—
130C22×C8—1271no—
131C2×C4⋊C4—2541no—
132M32—1271no—
133M32—2541no—
134M32C2×D12711yescharacteristic, complemented
135C2×C16—2541no—
136#106—1271no—
137#106—2541no—
138#106C2×D12711yescharacteristic, complemented
139#41—21no—
140#41—5081no—
141#41—5081no—
142#41—2541no—
143#41—2541no—
144#41—2541no—
145C2×Q8—2541no—
146C2×D4—5081no—
147C2×D4—2541no—
148C22×C4—1271no—
149C22×C4—5081no—
150C22×C4—5081no—
151Q16—1271no—
152Q16—2541no—
153Q16C4×D12711yescharacteristic, complemented
154SD16—21no—
155SD16—2541no—
156SD16—2541no—
157SD16—2541no—
158D8—1271no—
159D8—2541no—
160D8C4×D12711yescharacteristic, complemented
161C2×C8—1271no—
162C2×C8—1271no—
163C2×C8—1271no—
164C2×C8—1271no—
165C2×C8—1271no—
166C2×C8C22×D12711yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—5081no—
169C4⋊C4—2541no—
170C16—21no—
171C16—2541no—
172C23—1271no—
173C23—5081no—
174Q8—21no—
175Q8—5081no—
176Q8—2541no—
177D4—21no—
178D4—41no—
179D4—5081no—
180D4—5081no—
181D4—5081no—
182D4—5081no—
183D4—2541no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1271no—
187C2×C4—1271no—
188C2×C4—1271no—
189C2×C4—1271no—
190C2×C4—1271no—
191C2×C4—10162no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4D4×D12711yescharacteristic
198C8—1271no—
199C8—1271no—
200C8C2×C4×D12711yescharacteristic
201C8D4×D12711yescharacteristic
202C22—41no—
203C22—1271no—
204C22—1271no—
205C22—10162no—
206C22—5081no—
207C22—2542no—
208C22D8×D12711yescharacteristic
209C4—41no—
210C4—81no—
211C4—1271no—
212C4—1271no—
213C4—10161no—
214C4—5081no—
215C4C22⋊C4×D12711yescharacteristic
216C4SD16×D12711yescharacteristic
217C2—21no—
218C2—81no—
219C2—1272no—
220C2—10161no—
221C2—2541no—
222C2#102×D12711yescharacteristic
subgroupG/H
centerC2#102×D127
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#360×C127C2
socleC2×C127C2×#102
prime radical#135×D127C2
cosocleC2×C8×C127C23
hypercenter#360D127
hypocenterC127C2×#360
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. M32×D127C2semidirect
3. #106×D127C2semidirect
4. #22549687C2semidirect
5. #22549686C2semidirect
6. #22549685C2semidirect
7. #360×C127C2semidirect
9. #16861232C4semidirect
13. #16861228C4semidirect
14. Q16×D127C4semidirect
16. D8×D127C4semidirect
19. #16861149C22semidirect
22. #16861141C22semidirect
25. M32×C127C22semidirect
26. #106×C127C22semidirect
38. Dic508C2×C4semidirect
43. D1016C2×C4semidirect
54. Q16×C127C2×C4semidirect
56. D8×C127C2×C4semidirect
110. D127#360direct (N × H)
112. C127C2×#360semidirect
119. #360D127direct (N × H)
134. M32C2×D127semidirect
138. #106C2×D127semidirect
153. Q16C4×D127semidirect
160. D8C4×D127semidirect

Maximal subgroups

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

subgroupindex
#135×D1272
M32×D1272
#106×D1272
#225496872
#225496862
#225496852
#360×C1272
C2×#360127