Subgroups — C₂⁵×#117 #72295514

Order 1024.

1060245 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#117	C₂	1	248	yes	maximal, direct product factor
3	C₂⁵×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂⁵×C₂²⋊C₄	C₂	1	2	yes	maximal
5	C₂⁵×C₄²	C₂	1	2	yes	maximal, complemented
6	C₂⁸	C₂²	1	1	yes	characteristic
7	C₂⁶×C₄	C₄	1	128	yes	complemented
8	C₂⁶×C₄	C₂²	1	4	yes	—
9	C₂⁶×C₄	C₂²	1	1	yes	characteristic
10	C₂⁶×C₄	C₂²	1	124	yes	—
11	C₂⁶×C₄	C₂²	1	1	yes	characteristic
12	C₂³×#117	C₂²	1	9920	yes	direct product factor
13	C₂⁴×C₄⋊C₄	C₂²	1	248	yes	complemented
14	C₂⁴×C₂²⋊C₄	C₂²	1	248	yes	—
15	C₂⁴×C₄²	C₂²	1	248	yes	complemented
16	C₂⁷	—	2	64	no	—
17	C₂⁷	C₂×C₄	1	64	yes	—
18	C₂⁷	C₂³	1	62	yes	—
19	C₂⁷	C₂³	1	1	yes	characteristic
20	C₂⁵×C₄	—	2	128	no	—
21	C₂⁵×C₄	C₂×C₄	1	64	yes	—
22	C₂⁵×C₄	C₂×C₄	1	64	yes	—
23	C₂⁵×C₄	C₂×C₄	1	7936	yes	complemented
24	C₂⁵×C₄	C₂³	1	248	yes	—
25	C₂⁵×C₄	C₂³	1	62	yes	—
26	C₂⁵×C₄	C₂³	1	62	yes	—
27	C₂⁵×C₄	C₂³	1	2480	yes	—
28	C₂²×#117	C₂³	1	79360	yes	direct product factor
29	C₂³×C₄⋊C₄	C₂³	1	4960	yes	complemented
30	C₂³×C₂²⋊C₄	C₂³	1	4960	yes	—
31	C₂³×C₄²	C₂³	1	4960	yes	complemented
32	C₂⁶	—	2	2048	no	—
33	C₂⁶	—	2	1984	no	—
34	C₂⁶	C₂²×C₄	1	1984	yes	—
35	C₂⁶	C₂²×C₄	1	32	yes	—
36	C₂⁶	#41	1	64	yes	—
37	C₂⁶	C₂⁴	1	620	yes	—
38	C₂⁶	C₂⁴	1	31	yes	—
39	C₂⁴×C₄	—	2	3968	no	—
40	C₂⁴×C₄	C₂²×C₄	1	1984	yes	—
41	C₂⁴×C₄	C₂²×C₄	1	1984	yes	—
42	C₂⁴×C₄	C₂²×C₄	1	79360	yes	complemented
43	C₂⁴×C₄	C₂⁴	1	2480	yes	—
44	C₂⁴×C₄	C₂⁴	1	620	yes	—
45	C₂⁴×C₄	C₂⁴	1	620	yes	—
46	C₂⁴×C₄	C₂⁴	1	9920	yes	—
47	C₂×#117	C₂⁴	1	126976	yes	direct product factor
48	C₂²×C₄⋊C₄	C₂⁴	1	19840	yes	complemented
49	C₂²×C₂²⋊C₄	C₂⁴	1	19840	yes	—
50	C₂²×C₄²	C₂⁴	1	19840	yes	complemented
51	C₂⁵	—	2	31744	no	—
52	C₂⁵	—	2	9920	no	—
53	C₂⁵	#117	1	1024	yes	direct product factor
54	C₂⁵	C₂³×C₄	1	9920	yes	—
55	C₂⁵	C₂³×C₄	1	496	yes	—
56	C₂⁵	C₂×#41	1	992	yes	—
57	C₂⁵	C₂⁵	1	1240	yes	—
58	C₂⁵	C₂⁵	1	155	yes	—
59	C₂³×C₄	—	2	19840	no	—
60	C₂³×C₄	C₂³×C₄	1	9920	yes	—
61	C₂³×C₄	C₂³×C₄	1	9920	yes	—
62	C₂³×C₄	C₂³×C₄	1	158720	yes	complemented
63	C₂³×C₄	C₂⁵	1	4960	yes	—
64	C₂³×C₄	C₂⁵	1	1240	yes	—
65	C₂³×C₄	C₂⁵	1	1240	yes	—
66	C₂³×C₄	C₂⁵	1	7936	yes	—
67	#117	C₂⁵	1	32768	yes	direct product factor
68	C₂×C₄⋊C₄	C₂⁵	1	15872	yes	complemented
69	C₂×C₂²⋊C₄	C₂⁵	1	15872	yes	—
70	C₂×C₄²	C₂⁵	1	15872	yes	complemented
71	C₂⁴	—	2	9920	no	—
72	C₂⁴	—	2	79360	no	—
73	C₂⁴	C₂×#117	1	7936	yes	direct product factor
74	C₂⁴	C₂⁴×C₄	1	9920	yes	—
75	C₂⁴	C₂⁴×C₄	1	1240	yes	—
76	C₂⁴	C₂²×#41	1	2480	yes	—
77	C₂⁴	C₂⁶	1	496	yes	—
78	C₂⁴	C₂⁶	1	155	yes	—
79	C₂²×C₄	—	2	19840	no	—
80	C₂²×C₄	C₂⁴×C₄	1	9920	yes	—
81	C₂²×C₄	C₂⁴×C₄	1	9920	yes	—
82	C₂²×C₄	C₂⁴×C₄	1	63488	yes	complemented
83	C₂²×C₄	C₂⁶	1	1984	yes	—
84	C₂²×C₄	C₂⁶	1	496	yes	—
85	C₂²×C₄	C₂⁶	1	496	yes	—
86	C₂²×C₄	C₂⁶	1	1024	yes	—
87	C₄⋊C₄	C₂⁶	1	2048	yes	complemented
88	C₂²⋊C₄	C₂⁶	1	2048	yes	—
89	C₄²	C₂⁶	1	2048	yes	complemented
90	C₂³	—	2	1984	no	—
91	C₂³	—	2	39680	no	—
92	C₂³	C₂²×#117	1	9920	yes	direct product factor
93	C₂³	C₂⁵×C₄	1	1984	yes	—
94	C₂³	C₂⁵×C₄	1	620	yes	—
95	C₂³	C₂³×#41	1	1240	yes	—
96	C₂³	C₂⁷	1	32	yes	—
97	C₂³	C₂⁷	1	31	yes	—
98	C₂×C₄	—	2	3968	no	—
99	C₂×C₄	C₂⁵×C₄	1	1984	yes	—
100	C₂×C₄	C₂⁵×C₄	1	1984	yes	—
101	C₂×C₄	C₂⁵×C₄	1	4096	yes	complemented
102	C₂×C₄	C₂⁷	1	128	yes	—
103	C₂×C₄	C₂⁷	1	32	yes	—
104	C₂×C₄	C₂⁷	1	32	yes	—
105	C₂²	—	2	3968	no	—
106	C₂²	—	2	64	no	—
107	C₂²	C₂³×#117	1	2480	yes	direct product factor
108	C₂²	C₂⁶×C₄	1	64	yes	—
109	C₂²	C₂⁶×C₄	1	62	yes	—
110	C₂²	C₂⁴×#41	1	124	yes	—
111	C₂²	C₂⁸	1	1	yes	characteristic
112	C₄	—	2	128	no	—
113	C₄	C₂⁶×C₄	1	64	yes	—
114	C₄	C₂⁶×C₄	1	64	yes	—
115	C₂	—	2	64	no	—
116	C₂	C₂⁴×#117	1	124	yes	direct product factor
117	C₂	C₂⁷×C₄	1	1	yes	characteristic
118	C₂	C₂⁵×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁶×C₄	C₂²
commutator	C₂	C₂⁷×C₄
Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁴×#117	C₂	direct (N × H)
3. C₂⁵×C₄⋊C₄	C₂	semidirect
5. C₂⁵×C₄²	C₂	semidirect
7. C₂⁶×C₄	C₄	semidirect
12. C₂³×#117	C₂²	direct (N × H)
13. C₂⁴×C₄⋊C₄	C₂²	semidirect
15. C₂⁴×C₄²	C₂²	semidirect
23. C₂⁵×C₄	C₂×C₄	semidirect
28. C₂²×#117	C₂³	direct (N × H)
29. C₂³×C₄⋊C₄	C₂³	semidirect
31. C₂³×C₄²	C₂³	semidirect
42. C₂⁴×C₄	C₂²×C₄	semidirect
47. C₂×#117	C₂⁴	direct (N × H)
48. C₂²×C₄⋊C₄	C₂⁴	semidirect
50. C₂²×C₄²	C₂⁴	semidirect
53. C₂⁵	#117	direct (N × H)
62. C₂³×C₄	C₂³×C₄	semidirect
67. #117	C₂⁵	direct (N × H)
68. C₂×C₄⋊C₄	C₂⁵	semidirect
70. C₂×C₄²	C₂⁵	semidirect
73. C₂⁴	C₂×#117	direct (N × H)
82. C₂²×C₄	C₂⁴×C₄	semidirect
87. C₄⋊C₄	C₂⁶	semidirect
89. C₄²	C₂⁶	semidirect
92. C₂³	C₂²×#117	direct (N × H)
101. C₂×C₄	C₂⁵×C₄	semidirect
107. C₂²	C₂³×#117	direct (N × H)
116. C₂	C₂⁴×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷×C₄	2
C₂⁴×#117	2
C₂⁵×C₄⋊C₄	2
C₂⁵×C₂²⋊C₄	2
C₂⁵×C₄²	2

Subgroups — C25×#117 #72295514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×#117 #72295514