Finite Group Databasebeta

← C₂⁵×#124 #72291293

Subgroups — C₂⁵×#124 #72291293

Order 1024.

1060245 conjugacy classes of subgroups; only the 469973 normal subgroups are shown.

normal subgroups only

This group has 1060245 conjugacy classes of subgroups — too many to draw. Only its 469973 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁶×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×#124	C₂	1	248	yes	maximal, direct product factor
4	C₂⁵×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂⁵×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁸	C₂²	1	2	yes	—
7	C₂⁵×Q₈	—	2	64	no	—
8	C₂⁵×Q₈	C₂²	1	124	yes	complemented
9	C₂⁵×D₄	—	2	64	no	—
10	C₂⁵×D₄	C₂²	1	124	yes	—
11	C₂⁶×C₄	C₂²	1	2	yes	—
12	C₂⁶×C₄	C₂²	1	2	yes	—
13	C₂⁶×C₄	C₂²	1	1	yes	characteristic
14	C₂³×#124	C₂²	1	9920	yes	direct product factor
15	C₂⁴×C₂²⋊C₄	C₂²	1	496	yes	complemented
16	C₂⁴×C₄²	C₂²	1	124	yes	complemented
17	C₂⁷	—	2	64	no	—
18	C₂⁷	—	2	128	no	—
19	C₂⁷	C₂³	1	124	yes	—
20	C₂⁷	C₂³	1	1	yes	characteristic
21	C₂⁴×Q₈	—	2	3968	no	—
22	C₂⁴×Q₈	C₂³	1	2480	yes	complemented
23	C₂⁴×D₄	—	2	3968	no	—
24	C₂⁴×D₄	C₂³	1	2480	yes	—
25	C₂⁵×C₄	—	2	64	no	—
26	C₂⁵×C₄	—	2	64	no	—
27	C₂⁵×C₄	D₄	1	64	yes	—
28	C₂⁵×C₄	C₂³	1	124	yes	—
29	C₂⁵×C₄	C₂³	1	124	yes	—
30	C₂⁵×C₄	C₂³	1	62	yes	—
31	C₂²×#124	C₂³	1	79360	yes	direct product factor
32	C₂³×C₂²⋊C₄	C₂³	1	9920	yes	complemented
33	C₂³×C₄²	C₂³	1	2480	yes	complemented
34	C₂⁶	—	2	1984	no	—
35	C₂⁶	—	2	3968	no	—
36	C₂⁶	—	4	2048	no	—
37	C₂⁶	C₂×D₄	1	32	yes	—
38	C₂⁶	#41	1	64	yes	—
39	C₂⁶	C₂⁴	1	1240	yes	—
40	C₂⁶	C₂⁴	1	31	yes	—
41	C₂³×Q₈	—	2	39680	no	—
42	C₂³×Q₈	C₂⁴	1	9920	yes	complemented
43	C₂³×D₄	—	2	39680	no	—
44	C₂³×D₄	C₂⁴	1	9920	yes	—
45	C₂⁴×C₄	—	2	1984	no	—
46	C₂⁴×C₄	—	2	1984	no	—
47	C₂⁴×C₄	C₂×D₄	1	1984	yes	—
48	C₂⁴×C₄	C₂⁴	1	1240	yes	—
49	C₂⁴×C₄	C₂⁴	1	1240	yes	—
50	C₂⁴×C₄	C₂⁴	1	620	yes	—
51	C₂×#124	C₂⁴	1	126976	yes	direct product factor
52	C₂²×C₂²⋊C₄	C₂⁴	1	39680	yes	complemented
53	C₂²×C₄²	C₂⁴	1	9920	yes	complemented
54	C₂⁵	—	2	9920	no	—
55	C₂⁵	—	2	19840	no	—
56	C₂⁵	—	4	31744	no	—
57	C₂⁵	#124	1	1024	yes	direct product factor
58	C₂⁵	C₂²×D₄	1	496	yes	—
59	C₂⁵	C₂×#41	1	992	yes	—
60	C₂⁵	C₂⁵	1	2480	yes	—
61	C₂⁵	C₂⁵	1	155	yes	—
62	C₂²×Q₈	—	2	79360	no	—
63	C₂²×Q₈	C₂⁵	1	7936	yes	complemented
64	C₂²×D₄	—	2	79360	no	—
65	C₂²×D₄	C₂⁵	1	7936	yes	—
66	C₂³×C₄	—	2	9920	no	—
67	C₂³×C₄	—	2	9920	no	—
68	C₂³×C₄	C₂²×D₄	1	9920	yes	—
69	C₂³×C₄	C₂⁵	1	2480	yes	—
70	C₂³×C₄	C₂⁵	1	2480	yes	—
71	C₂³×C₄	C₂⁵	1	1240	yes	—
72	#124	C₂⁵	1	32768	yes	direct product factor
73	C₂×C₂²⋊C₄	C₂⁵	1	31744	yes	complemented
74	C₂×C₄²	C₂⁵	1	7936	yes	complemented
75	C₂⁴	—	2	9920	no	—
76	C₂⁴	—	2	19840	no	—
77	C₂⁴	—	4	79360	no	—
78	C₂⁴	C₂×#124	1	7936	yes	direct product factor
79	C₂⁴	C₂³×D₄	1	1240	yes	—
80	C₂⁴	C₂²×#41	1	2480	yes	—
81	C₂⁴	C₂⁶	1	992	yes	—
82	C₂⁴	C₂⁶	1	155	yes	—
83	C₂×Q₈	—	2	31744	no	—
84	C₂×Q₈	C₂⁶	1	1024	yes	complemented
85	C₂×D₄	—	2	31744	no	—
86	C₂×D₄	C₂⁶	1	1024	yes	—
87	C₂²×C₄	—	2	9920	no	—
88	C₂²×C₄	—	2	9920	no	—
89	C₂²×C₄	C₂³×D₄	1	9920	yes	—
90	C₂²×C₄	C₂⁶	1	992	yes	—
91	C₂²×C₄	C₂⁶	1	992	yes	—
92	C₂²×C₄	C₂⁶	1	496	yes	—
93	C₂²⋊C₄	C₂⁶	1	4096	yes	complemented
94	C₄²	C₂⁶	1	1024	yes	complemented
95	C₂³	—	2	1984	no	—
96	C₂³	—	2	3968	no	—
97	C₂³	—	4	39680	no	—
98	C₂³	C₂²×#124	1	9920	yes	direct product factor
99	C₂³	C₂⁴×D₄	1	620	yes	—
100	C₂³	C₂³×#41	1	1240	yes	—
101	C₂³	C₂⁷	1	64	yes	—
102	C₂³	C₂⁷	1	31	yes	—
103	Q₈	—	2	2048	no	—
104	D₄	—	2	2048	no	—
105	C₂×C₄	—	2	1984	no	—
106	C₂×C₄	—	2	1984	no	—
107	C₂×C₄	C₂⁴×D₄	1	1984	yes	—
108	C₂×C₄	C₂⁷	1	64	yes	—
109	C₂×C₄	C₂⁷	1	64	yes	—
110	C₂×C₄	C₂⁷	1	32	yes	—
111	C₂²	—	2	64	no	—
112	C₂²	—	2	128	no	—
113	C₂²	—	4	3968	no	—
114	C₂²	C₂³×#124	1	2480	yes	direct product factor
115	C₂²	C₂⁵×D₄	1	62	yes	—
116	C₂²	C₂⁴×#41	1	124	yes	—
117	C₂²	C₂⁸	1	1	yes	characteristic
118	C₄	—	2	64	no	—
119	C₄	—	2	64	no	—
120	C₄	C₂⁵×D₄	1	64	yes	—
121	C₂	—	4	64	no	—
122	C₂	C₂⁴×#124	1	124	yes	direct product factor
123	C₂	C₂⁶×D₄	1	1	yes	characteristic
124	C₂	C₂⁵×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁷	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁶×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁶×Q₈	C₂	semidirect
3. C₂⁴×#124	C₂	direct (N × H)
4. C₂⁵×C₂²⋊C₄	C₂	semidirect
5. C₂⁵×C₄²	C₂	semidirect
8. C₂⁵×Q₈	C₂²	semidirect
14. C₂³×#124	C₂²	direct (N × H)
15. C₂⁴×C₂²⋊C₄	C₂²	semidirect
16. C₂⁴×C₄²	C₂²	semidirect
22. C₂⁴×Q₈	C₂³	semidirect
31. C₂²×#124	C₂³	direct (N × H)
32. C₂³×C₂²⋊C₄	C₂³	semidirect
33. C₂³×C₄²	C₂³	semidirect
42. C₂³×Q₈	C₂⁴	semidirect
51. C₂×#124	C₂⁴	direct (N × H)
52. C₂²×C₂²⋊C₄	C₂⁴	semidirect
53. C₂²×C₄²	C₂⁴	semidirect
57. C₂⁵	#124	direct (N × H)
63. C₂²×Q₈	C₂⁵	semidirect
72. #124	C₂⁵	direct (N × H)
73. C₂×C₂²⋊C₄	C₂⁵	semidirect
74. C₂×C₄²	C₂⁵	semidirect
78. C₂⁴	C₂×#124	direct (N × H)
84. C₂×Q₈	C₂⁶	semidirect
93. C₂²⋊C₄	C₂⁶	semidirect
94. C₄²	C₂⁶	semidirect
98. C₂³	C₂²×#124	direct (N × H)
114. C₂²	C₂³×#124	direct (N × H)
122. C₂	C₂⁴×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×Q₈	2
C₂⁶×D₄	2
C₂⁴×#124	2
C₂⁵×C₂²⋊C₄	2
C₂⁵×C₄²	2