Finite Group Databasebeta

← C₂³×D₄×D₃₁ #16860502

Subgroups — C₂³×D₄×D₃₁ #16860502

Order 3968.

10552 conjugacy classes of subgroups; only the 3557 normal subgroups are shown.

normal subgroups only

This group has 10552 conjugacy classes of subgroups — too many to draw. Only its 3557 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	C₂⁵×D₃₁	C₂	1	2	yes	maximal, complemented
2	C₂³×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
4	C₂²×D₄×D₃₁	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₃₁	C₂²	1	2	yes	complemented
8	C₂⁴×D₃₁	—	2	32	no	—
9	C₂⁴×D₃₁	C₂²	1	1	yes	characteristic
10	C₂⁴×D₃₁	C₂²	1	56	yes	complemented
11	C₂⁴×D₃₁	C₂²	1	2	yes	complemented
12	C₂²×D₄×C₃₁	C₂²	1	28	yes	complemented
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₃₁	C₂²	1	56	yes	complemented
15	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₄×D₃₁	C₂²	1	448	yes	direct product factor
17	C₂²×D₁₂₄	C₂²	1	28	yes	complemented
18	C₂²×C₄×D₃₁	C₂²	1	28	yes	complemented
19	C₂⁴×C₃₁	—	2	16	no	—
20	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
21	C₂⁴×C₃₁	C₂³	1	28	yes	complemented
22	C₂³×D₃₁	—	2	16	no	—
23	C₂³×D₃₁	—	2	448	no	—
24	C₂³×D₃₁	D₄	1	16	yes	direct product factor
25	C₂³×D₃₁	C₂³	1	14	yes	—
26	C₂³×D₃₁	C₂³	1	224	yes	complemented
27	C₂³×D₃₁	C₂³	1	28	yes	complemented
28	C₂×D₄×C₃₁	C₂³	1	112	yes	complemented
29	C₂²×C₄×C₃₁	C₂³	1	14	yes	complemented
30	C₂×C₂²⋊D₃₁	C₂³	1	224	yes	complemented
31	C₂²×Dic₃₁	C₂³	1	14	yes	complemented
32	D₄×D₃₁	C₂³	1	512	yes	direct product factor
33	C₂×D₁₂₄	C₂³	1	112	yes	complemented
34	C₂×C₄×D₃₁	C₂³	1	112	yes	complemented
35	C₂³×C₃₁	—	2	112	no	—
36	C₂³×C₃₁	C₂×D₄	1	8	yes	complemented
37	C₂³×C₃₁	C₂⁴	1	7	yes	—
38	C₂³×C₃₁	C₂⁴	1	56	yes	complemented
39	C₂²×D₃₁	—	2	112	no	—
40	C₂²×D₃₁	—	2	896	no	—
41	C₂²×D₃₁	C₂×D₄	1	112	yes	direct product factor
42	C₂²×D₃₁	C₂⁴	1	28	yes	—
43	C₂²×D₃₁	C₂⁴	1	128	yes	complemented
44	C₂²×D₃₁	C₂⁴	1	56	yes	complemented
45	D₄×C₃₁	C₂⁴	1	64	yes	complemented
46	C₂×C₄×C₃₁	C₂⁴	1	28	yes	complemented
47	C₂²⋊D₃₁	C₂⁴	1	128	yes	complemented
48	C₂×Dic₃₁	C₂⁴	1	28	yes	complemented
49	D₁₂₄	C₂⁴	1	64	yes	complemented
50	C₄×D₃₁	C₂⁴	1	64	yes	complemented
51	C₂⁴×D₄	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	2	112	no	—
53	C₂²×C₃₁	C₂²×D₄	1	28	yes	complemented
54	C₂²×C₃₁	C₂⁵	1	7	yes	—
55	C₂²×C₃₁	C₂⁵	1	16	yes	complemented
56	C₂×D₃₁	—	2	112	no	—
57	C₂×D₃₁	—	2	256	no	—
58	C₂×D₃₁	C₂²×D₄	1	112	yes	direct product factor
59	C₂×D₃₁	C₂⁵	1	8	yes	—
60	C₂×D₃₁	C₂⁵	1	16	yes	complemented
61	C₄×C₃₁	C₂⁵	1	8	yes	complemented
62	Dic₃₁	C₂⁵	1	8	yes	complemented
63	C₂⁶	—	31	2	no	—
64	C₂³×D₄	—	31	2	no	—
65	C₂³×D₄	—	31	56	no	—
66	C₂³×D₄	—	31	1	no	—
67	C₂³×D₄	D₃₁	1	1	yes	characteristic, direct product factor
68	C₂⁴×C₄	—	31	1	no	—
69	C₂×C₃₁	—	2	16	no	—
70	C₂×C₃₁	C₂³×D₄	1	14	yes	complemented
71	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
72	D₃₁	—	2	16	no	—
73	D₃₁	C₂³×D₄	1	16	yes	direct product factor
74	C₂⁵	—	31	1	no	—
75	C₂⁵	—	31	2	no	—
76	C₂⁵	—	31	56	no	—
77	C₂⁵	—	62	32	no	—
78	C₂⁵	C₂×D₃₁	1	2	yes	complemented
79	C₂²×D₄	—	31	56	no	—
80	C₂²×D₄	—	31	448	no	—
81	C₂²×D₄	—	31	28	no	—
82	C₂²×D₄	C₂×D₃₁	1	28	yes	direct product factor
83	C₂³×C₄	—	31	1	no	—
84	C₂³×C₄	—	31	28	no	—
85	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic, complemented
86	C₃₁	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
87	C₂⁴	—	2	16	no	—
88	C₂⁴	—	31	16	no	—
89	C₂⁴	—	31	14	no	—
90	C₂⁴	—	31	28	no	—
91	C₂⁴	—	31	224	no	—
92	C₂⁴	—	62	448	no	—
93	C₂⁴	—	62	16	no	—
94	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
95	C₂⁴	C₂²×D₃₁	1	28	yes	complemented
96	C₂×D₄	—	31	224	no	—
97	C₂×D₄	—	31	512	no	—
98	C₂×D₄	—	31	112	no	—
99	C₂×D₄	C₂²×D₃₁	1	112	yes	direct product factor
100	C₂²×C₄	—	31	14	no	—
101	C₂²×C₄	—	31	112	no	—
102	C₂²×C₄	C₂²×D₃₁	1	14	yes	complemented
103	C₂³	—	2	112	no	—
104	C₂³	—	31	112	no	—
105	C₂³	—	31	28	no	—
106	C₂³	—	31	56	no	—
107	C₂³	—	31	128	no	—
108	C₂³	—	62	896	no	—
109	C₂³	—	62	112	no	—
110	C₂³	D₄×D₃₁	1	8	yes	direct product factor
111	C₂³	C₂³×D₃₁	1	7	yes	—
112	C₂³	C₂³×D₃₁	1	56	yes	complemented
113	D₄	—	31	128	no	—
114	D₄	—	31	64	no	—
115	D₄	C₂³×D₃₁	1	64	yes	direct product factor
116	C₂×C₄	—	31	28	no	—
117	C₂×C₄	—	31	64	no	—
118	C₂×C₄	C₂³×D₃₁	1	28	yes	complemented
119	C₂²	—	2	112	no	—
120	C₂²	—	31	112	no	—
121	C₂²	—	31	8	no	—
122	C₂²	—	31	16	no	—
123	C₂²	—	62	256	no	—
124	C₂²	—	62	112	no	—
125	C₂²	C₂×D₄×D₃₁	1	28	yes	direct product factor
126	C₂²	C₂⁴×D₃₁	1	7	yes	—
127	C₂²	C₂⁴×D₃₁	1	16	yes	complemented
128	C₄	—	31	8	no	—
129	C₄	C₂⁴×D₃₁	1	8	yes	complemented
130	C₂	—	2	16	no	—
131	C₂	—	31	16	no	—
132	C₂	—	62	16	no	—
133	C₂	C₂²×D₄×D₃₁	1	14	yes	direct product factor
134	C₂	C₂⁵×D₃₁	1	1	yes	characteristic

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₃₁	C₂	semidirect
2. C₂³×D₁₂₄	C₂	semidirect
3. C₂³×C₂²⋊D₃₁	C₂	semidirect
4. C₂²×D₄×D₃₁	C₂	direct (N × H)
5. C₂³×C₄×D₃₁	C₂	semidirect
6. C₂³×D₄×C₃₁	C₂	semidirect
7. C₂⁵×C₃₁	C₂²	semidirect
10. C₂⁴×D₃₁	C₂²	semidirect
11. C₂⁴×D₃₁	C₂²	semidirect
12. C₂²×D₄×C₃₁	C₂²	semidirect
13. C₂³×C₄×C₃₁	C₂²	semidirect
14. C₂²×C₂²⋊D₃₁	C₂²	semidirect
15. C₂³×Dic₃₁	C₂²	semidirect
16. C₂×D₄×D₃₁	C₂²	direct (N × H)
17. C₂²×D₁₂₄	C₂²	semidirect
18. C₂²×C₄×D₃₁	C₂²	semidirect
21. C₂⁴×C₃₁	C₂³	semidirect
24. C₂³×D₃₁	D₄	direct (N × H)
26. C₂³×D₃₁	C₂³	semidirect
27. C₂³×D₃₁	C₂³	semidirect
28. C₂×D₄×C₃₁	C₂³	semidirect
29. C₂²×C₄×C₃₁	C₂³	semidirect
30. C₂×C₂²⋊D₃₁	C₂³	semidirect
31. C₂²×Dic₃₁	C₂³	semidirect
32. D₄×D₃₁	C₂³	direct (N × H)
33. C₂×D₁₂₄	C₂³	semidirect
34. C₂×C₄×D₃₁	C₂³	semidirect
36. C₂³×C₃₁	C₂×D₄	semidirect
38. C₂³×C₃₁	C₂⁴	semidirect
41. C₂²×D₃₁	C₂×D₄	direct (N × H)
43. C₂²×D₃₁	C₂⁴	semidirect
44. C₂²×D₃₁	C₂⁴	semidirect
45. D₄×C₃₁	C₂⁴	semidirect
46. C₂×C₄×C₃₁	C₂⁴	semidirect
47. C₂²⋊D₃₁	C₂⁴	semidirect
48. C₂×Dic₃₁	C₂⁴	semidirect
49. D₁₂₄	C₂⁴	semidirect
50. C₄×D₃₁	C₂⁴	semidirect
53. C₂²×C₃₁	C₂²×D₄	semidirect
55. C₂²×C₃₁	C₂⁵	semidirect
58. C₂×D₃₁	C₂²×D₄	direct (N × H)
60. C₂×D₃₁	C₂⁵	semidirect
61. C₄×C₃₁	C₂⁵	semidirect
62. Dic₃₁	C₂⁵	semidirect
67. C₂³×D₄	D₃₁	direct (N × H)
70. C₂×C₃₁	C₂³×D₄	semidirect
73. D₃₁	C₂³×D₄	direct (N × H)
78. C₂⁵	C₂×D₃₁	semidirect
82. C₂²×D₄	C₂×D₃₁	direct (N × H)
85. C₂³×C₄	C₂×D₃₁	semidirect
86. C₃₁	C₂⁴×D₄	semidirect
95. C₂⁴	C₂²×D₃₁	semidirect
99. C₂×D₄	C₂²×D₃₁	direct (N × H)
102. C₂²×C₄	C₂²×D₃₁	semidirect
110. C₂³	D₄×D₃₁	direct (N × H)
112. C₂³	C₂³×D₃₁	semidirect
115. D₄	C₂³×D₃₁	direct (N × H)
118. C₂×C₄	C₂³×D₃₁	semidirect
125. C₂²	C₂×D₄×D₃₁	direct (N × H)
127. C₂²	C₂⁴×D₃₁	semidirect
129. C₄	C₂⁴×D₃₁	semidirect
133. C₂	C₂²×D₄×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₃₁	2
C₂³×D₁₂₄	2
C₂³×C₂²⋊D₃₁	2
C₂²×D₄×D₃₁	2
C₂³×C₄×D₃₁	2
C₂³×D₄×C₃₁	2
C₂⁴×D₄	31