Subgroups — #72319485

Order 1024.

Only the 34115 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 177 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587001	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#63091	C₂	1	1	yes	maximal, characteristic, complemented
3	#10585861	C₂	1	32	yes	maximal, complemented
4	#10585860	C₂	1	32	yes	maximal, complemented
5	#10585859	C₂	1	30	yes	maximal, complemented
6	C₂×#62636	C₂	1	30	yes	maximal, complemented
7	#10585848	C₂	1	1	yes	maximal, characteristic, complemented
8	#63093	C₂²	1	16	yes	complemented
9	#63091	C₂²	1	16	yes	complemented
10	#63091	C₂²	1	32	yes	complemented
11	C₂×#3584	C₂²	1	30	yes	complemented
12	C₂×#3584	C₂²	1	15	yes	complemented
13	C₂²×#576	C₂²	1	15	yes	complemented
14	C₂×#3573	C₂²	1	1	yes	characteristic
15	#62637	C₂²	1	80	yes	complemented
16	#62637	C₂²	1	240	yes	complemented
17	#62636	C₂²	1	240	yes	complemented
18	#62636	C₂²	1	480	yes	complemented
19	#62636	C₂²	1	240	yes	complemented
20	#62635	C₂²	1	480	yes	complemented
21	C₂×#3416	C₂²	1	240	yes	complemented
22	C₂×#3415	C₂²	1	240	yes	complemented
23	C₂×#3414	C₂²	1	180	yes	complemented
24	C₂²×#509	C₂²	1	60	yes	complemented
25	#62624	C₂²	1	32	yes	complemented
26	C₂×#3407	C₂²	1	30	yes	complemented
27	C₂×#585	D₄	1	1	yes	characteristic
28	#3584	C₂³	1	240	yes	complemented
29	#3584	C₂³	1	80	yes	complemented
30	#3584	C₂³	1	240	yes	complemented
31	C₂×#578	C₂³	1	20	yes	complemented
32	C₂×#578	C₂³	1	120	yes	complemented
33	C₂×#576	C₂³	1	60	yes	complemented
34	C₂×#576	C₂³	1	240	yes	complemented
35	C₂×#576	C₂³	1	120	yes	complemented
36	C₂²×#135	C₂³	1	45	yes	complemented
37	C₂²×#135	C₂³	1	60	yes	complemented
38	C₂³×D₈	C₂³	1	15	yes	complemented
39	#3573	D₄	1	16	yes	—
40	#3573	C₂³	1	16	yes	—
41	C₂×#568	D₄	1	15	yes	—
42	C₂²×#131	C₂³	1	15	yes	—
43	#3417	C₂³	1	640	yes	complemented
44	#3416	C₂³	1	1920	yes	complemented
45	#3416	C₂³	1	640	yes	complemented
46	#3415	C₂³	1	1920	yes	complemented
47	#3414	C₂³	1	480	yes	complemented
48	#3414	C₂³	1	1440	yes	complemented
49	C₂×#510	C₂³	1	120	yes	complemented
50	C₂×#510	C₂³	1	360	yes	complemented
51	C₂×#509	C₂³	1	360	yes	complemented
52	C₂×#509	C₂³	1	720	yes	complemented
53	C₂×#509	C₂³	1	360	yes	complemented
54	C₂×#508	C₂³	1	720	yes	complemented
55	C₂²×SD₃₂	C₂³	1	120	yes	complemented
56	C₂²×D₁₆	C₂³	1	120	yes	complemented
57	#3407	C₂³	1	80	yes	complemented
58	#3407	C₂³	1	240	yes	complemented
59	C₂×#504	C₂³	1	240	yes	complemented
60	C₂²×M₃₂	C₂³	1	60	yes	complemented
61	#585	D₈	1	16	yes	—
62	#585	C₂×D₄	1	16	yes	—
63	C₂×2₋¹⁺⁴	D₈	1	6	yes	—
64	C₂×H(5,𝔽₂)	D₈	1	10	yes	—
65	C₂²×#41	C₂×D₄	1	15	yes	—
66	#578	C₂⁴	1	160	yes	—
67	#578	C₂⁴	1	320	yes	complemented
68	#576	C₂⁴	1	480	yes	—
69	#576	C₂⁴	1	320	yes	complemented
70	C₂×#135	C₂⁴	1	360	yes	—
71	C₂×#135	C₂⁴	1	120	yes	—
72	C₂×#135	C₂⁴	1	60	yes	—
73	C₂×#135	C₂⁴	1	360	yes	complemented
74	C₂²×Q₁₆	C₂⁴	1	15	yes	—
75	C₂²×Q₁₆	C₂⁴	1	60	yes	complemented
76	C₂²×D₈	C₂⁴	1	120	yes	—
77	C₂²×D₈	C₂⁴	1	45	yes	—
78	C₂²×D₈	C₂⁴	1	60	yes	complemented
79	#568	C₂×D₄	1	240	yes	—
80	C₂×#131	C₂×D₄	1	60	yes	—
81	C₂×#131	C₂×D₄	1	120	yes	—
82	C₂×#131	C₂⁴	1	120	yes	—
83	C₂×#131	C₂⁴	1	20	yes	—
84	C₂²×M₁₆	C₂×D₄	1	45	yes	—
85	C₂³×C₈	C₂⁴	1	15	yes	—
86	#510	C₂⁴	1	96	yes	—
87	#510	C₂⁴	1	960	yes	complemented
88	#510	C₂⁴	1	480	yes	complemented
89	#509	C₂⁴	1	160	yes	—
90	#509	C₂⁴	1	1440	yes	complemented
91	#509	C₂⁴	1	960	yes	complemented
92	#508	C₂⁴	1	1920	yes	complemented
93	C₂×Q₃₂	C₂⁴	1	240	yes	complemented
94	C₂×SD₃₂	C₂⁴	1	720	yes	complemented
95	C₂×SD₃₂	C₂⁴	1	240	yes	complemented
96	C₂×D₁₆	C₂⁴	1	720	yes	complemented
97	#504	C₂⁴	1	640	yes	complemented
98	C₂×M₃₂	C₂⁴	1	120	yes	—
99	C₂×M₃₂	C₂⁴	1	360	yes	complemented
100	C₂²×C₁₆	C₂⁴	1	120	yes	complemented
101	2₋¹⁺⁴	C₂×D₈	1	96	yes	—
102	H(5,𝔽₂)	C₂×D₈	1	160	yes	—
103	C₂×#41	C₂×D₈	1	60	yes	—
104	C₂×#41	C₂×D₈	1	120	yes	—
105	C₂×#41	C₂²×D₄	1	20	yes	—
106	C₂×#41	C₂²×D₄	1	120	yes	—
107	C₂²×Q₈	C₂×D₈	1	15	yes	—
108	C₂²×D₄	C₂×D₈	1	45	yes	—
109	C₂³×C₄	C₂²×D₄	1	15	yes	—
110	#135	C₂⁵	1	16	yes	—
111	#135	C₂⁵	1	240	yes	—
112	#135	C₂⁵	1	240	yes	—
113	C₂×Q₁₆	C₂⁵	1	6	yes	—
114	C₂×Q₁₆	C₂⁵	1	60	yes	—
115	C₂×Q₁₆	C₂⁵	1	120	yes	—
116	C₂×D₈	C₂⁵	1	10	yes	—
117	C₂×D₈	C₂⁵	1	180	yes	—
118	C₂×D₈	C₂⁵	1	120	yes	—
119	#131	C₂²×D₄	1	80	yes	—
120	#131	C₂²×D₄	1	480	yes	—
121	#131	C₂⁵	1	80	yes	—
122	C₂×M₁₆	C₂²×D₄	1	60	yes	—
123	C₂×M₁₆	C₂²×D₄	1	360	yes	—
124	C₂²×C₈	C₂²×D₄	1	45	yes	—
125	C₂²×C₈	C₂²×D₄	1	60	yes	—
126	C₂²×C₈	C₂⁵	1	15	yes	—
127	C₂²×C₈	C₂⁵	1	60	yes	—
128	Q₃₂	C₂⁵	1	192	yes	—
129	SD₃₂	C₂⁵	1	320	yes	—
130	SD₃₂	C₂⁵	1	192	yes	—
131	D₁₆	C₂⁵	1	320	yes	—
132	M₃₂	C₂⁵	1	16	yes	—
133	M₃₂	C₂⁵	1	240	yes	—
134	C₂×C₁₆	C₂⁵	1	240	yes	—
135	C₂⁴	C₂²×D₈	1	15	yes	—
136	#41	C₂²×D₈	1	80	yes	—
137	#41	C₂²×D₈	1	480	yes	—
138	#41	C₂³×D₄	1	80	yes	—
139	C₂×Q₈	C₂²×D₈	1	20	yes	—
140	C₂×Q₈	C₂²×D₈	1	120	yes	—
141	C₂×D₄	C₂²×D₈	1	60	yes	—
142	C₂×D₄	C₂²×D₈	1	360	yes	—
143	C₂²×C₄	C₂²×D₈	1	45	yes	—
144	C₂²×C₄	C₂²×D₈	1	60	yes	—
145	C₂²×C₄	C₂³×D₄	1	15	yes	—
146	C₂²×C₄	C₂³×D₄	1	60	yes	—
147	Q₁₆	C₂⁶	1	16	yes	—
148	Q₁₆	C₂⁶	1	12	yes	—
149	D₈	C₂⁶	1	16	yes	—
150	D₈	C₂⁶	1	20	yes	—
151	M₁₆	C₂³×D₄	1	240	yes	—
152	C₂×C₈	C₂³×D₄	1	15	yes	—
153	C₂×C₈	C₂³×D₄	1	30	yes	—
154	C₂×C₈	C₂³×D₄	1	180	yes	—
155	C₂×C₈	C₂⁶	1	30	yes	—
156	C₂×C₈	C₂⁶	1	1	yes	characteristic
157	C₁₆	C₂⁶	1	32	yes	—
158	C₂³	C₂³×D₈	1	15	yes	—
159	C₂³	C₂³×D₈	1	60	yes	—
160	Q₈	C₂³×D₈	1	80	yes	—
161	D₄	C₂³×D₈	1	240	yes	—
162	C₂×C₄	C₂³×D₈	1	15	yes	—
163	C₂×C₄	C₂³×D₈	1	30	yes	—
164	C₂×C₄	C₂³×D₈	1	180	yes	—
165	C₂×C₄	C₂⁴×D₄	1	1	yes	characteristic
166	C₂×C₄	C₂⁴×D₄	1	30	yes	—
167	C₈	C₂⁴×D₄	1	1	yes	characteristic
168	C₈	C₂⁴×D₄	1	30	yes	—
169	C₈	C₂⁷	1	1	yes	characteristic
170	C₂²	C₂⁴×D₈	1	1	yes	characteristic
171	C₂²	C₂⁴×D₈	1	30	yes	—
172	C₄	C₂⁴×D₈	1	1	yes	characteristic
173	C₄	C₂⁴×D₈	1	30	yes	—
174	C₄	C₂⁵×D₄	1	1	yes	characteristic
175	C₂	C₂⁵×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂⁵×D₈
commutator, Frattini, cosocle	C₈	C₂⁷
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#585 ◁ C₂×#3573 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10587001	C₂	semidirect
2. C₂×#63091	C₂	semidirect
3. #10585861	C₂	semidirect
4. #10585860	C₂	semidirect
5. #10585859	C₂	semidirect
6. C₂×#62636	C₂	semidirect
7. #10585848	C₂	semidirect
8. #63093	C₂²	semidirect
9. #63091	C₂²	semidirect
10. #63091	C₂²	semidirect
11. C₂×#3584	C₂²	semidirect
12. C₂×#3584	C₂²	semidirect
13. C₂²×#576	C₂²	semidirect
15. #62637	C₂²	semidirect
16. #62637	C₂²	semidirect
17. #62636	C₂²	semidirect
18. #62636	C₂²	semidirect
19. #62636	C₂²	semidirect
20. #62635	C₂²	semidirect
21. C₂×#3416	C₂²	semidirect
22. C₂×#3415	C₂²	semidirect
23. C₂×#3414	C₂²	semidirect
24. C₂²×#509	C₂²	semidirect
25. #62624	C₂²	semidirect
26. C₂×#3407	C₂²	semidirect
28. #3584	C₂³	semidirect
29. #3584	C₂³	semidirect
30. #3584	C₂³	semidirect
31. C₂×#578	C₂³	semidirect
32. C₂×#578	C₂³	semidirect
33. C₂×#576	C₂³	semidirect
34. C₂×#576	C₂³	semidirect
35. C₂×#576	C₂³	semidirect
36. C₂²×#135	C₂³	semidirect
37. C₂²×#135	C₂³	semidirect
38. C₂³×D₈	C₂³	semidirect
43. #3417	C₂³	semidirect
44. #3416	C₂³	semidirect
45. #3416	C₂³	semidirect
46. #3415	C₂³	semidirect
47. #3414	C₂³	semidirect
48. #3414	C₂³	semidirect
49. C₂×#510	C₂³	semidirect
50. C₂×#510	C₂³	semidirect
51. C₂×#509	C₂³	semidirect
52. C₂×#509	C₂³	semidirect
53. C₂×#509	C₂³	semidirect
54. C₂×#508	C₂³	semidirect
55. C₂²×SD₃₂	C₂³	semidirect
56. C₂²×D₁₆	C₂³	semidirect
57. #3407	C₂³	semidirect
58. #3407	C₂³	semidirect
59. C₂×#504	C₂³	semidirect
60. C₂²×M₃₂	C₂³	semidirect
67. #578	C₂⁴	semidirect
69. #576	C₂⁴	semidirect
73. C₂×#135	C₂⁴	semidirect
75. C₂²×Q₁₆	C₂⁴	semidirect
78. C₂²×D₈	C₂⁴	semidirect
87. #510	C₂⁴	semidirect
88. #510	C₂⁴	semidirect
90. #509	C₂⁴	semidirect
91. #509	C₂⁴	semidirect
92. #508	C₂⁴	semidirect
93. C₂×Q₃₂	C₂⁴	semidirect
94. C₂×SD₃₂	C₂⁴	semidirect
95. C₂×SD₃₂	C₂⁴	semidirect
96. C₂×D₁₆	C₂⁴	semidirect
97. #504	C₂⁴	semidirect
99. C₂×M₃₂	C₂⁴	semidirect
100. C₂²×C₁₆	C₂⁴	semidirect

Maximal subgroups

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

subgroup	index
#10587001	2
C₂×#63091	2
#10585861	2
#10585860	2
#10585859	2
C₂×#62636	2
#10585848	2