Subgroups — C₄×#7511 #72234368

Order 1024.

359 conjugacy classes of subgroups.

The subgroup lattice has 135 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#7511	C₂	1	4	yes	maximal
2	C₄×M₁₂₈	C₂	1	2	yes	maximal
3	C₄×#1397	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₂₈	C₂²	1	4	yes	—
5	C₂×#1397	C₄	1	2	yes	—
6	C₂×#1397	C₂²	1	2	yes	—
7	C₂×C₄×C₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₃₂	C₂²	1	1	yes	characteristic
9	C₄²×C₁₆	C₄	1	1	yes	characteristic
10	#7511	C₄	1	16	yes	direct product factor
11	#7510	C₄	1	4	yes	complemented
12	C₄×C₆₄	C₄	1	4	yes	complemented
13	C₂²×C₃₂	C₂×C₄	1	2	yes	—
14	C₂²×C₃₂	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃₂	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₁₆	C₈	1	2	yes	—
17	C₂×C₄×C₁₆	C₈	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₈	C₈	1	1	yes	characteristic
21	M₁₂₈	C₂×C₄	1	16	yes	—
22	C₂×C₆₄	C₂×C₄	1	8	yes	—
23	#1397	C₈	1	8	yes	—
24	#1397	C₂×C₄	1	8	yes	—
25	C₄×C₃₂	C₈	1	2	yes	—
26	C₄×C₃₂	C₈	1	2	yes	—
27	C₄×C₃₂	C₂×C₄	1	2	yes	—
28	C₄×C₃₂	C₂×C₄	1	2	yes	—
29	C₂²×C₁₆	C₂×C₈	1	2	yes	—
30	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₁₆	1	2	yes	—
33	C₂×C₄×C₈	C₁₆	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
36	C₄³	C₁₆	1	1	yes	characteristic
37	C₂×C₃₂	C₄²	1	2	yes	—
38	C₂×C₃₂	C₄²	1	4	yes	—
39	C₂×C₃₂	C₂×C₈	1	4	yes	—
40	C₂×C₃₂	C₂×C₈	1	4	yes	—
41	C₂×C₃₂	C₂×C₈	1	4	yes	—
42	C₂×C₃₂	C₂²×C₄	1	2	yes	—
43	C₂×C₃₂	C₂²×C₄	1	4	yes	—
44	C₄×C₁₆	C₁₆	1	8	yes	—
45	C₄×C₁₆	C₁₆	1	2	yes	—
46	C₄×C₁₆	C₁₆	1	2	yes	—
47	C₄×C₁₆	C₄²	1	1	yes	characteristic
48	C₄×C₁₆	C₄²	1	4	yes	—
49	C₄×C₁₆	C₄²	1	1	yes	characteristic
50	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
51	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
52	C₄×C₁₆	C₂×C₈	1	4	yes	—
53	C₆₄	C₄²	1	16	yes	complemented
54	C₂²×C₈	C₂×C₁₆	1	2	yes	—
55	C₂²×C₈	C₂×C₁₆	1	1	yes	characteristic
56	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄²	—	2	2	no	—
58	C₂×C₄²	C₂×C₁₆	1	2	yes	—
59	C₂×C₄²	C₂×C₁₆	1	1	yes	characteristic
60	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
61	C₂×C₁₆	C₄×C₈	1	4	yes	—
62	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
63	C₂×C₁₆	C₂×C₁₆	1	4	yes	—
64	C₂×C₁₆	C₂×C₁₆	1	4	yes	—
65	C₂×C₁₆	C₂×C₁₆	1	4	yes	—
66	C₂×C₁₆	C₂×C₄²	1	2	yes	—
67	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₁₆	C₂²×C₈	1	2	yes	—
70	C₄×C₈	—	2	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	C₄×C₈	1	4	yes	—
73	C₄×C₈	C₄×C₈	1	1	yes	characteristic
74	C₄×C₈	C₄×C₈	1	1	yes	characteristic
75	C₄×C₈	C₂×C₁₆	1	4	yes	—
76	C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
77	C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
78	C₃₂	C₄×C₈	1	4	yes	—
79	C₃₂	C₄×C₈	1	8	yes	—
80	C₃₂	C₂×C₄²	1	4	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	M₆₄	1	2	yes	—
83	C₂²×C₄	C₂²×C₁₆	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	C₄×C₁₆	1	4	yes	—
87	C₂×C₈	C₄×C₁₆	1	1	yes	characteristic
88	C₂×C₈	C₄×C₁₆	1	1	yes	characteristic
89	C₂×C₈	M₆₄	1	4	yes	—
90	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
92	C₂×C₈	C₂²×C₁₆	1	1	yes	characteristic
93	C₂×C₈	C₂²×C₁₆	1	2	yes	—
94	C₄²	—	2	2	no	—
95	C₄²	—	4	4	no	—
96	C₄²	C₄×C₁₆	1	4	yes	—
97	C₄²	C₄×C₁₆	1	1	yes	characteristic
98	C₄²	C₄×C₁₆	1	1	yes	characteristic
99	C₄²	M₆₄	1	2	yes	—
100	C₁₆	C₄×C₁₆	1	2	yes	—
101	C₁₆	C₄×C₁₆	1	8	yes	—
102	C₁₆	C₄×C₁₆	1	2	yes	—
103	C₁₆	C₄³	1	1	yes	characteristic
104	C₁₆	C₂×C₄×C₈	1	2	yes	—
105	C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
106	C₂³	C₂×M₆₄	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	#1397	1	2	yes	—
111	C₂×C₄	#1397	1	2	yes	—
112	C₂×C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×C₁₆	1	2	yes	—
114	C₂×C₄	C₂×M₆₄	1	2	yes	—
115	C₂×C₄	C₂×M₆₄	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₈	#1397	1	2	yes	—
118	C₈	#1397	1	2	yes	—
119	C₈	C₄²×C₈	1	1	yes	characteristic
120	C₈	C₂×C₄×C₁₆	1	2	yes	—
121	C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	C₂×#1397	1	1	yes	characteristic
124	C₂²	C₄×M₆₄	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	#7511	1	4	yes	direct product factor
128	C₄	C₄²×C₁₆	1	1	yes	characteristic
129	C₄	C₂×#1397	1	1	yes	characteristic
130	C₄	C₄×M₆₄	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₄×#1397	1	1	yes	characteristic
133	C₂	C₂×#7511	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₄²
commutator	C₄	C₄²×C₁₆
Frattini, cosocle	C₂²×C₃₂	C₂³
socle	C₂²	C₂×#1397
prime radical	C₂³	C₂×M₆₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. #7511	C₄	direct (N × H)
11. #7510	C₄	semidirect
12. C₄×C₆₄	C₄	semidirect
53. C₆₄	C₄²	semidirect
127. C₄	#7511	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#7511	2
C₄×M₁₂₈	2
C₄×#1397	2

Subgroups — C4×#7511 #72234368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#7511 #72234368