Subgroups — C₂³×#533 #10585898

Order 512.

39730 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#533	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂³×#125	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊Q₈	C₂	1	4	yes	maximal
5	C₂³×C₄×Q₈	C₂	1	4	yes	maximal, complemented
6	C₂³×#117	C₂	1	2	yes	maximal
7	C₂⁴×C₄²	C₂	1	1	yes	maximal, characteristic
8	C₂⁴×Q₈	C₂²	1	4	yes	—
9	C₂⁵×C₄	C₂²	1	2	yes	—
10	C₂⁵×C₄	C₂²	1	1	yes	characteristic
11	C₂×#533	C₂²	1	1792	yes	direct product factor
12	C₂²×C₄⋊Q₈	C₂²	1	112	yes	complemented
13	C₂²×#125	C₂²	1	112	yes	complemented
14	C₂²×C₂²⋊Q₈	C₂²	1	224	yes	—
15	C₂²×C₄×Q₈	C₂²	1	224	yes	complemented
16	C₂²×#117	C₂²	1	112	yes	—
17	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂³×C₄⋊C₄	C₂²	1	8	yes	—
19	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂³×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂³×C₄²	C₂²	1	1	yes	characteristic
22	C₂³×C₄²	C₂²	1	1	yes	characteristic
23	C₂³×C₄²	C₂²	1	56	yes	—
24	C₂³×C₄²	C₂²	1	2	yes	—
25	C₂³×C₄²	C₂²	1	4	yes	—
26	C₂⁶	C₂³	1	1	yes	characteristic
27	C₂³×Q₈	—	2	64	no	—
28	C₂³×Q₈	C₂³	1	112	yes	—
29	C₂⁴×C₄	—	2	32	no	—
30	C₂⁴×C₄	Q₈	1	32	yes	complemented
31	C₂⁴×C₄	C₂³	1	2	yes	—
32	C₂⁴×C₄	C₂³	1	56	yes	—
33	C₂⁴×C₄	C₂³	1	8	yes	—
34	C₂⁴×C₄	C₂³	1	1	yes	characteristic
35	C₂⁴×C₄	C₂³	1	28	yes	—
36	C₂⁴×C₄	C₂³	1	2	yes	—
37	C₂⁴×C₄	C₂³	1	1	yes	characteristic
38	#533	C₂³	1	4096	yes	direct product factor
39	C₂×C₄⋊Q₈	C₂³	1	896	yes	complemented
40	C₂×#125	C₂³	1	896	yes	complemented
41	C₂×C₂²⋊Q₈	C₂³	1	1792	yes	—
42	C₂×C₄×Q₈	C₂³	1	1792	yes	complemented
43	C₂×#117	C₂³	1	896	yes	—
44	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
45	C₂²×C₄⋊C₄	C₂³	1	224	yes	—
46	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
47	C₂²×C₂²⋊C₄	C₂³	1	112	yes	—
48	C₂²×C₄²	C₂³	1	56	yes	—
49	C₂²×C₄²	C₂³	1	28	yes	—
50	C₂²×C₄²	C₂³	1	28	yes	—
51	C₂²×C₄²	C₂³	1	448	yes	—
52	C₂²×C₄²	C₂³	1	112	yes	—
53	C₂⁵	—	2	16	no	—
54	C₂⁵	C₂×Q₈	1	16	yes	—
55	C₂⁵	C₂⁴	1	14	yes	—
56	C₂⁵	C₂⁴	1	1	yes	characteristic
57	C₂²×Q₈	—	2	896	no	—
58	C₂²×Q₈	C₂⁴	1	448	yes	—
59	C₂³×C₄	—	2	16	no	—
60	C₂³×C₄	—	2	64	no	—
61	C₂³×C₄	—	2	448	no	—
62	C₂³×C₄	C₂×Q₈	1	16	yes	—
63	C₂³×C₄	C₂×Q₈	1	16	yes	—
64	C₂³×C₄	C₂×Q₈	1	448	yes	complemented
65	C₂³×C₄	#41	1	32	yes	—
66	C₂³×C₄	C₂⁴	1	112	yes	—
67	C₂³×C₄	C₂⁴	1	28	yes	—
68	C₂³×C₄	C₂⁴	1	224	yes	—
69	C₂³×C₄	C₂⁴	1	14	yes	—
70	C₂³×C₄	C₂⁴	1	14	yes	—
71	C₂³×C₄	C₂⁴	1	112	yes	—
72	C₂³×C₄	C₂⁴	1	28	yes	—
73	C₄⋊Q₈	C₂⁴	1	1024	yes	complemented
74	#125	C₂⁴	1	1024	yes	complemented
75	C₂²⋊Q₈	C₂⁴	1	2048	yes	—
76	C₄×Q₈	C₂⁴	1	2048	yes	complemented
77	#117	C₂⁴	1	1024	yes	—
78	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
79	C₂×C₄⋊C₄	C₂⁴	1	896	yes	—
80	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
81	C₂×C₂²⋊C₄	C₂⁴	1	448	yes	—
82	C₂×C₄²	C₂⁴	1	448	yes	—
83	C₂×C₄²	C₂⁴	1	224	yes	—
84	C₂×C₄²	C₂⁴	1	112	yes	—
85	C₂×C₄²	C₂⁴	1	112	yes	—
86	C₂×C₄²	C₂⁴	1	512	yes	—
87	C₂⁴	—	2	128	no	—
88	C₂⁴	—	2	112	no	—
89	C₂⁴	C₂²×Q₈	1	112	yes	—
90	C₂⁴	C₂²×Q₈	1	8	yes	—
91	C₂⁴	C₂×#41	1	16	yes	—
92	C₂⁴	C₂⁵	1	28	yes	—
93	C₂⁴	C₂⁵	1	7	yes	—
94	C₂×Q₈	—	2	1792	no	—
95	C₂×Q₈	C₂⁵	1	256	yes	—
96	C₂²×C₄	—	2	112	no	—
97	C₂²×C₄	—	2	448	no	—
98	C₂²×C₄	—	2	896	no	—
99	C₂²×C₄	C₂²×Q₈	1	112	yes	—
100	C₂²×C₄	C₂²×Q₈	1	112	yes	—
101	C₂²×C₄	C₂²×Q₈	1	896	yes	complemented
102	C₂²×C₄	C₂×#41	1	224	yes	—
103	C₂²×C₄	C₂⁵	1	56	yes	—
104	C₂²×C₄	C₂⁵	1	128	yes	—
105	C₂²×C₄	C₂⁵	1	224	yes	—
106	C₂²×C₄	C₂⁵	1	28	yes	—
107	C₂²×C₄	C₂⁵	1	28	yes	—
108	C₂²×C₄	C₂⁵	1	64	yes	—
109	C₂²×C₄	C₂⁵	1	56	yes	—
110	C₄⋊C₄	C₂⁵	1	256	yes	—
111	C₄⋊C₄	C₂⁵	1	512	yes	—
112	C₄⋊C₄	C₂⁵	1	256	yes	—
113	C₂²⋊C₄	C₂⁵	1	256	yes	—
114	C₄²	C₂⁵	1	128	yes	—
115	C₄²	C₂⁵	1	64	yes	—
116	C₄²	C₂⁵	1	64	yes	—
117	C₄²	C₂⁵	1	256	yes	—
118	C₂³	—	2	112	no	—
119	C₂³	—	2	448	no	—
120	C₂³	#533	1	64	yes	direct product factor
121	C₂³	C₂³×Q₈	1	112	yes	—
122	C₂³	C₂³×Q₈	1	28	yes	—
123	C₂³	C₂²×#41	1	56	yes	—
124	C₂³	C₂⁶	1	8	yes	—
125	C₂³	C₂⁶	1	7	yes	—
126	Q₈	—	2	512	no	—
127	C₂×C₄	—	2	112	no	—
128	C₂×C₄	—	2	448	no	—
129	C₂×C₄	—	2	256	no	—
130	C₂×C₄	C₂³×Q₈	1	112	yes	—
131	C₂×C₄	C₂³×Q₈	1	112	yes	—
132	C₂×C₄	C₂³×Q₈	1	256	yes	complemented
133	C₂×C₄	C₂²×#41	1	224	yes	—
134	C₂×C₄	C₂⁶	1	16	yes	—
135	C₂×C₄	C₂⁶	1	64	yes	—
136	C₂×C₄	C₂⁶	1	8	yes	—
137	C₂×C₄	C₂⁶	1	8	yes	—
138	C₂×C₄	C₂⁶	1	16	yes	—
139	C₂²	—	2	16	no	—
140	C₂²	—	2	224	no	—
141	C₂²	C₂×#533	1	112	yes	direct product factor
142	C₂²	C₂⁴×Q₈	1	16	yes	—
143	C₂²	C₂⁴×Q₈	1	14	yes	—
144	C₂²	C₂³×#41	1	28	yes	—
145	C₂²	C₂⁷	1	1	yes	characteristic
146	C₄	—	2	16	no	—
147	C₄	—	2	64	no	—
148	C₄	C₂⁴×Q₈	1	16	yes	—
149	C₄	C₂⁴×Q₈	1	16	yes	—
150	C₄	C₂³×#41	1	32	yes	—
151	C₂	—	2	16	no	—
152	C₂	C₂²×#533	1	28	yes	direct product factor
153	C₂	C₂⁵×Q₈	1	1	yes	characteristic
154	C₂	C₂⁴×#41	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁷
socle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#533	C₂	direct (N × H)
2. C₂³×C₄⋊Q₈	C₂	semidirect
3. C₂³×#125	C₂	semidirect
5. C₂³×C₄×Q₈	C₂	semidirect
11. C₂×#533	C₂²	direct (N × H)
12. C₂²×C₄⋊Q₈	C₂²	semidirect
13. C₂²×#125	C₂²	semidirect
15. C₂²×C₄×Q₈	C₂²	semidirect
30. C₂⁴×C₄	Q₈	semidirect
38. #533	C₂³	direct (N × H)
39. C₂×C₄⋊Q₈	C₂³	semidirect
40. C₂×#125	C₂³	semidirect
42. C₂×C₄×Q₈	C₂³	semidirect
64. C₂³×C₄	C₂×Q₈	semidirect
73. C₄⋊Q₈	C₂⁴	semidirect
74. #125	C₂⁴	semidirect
76. C₄×Q₈	C₂⁴	semidirect
101. C₂²×C₄	C₂²×Q₈	semidirect
120. C₂³	#533	direct (N × H)
132. C₂×C₄	C₂³×Q₈	semidirect
141. C₂²	C₂×#533	direct (N × H)
152. C₂	C₂²×#533	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#533	2
C₂³×C₄⋊Q₈	2
C₂³×#125	2
C₂³×C₂²⋊Q₈	2
C₂³×C₄×Q₈	2
C₂³×#117	2
C₂⁴×C₄²	2

Subgroups — C23×#533 #10585898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#533 #10585898