Subgroups — C₂³×#563 #10586113

Order 512.

35266 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#563	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂³×#126	C₂	1	4	yes	maximal, complemented
4	C₂³×#125	C₂	1	2	yes	maximal, complemented
5	C₂³×#124	C₂	1	1	yes	maximal, characteristic
6	C₂³×#123	C₂	1	2	yes	maximal
7	C₂³×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
8	C₂⁴×Q₈	C₂²	1	2	yes	—
9	C₂⁴×Q₈	C₂²	1	1	yes	characteristic
10	C₂⁴×D₄	C₂²	1	1	yes	characteristic
11	C₂⁵×C₄	C₂²	1	2	yes	—
12	C₂×#563	C₂²	1	1792	yes	direct product factor
13	C₂²×C₄⋊Q₈	C₂²	1	112	yes	complemented
14	C₂²×#126	C₂²	1	224	yes	complemented
15	C₂²×#125	C₂²	1	112	yes	complemented
16	C₂²×#124	C₂²	1	56	yes	—
17	C₂²×#123	C₂²	1	112	yes	—
18	C₂²×C₂²⋊Q₈	C₂²	1	224	yes	complemented
19	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
23	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂³×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂³×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂³×C₄²	C₂²	1	1	yes	characteristic
28	C₂³×C₄²	C₂²	1	2	yes	—
29	C₂⁶	C₂³	1	2	yes	—
30	C₂³×Q₈	—	4	16	no	—
31	C₂³×Q₈	—	4	8	no	—
32	C₂³×Q₈	C₂³	1	56	yes	—
33	C₂³×Q₈	C₂³	1	28	yes	—
34	C₂³×D₄	—	4	8	no	—
35	C₂³×D₄	C₂³	1	28	yes	—
36	C₂⁴×C₄	—	4	16	no	—
37	C₂⁴×C₄	C₂³	1	4	yes	—
38	C₂⁴×C₄	C₂³	1	1	yes	characteristic
39	C₂⁴×C₄	C₂³	1	2	yes	—
40	C₂⁴×C₄	C₂³	1	2	yes	—
41	C₂⁴×C₄	C₂³	1	4	yes	—
42	C₂⁴×C₄	C₂³	1	56	yes	—
43	#563	C₂³	1	4096	yes	direct product factor
44	C₂×C₄⋊Q₈	C₂³	1	896	yes	complemented
45	C₂×#126	C₂³	1	1792	yes	complemented
46	C₂×#125	C₂³	1	896	yes	complemented
47	C₂×#124	C₂³	1	448	yes	—
48	C₂×#123	C₂³	1	896	yes	—
49	C₂×C₂²⋊Q₈	C₂³	1	1792	yes	complemented
50	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
51	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
52	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
53	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
54	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
55	C₂²×C₂²⋊C₄	C₂³	1	112	yes	—
56	C₂²×C₂²⋊C₄	C₂³	1	112	yes	—
57	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
58	C₂²×C₄²	C₂³	1	28	yes	—
59	C₂²×C₄²	C₂³	1	56	yes	—
60	C₂⁵	—	2	16	no	—
61	C₂⁵	—	2	32	no	—
62	C₂⁵	C₂⁴	1	28	yes	—
63	C₂⁵	C₂⁴	1	1	yes	characteristic
64	C₂²×Q₈	—	4	224	no	—
65	C₂²×Q₈	—	4	112	no	—
66	C₂²×Q₈	C₂⁴	1	224	yes	—
67	C₂²×Q₈	C₂⁴	1	112	yes	—
68	C₂²×D₄	—	4	112	no	—
69	C₂²×D₄	C₂⁴	1	112	yes	—
70	C₂³×C₄	—	2	32	no	—
71	C₂³×C₄	—	2	8	no	—
72	C₂³×C₄	—	2	16	no	—
73	C₂³×C₄	—	2	16	no	—
74	C₂³×C₄	—	2	32	no	—
75	C₂³×C₄	—	4	224	no	—
76	C₂³×C₄	C₂⁴	1	56	yes	—
77	C₂³×C₄	C₂⁴	1	28	yes	—
78	C₂³×C₄	C₂⁴	1	28	yes	—
79	C₂³×C₄	C₂⁴	1	56	yes	—
80	C₂³×C₄	C₂⁴	1	14	yes	—
81	C₂³×C₄	C₂⁴	1	224	yes	—
82	C₄⋊Q₈	C₂⁴	1	1024	yes	complemented
83	#126	C₂⁴	1	2048	yes	complemented
84	#125	C₂⁴	1	1024	yes	complemented
85	#124	C₂⁴	1	512	yes	—
86	#123	C₂⁴	1	1024	yes	—
87	C₂²⋊Q₈	C₂⁴	1	2048	yes	complemented
88	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
89	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
90	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
91	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
92	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
93	C₂×C₂²⋊C₄	C₂⁴	1	448	yes	—
94	C₂×C₂²⋊C₄	C₂⁴	1	448	yes	—
95	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
96	C₂×C₄²	C₂⁴	1	112	yes	—
97	C₂×C₄²	C₂⁴	1	224	yes	—
98	C₂⁴	—	2	112	no	—
99	C₂⁴	—	2	224	no	—
100	C₂⁴	—	4	128	no	—
101	C₂⁴	H(5,𝔽₂)	1	8	yes	—
102	C₂⁴	2₋¹⁺⁴	1	16	yes	—
103	C₂⁴	C₂⁵	1	56	yes	—
104	C₂⁴	C₂⁵	1	7	yes	—
105	C₂×Q₈	—	4	448	no	—
106	C₂×Q₈	—	4	224	no	—
107	C₂×Q₈	C₂⁵	1	128	yes	—
108	C₂×Q₈	C₂⁵	1	64	yes	—
109	C₂×D₄	—	4	224	no	—
110	C₂×D₄	C₂⁵	1	64	yes	—
111	C₂²×C₄	—	2	224	no	—
112	C₂²×C₄	—	2	56	no	—
113	C₂²×C₄	—	2	112	no	—
114	C₂²×C₄	—	2	224	no	—
115	C₂²×C₄	—	2	112	no	—
116	C₂²×C₄	—	4	448	no	—
117	C₂²×C₄	C₂⁵	1	56	yes	—
118	C₂²×C₄	C₂⁵	1	112	yes	—
119	C₂²×C₄	C₂⁵	1	56	yes	—
120	C₂²×C₄	C₂⁵	1	28	yes	—
121	C₂²×C₄	C₂⁵	1	112	yes	—
122	C₂²×C₄	C₂⁵	1	128	yes	—
123	C₄⋊C₄	C₂⁵	1	128	yes	—
124	C₄⋊C₄	C₂⁵	1	256	yes	—
125	C₄⋊C₄	C₂⁵	1	256	yes	—
126	C₄⋊C₄	C₂⁵	1	256	yes	—
127	C₄⋊C₄	C₂⁵	1	128	yes	—
128	C₂²⋊C₄	C₂⁵	1	256	yes	—
129	C₂²⋊C₄	C₂⁵	1	256	yes	—
130	C₂²⋊C₄	C₂⁵	1	128	yes	—
131	C₄²	C₂⁵	1	64	yes	—
132	C₄²	C₂⁵	1	128	yes	—
133	C₂³	—	2	112	no	—
134	C₂³	—	2	224	no	—
135	C₂³	—	4	448	no	—
136	C₂³	#563	1	64	yes	direct product factor
137	C₂³	C₂×H(5,𝔽₂)	1	28	yes	—
138	C₂³	C₂×2₋¹⁺⁴	1	56	yes	—
139	C₂³	C₂⁶	1	16	yes	—
140	C₂³	C₂⁶	1	7	yes	—
141	Q₈	—	4	128	no	—
142	Q₈	—	4	64	no	—
143	D₄	—	4	64	no	—
144	C₂×C₄	—	2	224	no	—
145	C₂×C₄	—	2	56	no	—
146	C₂×C₄	—	2	112	no	—
147	C₂×C₄	—	2	112	no	—
148	C₂×C₄	—	2	224	no	—
149	C₂×C₄	—	4	128	no	—
150	C₂×C₄	C₂⁶	1	16	yes	—
151	C₂×C₄	C₂⁶	1	32	yes	—
152	C₂×C₄	C₂⁶	1	16	yes	—
153	C₂×C₄	C₂⁶	1	8	yes	—
154	C₂×C₄	C₂⁶	1	32	yes	—
155	C₂²	—	2	16	no	—
156	C₂²	—	2	32	no	—
157	C₂²	—	4	224	no	—
158	C₂²	C₂×#563	1	112	yes	direct product factor
159	C₂²	C₂²×H(5,𝔽₂)	1	14	yes	—
160	C₂²	C₂²×2₋¹⁺⁴	1	28	yes	—
161	C₂²	C₂⁷	1	1	yes	characteristic
162	C₄	—	2	32	no	—
163	C₄	—	2	8	no	—
164	C₄	—	2	16	no	—
165	C₄	—	2	16	no	—
166	C₄	—	2	32	no	—
167	C₂	—	4	16	no	—
168	C₂	C₂²×#563	1	28	yes	direct product factor
169	C₂	C₂³×H(5,𝔽₂)	1	1	yes	characteristic
170	C₂	C₂³×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#563	C₂	direct (N × H)
2. C₂³×C₄⋊Q₈	C₂	semidirect
3. C₂³×#126	C₂	semidirect
4. C₂³×#125	C₂	semidirect
7. C₂³×C₂²⋊Q₈	C₂	semidirect
12. C₂×#563	C₂²	direct (N × H)
13. C₂²×C₄⋊Q₈	C₂²	semidirect
14. C₂²×#126	C₂²	semidirect
15. C₂²×#125	C₂²	semidirect
18. C₂²×C₂²⋊Q₈	C₂²	semidirect
43. #563	C₂³	direct (N × H)
44. C₂×C₄⋊Q₈	C₂³	semidirect
45. C₂×#126	C₂³	semidirect
46. C₂×#125	C₂³	semidirect
49. C₂×C₂²⋊Q₈	C₂³	semidirect
82. C₄⋊Q₈	C₂⁴	semidirect
83. #126	C₂⁴	semidirect
84. #125	C₂⁴	semidirect
87. C₂²⋊Q₈	C₂⁴	semidirect
136. C₂³	#563	direct (N × H)
158. C₂²	C₂×#563	direct (N × H)
168. C₂	C₂²×#563	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×#563	2
C₂³×C₄⋊Q₈	2
C₂³×#126	2
C₂³×#125	2
C₂³×#124	2
C₂³×#123	2
C₂³×C₂²⋊Q₈	2

Subgroups — C23×#563 #10586113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#563 #10586113