Subgroups — C₂³×#431 #10574109

Order 512.

12018 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×#431	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂³×C₄×C₈	C₂	1	2	yes	maximal, complemented
6	C₂⁵×C₄	C₄	1	1	yes	characteristic
7	C₂⁵×C₄	C₄	1	1	yes	characteristic
8	C₂⁵×C₄	C₂²	1	1	yes	characteristic
9	C₂⁴×C₈	C₂²	1	4	yes	—
10	C₂³×C₄²	C₄	1	1	yes	characteristic
11	C₂³×C₄²	C₄	1	28	yes	—
12	C₂³×C₄²	C₄	1	1	yes	characteristic
13	C₂³×C₄²	C₂²	1	1	yes	characteristic
14	C₂³×C₄²	C₂²	1	28	yes	—
15	C₂³×C₄²	C₂²	1	1	yes	characteristic
16	C₂×#431	C₂²	1	448	yes	direct product factor
17	C₂²×C₄⋊C₈	C₂²	1	56	yes	complemented
18	C₂²×C₂²⋊C₈	C₂²	1	56	yes	—
19	C₂²×C₄×C₈	C₂²	1	56	yes	complemented
20	C₂⁶	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×C₄	—	2	16	no	—
22	C₂⁴×C₄	—	2	16	no	—
23	C₂⁴×C₄	C₈	1	32	yes	complemented
24	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂⁴×C₄	C₂×C₄	1	28	yes	—
26	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂⁴×C₄	C₂×C₄	1	14	yes	—
28	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂⁴×C₄	C₂×C₄	1	28	yes	—
31	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂⁴×C₄	C₂³	1	14	yes	—
33	C₂⁴×C₄	C₂³	1	1	yes	characteristic
34	C₂³×C₈	—	2	32	no	—
35	C₂³×C₈	C₂³	1	56	yes	—
36	C₂²×C₄²	C₂×C₄	1	28	yes	—
37	C₂²×C₄²	C₂×C₄	1	336	yes	—
38	C₂²×C₄²	C₂×C₄	1	14	yes	—
39	C₂²×C₄²	C₂×C₄	1	28	yes	—
40	C₂²×C₄²	C₂×C₄	1	14	yes	—
41	C₂²×C₄²	C₂³	1	14	yes	—
42	C₂²×C₄²	C₂³	1	14	yes	—
43	C₂²×C₄²	C₂³	1	112	yes	—
44	#431	C₂³	1	512	yes	direct product factor
45	C₂×C₄⋊C₈	C₂³	1	224	yes	complemented
46	C₂×C₂²⋊C₈	C₂³	1	224	yes	—
47	C₂×C₄×C₈	C₂³	1	224	yes	complemented
48	C₂⁵	—	2	8	no	—
49	C₂⁵	—	2	8	no	—
50	C₂⁵	C₂×C₈	1	16	yes	—
51	C₂⁵	C₂²×C₄	1	14	yes	—
52	C₂⁵	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₄	—	2	8	no	—
54	C₂³×C₄	—	2	8	no	—
55	C₂³×C₄	—	2	112	no	—
56	C₂³×C₄	—	2	224	no	—
57	C₂³×C₄	—	2	112	no	—
58	C₂³×C₄	C₂×C₈	1	16	yes	—
59	C₂³×C₄	C₂×C₈	1	16	yes	—
60	C₂³×C₄	C₂×C₈	1	448	yes	complemented
61	C₂³×C₄	M₁₆	1	16	yes	—
62	C₂³×C₄	C₂²×C₄	1	14	yes	—
63	C₂³×C₄	C₂²×C₄	1	112	yes	—
64	C₂³×C₄	C₂²×C₄	1	14	yes	—
65	C₂³×C₄	C₂²×C₄	1	84	yes	—
66	C₂³×C₄	C₂²×C₄	1	7	yes	—
67	C₂³×C₄	C₂²×C₄	1	14	yes	—
68	C₂³×C₄	C₂²×C₄	1	14	yes	—
69	C₂³×C₄	C₂²×C₄	1	112	yes	—
70	C₂³×C₄	C₂²×C₄	1	14	yes	—
71	C₂³×C₄	#41	1	16	yes	—
72	C₂³×C₄	C₂⁴	1	28	yes	—
73	C₂³×C₄	C₂⁴	1	7	yes	—
74	C₂²×C₈	—	2	224	no	—
75	C₂²×C₈	C₂⁴	1	112	yes	—
76	C₂×C₄²	C₂²×C₄	1	112	yes	—
77	C₂×C₄²	C₂²×C₄	1	448	yes	—
78	C₂×C₄²	C₂²×C₄	1	112	yes	—
79	C₂×C₄²	C₂²×C₄	1	84	yes	—
80	C₂×C₄²	C₂²×C₄	1	84	yes	—
81	C₂×C₄²	C₂⁴	1	28	yes	—
82	C₂×C₄²	C₂⁴	1	28	yes	—
83	C₂×C₄²	C₂⁴	1	64	yes	—
84	C₄⋊C₈	C₂⁴	1	128	yes	complemented
85	C₂²⋊C₈	C₂⁴	1	128	yes	—
86	C₄×C₈	C₂⁴	1	128	yes	complemented
87	C₂⁴	—	2	56	no	—
88	C₂⁴	—	2	56	no	—
89	C₂⁴	—	2	128	no	—
90	C₂⁴	#117	1	8	yes	—
91	C₂⁴	C₂²×C₈	1	112	yes	—
92	C₂⁴	C₂²×C₈	1	8	yes	—
93	C₂⁴	C₂×M₁₆	1	8	yes	—
94	C₂⁴	C₂³×C₄	1	28	yes	—
95	C₂⁴	C₂³×C₄	1	7	yes	—
96	C₂²×C₄	—	2	56	no	—
97	C₂²×C₄	—	2	56	no	—
98	C₂²×C₄	—	2	336	no	—
99	C₂²×C₄	—	2	448	no	—
100	C₂²×C₄	—	2	112	no	—
101	C₂²×C₄	#117	1	56	yes	—
102	C₂²×C₄	C₂²×C₈	1	112	yes	—
103	C₂²×C₄	C₂²×C₈	1	112	yes	—
104	C₂²×C₄	C₂²×C₈	1	896	yes	complemented
105	C₂²×C₄	C₂×M₁₆	1	112	yes	—
106	C₂²×C₄	C₂³×C₄	1	28	yes	—
107	C₂²×C₄	C₂³×C₄	1	56	yes	—
108	C₂²×C₄	C₂³×C₄	1	28	yes	—
109	C₂²×C₄	C₂³×C₄	1	64	yes	—
110	C₂²×C₄	C₂³×C₄	1	21	yes	—
111	C₂²×C₄	C₂³×C₄	1	28	yes	—
112	C₂²×C₄	C₂³×C₄	1	28	yes	—
113	C₂²×C₄	C₂³×C₄	1	64	yes	—
114	C₂²×C₄	C₂³×C₄	1	28	yes	—
115	C₂²×C₄	C₂×#41	1	56	yes	—
116	C₂²×C₄	C₂⁵	1	8	yes	—
117	C₂²×C₄	C₂⁵	1	7	yes	—
118	C₂×C₈	—	2	224	no	—
119	C₂×C₈	C₂⁵	1	32	yes	—
120	C₄²	C₂³×C₄	1	64	yes	—
121	C₄²	C₂³×C₄	1	64	yes	—
122	C₄²	C₂³×C₄	1	56	yes	—
123	C₄²	C₂³×C₄	1	56	yes	—
124	C₄²	C₂⁵	1	8	yes	—
125	C₄²	C₂⁵	1	8	yes	—
126	C₂³	—	2	56	no	—
127	C₂³	—	2	448	no	—
128	C₂³	—	2	56	no	—
129	C₂³	#431	1	64	yes	direct product factor
130	C₂³	C₂×#117	1	28	yes	—
131	C₂³	C₂³×C₈	1	112	yes	—
132	C₂³	C₂³×C₈	1	28	yes	—
133	C₂³	C₂²×M₁₆	1	28	yes	—
134	C₂³	C₂⁴×C₄	1	8	yes	—
135	C₂³	C₂⁴×C₄	1	7	yes	—
136	C₂×C₄	—	2	56	no	—
137	C₂×C₄	—	2	56	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	2	112	no	—
140	C₂×C₄	—	2	128	no	—
141	C₂×C₄	C₂×#117	1	84	yes	—
142	C₂×C₄	C₂³×C₈	1	112	yes	—
143	C₂×C₄	C₂³×C₈	1	112	yes	—
144	C₂×C₄	C₂³×C₈	1	256	yes	complemented
145	C₂×C₄	C₂²×M₁₆	1	112	yes	—
146	C₂×C₄	C₂⁴×C₄	1	8	yes	—
147	C₂×C₄	C₂⁴×C₄	1	8	yes	—
148	C₂×C₄	C₂⁴×C₄	1	7	yes	—
149	C₂×C₄	C₂⁴×C₄	1	8	yes	—
150	C₂×C₄	C₂⁴×C₄	1	8	yes	—
151	C₂×C₄	C₂⁴×C₄	1	8	yes	—
152	C₂×C₄	C₂²×#41	1	28	yes	—
153	C₂×C₄	C₂⁶	1	1	yes	characteristic
154	C₈	—	2	32	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	224	no	—
158	C₂²	C₂×#431	1	112	yes	direct product factor
159	C₂²	C₂²×#117	1	14	yes	—
160	C₂²	C₂⁴×C₈	1	16	yes	—
161	C₂²	C₂⁴×C₈	1	14	yes	—
162	C₂²	C₂³×M₁₆	1	14	yes	—
163	C₂²	C₂⁵×C₄	1	1	yes	characteristic
164	C₄	—	2	8	no	—
165	C₄	—	2	8	no	—
166	C₄	C₂²×#117	1	14	yes	—
167	C₄	C₂⁴×C₈	1	16	yes	—
168	C₄	C₂⁴×C₈	1	16	yes	—
169	C₄	C₂³×M₁₆	1	16	yes	—
170	C₄	C₂³×#41	1	2	yes	—
171	C₂	—	2	16	no	—
172	C₂	C₂²×#431	1	28	yes	direct product factor
173	C₂	C₂³×#117	1	1	yes	characteristic
174	C₂	C₂⁵×C₈	1	1	yes	characteristic
175	C₂	C₂⁴×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄²	C₂²
commutator	C₂	C₂⁵×C₈
Frattini, cosocle	C₂×C₄	C₂⁶
socle	C₂⁵	C₂²×C₄
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×#431	C₂	direct (N × H)
3. C₂³×C₄⋊C₈	C₂	semidirect
5. C₂³×C₄×C₈	C₂	semidirect
16. C₂×#431	C₂²	direct (N × H)
17. C₂²×C₄⋊C₈	C₂²	semidirect
19. C₂²×C₄×C₈	C₂²	semidirect
23. C₂⁴×C₄	C₈	semidirect
44. #431	C₂³	direct (N × H)
45. C₂×C₄⋊C₈	C₂³	semidirect
47. C₂×C₄×C₈	C₂³	semidirect
60. C₂³×C₄	C₂×C₈	semidirect
84. C₄⋊C₈	C₂⁴	semidirect
86. C₄×C₈	C₂⁴	semidirect
104. C₂²×C₄	C₂²×C₈	semidirect
129. C₂³	#431	direct (N × H)
144. C₂×C₄	C₂³×C₈	semidirect
158. C₂²	C₂×#431	direct (N × H)
172. C₂	C₂²×#431	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄²	2
C₂²×#431	2
C₂³×C₄⋊C₈	2
C₂³×C₂²⋊C₈	2
C₂³×C₄×C₈	2

Subgroups — C23×#431 #10574109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#431 #10574109