Subgroups — C₂⁴×#334 #72286586

Order 1024.

23349 conjugacy classes of subgroups.

The subgroup lattice has 157 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂⁴×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂³×#334	C₂	1	60	yes	maximal, direct product factor
4	C₂⁵×C₈	—	2	2	no	—
5	C₂⁵×C₈	C₄	1	1	yes	characteristic
6	C₂⁵×C₈	C₄	1	1	yes	characteristic
7	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
8	C₂³×C₄⋊C₈	—	2	30	no	—
9	C₂³×C₄⋊C₈	C₂²	1	60	yes	—
10	C₂³×C₄×C₈	C₄	1	30	yes	—
11	C₂³×C₄×C₈	C₂²	1	30	yes	—
12	C₂²×#334	C₂²	1	560	yes	direct product factor
13	C₂⁵×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂⁵×C₄	D₄	1	1	yes	characteristic
15	C₂⁵×C₄	Q₈	1	1	yes	characteristic
16	C₂⁴×C₈	—	2	16	no	—
17	C₂⁴×C₈	—	2	60	no	—
18	C₂⁴×C₈	—	4	32	no	—
19	C₂⁴×C₈	C₈	1	32	yes	complemented
20	C₂⁴×C₈	C₂×C₄	1	30	yes	—
21	C₂⁴×C₈	C₂×C₄	1	30	yes	—
22	C₂³×C₄²	C₂×C₄	1	15	yes	—
23	C₂³×C₄²	D₄	1	15	yes	—
24	C₂³×C₄²	Q₈	1	15	yes	—
25	C₂³×C₄²	C₂³	1	15	yes	—
26	C₂²×C₄⋊C₈	—	2	420	no	—
27	C₂²×C₄⋊C₈	C₂³	1	280	yes	—
28	C₂²×C₄×C₈	C₂×C₄	1	420	yes	—
29	C₂²×C₄×C₈	C₂³	1	140	yes	—
30	C₂×#334	C₂³	1	960	yes	direct product factor
31	C₂⁶	C₄⋊C₄	1	1	yes	characteristic
32	C₂⁴×C₄	—	2	16	no	—
33	C₂⁴×C₄	C₄⋊C₄	1	15	yes	—
34	C₂⁴×C₄	C₄⋊C₄	1	15	yes	—
35	C₂⁴×C₄	C₄⋊C₄	1	15	yes	—
36	C₂⁴×C₄	C₂×C₈	1	16	yes	—
37	C₂⁴×C₄	M₁₆	1	16	yes	—
38	C₂⁴×C₄	SD₁₆	1	32	yes	—
39	C₂⁴×C₄	C₂²×C₄	1	15	yes	—
40	C₂⁴×C₄	C₂×D₄	1	15	yes	—
41	C₂⁴×C₄	C₂×Q₈	1	15	yes	—
42	C₂³×C₈	—	2	240	no	—
43	C₂³×C₈	—	2	280	no	—
44	C₂³×C₈	—	4	480	no	—
45	C₂³×C₈	C₂×C₈	1	480	yes	complemented
46	C₂³×C₈	C₂²×C₄	1	140	yes	—
47	C₂³×C₈	C₂²×C₄	1	140	yes	—
48	C₂²×C₄²	C₄⋊C₄	1	210	yes	—
49	C₂²×C₄²	C₂²×C₄	1	105	yes	—
50	C₂²×C₄²	C₂×D₄	1	105	yes	—
51	C₂²×C₄²	C₂×Q₈	1	105	yes	—
52	C₂²×C₄²	C₂⁴	1	35	yes	—
53	C₂×C₄⋊C₈	—	2	840	no	—
54	C₂×C₄⋊C₈	C₂⁴	1	240	yes	—
55	C₂×C₄×C₈	C₂²×C₄	1	840	yes	—
56	C₂×C₄×C₈	C₂⁴	1	120	yes	—
57	#334	C₂⁴	1	256	yes	direct product factor
58	C₂⁵	C₄⋊C₈	1	16	yes	—
59	C₂⁵	#106	1	16	yes	—
60	C₂⁵	#108	1	16	yes	—
61	C₂⁵	C₂×C₄⋊C₄	1	15	yes	—
62	C₂³×C₄	—	2	120	no	—
63	C₂³×C₄	—	2	120	no	—
64	C₂³×C₄	C₄⋊C₈	1	240	yes	—
65	C₂³×C₄	#106	1	240	yes	—
66	C₂³×C₄	C₂×C₄⋊C₄	1	105	yes	—
67	C₂³×C₄	C₂×C₄⋊C₄	1	105	yes	—
68	C₂³×C₄	C₂×C₄⋊C₄	1	105	yes	—
69	C₂³×C₄	C₂²×C₈	1	120	yes	—
70	C₂³×C₄	C₂×M₁₆	1	120	yes	—
71	C₂³×C₄	C₂×SD₁₆	1	240	yes	—
72	C₂³×C₄	C₂³×C₄	1	35	yes	—
73	C₂³×C₄	C₂²×D₄	1	35	yes	—
74	C₂³×C₄	C₂²×Q₈	1	35	yes	—
75	C₂²×C₈	—	2	560	no	—
76	C₂²×C₈	—	2	240	no	—
77	C₂²×C₈	—	4	1120	no	—
78	C₂²×C₈	C₂²×C₈	1	1120	yes	complemented
79	C₂²×C₈	C₂³×C₄	1	120	yes	—
80	C₂²×C₈	C₂³×C₄	1	120	yes	—
81	C₂×C₄²	C₂×C₄⋊C₄	1	630	yes	—
82	C₂×C₄²	C₂³×C₄	1	105	yes	—
83	C₂×C₄²	C₂²×D₄	1	105	yes	—
84	C₂×C₄²	C₂²×Q₈	1	105	yes	—
85	C₂×C₄²	C₂⁵	1	15	yes	—
86	C₄⋊C₈	—	2	240	no	—
87	C₄⋊C₈	C₂⁵	1	32	yes	—
88	C₄×C₈	C₂³×C₄	1	240	yes	—
89	C₄×C₈	C₂⁵	1	16	yes	—
90	C₂⁴	#334	1	256	yes	direct product factor
91	C₂⁴	C₂×C₄⋊C₈	1	120	yes	—
92	C₂⁴	C₂×#106	1	120	yes	—
93	C₂⁴	C₂×#108	1	120	yes	—
94	C₂⁴	C₂²×C₄⋊C₄	1	35	yes	—
95	C₂²×C₄	—	2	420	no	—
96	C₂²×C₄	—	2	140	no	—
97	C₂²×C₄	C₂×C₄⋊C₈	1	840	yes	—
98	C₂²×C₄	C₂×#106	1	840	yes	—
99	C₂²×C₄	C₂²×C₄⋊C₄	1	105	yes	—
100	C₂²×C₄	C₂²×C₄⋊C₄	1	105	yes	—
101	C₂²×C₄	C₂²×C₄⋊C₄	1	105	yes	—
102	C₂²×C₄	C₂³×C₈	1	140	yes	—
103	C₂²×C₄	C₂²×M₁₆	1	140	yes	—
104	C₂²×C₄	C₂²×SD₁₆	1	280	yes	—
105	C₂²×C₄	C₂⁴×C₄	1	15	yes	—
106	C₂²×C₄	C₂³×D₄	1	15	yes	—
107	C₂²×C₄	C₂³×Q₈	1	15	yes	—
108	C₂×C₈	—	2	240	no	—
109	C₂×C₈	—	2	32	no	—
110	C₂×C₈	—	4	480	no	—
111	C₂×C₈	C₂³×C₈	1	480	yes	complemented
112	C₂×C₈	C₂⁴×C₄	1	16	yes	—
113	C₂×C₈	C₂⁴×C₄	1	16	yes	—
114	C₄²	C₂²×C₄⋊C₄	1	210	yes	—
115	C₄²	C₂⁴×C₄	1	15	yes	—
116	C₄²	C₂³×D₄	1	15	yes	—
117	C₄²	C₂³×Q₈	1	15	yes	—
118	C₄²	C₂⁶	1	1	yes	characteristic
119	C₂³	C₂×#334	1	960	yes	direct product factor
120	C₂³	C₂²×C₄⋊C₈	1	140	yes	—
121	C₂³	C₂²×#106	1	140	yes	—
122	C₂³	C₂²×#108	1	140	yes	—
123	C₂³	C₂³×C₄⋊C₄	1	15	yes	—
124	C₂×C₄	—	2	210	no	—
125	C₂×C₄	—	2	30	no	—
126	C₂×C₄	C₂²×C₄⋊C₈	1	420	yes	—
127	C₂×C₄	C₂²×#106	1	420	yes	—
128	C₂×C₄	C₂³×C₄⋊C₄	1	15	yes	—
129	C₂×C₄	C₂³×C₄⋊C₄	1	15	yes	—
130	C₂×C₄	C₂³×C₄⋊C₄	1	15	yes	—
131	C₂×C₄	C₂⁴×C₈	1	30	yes	—
132	C₂×C₄	C₂³×M₁₆	1	30	yes	—
133	C₂×C₄	C₂³×SD₁₆	1	60	yes	—
134	C₂×C₄	C₂⁵×C₄	1	1	yes	characteristic
135	C₂×C₄	C₂⁴×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂⁴×Q₈	1	1	yes	characteristic
137	C₈	—	2	16	no	—
138	C₈	—	4	32	no	—
139	C₈	C₂⁴×C₈	1	32	yes	complemented
140	C₂²	C₂²×#334	1	560	yes	direct product factor
141	C₂²	C₂³×C₄⋊C₈	1	30	yes	—
142	C₂²	C₂³×#106	1	30	yes	—
143	C₂²	C₂³×#108	1	30	yes	—
144	C₂²	C₂⁴×C₄⋊C₄	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	15	no	—
147	C₄	C₂³×C₄⋊C₈	1	30	yes	—
148	C₄	C₂³×#106	1	30	yes	—
149	C₄	C₂⁵×C₈	1	1	yes	characteristic
150	C₄	C₂⁴×M₁₆	1	1	yes	characteristic
151	C₄	C₂⁴×SD₁₆	1	2	yes	—
152	C₂	C₂³×#334	1	60	yes	direct product factor
153	C₂	C₂⁴×#106	1	1	yes	characteristic
154	C₂	C₂⁴×#108	1	1	yes	characteristic
155	C₂	C₂⁴×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵×C₄	D₄
commutator	C₄	C₂⁵×C₈
Frattini, cosocle	C₄²	C₂⁶
socle, prime radical	C₂⁶	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂³×#334	C₂	direct (N × H)
12. C₂²×#334	C₂²	direct (N × H)
19. C₂⁴×C₈	C₈	semidirect
30. C₂×#334	C₂³	direct (N × H)
45. C₂³×C₈	C₂×C₈	semidirect
57. #334	C₂⁴	direct (N × H)
78. C₂²×C₈	C₂²×C₈	semidirect
90. C₂⁴	#334	direct (N × H)
111. C₂×C₈	C₂³×C₈	semidirect
119. C₂³	C₂×#334	direct (N × H)
139. C₈	C₂⁴×C₈	semidirect
140. C₂²	C₂²×#334	direct (N × H)
152. C₂	C₂³×#334	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×C₄⋊C₈	2
C₂⁴×C₄×C₈	2
C₂³×#334	2

Subgroups — C24×#334 #72286586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#334 #72286586