Subgroups — C₂⁴×#365 #72286486

Order 1024.

16917 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#108	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×#106	C₂	1	1	yes	maximal, characteristic
4	C₂³×#365	C₂	1	60	yes	maximal, direct product factor
5	C₂⁴×M₁₆	—	2	1	no	—
6	C₂⁵×C₈	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄⋊C₄	—	2	1	no	—
8	C₂³×M₃₂	C₄	1	30	yes	complemented
9	C₂³×M₃₂	C₂²	1	30	yes	—
10	C₂⁴×C₁₆	C₄	1	2	yes	complemented
11	C₂³×#108	—	2	15	no	—
12	C₂³×#108	C₂²	1	30	yes	—
13	C₂³×#106	—	2	15	no	—
14	C₂³×#106	C₂²	1	30	yes	—
15	C₂²×#365	C₂²	1	560	yes	direct product factor
16	C₂⁵×C₄	—	4	1	no	—
17	C₂⁵×C₄	D₄	1	1	yes	characteristic
18	C₂³×M₁₆	—	2	30	no	—
19	C₂³×M₁₆	—	4	15	no	—
20	C₂⁴×C₈	—	4	1	no	—
21	C₂⁴×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂⁴×C₈	C₂×C₄	1	15	yes	—
23	C₂⁴×C₈	D₄	1	15	yes	—
24	C₂⁴×C₈	Q₈	1	15	yes	—
25	C₂⁴×C₈	Q₈	1	1	yes	characteristic
26	C₂⁴×C₈	C₂³	1	15	yes	—
27	C₂³×C₄⋊C₄	—	2	30	no	—
28	C₂³×C₄⋊C₄	—	4	15	no	—
29	C₂²×M₃₂	C₂×C₄	1	420	yes	complemented
30	C₂²×M₃₂	C₂³	1	140	yes	—
31	C₂³×C₁₆	C₂×C₄	1	60	yes	complemented
32	C₂²×#108	—	2	210	no	—
33	C₂²×#108	C₂³	1	140	yes	—
34	C₂²×#106	—	2	210	no	—
35	C₂²×#106	C₂³	1	140	yes	—
36	C₂×#365	C₂³	1	960	yes	direct product factor
37	C₂⁶	SD₁₆	1	1	yes	characteristic
38	C₂⁴×C₄	—	4	30	no	—
39	C₂⁴×C₄	—	8	16	no	—
40	C₂⁴×C₄	C₄⋊C₄	1	15	yes	—
41	C₂⁴×C₄	C₄⋊C₄	1	1	yes	characteristic
42	C₂⁴×C₄	SD₁₆	1	15	yes	—
43	C₂⁴×C₄	SD₁₆	1	15	yes	—
44	C₂⁴×C₄	SD₁₆	1	1	yes	characteristic
45	C₂⁴×C₄	C₂×D₄	1	15	yes	—
46	C₂²×M₁₆	—	2	140	no	—
47	C₂²×M₁₆	—	4	210	no	—
48	C₂³×C₈	—	4	30	no	—
49	C₂³×C₈	C₄⋊C₄	1	15	yes	—
50	C₂³×C₈	C₄⋊C₄	1	210	yes	—
51	C₂³×C₈	C₄⋊C₄	1	15	yes	—
52	C₂³×C₈	C₂²×C₄	1	15	yes	—
53	C₂³×C₈	C₂²×C₄	1	105	yes	—
54	C₂³×C₈	C₂×D₄	1	105	yes	—
55	C₂³×C₈	C₂×Q₈	1	105	yes	—
56	C₂³×C₈	C₂×Q₈	1	15	yes	—
57	C₂³×C₈	C₂⁴	1	35	yes	—
58	C₂²×C₄⋊C₄	—	2	140	no	—
59	C₂²×C₄⋊C₄	—	4	210	no	—
60	C₂×M₃₂	C₂²×C₄	1	840	yes	complemented
61	C₂×M₃₂	C₂⁴	1	120	yes	—
62	C₂²×C₁₆	C₂²×C₄	1	280	yes	complemented
63	C₂×#108	—	2	420	no	—
64	C₂×#108	C₂⁴	1	120	yes	—
65	C₂×#106	—	2	420	no	—
66	C₂×#106	C₂⁴	1	120	yes	—
67	#365	C₂⁴	1	256	yes	direct product factor
68	C₂⁵	—	2	16	no	—
69	C₂⁵	#106	1	15	yes	—
70	C₂⁵	#106	1	1	yes	characteristic
71	C₂⁵	C₂×SD₁₆	1	15	yes	—
72	C₂³×C₄	—	4	140	no	—
73	C₂³×C₄	—	8	240	no	—
74	C₂³×C₄	#106	1	210	yes	—
75	C₂³×C₄	#106	1	15	yes	—
76	C₂³×C₄	#106	1	15	yes	—
77	C₂³×C₄	C₂×C₄⋊C₄	1	105	yes	—
78	C₂³×C₄	C₂×C₄⋊C₄	1	15	yes	—
79	C₂³×C₄	C₂×SD₁₆	1	15	yes	—
80	C₂³×C₄	C₂×SD₁₆	1	105	yes	—
81	C₂³×C₄	C₂×SD₁₆	1	105	yes	—
82	C₂³×C₄	C₂²×D₄	1	35	yes	—
83	C₂×M₁₆	—	2	120	no	—
84	C₂×M₁₆	—	4	420	no	—
85	C₂²×C₈	—	4	140	no	—
86	C₂²×C₈	C₂×C₄⋊C₄	1	105	yes	—
87	C₂²×C₈	C₂×C₄⋊C₄	1	630	yes	—
88	C₂²×C₈	C₂×C₄⋊C₄	1	105	yes	—
89	C₂²×C₈	C₂³×C₄	1	35	yes	—
90	C₂²×C₈	C₂³×C₄	1	105	yes	—
91	C₂²×C₈	C₂²×D₄	1	105	yes	—
92	C₂²×C₈	C₂²×Q₈	1	105	yes	—
93	C₂²×C₈	C₂²×Q₈	1	35	yes	—
94	C₂²×C₈	C₂⁵	1	15	yes	—
95	C₂×C₄⋊C₄	—	2	120	no	—
96	C₂×C₄⋊C₄	—	4	420	no	—
97	M₃₂	C₂³×C₄	1	240	yes	complemented
98	M₃₂	C₂⁵	1	16	yes	—
99	C₂×C₁₆	C₂³×C₄	1	240	yes	complemented
100	#108	—	2	120	no	—
101	#108	C₂⁵	1	16	yes	—
102	#106	—	2	120	no	—
103	#106	C₂⁵	1	16	yes	—
104	C₂⁴	—	2	120	no	—
105	C₂⁴	—	2	120	no	—
106	C₂⁴	#365	1	16	yes	direct product factor
107	C₂⁴	C₂×#106	1	105	yes	—
108	C₂⁴	C₂×#106	1	15	yes	—
109	C₂⁴	C₂²×SD₁₆	1	35	yes	—
110	C₂²×C₄	—	4	120	no	—
111	C₂²×C₄	—	8	560	no	—
112	C₂²×C₄	C₂×#106	1	630	yes	—
113	C₂²×C₄	C₂×#106	1	105	yes	—
114	C₂²×C₄	C₂×#106	1	105	yes	—
115	C₂²×C₄	C₂²×C₄⋊C₄	1	105	yes	—
116	C₂²×C₄	C₂²×C₄⋊C₄	1	35	yes	—
117	C₂²×C₄	C₂²×SD₁₆	1	105	yes	—
118	C₂²×C₄	C₂²×SD₁₆	1	35	yes	—
119	C₂²×C₄	C₂²×SD₁₆	1	105	yes	—
120	C₂²×C₄	C₂³×D₄	1	15	yes	—
121	M₁₆	—	2	16	no	—
122	M₁₆	—	4	120	no	—
123	C₂×C₈	—	4	120	no	—
124	C₂×C₈	C₂²×C₄⋊C₄	1	105	yes	—
125	C₂×C₈	C₂²×C₄⋊C₄	1	210	yes	—
126	C₂×C₈	C₂²×C₄⋊C₄	1	105	yes	—
127	C₂×C₈	C₂⁴×C₄	1	15	yes	—
128	C₂×C₈	C₂⁴×C₄	1	15	yes	—
129	C₂×C₈	C₂³×D₄	1	15	yes	—
130	C₂×C₈	C₂³×Q₈	1	15	yes	—
131	C₂×C₈	C₂³×Q₈	1	15	yes	—
132	C₂×C₈	C₂⁶	1	1	yes	characteristic
133	C₄⋊C₄	—	2	16	no	—
134	C₄⋊C₄	—	4	120	no	—
135	C₁₆	C₂⁴×C₄	1	32	yes	complemented
136	C₂³	—	2	140	no	—
137	C₂³	—	2	420	no	—
138	C₂³	C₂×#365	1	120	yes	direct product factor
139	C₂³	C₂²×#106	1	105	yes	—
140	C₂³	C₂²×#106	1	35	yes	—
141	C₂³	C₂³×SD₁₆	1	15	yes	—
142	C₂×C₄	—	4	16	no	—
143	C₂×C₄	—	8	240	no	—
144	C₂×C₄	C₂²×#106	1	210	yes	—
145	C₂×C₄	C₂²×#106	1	105	yes	—
146	C₂×C₄	C₂²×#106	1	105	yes	—
147	C₂×C₄	C₂³×C₄⋊C₄	1	15	yes	—
148	C₂×C₄	C₂³×C₄⋊C₄	1	15	yes	—
149	C₂×C₄	C₂³×SD₁₆	1	15	yes	—
150	C₂×C₄	C₂³×SD₁₆	1	15	yes	—
151	C₂×C₄	C₂³×SD₁₆	1	15	yes	—
152	C₂×C₄	C₂⁴×D₄	1	1	yes	characteristic
153	C₈	—	4	16	no	—
154	C₈	C₂³×C₄⋊C₄	1	15	yes	—
155	C₈	C₂³×C₄⋊C₄	1	15	yes	—
156	C₈	C₂⁵×C₄	1	1	yes	characteristic
157	C₈	C₂⁴×Q₈	1	1	yes	characteristic
158	C₂²	—	2	30	no	—
159	C₂²	—	2	210	no	—
160	C₂²	C₂²×#365	1	140	yes	direct product factor
161	C₂²	C₂³×#106	1	15	yes	—
162	C₂²	C₂³×#106	1	15	yes	—
163	C₂²	C₂⁴×SD₁₆	1	1	yes	characteristic
164	C₄	—	8	16	no	—
165	C₄	C₂³×#106	1	15	yes	—
166	C₄	C₂³×#106	1	15	yes	—
167	C₄	C₂⁴×C₄⋊C₄	1	1	yes	characteristic
168	C₄	C₂⁴×SD₁₆	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	15	no	—
171	C₂	C₂³×#365	1	30	yes	direct product factor
172	C₂	C₂⁴×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	#106
commutator	C₈	C₂⁵×C₄
Frattini, cosocle	C₂×C₈	C₂⁶
prime radical	C₂⁶	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₂³×#365	C₂	direct (N × H)
8. C₂³×M₃₂	C₄	semidirect
10. C₂⁴×C₁₆	C₄	semidirect
15. C₂²×#365	C₂²	direct (N × H)
29. C₂²×M₃₂	C₂×C₄	semidirect
31. C₂³×C₁₆	C₂×C₄	semidirect
36. C₂×#365	C₂³	direct (N × H)
60. C₂×M₃₂	C₂²×C₄	semidirect
62. C₂²×C₁₆	C₂²×C₄	semidirect
67. #365	C₂⁴	direct (N × H)
97. M₃₂	C₂³×C₄	semidirect
99. C₂×C₁₆	C₂³×C₄	semidirect
106. C₂⁴	#365	direct (N × H)
135. C₁₆	C₂⁴×C₄	semidirect
138. C₂³	C₂×#365	direct (N × H)
160. C₂²	C₂²×#365	direct (N × H)
171. C₂	C₂³×#365	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×M₃₂	2
C₂⁴×#108	2
C₂⁴×#106	2
C₂³×#365	2

Subgroups — C24×#365 #72286486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#365 #72286486