Subgroups — C₄×#64772 #12004487

Order 1280.

910 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#681	C₂	1	2	yes	maximal
3	C₂×#64772	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×#681	C₂²	1	4	yes	—
7	C₂×C₄×C₅×C₈	—	2	4	no	—
8	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
9	C₄×#681	—	2	4	no	—
10	C₄×#681	C₄	1	8	yes	—
11	C₂×#3935	—	2	2	no	—
12	C₂×C₄×#168	—	2	2	no	—
13	#64772	C₄	1	16	yes	direct product factor
14	C₄×#341	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	Q₈	1	2	yes	—
16	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	4	yes	—
26	C₂×#681	—	2	4	no	—
27	C₂×#681	—	2	8	no	—
28	C₂×#681	C₂×C₄	1	16	yes	—
29	C₂×#681	C₂×C₄	1	8	yes	—
30	C₂²×#168	—	2	4	no	—
31	C₄×C₅×C₈	—	2	8	no	—
32	#3935	—	2	8	no	—
33	C₄×#168	—	2	8	no	—
34	C₂×C₄×M₁₆	—	5	2	no	—
35	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
36	C₂×#341	—	5	4	no	—
37	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₅	—	2	8	no	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅	#41	1	4	yes	—
48	C₂×C₅×C₈	—	2	4	no	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	2	16	no	—
51	C₂×C₅×C₈	C₄²	1	8	yes	—
52	C₄²×C₅	C₄²	1	4	yes	—
53	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
54	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
56	#681	—	2	8	no	—
57	#681	—	2	16	no	—
58	#681	C₄²	1	16	yes	—
59	C₂×#168	—	2	8	no	—
60	C₂×#168	—	2	16	no	—
61	C₂²×M₁₆	—	5	4	no	—
62	C₂³×C₈	C₂×D₅	1	2	yes	—
63	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
64	C₄×M₁₆	—	5	8	no	—
65	C₄×M₁₆	—	10	4	no	—
66	C₂×#97	—	10	2	no	—
67	C₂×C₄×C₈	—	2	4	no	—
68	C₂×C₄×C₈	—	10	2	no	—
69	C₂×C₄×C₈	Dic₅	1	4	yes	—
70	#341	—	5	16	no	—
71	C₂³×C₅	—	2	4	no	—
72	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₅	C₄×Q₈	1	4	yes	—
75	C₂×C₄×C₅	—	2	4	no	—
76	C₂×C₄×C₅	—	2	16	no	—
77	C₂×C₄×C₅	#95	1	4	yes	—
78	C₂×C₄×C₅	#95	1	4	yes	—
79	C₂×C₄×C₅	#108	1	8	yes	—
80	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
81	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₅	#117	1	4	yes	—
86	C₂×C₄×C₅	#117	1	2	yes	—
87	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
89	C₅×C₈	—	2	8	no	—
90	#168	—	2	16	no	—
91	C₂³×C₄	Dic₁₀	1	2	yes	—
92	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
93	C₂×M₁₆	—	5	16	no	—
94	C₂×M₁₆	—	5	8	no	—
95	C₂×M₁₆	—	10	8	no	—
96	C₂×M₁₆	—	10	4	no	—
97	C₂²×C₈	—	2	4	no	—
98	C₂²×C₈	—	2	4	no	—
99	C₂²×C₈	—	10	4	no	—
100	C₂²×C₈	C₄×D₅	1	8	yes	—
101	C₂²×C₈	C₂×Dic₅	1	4	yes	—
102	C₂×C₄²	C₄×D₅	1	4	yes	—
103	C₂×C₄²	D₂₀	1	1	yes	characteristic
104	C₂×C₄²	D₂₀	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₅	1	2	yes	—
106	C₂×C₄²	C₂²⋊D₅	1	4	yes	—
107	#97	—	10	8	no	—
108	C₄×C₈	—	2	8	no	—
109	C₄×C₈	—	10	8	no	—
110	C₂²×C₅	—	2	4	no	—
111	C₂²×C₅	—	2	8	no	—
112	C₂²×C₅	C₂×#95	1	1	yes	characteristic
113	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
114	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
115	C₂²×C₅	C₂×#108	1	4	yes	—
116	C₄×C₅	—	2	4	no	—
117	C₄×C₅	#341	1	8	yes	complemented
118	C₄×C₅	C₄³	1	1	yes	characteristic
119	C₄×C₅	#376	1	1	yes	characteristic
120	C₄×C₅	C₄×C₂²⋊C₄	1	4	yes	—
121	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
122	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	#683	1	8	yes	—
125	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
127	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
129	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
130	C₂²×C₄	#709	1	4	yes	—
131	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
133	M₁₆	—	5	16	no	—
134	M₁₆	—	10	8	no	—
135	M₁₆	—	10	16	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	16	no	—
139	C₂×C₈	—	10	16	no	—
140	C₂×C₈	—	10	8	no	—
141	C₂×C₈	C₄×Dic₅	1	8	yes	—
142	C₄²	C₄×Dic₅	1	4	yes	—
143	C₄²	#685	1	4	yes	—
144	C₄²	#685	1	4	yes	—
145	C₄²	C₂²⋊Dic₅	1	4	yes	—
146	C₂×C₅	—	2	4	no	—
147	C₂×C₅	C₄×#95	1	1	yes	characteristic
148	C₂×C₅	C₂×#341	1	2	yes	—
149	C₂×C₅	C₄×#108	1	4	yes	—
150	C₂³	—	2	4	no	—
151	C₂³	C₄×Dic₁₀	1	4	yes	—
152	C₂³	C₂×#683	1	2	yes	—
153	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	#3951	1	8	yes	—
157	C₂×C₄	#3963	1	4	yes	—
158	C₂×C₄	#3963	1	4	yes	—
159	C₂×C₄	C₄²×D₅	1	4	yes	—
160	C₂×C₄	#4018	1	4	yes	—
161	C₂×C₄	C₄×D₂₀	1	4	yes	—
162	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
164	C₂×C₄	#4072	1	2	yes	—
165	C₂×C₄	C₂×#685	1	2	yes	—
166	C₂×C₄	C₄×C₂²⋊D₅	1	8	yes	—
167	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
168	C₈	—	2	8	no	—
169	C₈	—	10	16	no	—
170	C₅	C₄×#341	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	4	no	—
172	C₂²	—	2	8	no	—
173	C₂²	C₄×#683	1	4	yes	—
174	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
175	C₂²	C₂×#3951	1	4	yes	—
176	C₂²	C₂×#3963	1	1	yes	characteristic
177	C₄	—	2	4	no	—
178	C₄	#64772	1	8	yes	direct product factor
179	C₄	C₄²×Dic₅	1	1	yes	characteristic
180	C₄	#65221	1	1	yes	characteristic
181	C₄	C₄×#685	1	4	yes	—
182	C₄	C₄×C₂²⋊Dic₅	1	2	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₄×#3951	1	4	yes	—
185	C₂	C₄×#3963	1	1	yes	characteristic
186	C₂	C₂×#64772	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₀
commutator	C₄×C₅	C₄³
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₄×#341
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
13. #64772	C₄	direct (N × H)
117. C₄×C₅	#341	semidirect
170. C₅	C₄×#341	semidirect
178. C₄	#64772	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×C₄×C₅×C₈	2
C₂×C₄×#681	2
C₂×#64772	2
C₄×#341	5

Subgroups — C4×#64772 #12004487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#64772 #12004487