Subgroups — C₂²×C₅×#335 #11940504

Order 1280.

942 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₈	—	2	2	no	—
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₈	—	2	6	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	12	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	6	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	6	yes	—
12	C₅×#335	C₂²	1	16	yes	direct product factor
13	C₂²×#335	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	12	no	—
19	C₂²×C₅×C₈	—	4	8	no	—
20	C₂²×C₅×C₈	C₈	1	8	yes	complemented
21	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
24	C₂×C₄²×C₅	D₄	1	3	yes	—
25	C₂×C₄²×C₅	Q₈	1	3	yes	—
26	C₂×C₄²×C₅	C₂³	1	3	yes	—
27	C₅×C₄⋊C₈	—	2	12	no	—
28	C₅×C₄⋊C₈	C₂³	1	8	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	12	yes	—
30	C₄×C₅×C₈	C₂³	1	4	yes	—
31	C₂²×C₄⋊C₈	C₂×C₅	1	2	yes	—
32	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
33	C₂×#335	C₂×C₅	1	12	yes	direct product factor
34	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	—	2	4	no	—
36	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
37	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
39	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₅	M₁₆	1	4	yes	—
41	C₂²×C₄×C₅	D₈	1	4	yes	—
42	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
45	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
46	C₂×C₅×C₈	—	2	12	no	—
47	C₂×C₅×C₈	—	2	8	no	—
48	C₂×C₅×C₈	—	4	24	no	—
49	C₂×C₅×C₈	C₂×C₈	1	24	yes	complemented
50	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
52	C₄²×C₅	C₄⋊C₄	1	6	yes	—
53	C₄²×C₅	C₂²×C₄	1	3	yes	—
54	C₄²×C₅	C₂×D₄	1	3	yes	—
55	C₄²×C₅	C₂×Q₈	1	3	yes	—
56	C₄²×C₅	C₂⁴	1	1	yes	characteristic
57	C₂³×C₈	—	2	2	no	—
58	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
59	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
60	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
61	C₂×C₄⋊C₈	—	2	6	no	—
62	C₂×C₄⋊C₈	C₂²×C₅	1	12	yes	—
63	C₂×C₄×C₈	C₄×C₅	1	6	yes	—
64	C₂×C₄×C₈	C₂²×C₅	1	6	yes	—
65	#335	C₂²×C₅	1	16	yes	direct product factor
66	C₂³×C₅	C₄⋊C₈	1	4	yes	—
67	C₂³×C₅	#107	1	4	yes	—
68	C₂³×C₅	#108	1	4	yes	—
69	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₅	—	2	6	no	—
71	C₂×C₄×C₅	—	2	6	no	—
72	C₂×C₄×C₅	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₅	#107	1	12	yes	—
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
78	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
79	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
80	C₂×C₄×C₅	C₂×Q₁₆	1	6	yes	—
81	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
84	C₅×C₈	—	2	4	no	—
85	C₅×C₈	—	4	8	no	—
86	C₅×C₈	C₂²×C₈	1	8	yes	complemented
87	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
88	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
89	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
90	C₂²×C₈	—	2	4	no	—
91	C₂²×C₈	—	2	12	no	—
92	C₂²×C₈	—	4	8	no	—
93	C₂²×C₈	C₅×C₈	1	8	yes	complemented
94	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
95	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
96	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
97	C₂×C₄²	C₅×D₄	1	3	yes	—
98	C₂×C₄²	C₅×Q₈	1	3	yes	—
99	C₂×C₄²	C₂³×C₅	1	3	yes	—
100	C₄⋊C₈	—	2	12	no	—
101	C₄⋊C₈	C₂³×C₅	1	8	yes	—
102	C₄×C₈	C₂×C₄×C₅	1	12	yes	—
103	C₄×C₈	C₂³×C₅	1	4	yes	—
104	C₂²×C₅	#335	1	16	yes	direct product factor
105	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
106	C₂²×C₅	C₂×#107	1	6	yes	—
107	C₂²×C₅	C₂×#108	1	6	yes	—
108	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₅	—	2	3	no	—
110	C₄×C₅	—	2	1	no	—
111	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
112	C₄×C₅	C₂×#107	1	6	yes	—
113	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
114	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
115	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
120	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
121	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
122	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
123	C₂²×C₄	C₅×M₁₆	1	4	yes	—
124	C₂²×C₄	C₅×D₈	1	4	yes	—
125	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
126	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
127	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
128	C₂²×C₄	C₂×C₅×Q₈	1	3	yes	—
129	C₂×C₈	—	2	12	no	—
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	4	24	no	—
132	C₂×C₈	C₂×C₅×C₈	1	24	yes	complemented
133	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
134	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
135	C₄²	C₅×C₄⋊C₄	1	6	yes	—
136	C₄²	C₂²×C₄×C₅	1	3	yes	—
137	C₄²	C₂×C₅×D₄	1	3	yes	—
138	C₄²	C₂×C₅×Q₈	1	3	yes	—
139	C₄²	C₂⁴×C₅	1	1	yes	characteristic
140	C₂×C₅	C₂×#335	1	12	yes	direct product factor
141	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₅	C₂²×#107	1	1	yes	characteristic
143	C₂×C₅	C₂²×#108	1	1	yes	characteristic
144	C₂³	C₅×C₄⋊C₈	1	4	yes	—
145	C₂³	C₅×#107	1	4	yes	—
146	C₂³	C₅×#108	1	4	yes	—
147	C₂³	C₂×C₅×C₄⋊C₄	1	3	yes	—
148	C₂×C₄	—	2	6	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	C₅×C₄⋊C₈	1	12	yes	—
151	C₂×C₄	C₅×#107	1	12	yes	—
152	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
153	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
154	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
155	C₂×C₄	C₂²×C₅×C₈	1	6	yes	—
156	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
157	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
158	C₂×C₄	C₂×C₅×Q₁₆	1	6	yes	—
159	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
162	C₈	—	2	4	no	—
163	C₈	—	4	8	no	—
164	C₈	C₂²×C₅×C₈	1	8	yes	complemented
165	C₅	C₂²×#335	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²	C₅×#335	1	16	yes	direct product factor
167	C₂²	C₂×C₅×C₄⋊C₈	1	6	yes	—
168	C₂²	C₂×C₅×#107	1	6	yes	—
169	C₂²	C₂×C₅×#108	1	6	yes	—
170	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
171	C₄	—	2	3	no	—
172	C₄	—	2	1	no	—
173	C₄	C₂×C₅×C₄⋊C₈	1	6	yes	—
174	C₄	C₂×C₅×#107	1	6	yes	—
175	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
176	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
177	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
178	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
179	C₂	C₂×C₅×#335	1	12	yes	direct product factor
180	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
181	C₂	C₂²×C₅×#107	1	1	yes	characteristic
182	C₂	C₂²×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₅	D₄
commutator	C₄	C₂³×C₅×C₈
Frattini, cosocle	C₄²	C₂⁴×C₅
socle, prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#335	C₂	direct (N × H)
12. C₅×#335	C₂²	direct (N × H)
13. C₂²×#335	C₅	direct (N × H)
20. C₂²×C₅×C₈	C₈	semidirect
33. C₂×#335	C₂×C₅	direct (N × H)
49. C₂×C₅×C₈	C₂×C₈	semidirect
65. #335	C₂²×C₅	direct (N × H)
86. C₅×C₈	C₂²×C₈	semidirect
93. C₂²×C₈	C₅×C₈	semidirect
104. C₂²×C₅	#335	direct (N × H)
132. C₂×C₈	C₂×C₅×C₈	semidirect
140. C₂×C₅	C₂×#335	direct (N × H)
164. C₈	C₂²×C₅×C₈	semidirect
165. C₅	C₂²×#335	direct (N × H)
166. C₂²	C₅×#335	direct (N × H)
179. C₂	C₂×C₅×#335	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×C₅×C₄⋊C₈	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#335	2
C₂²×#335	5

Subgroups — C22×C5×#335 #11940504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#335 #11940504