Subgroups — C₂²×C₅×#365 #11956122

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#365	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
8	C₂×C₅×M₃₂	C₄	1	6	yes	complemented
9	C₂×C₅×M₃₂	C₂²	1	6	yes	—
10	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
11	C₂×C₅×#108	—	2	3	no	—
12	C₂×C₅×#108	C₂²	1	6	yes	—
13	C₂×C₅×#106	—	2	3	no	—
14	C₂×C₅×#106	C₂²	1	6	yes	—
15	C₅×#365	C₂²	1	16	yes	direct product factor
16	C₂²×#365	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	—	4	1	no	—
18	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	6	no	—
20	C₂×C₅×M₁₆	—	4	3	no	—
21	C₂²×C₅×C₈	—	4	1	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
24	C₂²×C₅×C₈	D₄	1	3	yes	—
25	C₂²×C₅×C₈	Q₈	1	3	yes	—
26	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	3	yes	—
28	C₂×C₅×C₄⋊C₄	—	2	6	no	—
29	C₂×C₅×C₄⋊C₄	—	4	3	no	—
30	C₅×M₃₂	C₂×C₄	1	12	yes	complemented
31	C₅×M₃₂	C₂³	1	4	yes	—
32	C₂×C₅×C₁₆	C₂×C₄	1	12	yes	complemented
33	C₅×#108	—	2	6	no	—
34	C₅×#108	C₂³	1	4	yes	—
35	C₅×#106	—	2	6	no	—
36	C₅×#106	C₂³	1	4	yes	—
37	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
38	C₂²×#108	C₂×C₅	1	1	yes	characteristic
39	C₂²×#106	C₂×C₅	1	1	yes	characteristic
40	C₂×#365	C₂×C₅	1	12	yes	direct product factor
41	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	4	6	no	—
43	C₂²×C₄×C₅	—	8	4	no	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
47	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
48	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
50	C₅×M₁₆	—	2	4	no	—
51	C₅×M₁₆	—	4	6	no	—
52	C₂×C₅×C₈	—	4	6	no	—
53	C₂×C₅×C₈	C₄⋊C₄	1	3	yes	—
54	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
55	C₂×C₅×C₈	C₄⋊C₄	1	3	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
57	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
58	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
59	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
60	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
61	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	2	4	no	—
63	C₅×C₄⋊C₄	—	4	6	no	—
64	C₅×C₁₆	C₂²×C₄	1	8	yes	complemented
65	C₂²×M₁₆	—	2	1	no	—
66	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₂²×C₄⋊C₄	—	2	1	no	—
68	C₂×M₃₂	C₄×C₅	1	6	yes	complemented
69	C₂×M₃₂	C₂²×C₅	1	6	yes	—
70	C₂²×C₁₆	C₄×C₅	1	2	yes	complemented
71	C₂×#108	—	2	3	no	—
72	C₂×#108	C₂²×C₅	1	6	yes	—
73	C₂×#106	—	2	3	no	—
74	C₂×#106	C₂²×C₅	1	6	yes	—
75	#365	C₂²×C₅	1	16	yes	direct product factor
76	C₂³×C₅	—	2	4	no	—
77	C₂³×C₅	#106	1	3	yes	—
78	C₂³×C₅	#106	1	1	yes	characteristic
79	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	—	8	12	no	—
82	C₂×C₄×C₅	#106	1	6	yes	—
83	C₂×C₄×C₅	#106	1	3	yes	—
84	C₂×C₄×C₅	#106	1	3	yes	—
85	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
86	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
88	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
89	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
90	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
91	C₅×C₈	—	4	4	no	—
92	C₅×C₈	C₂×C₄⋊C₄	1	3	yes	—
93	C₅×C₈	C₂×C₄⋊C₄	1	3	yes	—
94	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
95	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
96	C₂³×C₄	—	4	1	no	—
97	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
98	C₂×M₁₆	—	2	6	no	—
99	C₂×M₁₆	—	4	3	no	—
100	C₂²×C₈	—	4	1	no	—
101	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
103	C₂²×C₈	C₅×D₄	1	3	yes	—
104	C₂²×C₈	C₅×Q₈	1	3	yes	—
105	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₅	1	3	yes	—
107	C₂×C₄⋊C₄	—	2	6	no	—
108	C₂×C₄⋊C₄	—	4	3	no	—
109	M₃₂	C₂×C₄×C₅	1	12	yes	complemented
110	M₃₂	C₂³×C₅	1	4	yes	—
111	C₂×C₁₆	C₂×C₄×C₅	1	12	yes	complemented
112	#108	—	2	6	no	—
113	#108	C₂³×C₅	1	4	yes	—
114	#106	—	2	6	no	—
115	#106	C₂³×C₅	1	4	yes	—
116	C₂²×C₅	—	2	6	no	—
117	C₂²×C₅	—	2	6	no	—
118	C₂²×C₅	#365	1	4	yes	direct product factor
119	C₂²×C₅	C₂×#106	1	3	yes	—
120	C₂²×C₅	C₂×#106	1	3	yes	—
121	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
122	C₄×C₅	—	8	4	no	—
123	C₄×C₅	C₂×#106	1	3	yes	—
124	C₄×C₅	C₂×#106	1	3	yes	—
125	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
127	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
128	C₂²×C₄	—	4	6	no	—
129	C₂²×C₄	—	8	4	no	—
130	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
131	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
133	C₂²×C₄	C₅×SD₁₆	1	3	yes	—
134	C₂²×C₄	C₅×SD₁₆	1	3	yes	—
135	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
136	M₁₆	—	2	4	no	—
137	M₁₆	—	4	6	no	—
138	C₂×C₈	—	4	6	no	—
139	C₂×C₈	C₅×C₄⋊C₄	1	3	yes	—
140	C₂×C₈	C₅×C₄⋊C₄	1	6	yes	—
141	C₂×C₈	C₅×C₄⋊C₄	1	3	yes	—
142	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
143	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
144	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
145	C₂×C₈	C₂×C₅×Q₈	1	3	yes	—
146	C₂×C₈	C₂×C₅×Q₈	1	3	yes	—
147	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	6	no	—
150	C₁₆	C₂²×C₄×C₅	1	8	yes	complemented
151	C₂×C₅	—	2	1	no	—
152	C₂×C₅	—	2	3	no	—
153	C₂×C₅	C₂×#365	1	6	yes	direct product factor
154	C₂×C₅	C₂²×#106	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	C₅×#106	1	3	yes	—
157	C₂³	C₅×#106	1	1	yes	characteristic
158	C₂³	C₂×C₅×SD₁₆	1	3	yes	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	12	no	—
161	C₂×C₄	C₅×#106	1	6	yes	—
162	C₂×C₄	C₅×#106	1	3	yes	—
163	C₂×C₄	C₅×#106	1	3	yes	—
164	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
165	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
166	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
167	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
168	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
169	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	C₂×C₅×C₄⋊C₄	1	3	yes	—
172	C₈	C₂×C₅×C₄⋊C₄	1	3	yes	—
173	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
174	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
175	C₅	C₂²×#365	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	C₅×#365	1	4	yes	direct product factor
179	C₂²	C₂×C₅×#106	1	3	yes	—
180	C₂²	C₂×C₅×#106	1	3	yes	—
181	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
182	C₄	—	8	4	no	—
183	C₄	C₂×C₅×#106	1	3	yes	—
184	C₄	C₂×C₅×#106	1	3	yes	—
185	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×C₅×#365	1	6	yes	direct product factor
190	C₂	C₂²×C₅×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#106
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#365	C₂	direct (N × H)
8. C₂×C₅×M₃₂	C₄	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
15. C₅×#365	C₂²	direct (N × H)
16. C₂²×#365	C₅	direct (N × H)
30. C₅×M₃₂	C₂×C₄	semidirect
32. C₂×C₅×C₁₆	C₂×C₄	semidirect
40. C₂×#365	C₂×C₅	direct (N × H)
64. C₅×C₁₆	C₂²×C₄	semidirect
68. C₂×M₃₂	C₄×C₅	semidirect
70. C₂²×C₁₆	C₄×C₅	semidirect
75. #365	C₂²×C₅	direct (N × H)
109. M₃₂	C₂×C₄×C₅	semidirect
111. C₂×C₁₆	C₂×C₄×C₅	semidirect
118. C₂²×C₅	#365	direct (N × H)
150. C₁₆	C₂²×C₄×C₅	semidirect
153. C₂×C₅	C₂×#365	direct (N × H)
175. C₅	C₂²×#365	direct (N × H)
178. C₂²	C₅×#365	direct (N × H)
189. C₂	C₂×C₅×#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
C₂²×C₅×M₃₂	2
C₂²×C₅×#108	2
C₂²×C₅×#106	2
C₂×C₅×#365	2
C₂²×#365	5

Subgroups — C22×C5×#365 #11956122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#365 #11956122