Subgroups — C₂²×C₅×#364 #11956119

Order 1280.

782 conjugacy classes of subgroups.

The subgroup lattice has 182 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	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#364	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	—	2	6	no	—
8	C₂×C₅×M₃₂	C₂²	1	12	yes	—
9	C₂²×C₅×C₁₆	—	2	2	no	—
10	C₂×C₄×C₅×C₈	C₄	1	6	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	6	yes	—
12	C₅×#364	C₂²	1	16	yes	direct product factor
13	C₂²×#364	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	—	2	1	no	—
15	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₈	1	6	yes	—
17	C₂²×C₅×C₈	C₈	1	2	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
19	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
20	C₂²×C₅×C₈	D₄	1	3	yes	—
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
23	C₂²×C₅×C₈	Q₈	1	3	yes	—
24	C₂²×C₅×C₈	C₂³	1	3	yes	—
25	C₂×C₄²×C₅	—	2	3	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
27	C₅×M₃₂	—	2	12	no	—
28	C₅×M₃₂	C₂³	1	8	yes	—
29	C₂×C₅×C₁₆	—	2	12	no	—
30	C₄×C₅×C₈	C₂×C₄	1	12	yes	—
31	C₄×C₅×C₈	C₂³	1	4	yes	—
32	C₂²×M₃₂	C₂×C₅	1	2	yes	—
33	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
34	C₂×#364	C₂×C₅	1	12	yes	direct product factor
35	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅	—	2	8	no	—
37	C₂²×C₄×C₅	—	2	6	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅	M₁₆	1	3	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
44	C₂×C₅×C₈	C₄⋊C₄	1	3	yes	—
45	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
46	C₂×C₅×C₈	C₄⋊C₄	1	3	yes	—
47	C₂×C₅×C₈	C₂×C₈	1	12	yes	—
48	C₂×C₅×C₈	C₂×C₈	1	12	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
51	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
52	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
53	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
54	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
55	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
56	C₄²×C₅	—	2	6	no	—
57	C₄²×C₅	C₂²×C₄	1	4	yes	—
58	C₅×C₁₆	—	2	8	no	—
59	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
60	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
61	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
62	C₂×M₃₂	—	2	6	no	—
63	C₂×M₃₂	C₂²×C₅	1	12	yes	—
64	C₂²×C₁₆	—	2	2	no	—
65	C₂×C₄×C₈	C₄×C₅	1	6	yes	—
66	C₂×C₄×C₈	C₂²×C₅	1	6	yes	—
67	#364	C₂²×C₅	1	16	yes	direct product factor
68	C₂³×C₅	—	2	4	no	—
69	C₂³×C₅	C₄⋊C₈	1	3	yes	—
70	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
71	C₂³×C₅	C₂×M₁₆	1	3	yes	—
72	C₂×C₄×C₅	—	2	24	no	—
73	C₂×C₄×C₅	—	2	4	no	—
74	C₂×C₄×C₅	C₄⋊C₈	1	6	yes	—
75	C₂×C₄×C₅	C₄⋊C₈	1	3	yes	—
76	C₂×C₄×C₅	C₄⋊C₈	1	3	yes	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
79	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
80	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
82	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
83	C₅×C₈	C₂×C₄⋊C₄	1	3	yes	—
84	C₅×C₈	C₂×C₄⋊C₄	1	3	yes	—
85	C₅×C₈	C₂²×C₈	1	8	yes	—
86	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
87	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
88	C₂³×C₄	—	2	1	no	—
89	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
90	C₂²×C₈	C₅×C₈	1	6	yes	—
91	C₂²×C₈	C₅×C₈	1	2	yes	—
92	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
93	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
94	C₂²×C₈	C₅×D₄	1	3	yes	—
95	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
97	C₂²×C₈	C₅×Q₈	1	3	yes	—
98	C₂²×C₈	C₂³×C₅	1	3	yes	—
99	C₂×C₄²	—	2	3	no	—
100	C₂×C₄²	C₂×C₄×C₅	1	6	yes	—
101	M₃₂	—	2	12	no	—
102	M₃₂	C₂³×C₅	1	8	yes	—
103	C₂×C₁₆	—	2	12	no	—
104	C₄×C₈	C₂×C₄×C₅	1	12	yes	—
105	C₄×C₈	C₂³×C₅	1	4	yes	—
106	C₂²×C₅	—	2	6	no	—
107	C₂²×C₅	—	2	6	no	—
108	C₂²×C₅	#364	1	4	yes	direct product factor
109	C₂²×C₅	C₂×C₄⋊C₈	1	3	yes	—
110	C₂²×C₅	C₂×C₄⋊C₈	1	3	yes	—
111	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
112	C₄×C₅	—	2	8	no	—
113	C₄×C₅	C₂×C₄⋊C₈	1	3	yes	—
114	C₄×C₅	C₂×C₄⋊C₈	1	3	yes	—
115	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
117	C₂⁴	C₅×M₁₆	1	1	yes	characteristic
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	6	no	—
120	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
121	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
123	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
124	C₂²×C₄	C₅×M₁₆	1	3	yes	—
125	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
126	C₂×C₈	C₅×C₄⋊C₄	1	6	yes	—
127	C₂×C₈	C₅×C₄⋊C₄	1	3	yes	—
128	C₂×C₈	C₅×C₄⋊C₄	1	3	yes	—
129	C₂×C₈	C₂×C₅×C₈	1	12	yes	—
130	C₂×C₈	C₂×C₅×C₈	1	12	yes	—
131	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
132	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
133	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
134	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
135	C₂×C₈	C₂×C₅×Q₈	1	3	yes	—
136	C₂×C₈	C₂×C₅×Q₈	1	3	yes	—
137	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
138	C₄²	—	2	6	no	—
139	C₄²	C₂²×C₄×C₅	1	4	yes	—
140	C₁₆	—	2	8	no	—
141	C₂×C₅	—	2	1	no	—
142	C₂×C₅	—	2	3	no	—
143	C₂×C₅	C₂×#364	1	6	yes	direct product factor
144	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
145	C₂³	—	2	4	no	—
146	C₂³	C₅×C₄⋊C₈	1	3	yes	—
147	C₂³	C₅×C₄⋊C₈	1	1	yes	characteristic
148	C₂³	C₂×C₅×M₁₆	1	3	yes	—
149	C₂×C₄	—	2	24	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	C₅×C₄⋊C₈	1	6	yes	—
152	C₂×C₄	C₅×C₄⋊C₈	1	3	yes	—
153	C₂×C₄	C₅×C₄⋊C₈	1	3	yes	—
154	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
155	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
156	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
157	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
158	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
159	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
160	C₈	C₂×C₅×C₄⋊C₄	1	3	yes	—
161	C₈	C₂×C₅×C₄⋊C₄	1	3	yes	—
162	C₈	C₂²×C₅×C₈	1	8	yes	—
163	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
164	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
165	C₅	C₂²×#364	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²	—	2	6	no	—
167	C₂²	—	2	6	no	—
168	C₂²	C₅×#364	1	4	yes	direct product factor
169	C₂²	C₂×C₅×C₄⋊C₈	1	3	yes	—
170	C₂²	C₂×C₅×C₄⋊C₈	1	3	yes	—
171	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
172	C₄	—	2	8	no	—
173	C₄	C₂×C₅×C₄⋊C₈	1	3	yes	—
174	C₄	C₂×C₅×C₄⋊C₈	1	3	yes	—
175	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
176	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×C₅×#364	1	6	yes	direct product factor
180	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₈	D₄
commutator	C₄	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₄⋊C₈
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#364	C₂	direct (N × H)
12. C₅×#364	C₂²	direct (N × H)
13. C₂²×#364	C₅	direct (N × H)
34. C₂×#364	C₂×C₅	direct (N × H)
67. #364	C₂²×C₅	direct (N × H)
108. C₂²×C₅	#364	direct (N × H)
143. C₂×C₅	C₂×#364	direct (N × H)
165. C₅	C₂²×#364	direct (N × H)
168. C₂²	C₅×#364	direct (N × H)
179. C₂	C₂×C₅×#364	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×C₄×C₅×C₈	2
C₂×C₅×#364	2
C₂²×#364	5

Subgroups — C22×C5×#364 #11956119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#364 #11956119