Subgroups — C₂²×C₅×#338 #11943059

Order 1280.

2110 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	C₂³×C₅×M₁₆	C₂	1	3	yes	maximal
2	C₂×C₅×#338	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
4	C₂²×C₅×M₁₆	—	2	6	no	—
5	C₂²×C₅×M₁₆	—	2	18	no	—
6	C₂²×C₅×M₁₆	C₄	1	18	yes	—
7	C₂²×C₅×M₁₆	C₂²	1	18	yes	—
8	C₂³×C₅×C₈	C₄	1	6	yes	—
9	C₅×#338	C₂²	1	16	yes	direct product factor
10	C₂²×#338	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁵×C₅	Q₈	1	1	yes	characteristic
12	C₂³×C₄×C₅	C₂×C₄	1	9	yes	—
13	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₅	D₄	1	3	yes	—
15	C₂³×C₄×C₅	D₄	1	9	yes	—
16	C₂³×C₄×C₅	Q₈	1	3	yes	—
17	C₂³×C₄×C₅	C₂³	1	3	yes	—
18	C₂×C₅×M₁₆	—	2	36	no	—
19	C₂×C₅×M₁₆	—	2	36	no	—
20	C₂×C₅×M₁₆	—	2	36	no	—
21	C₂×C₅×M₁₆	—	2	36	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	36	yes	—
23	C₂×C₅×M₁₆	C₂³	1	12	yes	—
24	C₂²×C₅×C₈	—	2	6	no	—
25	C₂²×C₅×C₈	—	2	18	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	36	yes	—
27	C₂³×M₁₆	C₂×C₅	1	3	yes	—
28	C₂×#338	C₂×C₅	1	12	yes	direct product factor
29	C₂⁴×C₅	—	4	4	no	—
30	C₂⁴×C₅	C₄⋊C₄	1	3	yes	—
31	C₂⁴×C₅	C₄⋊C₄	1	9	yes	—
32	C₂⁴×C₅	C₂×Q₈	1	3	yes	—
33	C₂²×C₄×C₅	C₄²	1	6	yes	—
34	C₂²×C₄×C₅	C₄²	1	9	yes	—
35	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
37	C₂²×C₄×C₅	C₂²⋊C₄	1	9	yes	—
38	C₂²×C₄×C₅	C₂²⋊C₄	1	18	yes	—
39	C₂²×C₄×C₅	C₂²⋊C₄	1	18	yes	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	9	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	18	yes	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	9	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	9	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	9	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	9	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	9	yes	—
47	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
48	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
49	C₅×M₁₆	—	2	72	no	—
50	C₅×M₁₆	—	2	24	no	—
51	C₂×C₅×C₈	—	2	36	no	—
52	C₂×C₅×C₈	—	2	36	no	—
53	C₂×C₅×C₈	C₂²×C₄	1	24	yes	—
54	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
55	C₂²×M₁₆	—	2	6	no	—
56	C₂²×M₁₆	—	2	18	no	—
57	C₂²×M₁₆	C₄×C₅	1	18	yes	—
58	C₂²×M₁₆	C₂²×C₅	1	18	yes	—
59	C₂³×C₈	C₄×C₅	1	6	yes	—
60	#338	C₂²×C₅	1	16	yes	direct product factor
61	C₂³×C₅	—	2	12	no	—
62	C₂³×C₅	—	4	18	no	—
63	C₂³×C₅	—	4	6	no	—
64	C₂³×C₅	#95	1	9	yes	—
65	C₂³×C₅	#95	1	6	yes	—
66	C₂³×C₅	#95	1	1	yes	characteristic
67	C₂³×C₅	C₂×C₄⋊C₄	1	9	yes	—
68	C₂³×C₅	C₂×C₄⋊C₄	1	9	yes	—
69	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅	#95	1	18	yes	—
71	C₂×C₄×C₅	#95	1	18	yes	—
72	C₂×C₄×C₅	#95	1	9	yes	—
73	C₂×C₄×C₅	#95	1	3	yes	—
74	C₂×C₄×C₅	C₂×C₄²	1	9	yes	—
75	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
76	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	9	yes	—
77	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	9	yes	—
78	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	18	yes	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	9	yes	—
80	C₂×C₄×C₅	C₂×C₄⋊C₄	1	9	yes	—
81	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
82	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
83	C₅×C₈	—	2	24	no	—
84	C₂⁵	C₅×Q₈	1	1	yes	characteristic
85	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
86	C₂³×C₄	C₂×C₄×C₅	1	9	yes	—
87	C₂³×C₄	C₅×D₄	1	3	yes	—
88	C₂³×C₄	C₅×D₄	1	9	yes	—
89	C₂³×C₄	C₅×Q₈	1	3	yes	—
90	C₂³×C₄	C₂³×C₅	1	3	yes	—
91	C₂×M₁₆	—	2	36	no	—
92	C₂×M₁₆	—	2	36	no	—
93	C₂×M₁₆	—	2	36	no	—
94	C₂×M₁₆	—	2	36	no	—
95	C₂×M₁₆	C₂×C₄×C₅	1	36	yes	—
96	C₂×M₁₆	C₂³×C₅	1	12	yes	—
97	C₂²×C₈	—	2	18	no	—
98	C₂²×C₈	—	2	6	no	—
99	C₂²×C₈	C₂×C₄×C₅	1	36	yes	—
100	C₂²×C₅	—	2	18	no	—
101	C₂²×C₅	—	2	18	no	—
102	C₂²×C₅	—	4	6	no	—
103	C₂²×C₅	—	4	9	no	—
104	C₂²×C₅	—	4	1	no	—
105	C₂²×C₅	#338	1	4	yes	direct product factor
106	C₂²×C₅	C₂×#95	1	9	yes	—
107	C₂²×C₅	C₂×#95	1	3	yes	—
108	C₂²×C₅	C₂²×C₄⋊C₄	1	3	yes	—
109	C₄×C₅	C₂×#95	1	9	yes	—
110	C₄×C₅	C₂×#95	1	3	yes	—
111	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₅	C₂²×C₂²⋊C₄	1	3	yes	—
113	C₂⁴	—	4	4	no	—
114	C₂⁴	C₅×C₄⋊C₄	1	3	yes	—
115	C₂⁴	C₅×C₄⋊C₄	1	9	yes	—
116	C₂⁴	C₂×C₅×Q₈	1	3	yes	—
117	C₂²×C₄	C₄²×C₅	1	9	yes	—
118	C₂²×C₄	C₄²×C₅	1	6	yes	—
119	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
120	C₂²×C₄	C₅×C₂²⋊C₄	1	9	yes	—
121	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
122	C₂²×C₄	C₅×C₂²⋊C₄	1	18	yes	—
123	C₂²×C₄	C₅×C₂²⋊C₄	1	18	yes	—
124	C₂²×C₄	C₅×C₄⋊C₄	1	9	yes	—
125	C₂²×C₄	C₅×C₄⋊C₄	1	9	yes	—
126	C₂²×C₄	C₅×C₄⋊C₄	1	18	yes	—
127	C₂²×C₄	C₂²×C₄×C₅	1	9	yes	—
128	C₂²×C₄	C₂²×C₄×C₅	1	9	yes	—
129	C₂²×C₄	C₂×C₅×D₄	1	9	yes	—
130	C₂²×C₄	C₂×C₅×D₄	1	9	yes	—
131	C₂²×C₄	C₂×C₅×Q₈	1	3	yes	—
132	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
133	M₁₆	—	2	72	no	—
134	M₁₆	—	2	24	no	—
135	C₂×C₈	—	2	36	no	—
136	C₂×C₈	—	2	36	no	—
137	C₂×C₈	C₂²×C₄×C₅	1	24	yes	—
138	C₂×C₅	—	2	3	no	—
139	C₂×C₅	—	2	9	no	—
140	C₂×C₅	C₂×#338	1	6	yes	direct product factor
141	C₂×C₅	C₂²×#95	1	1	yes	characteristic
142	C₂³	—	2	12	no	—
143	C₂³	—	4	18	no	—
144	C₂³	—	4	6	no	—
145	C₂³	C₅×#95	1	9	yes	—
146	C₂³	C₅×#95	1	6	yes	—
147	C₂³	C₅×#95	1	1	yes	characteristic
148	C₂³	C₂×C₅×C₄⋊C₄	1	9	yes	—
149	C₂³	C₂×C₅×C₄⋊C₄	1	9	yes	—
150	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
151	C₂×C₄	C₅×#95	1	18	yes	—
152	C₂×C₄	C₅×#95	1	18	yes	—
153	C₂×C₄	C₅×#95	1	9	yes	—
154	C₂×C₄	C₅×#95	1	3	yes	—
155	C₂×C₄	C₂×C₄²×C₅	1	9	yes	—
156	C₂×C₄	C₂×C₄²×C₅	1	3	yes	—
157	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	9	yes	—
158	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	9	yes	—
159	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	18	yes	—
160	C₂×C₄	C₂×C₅×C₄⋊C₄	1	9	yes	—
161	C₂×C₄	C₂×C₅×C₄⋊C₄	1	9	yes	—
162	C₂×C₄	C₂³×C₄×C₅	1	3	yes	—
163	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
164	C₈	—	2	24	no	—
165	C₅	C₂²×#338	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²	—	2	18	no	—
167	C₂²	—	2	18	no	—
168	C₂²	—	4	9	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	6	no	—
171	C₂²	C₅×#338	1	4	yes	direct product factor
172	C₂²	C₂×C₅×#95	1	9	yes	—
173	C₂²	C₂×C₅×#95	1	3	yes	—
174	C₂²	C₂²×C₅×C₄⋊C₄	1	3	yes	—
175	C₄	C₂×C₅×#95	1	3	yes	—
176	C₄	C₂×C₅×#95	1	9	yes	—
177	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
178	C₄	C₂²×C₅×C₂²⋊C₄	1	3	yes	—
179	C₂	—	2	9	no	—
180	C₂	—	2	3	no	—
181	C₂	C₂×C₅×#338	1	6	yes	direct product factor
182	C₂	C₂²×C₅×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₄²
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	#95
prime radical	C₂⁵×C₅	Q₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#338	C₂	direct (N × H)
9. C₅×#338	C₂²	direct (N × H)
10. C₂²×#338	C₅	direct (N × H)
28. C₂×#338	C₂×C₅	direct (N × H)
60. #338	C₂²×C₅	direct (N × H)
105. C₂²×C₅	#338	direct (N × H)
140. C₂×C₅	C₂×#338	direct (N × H)
165. C₅	C₂²×#338	direct (N × H)
171. C₂²	C₅×#338	direct (N × H)
181. C₂	C₂×C₅×#338	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₅×M₁₆	2
C₂×C₅×#338	2
C₂²×#338	5

Subgroups — C22×C5×#338 #11943059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#338 #11943059