Subgroups — C₂²×C₅×#487 #11983323

Order 1280.

2398 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#487	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×#103	C₂	1	4	yes	maximal
5	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₅×Q₈	—	2	1	no	—
7	C₂³×C₅×Q₈	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
10	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
11	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	—
13	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
14	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
15	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#487	C₂²	1	64	yes	direct product factor
19	C₂×C₅×#103	C₂²	1	48	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	12	yes	—
21	C₂²×#487	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	—	2	6	no	—
23	C₂²×C₅×Q₈	—	2	12	no	—
24	C₂²×C₅×Q₈	—	2	16	no	—
25	C₂²×C₅×Q₈	—	4	8	no	—
26	C₂²×C₅×Q₈	C₂³	1	12	yes	—
27	C₂³×C₄×C₅	—	2	2	no	—
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	—	2	8	no	—
33	C₂²×C₅×C₈	C₂³	1	12	yes	—
34	C₅×C₄⋊Q₈	—	2	24	no	—
35	C₅×C₄⋊Q₈	—	2	24	no	—
36	C₅×C₄⋊Q₈	C₂³	1	16	yes	—
37	C₅×C₄⋊Q₈	C₂³	1	16	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	12	no	—
39	C₂×C₅×C₄⋊C₄	—	2	12	no	—
40	C₂×C₅×C₄⋊C₄	—	2	12	no	—
41	C₂×C₅×C₄⋊C₄	—	2	6	no	—
42	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
43	C₂×C₄²×C₅	D₄	1	6	yes	—
44	C₂×C₄²×C₅	C₂³	1	6	yes	—
45	C₅×#103	C₂³	1	64	yes	—
46	C₄×C₅×C₈	C₂³	1	16	yes	—
47	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
48	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
49	C₂×#487	C₂×C₅	1	24	yes	direct product factor
50	C₂²×#103	C₂×C₅	1	4	yes	—
51	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
52	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅×Q₈	—	2	12	no	—
54	C₂×C₅×Q₈	—	2	16	no	—
55	C₂×C₅×Q₈	—	2	48	no	—
56	C₂×C₅×Q₈	—	4	24	no	—
57	C₂×C₅×Q₈	—	4	48	no	—
58	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₅	—	2	12	no	—
60	C₂²×C₄×C₅	—	2	12	no	—
61	C₂²×C₄×C₅	—	4	8	no	—
62	C₂²×C₄×C₅	—	4	8	no	—
63	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
64	C₂²×C₄×C₅	Q₁₆	1	8	yes	—
65	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₅	#41	1	8	yes	—
69	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
70	C₂×C₅×C₈	—	2	24	no	—
71	C₂×C₅×C₈	C₂⁴	1	8	yes	—
72	C₅×C₄⋊C₄	—	2	16	no	—
73	C₅×C₄⋊C₄	—	2	16	no	—
74	C₅×C₄⋊C₄	—	2	16	no	—
75	C₅×C₄⋊C₄	—	2	12	no	—
76	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
77	C₄²×C₅	C₂×D₄	1	12	yes	—
78	C₄²×C₅	C₂⁴	1	4	yes	—
79	C₂³×Q₈	—	2	1	no	—
80	C₂³×Q₈	C₂²×C₅	1	2	yes	—
81	C₂³×C₈	C₂²×C₅	1	2	yes	—
82	C₂×C₄⋊Q₈	—	2	6	no	—
83	C₂×C₄⋊Q₈	—	2	6	no	—
84	C₂×C₄⋊Q₈	C₂²×C₅	1	12	yes	—
85	C₂×C₄⋊Q₈	C₂²×C₅	1	12	yes	—
86	C₂²×C₄⋊C₄	—	2	1	no	—
87	C₂²×C₄⋊C₄	—	2	1	no	—
88	C₂²×C₄⋊C₄	—	2	1	no	—
89	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
91	#487	C₂²×C₅	1	64	yes	direct product factor
92	C₂×#103	C₂²×C₅	1	48	yes	—
93	C₂×C₄×C₈	C₂²×C₅	1	12	yes	—
94	C₂³×C₅	#124	1	4	yes	—
95	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
96	C₂³×C₅	C₂×Q₁₆	1	4	yes	—
97	C₂³×C₅	C₂²×D₄	1	3	yes	—
98	C₅×Q₈	—	2	16	no	—
99	C₅×Q₈	—	4	24	no	—
100	C₅×Q₈	—	4	32	no	—
101	C₂×C₄×C₅	—	2	8	no	—
102	C₂×C₄×C₅	—	2	8	no	—
103	C₂×C₄×C₅	—	4	24	no	—
104	C₂×C₄×C₅	—	4	24	no	—
105	C₂×C₄×C₅	#124	1	12	yes	—
106	C₂×C₄×C₅	C₂×SD₁₆	1	24	yes	—
107	C₂×C₄×C₅	C₂×Q₁₆	1	24	yes	—
108	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
109	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
110	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
111	C₂×C₄×C₅	C₂×#41	1	12	yes	—
112	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
113	C₅×C₈	—	2	8	no	—
114	C₂²×Q₈	—	2	6	no	—
115	C₂²×Q₈	—	2	12	no	—
116	C₂²×Q₈	—	2	16	no	—
117	C₂²×Q₈	—	4	8	no	—
118	C₂²×Q₈	C₂³×C₅	1	12	yes	—
119	C₂³×C₄	—	2	2	no	—
120	C₂³×C₄	—	2	2	no	—
121	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
122	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
123	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂²×C₈	—	2	8	no	—
125	C₂²×C₈	C₂³×C₅	1	12	yes	—
126	C₄⋊Q₈	—	2	24	no	—
127	C₄⋊Q₈	—	2	24	no	—
128	C₄⋊Q₈	C₂³×C₅	1	16	yes	—
129	C₄⋊Q₈	C₂³×C₅	1	16	yes	—
130	C₂×C₄⋊C₄	—	2	12	no	—
131	C₂×C₄⋊C₄	—	2	12	no	—
132	C₂×C₄⋊C₄	—	2	12	no	—
133	C₂×C₄⋊C₄	—	2	6	no	—
134	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
135	C₂×C₄²	C₅×D₄	1	6	yes	—
136	C₂×C₄²	C₂³×C₅	1	6	yes	—
137	#103	C₂³×C₅	1	64	yes	—
138	C₄×C₈	C₂³×C₅	1	16	yes	—
139	C₂²×C₅	#487	1	16	yes	direct product factor
140	C₂²×C₅	C₂×#124	1	6	yes	—
141	C₂²×C₅	C₂²×SD₁₆	1	6	yes	—
142	C₂²×C₅	C₂²×Q₁₆	1	6	yes	—
143	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₅	—	4	8	no	—
145	C₄×C₅	—	4	8	no	—
146	C₄×C₅	C₂×#124	1	6	yes	—
147	C₄×C₅	C₂²×SD₁₆	1	8	yes	—
148	C₄×C₅	C₂²×Q₁₆	1	8	yes	—
149	C₄×C₅	C₂²×#41	1	2	yes	—
150	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
151	C₂×Q₈	—	2	12	no	—
152	C₂×Q₈	—	2	16	no	—
153	C₂×Q₈	—	2	48	no	—
154	C₂×Q₈	—	4	24	no	—
155	C₂×Q₈	—	4	48	no	—
156	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
157	C₂²×C₄	—	2	12	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	4	8	no	—
161	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
162	C₂²×C₄	C₅×Q₁₆	1	8	yes	—
163	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
164	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
165	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
166	C₂²×C₄	C₅×#41	1	8	yes	—
167	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
168	C₂×C₈	—	2	24	no	—
169	C₂×C₈	C₂⁴×C₅	1	8	yes	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄⋊C₄	—	2	16	no	—
172	C₄⋊C₄	—	2	16	no	—
173	C₄⋊C₄	—	2	12	no	—
174	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
175	C₄²	C₂×C₅×D₄	1	12	yes	—
176	C₄²	C₂⁴×C₅	1	4	yes	—
177	C₂×C₅	C₂×#487	1	12	yes	direct product factor
178	C₂×C₅	C₂²×#124	1	1	yes	characteristic
179	C₂×C₅	C₂³×SD₁₆	1	1	yes	characteristic
180	C₂×C₅	C₂³×Q₁₆	1	1	yes	characteristic
181	C₂³	C₅×#124	1	4	yes	—
182	C₂³	C₂×C₅×SD₁₆	1	4	yes	—
183	C₂³	C₂×C₅×Q₁₆	1	4	yes	—
184	C₂³	C₂²×C₅×D₄	1	3	yes	—
185	Q₈	—	2	16	no	—
186	Q₈	—	4	24	no	—
187	Q₈	—	4	32	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	4	24	no	—
191	C₂×C₄	—	4	24	no	—
192	C₂×C₄	C₅×#124	1	12	yes	—
193	C₂×C₄	C₂×C₅×SD₁₆	1	24	yes	—
194	C₂×C₄	C₂×C₅×Q₁₆	1	24	yes	—
195	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
196	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
197	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
198	C₂×C₄	C₂×C₅×#41	1	12	yes	—
199	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
200	C₈	—	2	8	no	—
201	C₅	C₂²×#487	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²	C₅×#487	1	16	yes	direct product factor
203	C₂²	C₂×C₅×#124	1	6	yes	—
204	C₂²	C₂²×C₅×SD₁₆	1	6	yes	—
205	C₂²	C₂²×C₅×Q₁₆	1	6	yes	—
206	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
207	C₄	—	4	8	no	—
208	C₄	—	4	8	no	—
209	C₄	C₂×C₅×#124	1	6	yes	—
210	C₄	C₂²×C₅×SD₁₆	1	8	yes	—
211	C₄	C₂²×C₅×Q₁₆	1	8	yes	—
212	C₄	C₂²×C₅×#41	1	2	yes	—
213	C₂	C₂×C₅×#487	1	12	yes	direct product factor
214	C₂	C₂²×C₅×#124	1	1	yes	characteristic
215	C₂	C₂³×C₅×SD₁₆	1	1	yes	characteristic
216	C₂	C₂³×C₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#487	C₂	direct (N × H)
18. C₅×#487	C₂²	direct (N × H)
21. C₂²×#487	C₅	direct (N × H)
49. C₂×#487	C₂×C₅	direct (N × H)
91. #487	C₂²×C₅	direct (N × H)
139. C₂²×C₅	#487	direct (N × H)
177. C₂×C₅	C₂×#487	direct (N × H)
201. C₅	C₂²×#487	direct (N × H)
202. C₂²	C₅×#487	direct (N × H)
213. C₂	C₂×C₅×#487	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⋊Q₈	2
C₂×C₅×#487	2
C₂²×C₅×#103	2
C₂²×C₄×C₅×C₈	2
C₂²×#487	5

Subgroups — C22×C5×#487 #11983323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#487 #11983323