Subgroups — C₂×#10600633 #12097401

Order 1280.

1046 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	2	yes	maximal
3	C₂×C₄×#681	C₂	1	2	yes	maximal
4	C₂×#64772	C₂	1	2	yes	maximal
5	#10600633	C₂	1	8	yes	maximal, direct product factor
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×#681	C₂²	1	4	yes	—
10	C₂×C₅×#97	C₄	1	4	yes	—
11	C₂×C₅×#97	C₄	1	4	yes	—
12	C₂×C₅×#97	C₂²	1	4	yes	—
13	C₂×#3951	C₄	1	16	yes	complemented
14	C₂×#3951	C₂²	1	8	yes	—
15	C₄×#681	—	2	4	no	—
16	C₄×#681	C₂²	1	8	yes	—
17	C₂×#3935	—	2	2	no	—
18	C₂×C₄×#168	—	2	2	no	—
19	#64772	C₂²	1	8	yes	—
20	C₂×#1778	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	Q₈	1	2	yes	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	Q₈	1	2	yes	—
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₂×#681	—	2	4	no	—
38	C₂×#681	—	2	8	no	—
39	C₂×#681	C₂×C₄	1	16	yes	—
40	C₂×#681	C₂³	1	8	yes	—
41	C₂²×#168	—	2	4	no	—
42	C₅×#97	C₂×C₄	1	16	yes	—
43	#3951	C₂×C₄	1	32	yes	complemented
44	#3935	—	2	8	no	—
45	C₄×#168	—	2	8	no	—
46	C₂²×#108	—	5	2	no	—
47	C₂×C₄×M₁₆	—	5	2	no	—
48	C₂²×#97	D₅	1	1	yes	characteristic
49	#1778	—	5	8	no	—
50	C₂×#341	—	5	2	no	—
51	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	4	4	no	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	#41	1	4	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	C₄²	1	8	yes	—
68	C₂×C₅×C₈	C₄²	1	8	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
72	C₄²×C₅	C₄⋊C₄	1	4	yes	—
73	C₄²×C₅	C₄⋊C₄	1	4	yes	—
74	C₄²×C₅	C₂²×C₄	1	4	yes	—
75	C₄²×C₅	C₂×D₄	1	4	yes	—
76	#681	—	2	16	no	—
77	#681	—	2	8	no	—
78	#681	C₂²×C₄	1	16	yes	—
79	C₂×#168	—	2	16	no	—
80	C₂×#168	—	2	8	no	—
81	C₂²×M₁₆	—	5	4	no	—
82	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
84	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
85	C₂×#108	—	5	16	no	—
86	C₂×#108	—	5	8	no	—
87	C₄×M₁₆	—	5	8	no	—
88	C₄×M₁₆	—	10	4	no	—
89	C₂×#97	—	10	2	no	—
90	C₂×#97	Dic₅	1	4	yes	—
91	C₂×#97	Dic₅	1	4	yes	—
92	C₂×#97	C₂×D₅	1	4	yes	—
93	C₂×C₄×C₈	—	10	2	no	—
94	#341	—	5	8	no	—
95	C₂³×C₅	—	2	4	no	—
96	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₅	C₄×Q₈	1	4	yes	—
99	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	8	no	—
101	C₂×C₄×C₅	—	4	4	no	—
102	C₂×C₄×C₅	—	4	4	no	—
103	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
106	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	#117	1	2	yes	—
109	C₂×C₄×C₅	#117	1	2	yes	—
110	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×#41	1	2	yes	—
117	C₅×C₈	—	2	4	no	—
118	C₅×C₈	C₂×C₄²	1	8	yes	—
119	#168	—	2	16	no	—
120	C₂³×C₄	Dic₁₀	1	2	yes	—
121	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
122	C₂×M₁₆	—	5	16	no	—
123	C₂×M₁₆	—	5	8	no	—
124	C₂×M₁₆	—	10	4	no	—
125	C₂×M₁₆	—	10	8	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	10	4	no	—
128	C₂²×C₈	C₄×D₅	1	8	yes	—
129	C₂²×C₈	C₂×Dic₅	1	2	yes	—
130	C₂²×C₈	C₂×Dic₅	1	2	yes	—
131	C₂²×C₈	C₂×Dic₅	1	2	yes	—
132	C₂²×C₈	C₂×Dic₅	1	2	yes	—
133	C₂²×C₈	C₂²×D₅	1	2	yes	—
134	C₂²×C₈	C₂²×D₅	1	2	yes	—
135	C₂×C₄²	Dic₁₀	1	2	yes	—
136	C₂×C₄²	D₂₀	1	2	yes	—
137	C₂×C₄²	D₂₀	1	2	yes	—
138	C₂×C₄²	C₂×Dic₅	1	2	yes	—
139	C₂×C₄²	C₂×Dic₅	1	2	yes	—
140	C₂×C₄²	C₂²×D₅	1	2	yes	—
141	#108	—	5	32	no	—
142	#97	—	10	8	no	—
143	#97	C₂×Dic₅	1	16	yes	—
144	C₄×C₈	—	10	8	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	2	8	no	—
148	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
149	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
150	C₂²×C₅	#430	1	4	yes	—
151	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
153	C₄×C₅	—	2	8	no	—
154	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
155	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
156	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
157	C₄×C₅	C₂×#117	1	1	yes	characteristic
158	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
159	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
160	C₂²×C₄	—	4	4	no	—
161	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
162	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
163	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
164	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
166	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
168	C₂²×C₄	#709	1	4	yes	—
169	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
171	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
172	M₁₆	—	5	16	no	—
173	M₁₆	—	10	8	no	—
174	M₁₆	—	10	16	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	10	8	no	—
179	C₂×C₈	—	10	16	no	—
180	C₂×C₈	C₄×Dic₅	1	8	yes	—
181	C₂×C₈	C₄×Dic₅	1	8	yes	—
182	C₂×C₈	C₂×C₄×D₅	1	8	yes	—
183	C₂×C₈	C₂²×Dic₅	1	4	yes	—
184	C₂×C₈	C₂²×Dic₅	1	4	yes	—
185	C₄²	C₄⋊Dic₅	1	4	yes	—
186	C₄²	C₄⋊Dic₅	1	4	yes	—
187	C₄²	C₂×D₂₀	1	4	yes	—
188	C₄²	C₂²×Dic₅	1	4	yes	—
189	C₂×C₅	—	2	2	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	#1778	1	4	yes	complemented
192	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₅	C₂×#430	1	2	yes	—
194	C₂³	—	2	4	no	—
195	C₂³	C₄×Dic₁₀	1	4	yes	—
196	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
198	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
203	C₂×C₄	C₄×D₂₀	1	4	yes	—
204	C₂×C₄	C₄×D₂₀	1	4	yes	—
205	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
206	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₅	1	8	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
210	C₂×C₄	#4072	1	2	yes	—
211	C₂×C₄	#4072	1	2	yes	—
212	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
213	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
214	C₂×C₄	C₂×#709	1	2	yes	—
215	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
216	C₈	—	2	4	no	—
217	C₈	—	10	16	no	—
218	C₈	C₂×C₄×Dic₅	1	8	yes	—
219	C₅	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	8	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
224	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
225	C₂²	#65414	1	4	yes	—
226	C₂²	C₂×C₄×Dic₁₀	1	2	yes	—
227	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
230	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
231	C₄	C₂×C₄×D₂₀	1	2	yes	—
232	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
233	C₄	C₂×#4072	1	1	yes	characteristic
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	#10600633	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊Dic₅	1	1	yes	characteristic
238	C₂	C₂×#65414	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₅×#97	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×#97	D₅
hypocenter	C₅	C₂×#1778
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
5. #10600633	C₂	direct (N × H)
13. C₂×#3951	C₄	semidirect
43. #3951	C₂×C₄	semidirect
191. C₂×C₅	#1778	semidirect
219. C₅	C₂×#1778	semidirect
236. C₂	#10600633	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×#97	2
C₂²×#3951	2
C₂×C₄×#681	2
C₂×#64772	2
#10600633	2
C₂×#1778	5

Subgroups — C2×#10600633 #12097401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600633 #12097401