Subgroups — C₂×#10602177 #12254775

Order 1280.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#64971	C₂	1	1	yes	maximal, characteristic
3	#10602177	C₂	1	8	yes	maximal, direct product factor
4	C₂×#64709	C₂	1	4	yes	maximal
5	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
7	C₂×#4015	—	2	1	no	—
8	C₅×#500	—	2	2	no	—
9	C₅×#500	C₂²	1	4	yes	—
10	C₂×C₅×#107	—	2	1	no	—
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3950	—	2	1	no	—
14	C₂×#3950	C₂²	1	2	yes	—
15	C₂²×#672	C₂²	1	2	yes	—
16	#64971	—	2	2	no	—
17	#64971	C₂²	1	4	yes	—
18	#64709	C₂²	1	16	yes	—
19	C₄×#672	C₂²	1	4	yes	—
20	C₂×#2252	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	4	1	no	—
22	C₂²×Dic₁₀	—	4	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	—	2	4	no	—
26	C₅×C₄⋊Q₈	—	4	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	4	1	no	—
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅	—	2	2	no	—
31	C₂×C₄⋊Dic₅	—	4	1	no	—
32	#4015	—	2	4	no	—
33	#4015	—	4	2	no	—
34	C₅×#107	—	2	4	no	—
35	C₅×#107	—	2	2	no	—
36	C₅×#107	C₂³	1	4	yes	—
37	C₄×C₅×C₈	D₄	1	2	yes	—
38	C₄×C₅×C₈	C₂³	1	2	yes	—
39	#3950	—	2	4	no	—
40	#3950	—	2	2	no	—
41	#3950	C₂³	1	4	yes	—
42	C₂×#672	Q₈	1	8	yes	complemented
43	C₂×#672	C₂³	1	4	yes	—
44	C₂×#500	—	5	1	no	—
45	C₂×#500	D₅	1	1	yes	characteristic
46	#2252	—	5	8	no	—
47	C₂×#366	—	5	4	no	—
48	C₂×C₄×C₁₆	—	5	1	no	—
49	C₂×C₅×Q₈	—	4	4	no	—
50	C₂×C₅×Q₈	—	8	4	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅	—	4	2	no	—
56	C₂×Dic₁₀	—	4	4	no	—
57	C₂×Dic₁₀	—	8	4	no	—
58	C₂×C₅×C₈	Q₁₆	1	4	yes	—
59	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
62	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
63	C₅×C₄⋊C₄	—	2	4	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	4	no	—
66	C₄²×C₅	D₈	1	2	yes	—
67	C₄²×C₅	C₂×D₄	1	2	yes	—
68	C₄⋊Dic₅	—	2	4	no	—
69	C₄⋊Dic₅	—	4	4	no	—
70	C₄⋊Dic₅	—	4	2	no	—
71	#672	C₂×Q₈	1	8	yes	complemented
72	C₂×C₄⋊Q₈	—	2	1	no	—
73	C₂×C₄⋊Q₈	—	10	1	no	—
74	C₂²×C₁₆	—	5	2	no	—
75	#500	—	2	2	no	—
76	#500	—	5	4	no	—
77	#500	—	10	2	no	—
78	#500	C₂×D₅	1	4	yes	—
79	C₂×#107	—	2	1	no	—
80	C₂×#107	—	5	2	no	—
81	C₂×#107	—	10	1	no	—
82	C₂×#107	C₂×D₅	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
84	#366	—	5	16	no	—
85	C₄×C₁₆	—	5	4	no	—
86	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
87	C₅×Q₈	—	8	8	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	8	4	no	—
90	C₂×C₄×C₅	D₁₆	1	4	yes	—
91	C₂×C₄×C₅	Q₃₂	1	4	yes	—
92	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₅	—	4	4	no	—
99	C₂×Dic₅	—	8	4	no	—
100	Dic₁₀	—	8	8	no	—
101	C₅×C₈	C₄⋊Q₈	1	2	yes	—
102	C₅×C₈	C₂×Q₁₆	1	4	yes	—
103	C₅×C₈	C₂²×Q₈	1	2	yes	—
104	C₂²×Q₈	—	4	1	no	—
105	C₂²×Q₈	—	20	1	no	—
106	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
108	C₄⋊Q₈	—	2	4	no	—
109	C₄⋊Q₈	—	4	2	no	—
110	C₄⋊Q₈	—	10	4	no	—
111	C₄⋊Q₈	—	20	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	4	1	no	—
114	C₂×C₄⋊C₄	—	10	2	no	—
115	C₂×C₄⋊C₄	—	20	1	no	—
116	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
117	C₂×C₁₆	—	5	8	no	—
118	C₂×C₁₆	—	5	4	no	—
119	#107	—	2	4	no	—
120	#107	—	2	2	no	—
121	#107	—	5	4	no	—
122	#107	—	10	2	no	—
123	#107	—	10	4	no	—
124	#107	C₂²×D₅	1	4	yes	—
125	C₄×C₈	C₂²⋊D₅	1	2	yes	—
126	C₄×C₈	C₂²×D₅	1	2	yes	—
127	C₂²×C₅	#500	1	2	yes	—
128	C₂²×C₅	C₂×D₁₆	1	2	yes	—
129	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
130	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₅	—	8	4	no	—
132	C₄×C₅	#500	1	2	yes	—
133	C₄×C₅	C₂×D₁₆	1	4	yes	—
134	C₄×C₅	C₂×Q₃₂	1	4	yes	—
135	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
137	Dic₅	—	8	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂×Q₈	—	8	4	no	—
140	C₂×Q₈	—	20	4	no	—
141	C₂×Q₈	—	40	4	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	20	2	no	—
144	C₂²×C₄	#686	1	1	yes	characteristic
145	C₂²×C₄	#686	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
147	C₂×C₈	#689	1	4	yes	—
148	C₂×C₈	Q₈×D₅	1	4	yes	—
149	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
150	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	10	4	no	—
156	C₄⋊C₄	—	20	2	no	—
157	C₄⋊C₄	—	20	4	no	—
158	C₄²	#686	1	2	yes	—
159	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
160	C₁₆	—	5	8	no	—
161	C₂×C₅	#2252	1	4	yes	complemented
162	C₂×C₅	C₂×#500	1	1	yes	characteristic
163	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
164	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
165	C₂³	C₂×#686	1	1	yes	characteristic
166	Q₈	—	8	8	no	—
167	Q₈	—	40	8	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	20	4	no	—
171	C₂×C₄	—	40	4	no	—
172	C₂×C₄	#3958	1	4	yes	—
173	C₂×C₄	#3961	1	4	yes	—
174	C₂×C₄	C₂×#686	1	1	yes	characteristic
175	C₂×C₄	C₂×#686	1	2	yes	—
176	C₂×C₄	C₂×#686	1	2	yes	—
177	C₂×C₄	C₂×#689	1	2	yes	—
178	C₂×C₄	#4090	1	2	yes	—
179	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
180	C₈	C₂×#689	1	4	yes	—
181	C₈	#4090	1	2	yes	—
182	C₈	C₂×Q₈×D₅	1	2	yes	—
183	C₅	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
184	C₂²	#65367	1	2	yes	—
185	C₂²	C₂×#3958	1	2	yes	—
186	C₂²	C₂×#3961	1	2	yes	—
187	C₂²	C₂²×#686	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	40	4	no	—
190	C₄	#65367	1	2	yes	—
191	C₄	C₂×#3958	1	4	yes	—
192	C₄	C₂×#3961	1	4	yes	—
193	C₄	C₂²×#689	1	1	yes	characteristic
194	C₄	C₂×#4090	1	1	yes	characteristic
195	C₂	#10602177	1	4	yes	direct product factor
196	C₂	C₂×#65367	1	1	yes	characteristic
197	C₂	C₂²×#3958	1	1	yes	characteristic
198	C₂	C₂²×#3961	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#500	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
hypercenter	C₂×#500	D₅
hypocenter	C₅	C₂×#2252
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #10602177	C₂	direct (N × H)
42. C₂×#672	Q₈	semidirect
71. #672	C₂×Q₈	semidirect
161. C₂×C₅	#2252	semidirect
183. C₅	C₂×#2252	semidirect
195. C₂	#10602177	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#500	2
C₂×#64971	2
#10602177	2
C₂×#64709	2
C₂×C₄×#672	2
C₂×#2252	5

Subgroups — C2×#10602177 #12254775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602177 #12254775