Subgroups — C₂×#10614166 #12964239

Order 1280.

2234 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	#10614166	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×#168	C₂	1	8	yes	maximal, complemented
4	C₂×#65214	C₂	1	6	yes	maximal
5	C₂³×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₄	1	4	yes	—
7	C₂×C₄×C₅×Q₈	C₄	1	8	yes	—
8	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
9	C₂×C₄×C₅×Q₈	C₂²	1	8	yes	—
10	C₂²×C₅×C₄⋊C₄	C₄	1	3	yes	—
11	C₂²×C₄²×C₅	C₂²	1	3	yes	—
12	C₂×C₂²⋊#168	C₂²	1	4	yes	—
13	Q₈×#168	C₂²	1	64	yes	complemented
14	#65214	C₂²	1	48	yes	—
15	C₂×C₄⋊#168	C₂²	1	12	yes	—
16	C₂×C₄×#168	C₂²	1	12	yes	—
17	C₂×#2982	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	C₈	1	16	yes	complemented
19	C₂²×C₅×Q₈	C₂×C₄	1	4	yes	—
20	C₂²×C₅×Q₈	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₄×C₅×Q₈	C₂×C₄	1	16	yes	—
25	C₄×C₅×Q₈	C₂×C₄	1	32	yes	—
26	C₄×C₅×Q₈	C₂³	1	16	yes	—
27	C₂×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	24	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₅	C₂³	1	6	yes	—
32	C₂×C₄²×C₅	C₂³	1	12	yes	—
33	C₂²×#168	C₂³	1	8	yes	—
34	C₂²⋊#168	C₂³	1	16	yes	—
35	C₄⋊#168	C₂³	1	48	yes	—
36	C₄×#168	C₂³	1	48	yes	—
37	C₂²×C₄×Q₈	D₅	1	1	yes	characteristic
38	#2982	—	5	16	no	—
39	C₂×C₈×Q₈	—	5	8	no	—
40	C₂×#431	—	5	6	no	—
41	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₅×Q₈	C₂×C₈	1	32	yes	—
43	C₂×C₅×Q₈	C₂×C₈	1	64	yes	complemented
44	C₂×C₅×Q₈	C₂²×C₄	1	16	yes	—
45	C₂²×C₄×C₅	—	2	4	no	—
46	C₂²×C₄×C₅	—	4	6	no	—
47	C₂²×C₄×C₅	C₂×C₈	1	24	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
55	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
56	C₅×C₄⋊C₄	C₂²×C₄	1	48	yes	—
57	C₄²×C₅	C₂²×C₄	1	24	yes	—
58	C₄²×C₅	C₂²×C₄	1	12	yes	—
59	C₄²×C₅	C₂⁴	1	12	yes	—
60	C₂×#168	—	2	16	no	—
61	C₂×#168	C₂⁴	1	16	yes	—
62	C₂³×Q₈	Dic₅	1	1	yes	characteristic
63	C₂×C₄×Q₈	Dic₅	1	4	yes	—
64	C₂×C₄×Q₈	Dic₅	1	8	yes	—
65	C₂×C₄×Q₈	C₂×D₅	1	4	yes	—
66	C₂×C₄×Q₈	C₂×D₅	1	8	yes	—
67	C₂²×C₄⋊C₄	Dic₅	1	3	yes	—
68	C₂²×C₄²	C₂×D₅	1	3	yes	—
69	C₈×Q₈	—	5	64	no	—
70	#431	—	5	48	no	—
71	C₂×C₄⋊C₈	—	5	12	no	—
72	C₂×C₂²⋊C₈	—	5	4	no	—
73	C₂×C₄×C₈	—	5	12	no	—
74	C₂³×C₅	—	2	2	no	—
75	C₂³×C₅	—	2	2	no	—
76	C₂³×C₅	C₂²×C₈	1	4	yes	—
77	C₂³×C₅	C₂³×C₄	1	2	yes	—
78	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
79	C₅×Q₈	C₂²×C₈	1	64	yes	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	12	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	4	12	no	—
85	C₂×C₄×C₅	C₂²×C₈	1	24	yes	—
86	C₂×C₄×C₅	C₂²×C₈	1	48	yes	—
87	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
91	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
92	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
93	#168	—	2	16	no	—
94	C₂²×Q₈	#168	1	16	yes	complemented
95	C₂²×Q₈	C₂×Dic₅	1	4	yes	—
96	C₂²×Q₈	C₂×Dic₅	1	8	yes	—
97	C₂³×C₄	C₂×Dic₅	1	3	yes	—
98	C₂³×C₄	C₂×Dic₅	1	3	yes	—
99	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
100	C₂²×C₈	—	5	8	no	—
101	C₄×Q₈	C₂×Dic₅	1	16	yes	—
102	C₄×Q₈	C₂×Dic₅	1	32	yes	—
103	C₄×Q₈	C₂²×D₅	1	16	yes	—
104	C₂×C₄⋊C₄	C₂×Dic₅	1	12	yes	—
105	C₂×C₄⋊C₄	C₂×Dic₅	1	24	yes	—
106	C₂×C₄²	C₂×Dic₅	1	6	yes	—
107	C₂×C₄²	C₂×Dic₅	1	12	yes	—
108	C₂×C₄²	C₂²×D₅	1	12	yes	—
109	C₂×C₄²	C₂²×D₅	1	6	yes	—
110	C₄⋊C₈	—	5	48	no	—
111	C₂²⋊C₈	—	5	16	no	—
112	C₄×C₈	—	5	48	no	—
113	C₂²×C₅	—	2	2	no	—
114	C₂²×C₅	—	2	8	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	#519	1	2	yes	—
117	C₂²×C₅	C₂³×C₈	1	4	yes	—
118	C₂²×C₅	C₂³×C₈	1	2	yes	—
119	C₂²×C₅	#568	1	2	yes	—
120	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	2	12	no	—
123	C₄×C₅	#519	1	2	yes	—
124	C₄×C₅	C₂³×C₈	1	24	yes	—
125	C₄×C₅	C₂×2₋¹⁺⁴	1	2	yes	—
126	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
127	C₂×Q₈	C₂×#168	1	32	yes	—
128	C₂×Q₈	C₂×#168	1	64	yes	complemented
129	C₂×Q₈	C₂²×Dic₅	1	16	yes	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	4	6	no	—
132	C₂²×C₄	C₂×#168	1	24	yes	—
133	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
136	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
137	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
138	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
139	C₂²×C₄	C₂³×D₅	1	2	yes	—
140	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
141	C₂×C₈	—	5	16	no	—
142	C₂×C₈	—	10	16	no	—
143	C₄⋊C₄	C₂²×Dic₅	1	48	yes	—
144	C₄²	C₂²×Dic₅	1	24	yes	—
145	C₄²	C₂²×Dic₅	1	12	yes	—
146	C₄²	C₂³×D₅	1	12	yes	—
147	C₂×C₅	—	2	4	no	—
148	C₂×C₅	#2982	1	4	yes	complemented
149	C₂×C₅	C₂×#519	1	1	yes	characteristic
150	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
151	C₂×C₅	C₂×#568	1	1	yes	characteristic
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	C₂²×#168	1	4	yes	—
155	C₂³	C₂³×Dic₅	1	2	yes	—
156	C₂³	C₂³×Dic₅	1	1	yes	characteristic
157	Q₈	C₂²×#168	1	64	yes	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	12	no	—
163	C₂×C₄	C₂²×#168	1	24	yes	—
164	C₂×C₄	C₂²×#168	1	48	yes	—
165	C₂×C₄	#4147	1	4	yes	—
166	C₂×C₄	C₂³×Dic₅	1	2	yes	—
167	C₂×C₄	C₂³×Dic₅	1	12	yes	—
168	C₂×C₄	C₂³×Dic₅	1	12	yes	—
169	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
170	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
171	C₈	—	10	16	no	—
172	C₅	C₂×#2982	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	8	no	—
176	C₂²	C₂³×#168	1	4	yes	—
177	C₂²	C₂³×#168	1	2	yes	—
178	C₂²	#66146	1	2	yes	—
179	C₂²	#66153	1	2	yes	—
180	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	12	no	—
183	C₄	C₂³×#168	1	24	yes	—
184	C₄	#66146	1	2	yes	—
185	C₄	C₂×#4147	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	#10614166	1	4	yes	direct product factor
188	C₂	C₂⁴×#168	1	1	yes	characteristic
189	C₂	C₂×#66146	1	1	yes	characteristic
190	C₂	C₂×#66153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅
commutator	C₂×C₅	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×Q₈	C₂
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₅
hypocenter	C₅	C₂×#2982
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #10614166	C₂	direct (N × H)
3. C₂×Q₈×#168	C₂	semidirect
13. Q₈×#168	C₂²	semidirect
18. C₂²×C₅×Q₈	C₈	semidirect
43. C₂×C₅×Q₈	C₂×C₈	semidirect
94. C₂²×Q₈	#168	semidirect
128. C₂×Q₈	C₂×#168	semidirect
148. C₂×C₅	#2982	semidirect
172. C₅	C₂×#2982	semidirect
187. C₂	#10614166	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×C₄×C₅×Q₈	2
#10614166	2
C₂×Q₈×#168	2
C₂×#65214	2
C₂×#2982	5

Subgroups — C2×#10614166 #12964239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10614166 #12964239