Subgroups — C₂×#10600674 #12251935

Order 1280.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
3	C₂³×#672	C₂	1	1	yes	maximal, characteristic
4	#10600674	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64681	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#672	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₄	1	1	yes	characteristic
10	C₅×#405	C₄	1	4	yes	—
11	C₅×#405	C₂²	1	4	yes	—
12	C₂×C₅×#97	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	C₂×#3944	C₄	1	8	yes	complemented
15	C₂×#3944	C₂²	1	4	yes	—
16	C₂²×#672	C₄	1	8	yes	complemented
17	C₂²×#672	C₂²	1	1	yes	characteristic
18	C₂²×#672	C₂²	1	2	yes	—
19	C₂²×#672	C₂²	1	1	yes	characteristic
20	C₂²×#672	C₂²	1	4	yes	—
21	#64681	C₂²	1	8	yes	complemented
22	C₄×#672	C₂²	1	8	yes	complemented
23	C₂×#2108	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	C₈	1	8	yes	—
26	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×#117	C₈	1	4	yes	—
35	C₅×#117	C₂×C₄	1	4	yes	—
36	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
37	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₅×#97	C₂×C₄	1	8	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₅×C₈	C₂³	1	4	yes	—
42	#3944	C₂×C₄	1	16	yes	complemented
43	C₂×#672	C₂×C₄	1	4	yes	—
44	C₂×#672	C₂×C₄	1	4	yes	—
45	C₂×#672	C₂×C₄	1	8	yes	—
46	C₂×#672	C₂×C₄	1	16	yes	complemented
47	C₂×#672	C₂³	1	2	yes	—
48	C₂×#672	C₂³	1	2	yes	—
49	C₂×#672	C₂³	1	4	yes	—
50	C₂²×M₃₂	—	5	1	no	—
51	C₂³×C₁₆	—	5	1	no	—
52	C₂×#405	D₅	1	1	yes	characteristic
53	#2108	—	5	8	no	—
54	C₂×#346	—	5	2	no	—
55	C₂×C₄×C₁₆	—	5	2	no	—
56	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄²	1	4	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
64	C₅×M₁₆	C₂×C₈	1	16	yes	—
65	C₂×C₅×C₈	C₄²	1	4	yes	—
66	C₂×C₅×C₈	C₄²	1	8	yes	—
67	C₂×C₅×C₈	C₂×C₈	1	8	yes	—
68	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
75	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
76	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
77	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
78	C₄²×C₅	C₂×C₈	1	4	yes	—
79	C₄²×C₅	C₂²×C₄	1	4	yes	—
80	#672	C₂²×C₄	1	4	yes	—
81	#672	C₂²×C₄	1	4	yes	—
82	#672	C₂²×C₄	1	8	yes	—
83	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
84	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
85	C₂×#117	Dic₅	1	1	yes	characteristic
86	C₂×M₃₂	—	5	8	no	—
87	C₂×M₃₂	—	5	4	no	—
88	C₂²×C₁₆	—	5	8	no	—
89	C₂²×C₁₆	—	5	4	no	—
90	C₂²×C₁₆	—	5	1	no	—
91	C₂²×C₁₆	—	5	2	no	—
92	C₂²×C₁₆	—	5	1	no	—
93	#405	Dic₅	1	4	yes	—
94	#405	C₂×D₅	1	4	yes	—
95	C₂×#97	Dic₅	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
97	#346	—	5	8	no	—
98	C₄×C₁₆	—	5	8	no	—
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	C₄×C₈	1	4	yes	—
101	C₂³×C₅	C₂²×C₈	1	2	yes	—
102	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	8	no	—
104	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
105	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
106	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
109	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
114	C₅×C₈	C₂×C₄²	1	2	yes	—
115	C₅×C₈	C₂×C₄²	1	4	yes	—
116	C₅×C₈	C₂²×C₈	1	8	yes	—
117	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
118	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
119	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
120	C₂×M₁₆	#168	1	8	yes	—
121	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
122	C₂²×C₈	C₄×D₅	1	4	yes	—
123	C₂²×C₈	C₄×D₅	1	4	yes	—
124	C₂²×C₈	C₂×Dic₅	1	2	yes	—
125	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
126	C₂²×C₈	C₂×Dic₅	1	2	yes	—
127	C₂²×C₈	C₂²×D₅	1	2	yes	—
128	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
129	#117	#168	1	4	yes	—
130	#117	C₂×Dic₅	1	4	yes	—
131	C₂×C₄⋊C₄	#168	1	2	yes	—
132	C₂×C₂²⋊C₄	#168	1	2	yes	—
133	C₂×C₄²	C₂×Dic₅	1	2	yes	—
134	M₃₂	—	5	16	no	—
135	C₂×C₁₆	—	5	4	no	—
136	C₂×C₁₆	—	5	4	no	—
137	C₂×C₁₆	—	5	8	no	—
138	C₂×C₁₆	—	5	16	no	—
139	C₂×C₁₆	—	5	2	no	—
140	C₂×C₁₆	—	5	4	no	—
141	C₂×C₁₆	—	5	2	no	—
142	#97	C₂×Dic₅	1	8	yes	—
143	C₄×C₈	C₂×Dic₅	1	4	yes	—
144	C₄×C₈	C₂²×D₅	1	4	yes	—
145	C₂²×C₅	—	2	8	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
148	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
149	C₂²×C₅	#504	1	4	yes	—
150	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
151	C₄×C₅	—	2	8	no	—
152	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
153	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
154	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
155	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
156	C₂⁴	C₂×#168	1	1	yes	characteristic
157	C₂²×C₄	C₂×#168	1	4	yes	—
158	C₂²×C₄	C₂×#168	1	1	yes	characteristic
159	C₂²×C₄	C₂×#168	1	2	yes	—
160	C₂²×C₄	C₂×#168	1	4	yes	—
161	C₂²×C₄	C₄×Dic₅	1	4	yes	—
162	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
163	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
164	M₁₆	C₂×#168	1	16	yes	—
165	C₂×C₈	C₂×#168	1	8	yes	—
166	C₂×C₈	C₄×Dic₅	1	4	yes	—
167	C₂×C₈	C₄×Dic₅	1	8	yes	—
168	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
169	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
171	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
172	C₂×C₈	C₂²×Dic₅	1	4	yes	—
173	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₅	1	2	yes	—
175	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
176	C₄⋊C₄	C₂×#168	1	8	yes	—
177	C₂²⋊C₄	C₂×#168	1	8	yes	—
178	C₄²	C₂×#168	1	4	yes	—
179	C₄²	C₂²×Dic₅	1	4	yes	—
180	C₁₆	—	5	4	no	—
181	C₁₆	—	5	4	no	—
182	C₁₆	—	5	8	no	—
183	C₂×C₅	—	2	4	no	—
184	C₂×C₅	#2108	1	4	yes	complemented
185	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
186	C₂×C₅	C₂×#504	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₄×#168	1	4	yes	—
189	C₂³	C₂²×#168	1	2	yes	—
190	C₂³	C₂²×#168	1	1	yes	characteristic
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	C₄×#168	1	8	yes	—
193	C₂×C₄	C₄×#168	1	4	yes	—
194	C₂×C₄	C₂²×#168	1	8	yes	—
195	C₂×C₄	C₂²×#168	1	2	yes	—
196	C₂×C₄	C₂²×#168	1	4	yes	—
197	C₂×C₄	C₂²×#168	1	1	yes	characteristic
198	C₂×C₄	C₂²×#168	1	2	yes	—
199	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
200	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
201	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
202	C₈	C₂²×#168	1	8	yes	—
203	C₈	C₂×C₄×Dic₅	1	4	yes	—
204	C₈	C₂×C₄×Dic₅	1	2	yes	—
205	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
206	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
207	C₅	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₄×#168	1	4	yes	—
211	C₂²	C₂×C₄×#168	1	2	yes	—
212	C₂²	#65429	1	4	yes	—
213	C₂²	C₂³×#168	1	1	yes	characteristic
214	C₄	—	2	8	no	—
215	C₄	C₂×C₄×#168	1	2	yes	—
216	C₄	C₂×C₄×#168	1	4	yes	—
217	C₄	C₂³×#168	1	1	yes	characteristic
218	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#10600674	1	4	yes	direct product factor
221	C₂	C₂²×C₄×#168	1	1	yes	characteristic
222	C₂	C₂×#65429	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#405	D₅
hypocenter	C₅	C₂×#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #10600674	C₂	direct (N × H)
5. C₂×#64681	C₂	semidirect
6. C₂×C₄×#672	C₂	semidirect
14. C₂×#3944	C₄	semidirect
16. C₂²×#672	C₄	semidirect
21. #64681	C₂²	semidirect
22. C₄×#672	C₂²	semidirect
42. #3944	C₂×C₄	semidirect
46. C₂×#672	C₂×C₄	semidirect
184. C₂×C₅	#2108	semidirect
207. C₅	C₂×#2108	semidirect
220. C₂	#10600674	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×#405	2
C₂²×#3944	2
C₂³×#672	2
#10600674	2
C₂×#64681	2
C₂×C₄×#672	2
C₂×#2108	5

Subgroups — C2×#10600674 #12251935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600674 #12251935