Subgroups — C₂×#65007 #10610312

Order 640.

528 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×#3946	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₅	C₂	1	2	yes	maximal, complemented
6	#65007	C₂	1	8	yes	maximal, direct product factor
7	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂³×Dic₅	C₄	1	1	yes	characteristic
9	C₂³×Dic₅	C₄	1	1	yes	characteristic
10	C₂²×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₄×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₅	C₄	1	4	yes	—
13	C₂×C₄×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₅	C₂²	1	4	yes	—
17	C₂²×#168	C₂²	1	2	yes	—
18	C₅×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊#168	C₂²	1	4	yes	—
20	#3946	C₂²	1	8	yes	complemented
21	C₈×Dic₅	C₂²	1	8	yes	complemented
22	C₂×#431	—	5	1	no	maximal, Sylow
23	C₂⁴×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅	—	2	4	no	—
25	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅	C₂³	1	2	yes	—
28	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅	—	2	4	no	—
30	C₂²×Dic₅	C₈	1	8	yes	complemented
31	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₅	C₂×C₄	1	4	yes	—
33	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₅	C₂×C₄	1	4	yes	—
36	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₈	—	2	4	no	—
38	C₂×C₅×C₈	C₂³	1	4	yes	—
39	C₄×Dic₅	C₂×C₄	1	4	yes	—
40	C₄×Dic₅	C₂×C₄	1	2	yes	—
41	C₄×Dic₅	C₂×C₄	1	4	yes	—
42	C₄×Dic₅	C₂×C₄	1	2	yes	—
43	C₄×Dic₅	C₂³	1	2	yes	—
44	C₄×Dic₅	C₂³	1	2	yes	—
45	C₂×#168	—	2	4	no	—
46	C₂×#168	C₂³	1	4	yes	—
47	C₂²×C₄²	—	5	1	no	—
48	#431	—	5	8	no	—
49	C₂×C₄⋊C₈	—	5	2	no	—
50	C₂×C₂²⋊C₈	—	5	1	no	—
51	C₂×C₂²⋊C₈	D₅	1	1	yes	characteristic
52	C₂×C₄×C₈	—	5	2	no	—
53	C₂³×C₅	—	2	2	no	—
54	C₂³×C₅	—	2	2	no	—
55	C₂³×C₅	C₂×C₈	1	4	yes	—
56	C₂³×C₅	C₂²×C₄	1	2	yes	—
57	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	4	no	—
60	C₂×C₄×C₅	—	2	4	no	—
61	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₅	#41	1	4	yes	—
64	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₅	—	2	2	no	—
66	C₂×Dic₅	—	2	8	no	—
67	C₂×Dic₅	C₂×C₈	1	4	yes	—
68	C₂×Dic₅	C₂×C₈	1	4	yes	—
69	C₂×Dic₅	C₂×C₈	1	16	yes	complemented
70	C₂×Dic₅	M₁₆	1	4	yes	—
71	C₂×Dic₅	C₂²×C₄	1	2	yes	—
72	C₂×Dic₅	C₂²×C₄	1	2	yes	—
73	C₂×Dic₅	C₂²×C₄	1	2	yes	—
74	C₂×Dic₅	C₂²×C₄	1	2	yes	—
75	C₅×C₈	—	2	4	no	—
76	#168	—	2	4	no	—
77	C₂³×C₄	—	5	1	no	—
78	C₂³×C₄	—	5	1	no	—
79	C₂³×C₄	C₂×D₅	1	1	yes	characteristic
80	C₂²×C₈	—	5	2	no	—
81	C₂²×C₈	C₂×D₅	1	2	yes	—
82	C₂×C₄²	—	5	1	no	—
83	C₂×C₄²	—	5	4	no	—
84	C₂×C₄²	—	5	1	no	—
85	C₂×C₄²	—	5	4	no	—
86	C₂×C₄²	—	5	1	no	—
87	C₂×C₄²	—	5	1	no	—
88	C₄⋊C₈	—	5	8	no	—
89	C₂²⋊C₈	—	5	4	no	—
90	C₂²⋊C₈	C₂×D₅	1	4	yes	—
91	C₄×C₈	—	5	8	no	—
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	—	2	8	no	—
95	C₂²×C₅	#117	1	2	yes	—
96	C₂²×C₅	C₂²×C₈	1	4	yes	—
97	C₂²×C₅	C₂²×C₈	1	2	yes	—
98	C₂²×C₅	C₂×M₁₆	1	2	yes	—
99	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	#117	1	2	yes	—
102	C₄×C₅	C₂×#41	1	2	yes	—
103	Dic₅	—	2	2	no	—
104	Dic₅	C₂²×C₈	1	4	yes	—
105	Dic₅	C₂²×C₈	1	4	yes	—
106	Dic₅	C₂×M₁₆	1	4	yes	—
107	C₂⁴	C₄×D₅	1	1	yes	characteristic
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	5	8	no	—
110	C₂²×C₄	—	5	1	no	—
111	C₂²×C₄	—	5	1	no	—
112	C₂²×C₄	—	5	4	no	—
113	C₂²×C₄	—	5	1	no	—
114	C₂²×C₄	—	5	4	no	—
115	C₂²×C₄	—	5	1	no	—
116	C₂²×C₄	—	10	4	no	—
117	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₅	1	2	yes	—
119	C₂²×C₄	C₂²×D₅	1	2	yes	—
120	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	5	4	no	—
123	C₂×C₈	—	10	4	no	—
124	C₂×C₈	C₂²×D₅	1	4	yes	—
125	C₄²	—	5	4	no	—
126	C₄²	—	5	2	no	—
127	C₄²	—	5	4	no	—
128	C₄²	—	5	2	no	—
129	C₄²	—	5	2	no	—
130	C₄²	—	5	2	no	—
131	C₂×C₅	—	2	4	no	—
132	C₂×C₅	#431	1	4	yes	complemented
133	C₂×C₅	C₂×#117	1	1	yes	characteristic
134	C₂×C₅	C₂³×C₈	1	1	yes	characteristic
135	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₈×D₅	1	4	yes	—
139	C₂³	C₂×C₄×D₅	1	2	yes	—
140	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	5	2	no	—
145	C₂×C₄	—	5	2	no	—
146	C₂×C₄	—	5	2	no	—
147	C₂×C₄	—	5	4	no	—
148	C₂×C₄	—	5	4	no	—
149	C₂×C₄	—	5	16	no	—
150	C₂×C₄	—	5	4	no	—
151	C₂×C₄	—	5	2	no	—
152	C₂×C₄	—	10	8	no	—
153	C₂×C₄	—	10	2	no	—
154	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
156	C₂×C₄	#711	1	4	yes	—
157	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	10	4	no	—
160	C₅	C₂×#431	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	#4023	1	2	yes	—
165	C₂²	C₂×C₈×D₅	1	4	yes	—
166	C₂²	C₂×C₈×D₅	1	2	yes	—
167	C₂²	D₅×M₁₆	1	2	yes	—
168	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	5	4	no	—
171	C₄	—	5	4	no	—
172	C₄	—	5	4	no	—
173	C₄	—	10	2	no	—
174	C₄	#4023	1	2	yes	—
175	C₄	C₂×#711	1	2	yes	—
176	C₂	—	2	4	no	—
177	C₂	#65007	1	4	yes	direct product factor
178	C₂	C₂×#4023	1	1	yes	characteristic
179	C₂	C₂²×C₈×D₅	1	1	yes	characteristic
180	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅
commutator	C₂×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₂×C₅×C₂²⋊C₈	C₂
socle	C₂³×C₅	C₂²×C₄
prime radical	C₂⁴×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₅
hypocenter	C₅	C₂×#431
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₂×#3946	C₂	semidirect
5. C₂×C₈×Dic₅	C₂	semidirect
6. #65007	C₂	direct (N × H)
20. #3946	C₂²	semidirect
21. C₈×Dic₅	C₂²	semidirect
30. C₂²×Dic₅	C₈	semidirect
69. C₂×Dic₅	C₂×C₈	semidirect
132. C₂×C₅	#431	semidirect
160. C₅	C₂×#431	semidirect
177. C₂	#65007	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×Dic₅	2
C₂×C₅×C₂²⋊C₈	2
C₂×C₂²⋊#168	2
C₂×#3946	2
C₂×C₈×Dic₅	2
#65007	2
C₂×#431	5

Subgroups — C2×#65007 #10610312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#65007 #10610312