Subgroups — #10617153

Order 640.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#711	C₂	1	1	yes	maximal, characteristic
2	D₄×#170	C₂	1	2	yes	maximal, complemented
3	#65754	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#170	C₂	1	2	yes	maximal, complemented
5	#65703	C₂	1	2	yes	maximal, complemented
6	#65692	C₂	1	4	yes	maximal, complemented
7	#65680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#711	C₄	1	1	yes	characteristic
10	C₂²×#170	C₂²	1	4	yes	—
11	C₂×D₅⋊C₈	C₂²	1	2	yes	—
12	C₄×C₅×D₄	C₄	1	1	yes	characteristic
13	D₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₅	C₄	1	2	yes	—
15	C₂×C₄×Dic₅	C₂²	1	2	yes	—
16	#4038	C₄	1	1	yes	characteristic
17	#4033	C₂²	1	1	yes	characteristic
18	#4027	C₂²	1	2	yes	—
19	#4023	C₄	1	2	yes	—
20	C₄²×D₅	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₀	C₄	1	1	yes	characteristic
22	C₂²⋊#170	C₂²	1	4	yes	—
23	#4004	C₂²	1	4	yes	—
24	#4003	C₂²	1	2	yes	—
25	C₄⋊#170	C₂²	1	2	yes	—
26	C₄×#170	C₂²	1	2	yes	—
27	C₄×#170	C₂²	1	1	yes	characteristic
28	C₄×#170	C₂²	1	1	yes	characteristic
29	C₄×#170	C₂²	1	2	yes	—
30	C₄×#170	C₂²	1	4	yes	—
31	C₈×#41	—	5	1	no	maximal, Sylow
32	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
35	C₂²×Dic₅	C₂×C₄	1	2	yes	—
36	C₂²×Dic₅	C₂³	1	2	yes	—
37	#711	C₈	1	8	yes	complemented
38	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
42	C₂×#170	—	2	8	no	—
43	C₂×#170	C₂³	1	2	yes	—
44	C₂×#170	C₂³	1	4	yes	—
45	C₂×#170	C₂³	1	2	yes	—
46	D₅⋊C₈	—	2	4	no	—
47	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
49	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
51	#685	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
53	#683	C₂×C₄	1	2	yes	—
54	#683	C₂×C₄	1	2	yes	—
55	C₄×Dic₅	C₂×C₄	1	2	yes	—
56	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₅	C₂×C₄	1	2	yes	—
58	C₄×Dic₅	C₂³	1	1	yes	characteristic
59	C₄×Dic₅	C₂³	1	1	yes	characteristic
60	C₄×Dic₅	C₂³	1	2	yes	—
61	C₄×#41	—	5	1	no	—
62	C₈×Q₈	—	5	2	no	—
63	C₈×D₄	—	5	4	no	—
64	C₈×D₄	—	5	2	no	—
65	#431	—	5	1	no	—
66	#431	—	5	2	no	—
67	C₂×C₄×C₈	—	5	1	no	—
68	C₂×C₄×C₈	—	5	2	no	—
69	C₂³×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
71	C₅×D₄	C₂×C₈	1	4	yes	complemented
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²⋊D₅	C₂×C₈	1	8	yes	complemented
78	C₂×Dic₅	—	2	4	no	—
79	C₂×Dic₅	C₂×C₈	1	8	yes	complemented
80	C₂×Dic₅	C₂²×C₄	1	2	yes	—
81	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₅	C₂²×C₄	1	2	yes	—
84	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
85	C₄×D₅	—	2	2	no	—
86	C₄×D₅	—	2	2	no	—
87	C₄×D₅	C₂×C₈	1	4	yes	complemented
88	Dic₁₀	C₂×C₈	1	4	yes	complemented
89	#170	—	2	2	no	—
90	#170	—	2	4	no	—
91	#170	#41	1	4	yes	complemented
92	C₂×#41	—	5	1	no	—
93	C₂²×C₈	—	5	2	no	—
94	C₂²×C₈	—	5	4	no	—
95	C₄×Q₈	—	5	1	no	—
96	C₄×Q₈	—	5	1	no	—
97	C₄×D₄	—	5	1	no	—
98	C₄×D₄	—	5	2	no	—
99	C₄×D₄	—	5	2	no	—
100	C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
101	#117	—	5	1	no	—
102	#117	—	5	2	no	—
103	C₂×C₄²	—	5	2	no	—
104	C₂×C₄²	—	5	1	no	—
105	C₄⋊C₈	—	5	2	no	—
106	C₄⋊C₈	—	5	4	no	—
107	C₂²⋊C₈	—	5	2	no	—
108	C₂²⋊C₈	—	5	4	no	—
109	C₄×C₈	—	5	1	no	—
110	C₄×C₈	—	5	1	no	—
111	C₄×C₈	—	5	2	no	—
112	C₄×C₈	—	5	2	no	—
113	C₄×C₈	—	5	4	no	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	C₂²×C₈	1	4	yes	—
117	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
118	C₂×D₅	—	2	1	no	—
119	C₂×D₅	—	2	1	no	—
120	C₂×D₅	C₂²×C₈	1	2	yes	—
121	C₄×C₅	—	2	1	no	—
122	C₄×C₅	—	2	2	no	—
123	C₄×C₅	C₂²×C₈	1	2	yes	—
124	C₄×C₅	#131	1	2	yes	—
125	Dic₅	—	2	2	no	—
126	Dic₅	—	2	1	no	—
127	Dic₅	C₂²×C₈	1	4	yes	—
128	Dic₅	C₂²×C₈	1	2	yes	—
129	Dic₅	C₂×#41	1	2	yes	—
130	#41	—	5	8	no	—
131	C₂×Q₈	—	5	1	no	—
132	C₂×D₄	—	5	2	no	—
133	C₂×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂²×C₄	—	5	2	no	—
135	C₂²×C₄	—	5	1	no	—
136	C₂²×C₄	—	5	2	no	—
137	C₂²×C₄	—	5	1	no	—
138	C₂²×C₄	—	5	1	no	—
139	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
140	C₂×C₈	—	5	2	no	—
141	C₂×C₈	—	5	4	no	—
142	C₂×C₈	—	5	2	no	—
143	C₂×C₈	—	10	4	no	—
144	C₂×C₈	—	10	8	no	—
145	C₄⋊C₄	—	5	2	no	—
146	C₄⋊C₄	—	5	2	no	—
147	C₄⋊C₄	—	5	1	no	—
148	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₂²⋊C₄	—	5	2	no	—
150	C₂²⋊C₄	—	5	2	no	—
151	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
152	C₄²	—	5	2	no	—
153	C₄²	—	5	1	no	—
154	C₄²	—	5	2	no	—
155	C₄²	—	5	1	no	—
156	C₄²	—	5	1	no	—
157	C₄²	—	5	2	no	—
158	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₅	—	2	4	no	—
160	C₂×C₅	C₄×#41	1	1	yes	characteristic
161	C₂×C₅	C₂³×C₈	1	1	yes	characteristic
162	C₂×C₅	C₂×#131	1	1	yes	characteristic
163	D₅	—	2	2	no	—
164	C₂³	—	5	1	no	—
165	C₂³	C₂²×AGL(1,𝔽₅)	1	2	yes	—
166	Q₈	—	5	4	no	—
167	D₄	—	5	8	no	—
168	D₄	D₅⋊C₈	1	4	yes	complemented
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	5	8	no	—
171	C₂×C₄	—	5	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	2	no	—
174	C₂×C₄	—	5	4	no	—
175	C₂×C₄	—	5	1	no	—
176	C₂×C₄	—	5	2	no	—
177	C₂×C₄	—	10	2	no	—
178	C₂×C₄	—	10	4	no	—
179	C₂×C₄	—	10	2	no	—
180	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
181	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₈	—	5	4	no	—
185	C₈	—	10	2	no	—
186	C₈	—	10	4	no	—
187	C₅	C₈×#41	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	5	2	no	—
191	C₂²	—	10	1	no	—
192	C₂²	—	10	1	no	—
193	C₂²	C₂×D₅⋊C₈	1	4	yes	—
194	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	5	2	no	—
198	C₄	—	5	4	no	—
199	C₄	—	5	2	no	—
200	C₄	—	10	1	no	—
201	C₄	—	10	2	no	—
202	C₄	C₂×D₅⋊C₈	1	2	yes	—
203	C₄	#4131	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	10	2	no	—
206	C₂	C₂²×D₅⋊C₈	1	1	yes	characteristic
207	C₂	C₂×#4131	1	1	yes	characteristic
208	C₂	#41×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₈
Frattini	C₂²	C₂³×AGL(1,𝔽₅)
Fitting	C₄×C₅×D₄	C₄
socle	C₂²×C₅	C₂³×C₄
prime radical	C₂×#711	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₈×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. D₄×#170	C₂	semidirect
3. #65754	C₂	semidirect
4. C₂×C₄×#170	C₂	semidirect
5. #65703	C₂	semidirect
6. #65692	C₂	semidirect
7. #65680	C₂	semidirect
8. C₄×D₅⋊C₈	C₂	semidirect
37. #711	C₈	semidirect
71. C₅×D₄	C₂×C₈	semidirect
77. C₂²⋊D₅	C₂×C₈	semidirect
79. C₂×Dic₅	C₂×C₈	semidirect
87. C₄×D₅	C₂×C₈	semidirect
88. Dic₁₀	C₂×C₈	semidirect
91. #170	#41	semidirect
168. D₄	D₅⋊C₈	semidirect
187. C₅	C₈×#41	semidirect

Maximal subgroups

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

subgroup	index
C₄×#711	2
D₄×#170	2
#65754	2
C₂×C₄×#170	2
#65703	2
#65692	2
#65680	2
C₄×D₅⋊C₈	2
C₈×#41	5