Subgroups — #10614213

Order 640.

484 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	#65519	C₂	1	2	yes	maximal
4	#65273	C₂	1	4	yes	maximal
5	#65262	C₂	1	2	yes	maximal
6	#65224	C₂	1	1	yes	maximal, characteristic
7	#65218	C₂	1	2	yes	maximal
8	#65217	C₂	1	2	yes	maximal
9	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀	C₂²	1	1	yes	characteristic
11	C₄×C₅×Q₈	—	2	4	no	—
12	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₅	C₂²	1	2	yes	—
15	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
20	C₂×#683	C₂²	1	2	yes	—
21	C₂×#683	C₂²	1	4	yes	—
22	C₂×C₄×Dic₅	C₂²	1	2	yes	—
23	C₄×Dic₁₀	—	2	4	no	—
24	#3963	C₂²	1	4	yes	—
25	#3963	C₂²	1	4	yes	—
26	#3963	C₂²	1	4	yes	—
27	#2582	—	5	1	no	maximal, Sylow
28	C₂×C₅×Q₈	—	2	2	no	—
29	C₂×C₅×Q₈	—	2	2	no	—
30	C₂×C₅×Q₈	Q₈	1	4	yes	complemented
31	C₂²×C₄×C₅	C₂³	1	2	yes	—
32	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂³	1	2	yes	—
34	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×Dic₅	C₂³	1	4	yes	—
37	C₂²×Dic₅	C₂³	1	2	yes	—
38	C₂²×Dic₅	C₂³	1	2	yes	—
39	C₂×Dic₁₀	—	2	2	no	—
40	C₂×Dic₁₀	—	2	2	no	—
41	C₂×Dic₁₀	Q₈	1	4	yes	complemented
42	C₅×C₄⋊C₄	—	2	4	no	—
43	C₅×C₄⋊C₄	—	2	2	no	—
44	C₄²×C₅	—	2	4	no	—
45	C₄²×C₅	D₄	1	4	yes	—
46	C₄⋊Dic₅	—	2	2	no	—
47	C₄⋊Dic₅	—	2	4	no	—
48	C₄⋊Dic₅	—	2	4	no	—
49	C₄⋊Dic₅	—	4	2	no	—
50	#683	—	2	4	no	—
51	#683	—	4	4	no	—
52	C₄×Dic₅	—	2	4	no	—
53	C₂×C₄×Q₈	—	5	1	no	—
54	C₂×C₄×Q₈	D₅	1	1	yes	characteristic
55	#398	—	5	4	no	—
56	#391	—	5	2	no	—
57	#391	—	5	2	no	—
58	#389	—	5	2	no	—
59	#389	—	5	2	no	—
60	#384	—	5	1	no	—
61	C₂³×C₅	C₂⁴	1	1	yes	characteristic
62	C₅×Q₈	—	2	8	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	1	no	—
66	C₂×C₄×C₅	—	2	1	no	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
74	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
76	C₂×C₄×C₅	#41	1	2	yes	—
77	C₂×C₄×C₅	#41	1	2	yes	—
78	C₂×Dic₅	—	2	2	no	—
79	C₂×Dic₅	—	2	2	no	—
80	C₂×Dic₅	—	2	2	no	—
81	C₂×Dic₅	—	2	4	no	—
82	C₂×Dic₅	—	2	4	no	—
83	C₂×Dic₅	—	2	2	no	—
84	C₂×Dic₅	—	2	4	no	—
85	C₂×Dic₅	—	2	2	no	—
86	C₂×Dic₅	C₂×Q₈	1	4	yes	—
87	Dic₁₀	—	2	8	no	—
88	C₂²×Q₈	—	5	1	no	—
89	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic
90	C₄×Q₈	—	2	4	no	—
91	C₄×Q₈	—	10	4	no	—
92	C₂×C₄⋊C₄	—	5	1	no	—
93	C₂×C₄⋊C₄	—	5	2	no	—
94	C₂×C₄⋊C₄	—	5	2	no	—
95	C₂×C₄⋊C₄	—	5	2	no	—
96	C₂×C₄⋊C₄	—	5	2	no	—
97	C₂×C₄⋊C₄	—	5	4	no	—
98	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
100	C₂×C₄²	—	5	2	no	—
101	C₂×C₄²	C₂×D₅	1	2	yes	—
102	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
103	#95	—	5	4	no	—
104	#95	—	5	4	no	—
105	#95	—	5	4	no	—
106	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₅	C₂×#41	1	1	yes	characteristic
110	C₂²×C₅	C₂×#41	1	1	yes	characteristic
111	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₅	—	2	4	no	—
114	C₄×C₅	—	2	2	no	—
115	C₄×C₅	—	4	2	no	—
116	C₄×C₅	C₂²⋊Q₈	1	4	yes	—
117	C₄×C₅	C₂²⋊Q₈	1	4	yes	—
118	Dic₅	—	2	4	no	—
119	Dic₅	—	4	2	no	—
120	Dic₅	—	4	4	no	—
121	C₂×Q₈	—	2	2	no	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	5	4	no	—
124	C₂×Q₈	—	10	2	no	—
125	C₂×Q₈	—	10	2	no	—
126	C₂×Q₈	Dic₁₀	1	4	yes	complemented
127	C₂²×C₄	—	5	2	no	—
128	C₂²×C₄	—	5	4	no	—
129	C₂²×C₄	—	5	2	no	—
130	C₂²×C₄	C₂²×D₅	1	2	yes	—
131	C₂²×C₄	C₂²×D₅	1	2	yes	—
132	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	10	4	no	—
138	C₄⋊C₄	—	10	4	no	—
139	C₄⋊C₄	—	10	4	no	—
140	C₄⋊C₄	—	10	2	no	—
141	C₄⋊C₄	—	20	4	no	—
142	C₄⋊C₄	—	20	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	10	4	no	—
145	C₄²	C₂²⋊D₅	1	4	yes	—
146	C₂×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₅	#537	1	1	yes	characteristic
149	C₂×C₅	#557	1	1	yes	characteristic
150	C₂×C₅	#557	1	1	yes	characteristic
151	C₂×C₅	Q₈²	1	1	yes	characteristic
152	C₂×C₅	#559	1	1	yes	characteristic
153	C₂³	C₂³×D₅	1	1	yes	characteristic
154	Q₈	—	2	8	no	—
155	Q₈	—	10	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	5	4	no	—
164	C₂×C₄	—	10	2	no	—
165	C₂×C₄	—	10	2	no	—
166	C₂×C₄	—	10	4	no	—
167	C₂×C₄	—	10	4	no	—
168	C₂×C₄	—	10	2	no	—
169	C₂×C₄	—	10	4	no	—
170	C₂×C₄	—	10	2	no	—
171	C₂×C₄	—	10	2	no	—
172	C₂×C₄	C₂×Dic₁₀	1	4	yes	—
173	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
174	C₂×C₄	#709	1	2	yes	—
175	C₂×C₄	Q₈×D₅	1	2	yes	—
176	C₂×C₄	#713	1	2	yes	—
177	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
180	C₅	#2582	1	1	yes	characteristic, complemented, Sylow
181	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
182	C₂²	C₂×#709	1	1	yes	characteristic
183	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
184	C₂²	C₂×#713	1	1	yes	characteristic
185	C₂²	#4149	1	1	yes	characteristic
186	C₂²	#4150	1	1	yes	characteristic
187	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	10	4	no	—
192	C₄	—	20	2	no	—
193	C₄	—	20	4	no	—
194	C₄	#4070	1	4	yes	—
195	C₄	#4092	1	4	yes	—
196	C₂	Q₈×Dic₁₀	1	1	yes	characteristic
197	C₂	#65901	1	1	yes	characteristic
198	C₂	#65904	1	1	yes	characteristic
199	C₂	#65919	1	1	yes	characteristic
200	C₂	C₂×#4070	1	1	yes	characteristic
201	C₂	C₂×#4092	1	1	yes	characteristic
202	C₂	#66167	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₄×C₅×Q₈	C₂
hypercenter	C₂×C₄×Q₈	D₅
hypocenter	C₅	#2582
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
30. C₂×C₅×Q₈	Q₈	semidirect
41. C₂×Dic₁₀	Q₈	semidirect
126. C₂×Q₈	Dic₁₀	semidirect
180. C₅	#2582	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₈	2
C₂×C₄×Dic₁₀	2
#65519	2
#65273	2
#65262	2
#65224	2
#65218	2
#65217	2
#2582	5