Subgroups — C₅×#3044 #10616639

Order 640.

458 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×#137	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#536	C₂	1	1	yes	maximal, characteristic
4	C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#464	C₂	1	4	yes	maximal, complemented
6	C₅×#462	C₂	1	4	yes	maximal, complemented
7	C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#416	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₅×#137	—	2	4	no	—
14	C₂×C₅×Q₁₆	—	2	4	no	—
15	C₂×C₅×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₅×Q₁₆	C₂²	1	4	yes	—
17	C₂×C₅×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₅×#124	C₂²	1	2	yes	—
22	C₅×#123	—	2	2	no	—
23	C₅×C₂²⋊Q₈	—	2	2	no	—
24	C₄×C₅×Q₈	—	2	2	no	—
25	C₄×C₅×Q₈	C₂²	1	4	yes	—
26	C₅×#117	—	2	2	no	—
27	C₅×#117	C₂²	1	1	yes	characteristic
28	C₅×#117	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊C₈	C₂²	1	4	yes	—
30	C₅×#103	C₂²	1	2	yes	—
31	C₅×#103	C₂²	1	4	yes	—
32	C₅×#102	C₂²	1	2	yes	—
33	#3044	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₅×#41	—	2	2	no	—
35	C₅×#41	—	2	2	no	—
36	C₂×C₅×Q₈	—	2	1	no	—
37	C₂×C₅×Q₈	—	2	1	no	—
38	C₂×C₅×Q₈	—	2	2	no	—
39	C₂×C₅×Q₈	—	2	2	no	—
40	C₂×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₈	D₄	1	4	yes	—
42	C₂×C₅×Q₈	C₂³	1	2	yes	—
43	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
44	C₂×C₅×Q₈	C₂³	1	2	yes	—
45	C₂×C₅×D₄	—	2	1	no	—
46	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	1	no	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₅×Q₁₆	—	2	8	no	—
52	C₅×Q₁₆	—	2	4	no	—
53	C₅×SD₁₆	—	2	4	no	—
54	C₅×M₁₆	—	2	2	no	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	C₂³	1	2	yes	—
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	D₄	1	2	yes	—
61	C₅×C₄⋊C₄	C₂³	1	2	yes	—
62	C₅×C₂²⋊C₄	—	2	1	no	—
63	C₅×C₂²⋊C₄	—	2	4	no	—
64	C₅×C₂²⋊C₄	D₄	1	2	yes	—
65	C₄²×C₅	—	2	4	no	—
66	C₄²×C₅	—	2	1	no	—
67	C₄²×C₅	C₂³	1	2	yes	—
68	C₂×#137	C₂×C₅	1	1	yes	characteristic
69	C₂²×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
70	#536	C₂×C₅	1	1	yes	characteristic
71	#519	C₂×C₅	1	1	yes	characteristic, complemented
72	#464	C₂×C₅	1	4	yes	complemented
73	#462	C₂×C₅	1	4	yes	complemented
74	#424	C₂×C₅	1	1	yes	characteristic, complemented
75	#419	C₂×C₅	1	1	yes	characteristic, complemented
76	#416	C₂×C₅	1	1	yes	characteristic
77	C₂³×C₅	—	2	1	no	—
78	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×Q₈	—	2	4	no	—
80	C₅×Q₈	—	2	2	no	—
81	C₅×Q₈	—	2	2	no	—
82	C₅×Q₈	—	2	1	no	—
83	C₅×Q₈	—	2	2	no	—
84	C₅×Q₈	—	2	1	no	—
85	C₅×Q₈	—	4	2	no	—
86	C₅×Q₈	C₂×D₄	1	4	yes	—
87	C₅×D₄	—	2	1	no	—
88	C₅×D₄	—	2	1	no	—
89	C₅×D₄	—	4	2	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
98	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₅	#41	1	2	yes	—
101	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
102	C₅×C₈	—	2	2	no	—
103	C₅×C₈	—	2	2	no	—
104	C₂×#41	C₂²×C₅	1	1	yes	characteristic
105	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
106	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
107	#137	—	2	4	no	—
108	C₂×Q₁₆	—	2	4	no	—
109	C₂×Q₁₆	C₂²×C₅	1	4	yes	—
110	C₂×Q₁₆	C₂²×C₅	1	2	yes	—
111	C₂×SD₁₆	C₂²×C₅	1	2	yes	—
112	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₄⋊Q₈	C₂²×C₅	1	2	yes	—
115	#124	C₂²×C₅	1	2	yes	—
116	#123	—	2	2	no	—
117	C₂²⋊Q₈	—	2	2	no	—
118	C₄×Q₈	—	2	2	no	—
119	C₄×Q₈	C₂²×C₅	1	4	yes	—
120	#117	—	2	2	no	—
121	#117	C₂²×C₅	1	1	yes	characteristic
122	#117	C₂²×C₅	1	1	yes	characteristic
123	C₄⋊C₈	C₂²×C₅	1	4	yes	—
124	#103	C₂²×C₅	1	2	yes	—
125	#103	C₂²×C₅	1	4	yes	—
126	#102	C₂²×C₅	1	2	yes	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	4	1	no	—
129	C₂²×C₅	—	4	2	no	—
130	C₂²×C₅	#121	1	2	yes	—
131	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₅	—	2	4	no	—
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	—	4	1	no	—
135	C₄×C₅	—	4	2	no	—
136	C₄×C₅	—	4	2	no	—
137	C₄×C₅	#121	1	2	yes	—
138	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₅	C₂×#41	1	1	yes	characteristic
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	2	1	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	C₅×D₄	1	4	yes	—
148	C₂×Q₈	C₂³×C₅	1	2	yes	—
149	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
150	C₂×Q₈	C₂³×C₅	1	2	yes	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
157	Q₁₆	—	2	8	no	—
158	Q₁₆	—	2	4	no	—
159	SD₁₆	—	2	4	no	—
160	M₁₆	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂³×C₅	1	2	yes	—
163	C₂×C₈	C₂³×C₅	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	C₅×D₄	1	2	yes	—
167	C₄⋊C₄	C₂³×C₅	1	2	yes	—
168	C₂²⋊C₄	—	2	1	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	C₅×D₄	1	2	yes	—
171	C₄²	—	2	4	no	—
172	C₄²	—	2	1	no	—
173	C₄²	C₂³×C₅	1	2	yes	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	8	1	no	—
176	C₂×C₅	C₂×#121	1	1	yes	characteristic
177	C₂×C₅	#578	1	2	yes	—
178	C₂³	—	2	1	no	—
179	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
180	Q₈	—	2	4	no	—
181	Q₈	—	2	2	no	—
182	Q₈	—	2	2	no	—
183	Q₈	—	2	1	no	—
184	Q₈	—	2	2	no	—
185	Q₈	—	2	1	no	—
186	Q₈	—	4	2	no	—
187	Q₈	C₂×C₅×D₄	1	4	yes	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	—	4	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₂×C₅×D₄	1	4	yes	—
199	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
201	C₂×C₄	C₅×#41	1	2	yes	—
202	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₅	#3044	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	—	2	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₅×#121	1	2	yes	—
210	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₅×#121	1	2	yes	—
217	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
218	C₄	C₂×C₅×#41	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	8	1	no	—
221	C₂	C₂×C₅×#121	1	1	yes	characteristic
222	C₂	C₅×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Q₁₆	C₂	semidirect
4. C₅×#519	C₂	semidirect
5. C₅×#464	C₂	semidirect
6. C₅×#462	C₂	semidirect
7. C₅×#424	C₂	semidirect
8. C₅×#419	C₂	semidirect
33. #3044	C₅	direct (N × H)
69. C₂²×Q₁₆	C₂×C₅	semidirect
71. #519	C₂×C₅	semidirect
72. #464	C₂×C₅	semidirect
73. #462	C₂×C₅	semidirect
74. #424	C₂×C₅	semidirect
75. #419	C₂×C₅	semidirect
205. C₅	#3044	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×#137	2
C₂²×C₅×Q₁₆	2
C₅×#536	2
C₅×#519	2
C₅×#464	2
C₅×#462	2
C₅×#424	2
C₅×#419	2
C₅×#416	2
#3044	5

Subgroups — C5×#3044 #10616639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3044 #10616639