Subgroups — C₅×#3104 #10616699

Order 640.

384 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#484	C₂	1	2	yes	maximal
4	C₅×#477	C₂	1	2	yes	maximal, complemented
5	C₅×#462	C₂	1	2	yes	maximal, complemented
6	C₅×#451	C₂	1	2	yes	maximal
7	C₄×C₅×Q₁₆	C₂	1	4	yes	maximal, complemented
8	C₅×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×Q₁₆	C₂²	1	4	yes	—
11	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₅×#125	—	2	1	no	—
14	C₅×C₂²⋊Q₈	—	2	1	no	—
15	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₄×C₅×Q₈	—	2	1	no	—
17	C₄×C₅×Q₈	—	2	1	no	—
18	C₄×C₅×Q₈	—	2	1	no	—
19	C₄×C₅×Q₈	C₂²	1	2	yes	—
20	C₄×C₅×Q₈	C₂²	1	2	yes	—
21	C₄×C₅×Q₈	C₂²	1	2	yes	—
22	C₅×#117	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	—	2	1	no	—
24	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₅	—	2	1	no	—
26	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₅×#107	C₂²	1	4	yes	—
28	C₅×C₄⋊C₈	C₂²	1	2	yes	—
29	C₅×#103	C₂²	1	4	yes	—
30	C₅×#103	C₂²	1	4	yes	—
31	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₄×C₅×C₈	C₂²	1	2	yes	—
33	#3104	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×Q₈	—	2	4	no	—
35	C₂×C₅×Q₈	—	2	1	no	—
36	C₂×C₅×Q₈	—	2	1	no	—
37	C₂×C₅×Q₈	C₂³	1	2	yes	—
38	C₂×C₅×Q₈	C₂³	1	2	yes	—
39	C₂²×C₄×C₅	—	2	1	no	—
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₅×Q₁₆	—	2	8	no	—
45	C₂×C₅×C₈	C₂³	1	4	yes	—
46	C₅×C₄⋊C₄	—	2	2	no	—
47	C₅×C₄⋊C₄	—	2	2	no	—
48	C₅×C₄⋊C₄	—	2	1	no	—
49	C₅×C₄⋊C₄	—	2	1	no	—
50	C₅×C₄⋊C₄	—	2	1	no	—
51	C₅×C₄⋊C₄	—	2	1	no	—
52	C₅×C₄⋊C₄	—	2	2	no	—
53	C₅×C₄⋊C₄	—	2	2	no	—
54	C₅×C₄⋊C₄	C₂³	1	2	yes	—
55	C₅×C₄⋊C₄	C₂³	1	2	yes	—
56	C₅×C₂²⋊C₄	—	2	2	no	—
57	C₄²×C₅	—	2	2	no	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	—	2	2	no	—
60	C₄²×C₅	D₄	1	1	yes	characteristic
61	C₄²×C₅	D₄	1	1	yes	characteristic
62	C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₄²×C₅	C₂³	1	1	yes	characteristic
64	#533	C₂×C₅	1	1	yes	characteristic
65	C₂×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
66	#484	C₂×C₅	1	2	yes	—
67	#477	C₂×C₅	1	2	yes	complemented
68	#462	C₂×C₅	1	2	yes	complemented
69	#451	C₂×C₅	1	2	yes	—
70	C₄×Q₁₆	C₂×C₅	1	4	yes	complemented
71	#431	C₂×C₅	1	1	yes	characteristic
72	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
73	C₅×Q₈	—	2	4	no	—
74	C₅×Q₈	—	2	2	no	—
75	C₅×Q₈	—	2	4	no	—
76	C₅×Q₈	—	4	2	no	—
77	C₅×Q₈	#41	1	4	yes	—
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	4	1	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	Q₁₆	1	4	yes	complemented
87	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
93	C₅×C₈	—	2	4	no	—
94	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×Q₁₆	C₂²×C₅	1	4	yes	—
96	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
97	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	1	no	—
100	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
101	C₄×Q₈	—	2	1	no	—
102	C₄×Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	C₂²×C₅	1	2	yes	—
105	C₄×Q₈	C₂²×C₅	1	2	yes	—
106	C₄×Q₈	C₂²×C₅	1	2	yes	—
107	#117	—	2	1	no	—
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
112	#107	C₂²×C₅	1	4	yes	—
113	C₄⋊C₈	C₂²×C₅	1	2	yes	—
114	#103	C₂²×C₅	1	4	yes	—
115	#103	C₂²×C₅	1	4	yes	—
116	C₂²⋊C₈	C₂²×C₅	1	2	yes	—
117	C₄×C₈	C₂²×C₅	1	2	yes	—
118	C₂²×C₅	—	2	1	no	—
119	C₂²×C₅	—	2	1	no	—
120	C₂²×C₅	C₂×Q₁₆	1	2	yes	—
121	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₅	—	2	4	no	—
123	C₄×C₅	—	2	1	no	—
124	C₄×C₅	—	2	1	no	—
125	C₄×C₅	—	2	1	no	—
126	C₄×C₅	—	4	2	no	—
127	C₄×C₅	—	4	2	no	—
128	C₄×C₅	—	4	2	no	—
129	C₄×C₅	C₂×Q₁₆	1	2	yes	—
130	C₄×C₅	C₂×Q₁₆	1	2	yes	—
131	C₄×C₅	C₂×#41	1	2	yes	—
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	C₂³×C₅	1	2	yes	—
136	C₂×Q₈	C₂³×C₅	1	2	yes	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
142	Q₁₆	—	2	8	no	—
143	C₂×C₈	C₂³×C₅	1	4	yes	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	C₂³×C₅	1	2	yes	—
153	C₄⋊C₄	C₂³×C₅	1	2	yes	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	C₅×D₄	1	1	yes	characteristic
159	C₄²	C₅×D₄	1	1	yes	characteristic
160	C₄²	C₂³×C₅	1	1	yes	characteristic
161	C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₅	—	2	2	no	—
163	C₂×C₅	#525	1	1	yes	characteristic
164	C₂×C₅	C₂²×Q₁₆	1	1	yes	characteristic
165	C₂×C₅	#575	1	1	yes	characteristic
166	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	2	no	—
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	Q₈	C₅×#41	1	4	yes	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₅×Q₁₆	1	4	yes	complemented
181	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
187	C₈	—	2	4	no	—
188	C₅	#3104	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂×C₅×Q₁₆	1	2	yes	—
192	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₂×C₅×Q₁₆	1	2	yes	—
201	C₄	C₂×C₅×Q₁₆	1	2	yes	—
202	C₄	C₂×C₅×#41	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₅×#525	1	1	yes	characteristic
205	C₂	C₂²×C₅×Q₁₆	1	1	yes	characteristic
206	C₂	C₅×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×D₄
prime radical	C₂³×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#477	C₂	semidirect
5. C₅×#462	C₂	semidirect
7. C₄×C₅×Q₁₆	C₂	semidirect
33. #3104	C₅	direct (N × H)
67. #477	C₂×C₅	semidirect
68. #462	C₂×C₅	semidirect
70. C₄×Q₁₆	C₂×C₅	semidirect
86. C₂×C₄×C₅	Q₁₆	semidirect
180. C₂×C₄	C₅×Q₁₆	semidirect
188. C₅	#3104	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#533	2
C₂×C₄×C₅×Q₈	2
C₅×#484	2
C₅×#477	2
C₅×#462	2
C₅×#451	2
C₄×C₅×Q₁₆	2
C₅×#431	2
#3104	5

Subgroups — C5×#3104 #10616699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3104 #10616699