Subgroups — C₂×C₅×#3951 #13102367

Order 1600.

288 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#681	C₂	1	2	yes	maximal
3	C₅×#3951	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
5	C₂×C₅²×C₈	C₄	1	2	yes	—
6	C₂×C₅²×C₈	C₄	1	2	yes	—
7	C₂×C₅²×C₈	C₂²	1	2	yes	—
8	C₅×#681	—	2	2	no	—
9	C₅×#681	C₂²	1	4	yes	—
10	C₂×C₅×#168	—	2	2	no	—
11	C₂×C₅×#108	—	5	1	no	maximal
12	C₂×#3951	C₅	1	1	yes	maximal, characteristic, direct product factor
13	C₂³×C₅²	Q₈	1	1	yes	characteristic
14	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	Q₈	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₅²×C₈	C₂×C₄	1	4	yes	—
21	C₅×#168	—	2	4	no	—
22	C₂×C₅×M₁₆	—	5	2	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	D₅	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₅	1	1	yes	characteristic
26	C₂×#681	C₂×C₅	1	2	yes	—
27	C₅×#108	—	5	4	no	—
28	#3951	C₂×C₅	1	4	yes	direct product factor
29	C₂²×C₅²	—	2	2	no	—
30	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅	—	2	2	no	—
38	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
40	C₅×M₁₆	—	5	4	no	—
41	C₅×M₁₆	—	10	2	no	—
42	C₂×C₅×C₈	—	2	4	no	—
43	C₂×C₅×C₈	—	2	4	no	—
44	C₂×C₅×C₈	—	2	4	no	—
45	C₂×C₅×C₈	—	10	2	no	—
46	C₂×C₅×C₈	Dic₅	1	2	yes	—
47	C₂×C₅×C₈	Dic₅	1	2	yes	—
48	C₂×C₅×C₈	C₄×C₅	1	2	yes	—
49	C₂×C₅×C₈	C₄×C₅	1	2	yes	—
50	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
52	#681	—	2	2	no	—
53	#681	C₂²×C₅	1	4	yes	—
54	C₂×#168	—	2	2	no	—
55	C₂×#108	—	5	1	no	Sylow
56	C₂×C₅²	—	2	1	no	—
57	C₂×C₅²	—	2	1	no	—
58	C₂×C₅²	#108	1	2	yes	complemented
59	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₅	—	2	2	no	—
61	C₂³×C₅	Dic₁₀	1	1	yes	characteristic
62	C₂³×C₅	C₅×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	Dic₁₀	1	1	yes	characteristic
70	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
71	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
76	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₅×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
81	C₅×C₈	—	2	8	no	—
82	C₅×C₈	—	10	4	no	—
83	C₅×C₈	C₂×Dic₅	1	4	yes	—
84	C₅×C₈	C₂×C₄×C₅	1	4	yes	—
85	#168	—	2	4	no	—
86	C₂×M₁₆	—	5	2	no	—
87	C₂²×C₈	C₅×D₅	1	1	yes	characteristic
88	#108	—	5	4	no	—
89	C₅²	C₂×#108	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₅	—	2	8	no	—
91	C₂²×C₅	—	2	2	no	—
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	—	2	2	no	—
95	C₂²×C₅	—	2	2	no	—
96	C₂²×C₅	C₄⋊Dic₅	1	1	yes	characteristic
97	C₂²×C₅	C₄⋊Dic₅	1	1	yes	characteristic
98	C₂²×C₅	C₅×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₅	C₅×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
101	C₂²×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
102	C₄×C₅	—	2	2	no	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	—	2	2	no	—
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	C₄⋊Dic₅	1	1	yes	characteristic
107	C₄×C₅	C₄⋊Dic₅	1	1	yes	characteristic
108	C₄×C₅	C₅×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₅	C₅×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
111	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
112	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
113	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
115	M₁₆	—	5	4	no	—
116	M₁₆	—	10	2	no	—
117	C₂×C₈	—	10	2	no	—
118	C₂×C₈	C₅×Dic₅	1	2	yes	—
119	C₂×C₈	C₅×Dic₅	1	2	yes	—
120	C₂×C₈	C₂×C₅×D₅	1	2	yes	—
121	C₂×C₅	—	2	4	no	—
122	C₂×C₅	—	2	4	no	—
123	C₂×C₅	—	2	1	no	—
124	C₂×C₅	—	2	1	no	—
125	C₂×C₅	—	2	1	no	—
126	C₂×C₅	—	2	1	no	—
127	C₂×C₅	—	2	4	no	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	#3951	1	2	yes	direct product factor
130	C₂×C₅	C₅×#108	1	2	yes	complemented
131	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
132	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
133	C₂³	C₅×Dic₁₀	1	1	yes	characteristic
134	C₂×C₄	C₅×Dic₁₀	1	1	yes	characteristic
135	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
136	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
137	C₂×C₄	C₂×C₅×Dic₅	1	1	yes	characteristic
138	C₂×C₄	C₂×C₅×Dic₅	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
140	C₈	—	10	4	no	—
141	C₈	C₂×C₅×Dic₅	1	4	yes	—
142	C₅	—	2	2	no	—
143	C₅	C₂×#3951	1	1	yes	characteristic, direct product factor
144	C₅	C₂×C₅×#108	1	1	yes	characteristic, complemented
145	C₂²	—	2	2	no	—
146	C₂²	C₅×C₄⋊Dic₅	1	1	yes	characteristic
147	C₂²	C₅×C₄⋊Dic₅	1	1	yes	characteristic
148	C₂²	C₂×C₅×Dic₁₀	1	1	yes	characteristic
149	C₄	C₅×C₄⋊Dic₅	1	1	yes	characteristic
150	C₄	C₅×C₄⋊Dic₅	1	1	yes	characteristic
151	C₄	C₂×C₅×D₂₀	1	1	yes	characteristic
152	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	C₅×#3951	1	2	yes	direct product factor
156	C₂	C₂×C₅×C₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₂₀
commutator	C₄×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂²×C₅²×C₈	C₂
socle	C₂²×C₅²	C₄⋊C₄
prime radical	C₂³×C₅²	Q₈
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂²×C₅×C₈	D₅
hypocenter	C₅	C₂×C₅×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈

Complements

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

N	H ≅ G/N	type
3. C₅×#3951	C₂	direct (N × H)
12. C₂×#3951	C₅	direct (N × H)
28. #3951	C₂×C₅	direct (N × H)
58. C₂×C₅²	#108	semidirect
89. C₅²	C₂×#108	semidirect
129. C₂×C₅	#3951	direct (N × H)
130. C₂×C₅	C₅×#108	semidirect
143. C₅	C₂×#3951	direct (N × H)
144. C₅	C₂×C₅×#108	semidirect
155. C₂	C₅×#3951	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×C₅²×C₈	2
C₂×C₅×#681	2
C₅×#3951	2
C₂×C₅×#108	5
C₂×#3951	5

Subgroups — C2×C5×#3951 #13102367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3951 #13102367