Subgroups — C₂³×#3951 #12962989

Order 1280.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×#681	C₂	1	2	yes	maximal
3	C₂²×#3951	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₄	1	14	yes	—
6	C₂³×C₅×C₈	C₄	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	14	yes	—
8	C₂²×#681	—	2	14	no	—
9	C₂²×#681	C₂²	1	28	yes	—
10	C₂³×#168	—	2	2	no	—
11	C₂×#3951	C₂²	1	112	yes	direct product factor
12	C₂³×#108	—	5	1	no	maximal, Sylow
13	C₂⁵×C₅	Q₈	1	1	yes	characteristic
14	C₂³×C₄×C₅	C₂×C₄	1	7	yes	—
15	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	D₄	1	7	yes	—
17	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₅	Q₈	1	7	yes	—
19	C₂³×C₄×C₅	C₂³	1	7	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	84	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	28	yes	—
22	C₂²×C₅×C₈	C₂³	1	28	yes	—
23	C₂×#681	—	2	84	no	—
24	C₂×#681	C₂³	1	56	yes	—
25	C₂²×#168	—	2	28	no	—
26	#3951	C₂³	1	64	yes	direct product factor
27	C₂³×M₁₆	—	5	2	no	—
28	C₂⁴×C₈	D₅	1	1	yes	characteristic
29	C₂²×#108	—	5	28	no	—
30	C₂⁴×C₅	—	2	8	no	—
31	C₂⁴×C₅	C₄⋊C₄	1	7	yes	—
32	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂⁴×C₅	C₂×Q₈	1	7	yes	—
34	C₂²×C₄×C₅	C₄⋊C₄	1	42	yes	—
35	C₂²×C₄×C₅	C₄⋊C₄	1	7	yes	—
36	C₂²×C₄×C₅	C₄⋊C₄	1	7	yes	—
37	C₂²×C₄×C₅	C₂²×C₄	1	21	yes	—
38	C₂²×C₄×C₅	C₂²×C₄	1	7	yes	—
39	C₂²×C₄×C₅	C₂×D₄	1	7	yes	—
40	C₂²×C₄×C₅	C₂×D₄	1	21	yes	—
41	C₂²×C₄×C₅	C₂×Q₈	1	21	yes	—
42	C₂²×C₄×C₅	C₂⁴	1	7	yes	—
43	C₂×C₅×C₈	C₂²×C₄	1	56	yes	—
44	C₂×C₅×C₈	C₂²×C₄	1	56	yes	—
45	C₂×C₅×C₈	C₂⁴	1	8	yes	—
46	#681	—	2	56	no	—
47	#681	C₂⁴	1	16	yes	—
48	C₂×#168	—	2	56	no	—
49	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
50	C₂²×M₁₆	—	5	28	no	—
51	C₂²×M₁₆	—	10	14	no	—
52	C₂³×C₈	—	10	2	no	—
53	C₂³×C₈	Dic₅	1	2	yes	—
54	C₂³×C₈	Dic₅	1	14	yes	—
55	C₂³×C₈	C₂×D₅	1	14	yes	—
56	C₂×#108	—	5	112	no	—
57	C₂³×C₅	—	2	28	no	—
58	C₂³×C₅	—	2	28	no	—
59	C₂³×C₅	#108	1	8	yes	complemented
60	C₂³×C₅	C₂×C₄⋊C₄	1	21	yes	—
61	C₂³×C₅	C₂×C₄⋊C₄	1	7	yes	—
62	C₂³×C₅	C₂²×Q₈	1	7	yes	—
63	C₂×C₄×C₅	C₂×C₄⋊C₄	1	42	yes	—
64	C₂×C₄×C₅	C₂×C₄⋊C₄	1	21	yes	—
65	C₂×C₄×C₅	C₂×C₄⋊C₄	1	21	yes	—
66	C₂×C₄×C₅	C₂³×C₄	1	7	yes	—
67	C₂×C₄×C₅	C₂³×C₄	1	7	yes	—
68	C₂×C₄×C₅	C₂²×D₄	1	7	yes	—
69	C₂×C₄×C₅	C₂²×D₄	1	7	yes	—
70	C₂×C₄×C₅	C₂²×Q₈	1	7	yes	—
71	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
72	C₅×C₈	C₂³×C₄	1	16	yes	—
73	#168	—	2	16	no	—
74	C₂⁵	Dic₁₀	1	1	yes	characteristic
75	C₂³×C₄	Dic₁₀	1	7	yes	—
76	C₂³×C₄	D₂₀	1	7	yes	—
77	C₂³×C₄	D₂₀	1	1	yes	characteristic
78	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
79	C₂³×C₄	C₂×Dic₅	1	7	yes	—
80	C₂³×C₄	C₂²×D₅	1	7	yes	—
81	C₂×M₁₆	—	5	56	no	—
82	C₂×M₁₆	—	10	84	no	—
83	C₂²×C₈	—	10	28	no	—
84	C₂²×C₈	C₂×Dic₅	1	84	yes	—
85	C₂²×C₈	C₂×Dic₅	1	28	yes	—
86	C₂²×C₈	C₂²×D₅	1	28	yes	—
87	#108	—	5	64	no	—
88	C₂²×C₅	—	2	42	no	—
89	C₂²×C₅	—	2	14	no	—
90	C₂²×C₅	C₂×#108	1	28	yes	complemented
91	C₂²×C₅	C₂²×C₄⋊C₄	1	7	yes	—
92	C₂²×C₅	C₂²×C₄⋊C₄	1	7	yes	—
93	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
94	C₄×C₅	C₂²×C₄⋊C₄	1	7	yes	—
95	C₄×C₅	C₂²×C₄⋊C₄	1	7	yes	—
96	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
97	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
98	C₂⁴	—	2	8	no	—
99	C₂⁴	C₄⋊Dic₅	1	7	yes	—
100	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
101	C₂⁴	C₂×Dic₁₀	1	7	yes	—
102	C₂²×C₄	C₄⋊Dic₅	1	42	yes	—
103	C₂²×C₄	C₄⋊Dic₅	1	7	yes	—
104	C₂²×C₄	C₄⋊Dic₅	1	7	yes	—
105	C₂²×C₄	C₂×Dic₁₀	1	21	yes	—
106	C₂²×C₄	C₂×D₂₀	1	7	yes	—
107	C₂²×C₄	C₂×D₂₀	1	21	yes	—
108	C₂²×C₄	C₂²×Dic₅	1	21	yes	—
109	C₂²×C₄	C₂²×Dic₅	1	7	yes	—
110	C₂²×C₄	C₂³×D₅	1	7	yes	—
111	M₁₆	—	5	16	no	—
112	M₁₆	—	10	56	no	—
113	C₂×C₈	—	10	56	no	—
114	C₂×C₈	C₂²×Dic₅	1	56	yes	—
115	C₂×C₈	C₂²×Dic₅	1	56	yes	—
116	C₂×C₈	C₂³×D₅	1	8	yes	—
117	C₂×C₅	—	2	7	no	—
118	C₂×C₅	—	2	1	no	—
119	C₂×C₅	C₂²×#108	1	14	yes	complemented
120	C₂×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic
121	C₂³	—	2	28	no	—
122	C₂³	—	2	28	no	—
123	C₂³	#3951	1	8	yes	direct product factor
124	C₂³	C₂×C₄⋊Dic₅	1	21	yes	—
125	C₂³	C₂×C₄⋊Dic₅	1	7	yes	—
126	C₂³	C₂²×Dic₁₀	1	7	yes	—
127	C₂×C₄	C₂×C₄⋊Dic₅	1	42	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₅	1	21	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₅	1	21	yes	—
130	C₂×C₄	C₂²×Dic₁₀	1	7	yes	—
131	C₂×C₄	C₂²×D₂₀	1	7	yes	—
132	C₂×C₄	C₂²×D₂₀	1	7	yes	—
133	C₂×C₄	C₂³×Dic₅	1	7	yes	—
134	C₂×C₄	C₂³×Dic₅	1	7	yes	—
135	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
136	C₈	—	10	16	no	—
137	C₈	C₂³×Dic₅	1	16	yes	—
138	C₅	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	2	14	no	—
140	C₂²	—	2	42	no	—
141	C₂²	C₂×#3951	1	28	yes	direct product factor
142	C₂²	C₂²×C₄⋊Dic₅	1	7	yes	—
143	C₂²	C₂²×C₄⋊Dic₅	1	7	yes	—
144	C₂²	C₂³×Dic₁₀	1	1	yes	characteristic
145	C₄	C₂²×C₄⋊Dic₅	1	7	yes	—
146	C₄	C₂²×C₄⋊Dic₅	1	7	yes	—
147	C₄	C₂³×D₂₀	1	1	yes	characteristic
148	C₄	C₂⁴×Dic₅	1	1	yes	characteristic
149	C₂	—	2	7	no	—
150	C₂	—	2	1	no	—
151	C₂	C₂²×#3951	1	14	yes	direct product factor
152	C₂	C₂³×C₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₀
commutator	C₄×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂⁴×C₅×C₈	C₂
socle	C₂⁴×C₅	C₄⋊C₄
prime radical	C₂⁵×C₅	Q₈
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂⁴×C₈	D₅
hypocenter	C₅	C₂³×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#3951	C₂	direct (N × H)
11. C₂×#3951	C₂²	direct (N × H)
26. #3951	C₂³	direct (N × H)
59. C₂³×C₅	#108	semidirect
90. C₂²×C₅	C₂×#108	semidirect
119. C₂×C₅	C₂²×#108	semidirect
123. C₂³	#3951	direct (N × H)
138. C₅	C₂³×#108	semidirect
141. C₂²	C₂×#3951	direct (N × H)
151. C₂	C₂²×#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
C₂⁴×C₅×C₈	2
C₂³×#681	2
C₂²×#3951	2
C₂³×#108	5

Subgroups — C23×#3951 #12962989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#3951 #12962989