Subgroups — C₂³×#3947 #12962682

Order 1280.

4126 conjugacy classes of subgroups.

The subgroup lattice has 138 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×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×#168	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3947	C₂	1	28	yes	maximal, direct product factor
5	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₅	C₄	1	2	yes	—
7	C₂³×C₅×C₈	C₄	1	16	yes	complemented
8	C₂³×C₅×C₈	C₂²	1	14	yes	—
9	C₂²×C₄×Dic₅	C₄	1	14	yes	—
10	C₂²×C₄×Dic₅	C₂²	1	14	yes	—
11	C₂³×#168	C₄	1	16	yes	complemented
12	C₂³×#168	C₂²	1	14	yes	—
13	C₂×#3947	C₂²	1	112	yes	direct product factor
14	C₂³×#97	—	5	1	no	maximal, Sylow
15	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
18	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₅	C₂³	1	7	yes	—
20	C₂³×Dic₅	—	2	16	no	—
21	C₂³×Dic₅	C₂×C₄	1	28	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	112	yes	complemented
23	C₂²×C₅×C₈	C₂³	1	28	yes	—
24	C₂×C₄×Dic₅	C₂×C₄	1	84	yes	—
25	C₂×C₄×Dic₅	C₂³	1	28	yes	—
26	C₂²×#168	C₂×C₄	1	112	yes	complemented
27	C₂²×#168	C₂³	1	28	yes	—
28	#3947	C₂³	1	64	yes	direct product factor
29	C₂⁴×C₈	—	5	1	no	—
30	C₂⁴×C₈	D₅	1	1	yes	characteristic
31	C₂³×C₄²	—	5	1	no	—
32	C₂²×#97	—	5	28	no	—
33	C₂⁴×C₅	C₄²	1	8	yes	—
34	C₂⁴×C₅	M₁₆	1	16	yes	—
35	C₂⁴×C₅	C₂²×C₄	1	7	yes	—
36	C₂²×C₄×C₅	C₄²	1	56	yes	—
37	C₂²×C₄×C₅	C₂²×C₄	1	21	yes	—
38	C₂²×C₄×C₅	C₂²×C₄	1	28	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	28	yes	—
40	C₂²×C₄×C₅	C₂⁴	1	7	yes	—
41	C₂²×Dic₅	—	2	112	no	—
42	C₂²×Dic₅	C₂²×C₄	1	56	yes	—
43	C₂×C₅×C₈	C₂²×C₄	1	112	yes	complemented
44	C₂×C₅×C₈	C₂⁴	1	8	yes	—
45	C₄×Dic₅	C₂²×C₄	1	56	yes	—
46	C₄×Dic₅	C₂⁴	1	8	yes	—
47	C₂×#168	C₂²×C₄	1	112	yes	complemented
48	C₂×#168	C₂⁴	1	8	yes	—
49	C₂⁴×C₄	—	5	2	no	—
50	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
51	C₂³×C₈	—	5	16	no	—
52	C₂³×C₈	—	5	14	no	—
53	C₂³×C₈	Dic₅	1	16	yes	complemented
54	C₂³×C₈	C₂×D₅	1	14	yes	—
55	C₂²×C₄²	—	5	14	no	—
56	C₂²×C₄²	—	5	14	no	—
57	C₂×#97	—	5	112	no	—
58	C₂³×C₅	#97	1	64	yes	complemented
59	C₂³×C₅	C₂×C₄²	1	28	yes	—
60	C₂³×C₅	C₂×M₁₆	1	56	yes	—
61	C₂³×C₅	C₂³×C₄	1	7	yes	—
62	C₂×C₄×C₅	C₂×C₄²	1	84	yes	—
63	C₂×C₄×C₅	C₂³×C₄	1	14	yes	—
64	C₂×C₄×C₅	C₂³×C₄	1	7	yes	—
65	C₂×C₄×C₅	C₂³×C₄	1	14	yes	—
66	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
67	C₂×Dic₅	—	2	112	no	—
68	C₂×Dic₅	C₂³×C₄	1	16	yes	—
69	C₅×C₈	C₂³×C₄	1	16	yes	complemented
70	#168	C₂³×C₄	1	16	yes	complemented
71	C₂⁵	C₄×D₅	1	1	yes	characteristic
72	C₂³×C₄	—	5	28	no	—
73	C₂³×C₄	—	10	16	no	—
74	C₂³×C₄	C₄×D₅	1	7	yes	—
75	C₂³×C₄	C₄×D₅	1	8	yes	—
76	C₂³×C₄	C₂×Dic₅	1	8	yes	—
77	C₂³×C₄	C₂²×D₅	1	7	yes	—
78	C₂²×C₈	—	5	112	no	—
79	C₂²×C₈	—	5	28	no	—
80	C₂²×C₈	C₂×Dic₅	1	112	yes	complemented
81	C₂²×C₈	C₂²×D₅	1	28	yes	—
82	C₂×C₄²	—	5	84	no	—
83	C₂×C₄²	—	5	28	no	—
84	#97	—	5	64	no	—
85	C₂²×C₅	C₂×#97	1	112	yes	complemented
86	C₂²×C₅	C₂²×C₄²	1	14	yes	—
87	C₂²×C₅	C₂²×M₁₆	1	28	yes	—
88	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
89	C₄×C₅	C₂²×C₄²	1	14	yes	—
90	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
91	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
92	Dic₅	—	2	16	no	—
93	C₂⁴	#676	1	16	yes	—
94	C₂⁴	C₄×Dic₅	1	8	yes	—
95	C₂⁴	C₂×C₄×D₅	1	7	yes	—
96	C₂²×C₄	—	5	56	no	—
97	C₂²×C₄	—	10	112	no	—
98	C₂²×C₄	C₄×Dic₅	1	56	yes	—
99	C₂²×C₄	C₂×C₄×D₅	1	28	yes	—
100	C₂²×C₄	C₂×C₄×D₅	1	21	yes	—
101	C₂²×C₄	C₂²×Dic₅	1	28	yes	—
102	C₂²×C₄	C₂³×D₅	1	7	yes	—
103	C₂×C₈	—	5	112	no	—
104	C₂×C₈	—	5	8	no	—
105	C₂×C₈	C₂²×Dic₅	1	112	yes	complemented
106	C₂×C₈	C₂³×D₅	1	8	yes	—
107	C₄²	—	5	56	no	—
108	C₄²	—	5	8	no	—
109	C₂×C₅	C₂²×#97	1	28	yes	complemented
110	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
111	C₂×C₅	C₂³×M₁₆	1	2	yes	—
112	C₂³	#3947	1	64	yes	direct product factor
113	C₂³	C₂×#676	1	56	yes	—
114	C₂³	C₂×C₄×Dic₅	1	28	yes	—
115	C₂³	C₂²×C₄×D₅	1	7	yes	—
116	C₂×C₄	—	5	16	no	—
117	C₂×C₄	—	10	112	no	—
118	C₂×C₄	C₂×C₄×Dic₅	1	84	yes	—
119	C₂×C₄	C₂²×C₄×D₅	1	7	yes	—
120	C₂×C₄	C₂²×C₄×D₅	1	14	yes	—
121	C₂×C₄	C₂³×Dic₅	1	14	yes	—
122	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
123	C₈	—	5	16	no	—
124	C₈	C₂³×Dic₅	1	16	yes	complemented
125	C₅	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
126	C₂²	C₂×#3947	1	112	yes	direct product factor
127	C₂²	C₂²×#676	1	28	yes	—
128	C₂²	C₂²×C₄×Dic₅	1	14	yes	—
129	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
130	C₄	—	10	16	no	—
131	C₄	C₂²×C₄×Dic₅	1	14	yes	—
132	C₄	C₂³×C₄×D₅	1	1	yes	characteristic
133	C₄	C₂⁴×Dic₅	1	1	yes	characteristic
134	C₂	C₂²×#3947	1	28	yes	direct product factor
135	C₂	C₂³×#676	1	2	yes	—
136	C₂	C₂³×C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅
commutator	C₂×C₅	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂⁴×C₅×C₈	C₂
socle, prime radical	C₂⁵×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂⁴×C₈	D₅
hypocenter	C₅	C₂³×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
4. C₂²×#3947	C₂	direct (N × H)
7. C₂³×C₅×C₈	C₄	semidirect
11. C₂³×#168	C₄	semidirect
13. C₂×#3947	C₂²	direct (N × H)
22. C₂²×C₅×C₈	C₂×C₄	semidirect
26. C₂²×#168	C₂×C₄	semidirect
28. #3947	C₂³	direct (N × H)
43. C₂×C₅×C₈	C₂²×C₄	semidirect
47. C₂×#168	C₂²×C₄	semidirect
53. C₂³×C₈	Dic₅	semidirect
58. C₂³×C₅	#97	semidirect
69. C₅×C₈	C₂³×C₄	semidirect
70. #168	C₂³×C₄	semidirect
80. C₂²×C₈	C₂×Dic₅	semidirect
85. C₂²×C₅	C₂×#97	semidirect
105. C₂×C₈	C₂²×Dic₅	semidirect
109. C₂×C₅	C₂²×#97	semidirect
112. C₂³	#3947	direct (N × H)
124. C₈	C₂³×Dic₅	semidirect
125. C₅	C₂³×#97	semidirect
126. C₂²	C₂×#3947	direct (N × H)
134. C₂	C₂²×#3947	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×C₄×Dic₅	2
C₂⁴×#168	2
C₂²×#3947	2
C₂³×#97	5

Subgroups — C23×#3947 #12962682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#3947 #12962682