Subgroups — C₄×#16837178 #28503511

Order 15104.

510 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837178	C₂	1	4	yes	maximal
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	2	yes	maximal
3	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
5	#16837178	C₄	1	16	yes	direct product factor
6	C₄×C₅₉⋊C₁₆	C₄	1	8	yes	complemented
7	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₄³×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₅₉⋊C₁₆	C₂×C₄	1	8	yes	—
13	C₂×C₅₉⋊C₁₆	C₂×C₄	1	16	yes	—
14	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	C₈	1	4	yes	—
18	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₉	C₈	1	8	yes	—
21	C₄×C₈×C₅₉	C₈	1	4	yes	—
22	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₅₉	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₄²	1	2	yes	—
28	C₂×C₈×C₅₉	C₄²	1	4	yes	—
29	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
30	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
34	C₄²×C₅₉	—	2	8	no	—
35	C₄²×C₅₉	C₄²	1	4	yes	—
36	C₄²×C₅₉	C₄²	1	2	yes	—
37	C₄²×C₅₉	C₂×C₈	1	4	yes	—
38	C₄²×C₅₉	C₂×C₈	1	2	yes	—
39	C₅₉⋊C₁₆	C₄²	1	16	yes	complemented
40	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	—	2	8	no	—
42	C₂×C₄×C₅₉	C₄×C₈	1	8	yes	—
43	C₂×C₄×C₅₉	C₄×C₈	1	4	yes	—
44	C₂×C₄×C₅₉	M₃₂	1	8	yes	—
45	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
47	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
49	C₈×C₅₉	C₄×C₈	1	4	yes	—
50	C₈×C₅₉	C₄×C₈	1	8	yes	—
51	C₈×C₅₉	C₂×C₄²	1	4	yes	—
52	C₄×#346	—	59	1	no	maximal, Sylow
53	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂²×C₅₉	C₂×C₄×C₈	1	2	yes	—
55	C₂²×C₅₉	C₂×M₃₂	1	4	yes	—
56	C₄×C₅₉	—	2	8	no	—
57	C₄×C₅₉	#346	1	8	yes	complemented
58	C₄×C₅₉	C₄³	1	1	yes	characteristic
59	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₅₉	C₂×C₄×C₈	1	2	yes	—
61	C₂×#346	—	59	4	no	—
62	C₂×C₄×C₁₆	—	59	2	no	—
63	C₄²×C₈	D₅₉	1	1	yes	characteristic
64	C₂×C₅₉	C₄²×C₈	1	1	yes	characteristic
65	C₂×C₅₉	C₂×#346	1	2	yes	—
66	C₂×C₅₉	C₄×M₃₂	1	4	yes	—
67	C₂²×C₁₆	—	59	4	no	—
68	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
69	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
71	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
72	C₄³	Dic₅₉	1	1	yes	characteristic
73	#346	—	59	16	no	—
74	C₄×C₁₆	—	59	8	no	—
75	C₅₉	C₄×#346	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
77	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
79	C₂×C₄²	C₅₉⋊C₈	1	4	yes	—
80	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
81	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
82	C₂×C₁₆	—	59	8	no	—
83	C₂×C₁₆	—	59	16	no	—
84	C₄×C₈	C₅₉⋊C₈	1	8	yes	—
85	C₄×C₈	C₅₉⋊C₈	1	4	yes	—
86	C₄×C₈	C₄×D₅₉	1	4	yes	—
87	C₄×C₈	C₄×D₅₉	1	8	yes	—
88	C₂²×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
90	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
91	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
92	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
93	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
94	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
95	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
96	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
97	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	C₂×C₅₉⋊C₈	1	4	yes	—
100	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
101	C₄²	C₄×Dic₅₉	1	4	yes	—
102	C₄²	C₄×Dic₅₉	1	2	yes	—
103	C₁₆	—	59	16	no	—
104	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
105	C₂×C₄	—	2	8	no	—
106	C₂×C₄	C₄×C₅₉⋊C₈	1	8	yes	—
107	C₂×C₄	C₄×C₅₉⋊C₈	1	4	yes	—
108	C₂×C₄	#14273667	1	8	yes	—
109	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
111	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
113	C₈	C₄×C₅₉⋊C₈	1	8	yes	—
114	C₈	C₄×C₅₉⋊C₈	1	4	yes	—
115	C₈	C₄²×D₅₉	1	4	yes	—
116	C₂²	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
117	C₂²	C₂×C₄×C₅₉⋊C₈	1	2	yes	—
118	C₂²	C₂×#14273667	1	4	yes	—
119	C₄	—	2	8	no	—
120	C₄	#16837178	1	8	yes	direct product factor
121	C₄	C₄²×Dic₅₉	1	1	yes	characteristic
122	C₄	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
123	C₄	C₂×C₄×C₅₉⋊C₈	1	2	yes	—
124	C₂	C₄²×C₅₉⋊C₈	1	1	yes	characteristic
125	C₂	C₂×#16837178	1	2	yes	—
126	C₂	C₄×#14273667	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₉
commutator	C₂×C₅₉	C₄²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	C₄×#346
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
5. #16837178	C₄	direct (N × H)
6. C₄×C₅₉⋊C₁₆	C₄	semidirect
39. C₅₉⋊C₁₆	C₄²	semidirect
57. C₄×C₅₉	#346	semidirect
75. C₅₉	C₄×#346	semidirect
120. C₄	#16837178	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×#16837178	2
C₂×C₄×C₅₉⋊C₁₆	2
C₄²×C₈×C₅₉	2
C₄×#346	59

Subgroups — C4×#16837178 #28503511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837178 #28503511