Subgroups — C₅₉×#33410 #28445519

Order 15104.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2913	C₂	1	2	yes	maximal
3	C₅₉×#2159	C₂	1	6	yes	maximal, complemented
4	C₅₉×#2158	C₂	1	3	yes	maximal, complemented
5	C₂×C₅₉×#366	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
7	C₅₉×#520	—	2	2	no	—
8	C₅₉×#504	C₄	1	8	yes	complemented
9	C₂×M₃₂×C₅₉	C₂²	1	3	yes	—
10	C₂²×C₁₆×C₅₉	C₂²	1	3	yes	—
11	C₅₉×#428	—	2	6	no	—
12	C₅₉×#427	C₂²	1	6	yes	—
13	C₂×#107×C₅₉	C₂²	1	6	yes	—
14	C₅₉×#367	C₂²	1	6	yes	—
15	C₅₉×#366	C₂²	1	3	yes	—
16	C₅₉×#366	C₂²	1	6	yes	—
17	C₅₉×#366	C₂²	1	1	yes	characteristic
18	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
19	#131×C₅₉	C₂×C₄	1	4	yes	—
20	#131×C₅₉	Q₈	1	4	yes	—
21	C₂×M₁₆×C₅₉	D₄	1	3	yes	—
22	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
23	C₄×Q₈×C₅₉	—	4	2	no	—
24	C₄×D₄×C₅₉	—	4	6	no	—
25	#117×C₅₉	—	2	6	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	6	no	—
27	M₃₂×C₅₉	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₆×C₅₉	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	C₂³	1	3	yes	—
31	#107×C₅₉	—	2	6	no	—
32	#107×C₅₉	C₂³	1	6	yes	—
33	#107×C₅₉	C₂³	1	2	yes	—
34	#106×C₅₉	—	2	6	no	—
35	#41×C₅₉	C₄⋊C₄	1	4	yes	—
36	#41×C₅₉	Q₁₆	1	4	yes	—
37	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
38	C₂×D₄×C₅₉	D₈	1	3	yes	—
39	C₂²×C₄×C₅₉	—	4	6	no	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
41	M₁₆×C₅₉	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₅₉	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₅₉	—	2	2	no	—
47	C₄⋊C₄×C₅₉	—	2	6	no	—
48	C₄⋊C₄×C₅₉	—	4	6	no	—
49	C₂²⋊C₄×C₅₉	—	4	6	no	—
50	C₄²×C₅₉	—	4	6	no	—
51	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
52	C₁₆×C₅₉	C₂²×C₄	1	6	yes	—
53	C₂³×C₅₉	C₂×D₈	1	3	yes	—
54	Q₈×C₅₉	#107	1	4	yes	—
55	D₄×C₅₉	#107	1	12	yes	—
56	C₂×C₄×C₅₉	—	4	2	no	—
57	C₂×C₄×C₅₉	—	4	6	no	—
58	C₂×C₄×C₅₉	—	16	6	no	—
59	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	6	yes	—
60	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
61	C₂×C₄×C₅₉	C₂×Q₁₆	1	6	yes	—
62	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
63	C₈×C₅₉	C₂×C₄⋊C₄	1	6	yes	—
64	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
66	#33410	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₉	—	2	3	no	—
68	C₂²×C₅₉	—	2	3	no	—
69	C₂²×C₅₉	C₂×#107	1	6	yes	—
70	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
71	C₄×C₅₉	—	8	6	no	—
72	C₄×C₅₉	—	8	2	no	—
73	C₄×C₅₉	C₂×#107	1	6	yes	—
74	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×#504	C₂×C₅₉	1	1	yes	characteristic
77	#2913	C₂×C₅₉	1	2	yes	—
78	#2159	C₂×C₅₉	1	6	yes	complemented
79	#2158	C₂×C₅₉	1	3	yes	complemented
80	C₂×#366	C₂×C₅₉	1	3	yes	complemented
81	C₂×C₅₉	—	2	6	no	—
82	C₂×C₅₉	C₂²×#107	1	1	yes	characteristic
83	C₂×C₅₉	#3415	1	1	yes	characteristic
84	C₂×C₅₉	#3417	1	1	yes	characteristic
85	C₂×#131	C₂²×C₅₉	1	1	yes	characteristic
86	#520	—	2	2	no	—
87	#504	C₄×C₅₉	1	8	yes	complemented
88	C₂×M₃₂	C₂²×C₅₉	1	3	yes	—
89	C₂²×C₁₆	C₂²×C₅₉	1	3	yes	—
90	#428	—	2	6	no	—
91	#427	C₂²×C₅₉	1	6	yes	—
92	C₂×#107	C₂²×C₅₉	1	6	yes	—
93	#367	C₂²×C₅₉	1	6	yes	—
94	#366	C₂²×C₅₉	1	3	yes	—
95	#366	C₂²×C₅₉	1	6	yes	—
96	#366	C₂²×C₅₉	1	1	yes	characteristic
97	C₅₉	#33410	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#41	D₄×C₅₉	1	1	yes	characteristic
99	#131	C₂×C₄×C₅₉	1	4	yes	—
100	#131	Q₈×C₅₉	1	4	yes	—
101	C₂×M₁₆	D₄×C₅₉	1	3	yes	—
102	C₂²×C₈	C₂³×C₅₉	1	3	yes	—
103	C₄×Q₈	—	4	2	no	—
104	C₄×D₄	—	4	6	no	—
105	#117	—	2	6	no	—
106	C₂×C₄⋊C₄	—	2	6	no	—
107	M₃₂	C₂×C₄×C₅₉	1	12	yes	complemented
108	C₂×C₁₆	C₂×C₄×C₅₉	1	12	yes	complemented
109	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
110	C₂×C₁₆	C₂³×C₅₉	1	3	yes	—
111	#107	—	2	6	no	—
112	#107	C₂³×C₅₉	1	6	yes	—
113	#107	C₂³×C₅₉	1	2	yes	—
114	#106	—	2	6	no	—
115	#41	C₄⋊C₄×C₅₉	1	4	yes	—
116	#41	Q₁₆×C₅₉	1	4	yes	—
117	C₂×Q₈	D₈×C₅₉	1	1	yes	characteristic
118	C₂×D₄	D₈×C₅₉	1	3	yes	—
119	C₂²×C₄	—	4	6	no	—
120	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
121	M₁₆	C₄⋊C₄×C₅₉	1	12	yes	—
122	C₂×C₈	C₂²×C₄×C₅₉	1	6	yes	—
123	C₂×C₈	C₂×D₄×C₅₉	1	3	yes	—
124	C₂×C₈	C₂×Q₈×C₅₉	1	6	yes	—
125	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	6	no	—
128	C₄⋊C₄	—	4	6	no	—
129	C₂²⋊C₄	—	4	6	no	—
130	C₄²	—	4	6	no	—
131	C₁₆	C₂²×C₄×C₅₉	1	2	yes	—
132	C₁₆	C₂²×C₄×C₅₉	1	6	yes	—
133	C₂³	C₂×D₈×C₅₉	1	3	yes	—
134	Q₈	#107×C₅₉	1	4	yes	—
135	D₄	#107×C₅₉	1	12	yes	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	16	6	no	—
139	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	6	yes	—
140	C₂×C₄	C₂×D₈×C₅₉	1	3	yes	—
141	C₂×C₄	C₂×Q₁₆×C₅₉	1	6	yes	—
142	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
143	C₈	C₂×C₄⋊C₄×C₅₉	1	6	yes	—
144	C₈	C₂³×C₄×C₅₉	1	1	yes	characteristic
145	C₈	C₂²×Q₈×C₅₉	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×#107×C₅₉	1	6	yes	—
149	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
150	C₄	—	8	6	no	—
151	C₄	—	8	2	no	—
152	C₄	C₂×#107×C₅₉	1	6	yes	—
153	C₄	C₂²×C₄⋊C₄×C₅₉	1	1	yes	characteristic
154	C₄	C₂²×Q₁₆×C₅₉	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×#107×C₅₉	1	1	yes	characteristic
157	C₂	C₅₉×#3415	1	1	yes	characteristic
158	C₂	C₅₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂×#41×C₅₉	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×#41×C₅₉ ◁ C₂×#131×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₉×#2159	C₂	semidirect
4. C₅₉×#2158	C₂	semidirect
5. C₂×C₅₉×#366	C₂	semidirect
8. C₅₉×#504	C₄	semidirect
27. M₃₂×C₅₉	C₂×C₄	semidirect
28. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
66. #33410	C₅₉	direct (N × H)
78. #2159	C₂×C₅₉	semidirect
79. #2158	C₂×C₅₉	semidirect
80. C₂×#366	C₂×C₅₉	semidirect
87. #504	C₄×C₅₉	semidirect
97. C₅₉	#33410	direct (N × H)
107. M₃₂	C₂×C₄×C₅₉	semidirect
108. C₂×C₁₆	C₂×C₄×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#504	2
C₅₉×#2913	2
C₅₉×#2159	2
C₅₉×#2158	2
C₂×C₅₉×#366	2
#33410	59

Subgroups — C59×#33410 #28445519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33410 #28445519