Subgroups — #368×Dic₅₉ #28504103

Order 15104.

314 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837232	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₅₉	C₂	1	2	yes	maximal
4	#21422189	C₂	1	2	yes	maximal
5	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	M₁₆×Dic₅₉	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837232	C₄	1	4	yes	complemented
11	C₁₆×Dic₅₉	C₄	1	4	yes	—
12	#16837166	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#368	C₄	1	4	yes	complemented
16	C₂×#11745664	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	#14273663	—	2	2	no	—
23	#14273663	C₂×C₄	1	4	yes	—
24	#14273637	—	4	2	no	—
25	C₈×Dic₅₉	—	4	2	no	—
26	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
27	C₈×Dic₅₉	Q₈	1	2	yes	—
28	C₂×M₁₆×C₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	#108×C₅₉	—	2	2	no	—
34	#108×C₅₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₉	Q₁₆	1	2	yes	—
37	M₁₆×C₅₉	—	2	4	no	—
38	M₁₆×C₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
46	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₅₉	D₈	1	1	yes	characteristic
48	C₄×Dic₅₉	D₈	1	1	yes	characteristic
49	#11745664	—	2	4	no	—
50	#11745664	—	4	2	no	—
51	C₂×C₅₉⋊C₈	—	2	1	no	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₁₆×C₅₉	C₄²	1	4	yes	—
56	C₅₉⋊C₁₆	—	2	2	no	—
57	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
64	C₂×Dic₅₉	—	2	4	no	—
65	C₂×Dic₅₉	#107	1	2	yes	—
66	C₂×Dic₅₉	#107	1	2	yes	—
67	C₈×C₅₉	—	4	4	no	—
68	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₅₉	#117	1	1	yes	characteristic
70	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
72	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	—	4	4	no	—
74	C₄×#368	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	2	no	—
76	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
77	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅₉	#427	1	1	yes	characteristic
81	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
82	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
83	Dic₅₉	—	2	2	no	—
84	Dic₅₉	#368	1	4	yes	direct product factor
85	C₂×#368	—	59	1	no	—
86	C₂×#368	D₅₉	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	59	1	no	—
88	C₄×#108	—	59	2	no	—
89	#1381	—	59	2	no	—
90	C₂×C₅₉	—	2	1	no	—
91	C₂×C₅₉	—	2	1	no	—
92	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
93	C₂×C₅₉	C₂×#368	1	2	yes	—
94	C₂²×C₁₆	—	59	1	no	—
95	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
96	C₂×#108	—	59	2	no	—
97	C₂×#108	C₂×D₅₉	1	2	yes	—
98	C₄×M₁₆	—	118	2	no	—
99	C₂×C₄×C₈	—	59	1	no	—
100	#368	—	59	4	no	—
101	#368	Dic₅₉	1	4	yes	direct product factor
102	C₄×C₁₆	—	59	4	no	—
103	#341	—	118	2	no	—
104	C₅₉	C₄×#368	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	118	2	no	—
107	C₂²×C₈	—	59	1	no	—
108	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	C₂×C₄²	—	59	1	no	—
110	C₂×C₁₆	—	59	2	no	—
111	C₂×C₁₆	—	118	1	no	—
112	C₂×C₁₆	—	118	1	no	—
113	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
114	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	—	59	4	no	—
118	#108	—	118	2	no	—
119	#108	C₂×Dic₅₉	1	4	yes	—
120	#97	—	236	2	no	—
121	C₄×C₈	—	59	2	no	—
122	C₄×C₈	—	59	2	no	—
123	C₄×C₈	—	236	2	no	—
124	C₂²×C₄	—	59	2	no	—
125	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	118	4	no	—
129	M₁₆	—	236	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	59	2	no	—
132	C₂×C₈	—	118	1	no	—
133	C₂×C₈	—	118	1	no	—
134	C₂×C₈	—	236	2	no	—
135	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
137	C₂×C₈	#11745687	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
139	C₂×C₈	#11745689	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
141	C₄²	—	59	2	no	—
142	C₄²	—	59	1	no	—
143	C₄²	—	59	1	no	—
144	C₁₆	—	118	2	no	—
145	C₁₆	C₄×Dic₅₉	1	4	yes	—
146	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
147	C₂×C₄	—	59	2	no	—
148	C₂×C₄	—	59	2	no	—
149	C₂×C₄	—	118	4	no	—
150	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
151	C₂×C₄	#14273689	1	1	yes	characteristic
152	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
153	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
154	C₂×C₄	#14273746	1	1	yes	characteristic
155	C₂×C₄	#14273750	1	1	yes	characteristic
156	C₈	—	4	4	no	—
157	C₈	—	118	2	no	—
158	C₈	—	236	4	no	—
159	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
160	C₈	#14273685	1	1	yes	characteristic
161	C₈	#14273692	1	1	yes	characteristic
162	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#107×D₅₉	1	1	yes	characteristic
165	C₂²	#16837496	1	1	yes	characteristic
166	C₂²	Q₁₆×Dic₅₉	1	1	yes	characteristic
167	C₄	—	59	4	no	—
168	C₄	—	118	2	no	—
169	C₄	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
170	C₄	#16837439	1	1	yes	characteristic
171	C₄	#16837484	1	1	yes	characteristic
172	C₄	D₈×Dic₅₉	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#107×Dic₅₉	1	1	yes	characteristic
176	C₂	#368×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	C₄×#368
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
10. #16837232	C₄	semidirect
15. C₅₉×#368	C₄	semidirect
84. Dic₅₉	#368	direct (N × H)
101. #368	Dic₅₉	direct (N × H)
104. C₅₉	C₄×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837232	2
C₂×C₁₆×Dic₅₉	2
#108×Dic₅₉	2
#21422189	2
C₂×C₅₉×#368	2
C₄×#368	59

Subgroups — #368×Dic59 #28504103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #368×Dic₅₉ #28504103