Subgroups — #28503569

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837440	C₂	1	2	yes	maximal
2	#21422188	C₂	1	4	yes	maximal, complemented
3	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273686	—	2	2	no	—
5	#16837440	—	2	4	no	—
6	#16837440	C₄	1	8	yes	complemented
7	C₂×#14273661	C₂²	1	4	yes	—
8	C₂×#14273659	—	2	2	no	—
9	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	—	2	4	no	—
11	C₂²×C₁₆×C₅₉	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
13	#14273686	—	2	8	no	—
14	#14273686	—	4	4	no	—
15	C₂×C₄⋊Dic₅₉	—	2	4	no	—
16	C₂×C₂²⋊Dic₅₉	—	4	2	no	—
17	C₂×C₄×Dic₅₉	—	4	2	no	—
18	#14273661	—	2	4	no	—
19	#14273661	C₂×C₄	1	8	yes	—
20	#14273659	—	2	8	no	—
21	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	D₄	1	4	yes	—
26	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	—	2	8	no	—
29	C₂×C₁₆×C₅₉	—	2	4	no	—
30	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
31	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	4	4	no	—
39	C₂×C₈×C₅₉	C₄²	1	4	yes	—
40	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
44	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²⋊Dic₅₉	—	4	8	no	—
49	C₄⋊Dic₅₉	—	2	8	no	—
50	C₄⋊Dic₅₉	—	4	4	no	—
51	C₄×Dic₅₉	—	4	8	no	—
52	C₁₆×C₅₉	—	2	4	no	—
53	C₂³×C₅₉	—	2	4	no	—
54	C₂³×C₅₉	#102	1	4	yes	—
55	C₂³×C₅₉	#107	1	2	yes	—
56	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#95	1	4	yes	—
58	C₂×C₄×C₅₉	#102	1	4	yes	—
59	C₂×C₄×C₅₉	#103	1	8	yes	—
60	C₂×C₄×C₅₉	#106	1	4	yes	—
61	C₂×C₄×C₅₉	#107	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₅₉	—	4	8	no	—
67	C₂×Dic₅₉	—	8	8	no	—
68	C₈×C₅₉	#95	1	4	yes	—
69	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
72	#12556	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	—	2	8	no	—
74	C₂²×C₅₉	—	2	4	no	—
75	C₂²×C₅₉	#340	1	4	yes	—
76	C₂²×C₅₉	C₂×#102	1	2	yes	—
77	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₉	#508	1	4	yes	—
79	C₄×C₅₉	#340	1	4	yes	—
80	C₄×C₅₉	C₂×#95	1	1	yes	characteristic
81	C₄×C₅₉	C₂×#103	1	2	yes	—
82	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
83	Dic₅₉	—	8	8	no	—
84	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
85	C₂×#427	—	59	2	no	—
86	#1380	—	59	4	no	—
87	C₂×C₅₉	—	2	4	no	—
88	C₂×C₅₉	C₂×#340	1	1	yes	characteristic
89	C₂×C₅₉	#2138	1	4	yes	—
90	C₂×C₅₉	#2158	1	2	yes	—
91	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
92	C₂×#117	—	118	2	no	—
93	C₂²×C₁₆	—	2	4	no	—
94	C₂²×C₁₆	Dic₅₉	1	4	yes	complemented
95	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
96	#427	—	59	8	no	—
97	#427	—	118	4	no	—
98	C₂×#107	—	59	4	no	—
99	C₂×#106	—	118	2	no	—
100	C₅₉	#12556	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
102	C₂²×C₈	Dic₁₁₈	1	2	yes	—
103	C₂²×C₈	C₄×D₅₉	1	4	yes	—
104	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
106	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
107	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
108	#117	—	118	8	no	—
109	#117	—	236	4	no	—
110	C₂×C₄⋊C₄	—	118	4	no	—
111	C₂×C₂²⋊C₄	—	236	2	no	—
112	C₂×C₄²	—	236	2	no	—
113	C₂×C₁₆	—	2	8	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
116	#107	—	59	8	no	—
117	#107	—	118	4	no	—
118	#106	—	118	8	no	—
119	C₂⁴	D₄₇₂	1	1	yes	characteristic
120	C₂²×C₄	—	236	4	no	—
121	C₂²×C₄	#11745660	1	4	yes	—
122	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
123	C₂²×C₄	Dic₂₃₆	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
125	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
126	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
127	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
128	C₂×C₈	Dic₅₉⋊C₄	1	8	yes	—
129	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
130	C₂×C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
131	C₂×C₈	C₂²⋊Dic₅₉	1	4	yes	—
132	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
134	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
135	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
136	C₄⋊C₄	—	118	8	no	—
137	C₄⋊C₄	—	236	4	no	—
138	C₂²⋊C₄	—	236	8	no	—
139	C₄²	—	236	8	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	—	2	4	no	—
142	C₂³	#14273649	1	4	yes	—
143	C₂³	#14273661	1	2	yes	—
144	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
145	C₂×C₄	—	472	8	no	—
146	C₂×C₄	—	236	8	no	—
147	C₂×C₄	C₅₉⋊#95	1	4	yes	—
148	C₂×C₄	#14273649	1	4	yes	—
149	C₂×C₄	#14273652	1	8	yes	—
150	C₂×C₄	#14273659	1	4	yes	—
151	C₂×C₄	#14273661	1	2	yes	—
152	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
154	C₂×C₄	C₂×#11745660	1	2	yes	—
155	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
156	C₈	C₅₉⋊#95	1	4	yes	—
157	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
158	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
159	C₈	C₂×Dic₅₉⋊C₄	1	2	yes	—
160	C₂²	—	2	8	no	—
161	C₂²	—	2	4	no	—
162	C₂²	#16837165	1	4	yes	—
163	C₂²	C₂×#14273649	1	2	yes	—
164	C₂²	C₂×#14273661	1	1	yes	characteristic
165	C₂²	#16837834	1	4	yes	—
166	C₄	—	472	8	no	—
167	C₄	#16837165	1	4	yes	—
168	C₄	C₂×C₅₉⋊#95	1	1	yes	characteristic
169	C₄	C₂×#14273652	1	2	yes	—
170	C₄	C₂×#14273659	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×#16837165	1	1	yes	characteristic
173	C₂	#21425804	1	4	yes	—
174	C₂	#21425880	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂⁴×C₅₉	D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂³×C₁₆	D₅₉
hypocenter	C₅₉	#12556
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21422188	C₂	semidirect
6. #16837440	C₄	semidirect
11. C₂²×C₁₆×C₅₉	C₄	semidirect
94. C₂²×C₁₆	Dic₅₉	semidirect
100. C₅₉	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837440	2
#21422188	2
C₂³×C₁₆×C₅₉	2
#12556	59