Subgroups — #28500237

Order 15104.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#21422160	C₂	1	4	yes	maximal, complemented
4	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	—	2	1	no	—
6	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
11	C₅₉×#431	C₄	1	1	yes	characteristic
12	C₅₉×#431	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#322	C₄	1	2	yes	—
15	C₈²×C₅₉	C₂²	1	2	yes	—
16	#14273667	—	4	2	no	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	4	2	no	—
20	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	—	2	1	no	—
24	C₄⋊C₈×C₅₉	C₈	1	2	yes	—
25	C₂²⋊C₈×C₅₉	—	2	1	no	—
26	C₂²⋊C₈×C₅₉	C₈	1	2	yes	—
27	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
48	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₅₉⋊C₁₆	—	4	2	no	—
50	C₅₉⋊C₁₆	—	4	2	no	—
51	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#106	1	2	yes	—
58	C₂×C₄×C₅₉	#108	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	—	2	1	no	—
64	C₈×C₅₉	—	2	4	no	—
65	C₈×C₅₉	—	4	2	no	—
66	C₈×C₅₉	#106	1	2	yes	—
67	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
69	#11997	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	—	2	1	no	—
71	C₂²×C₅₉	—	2	1	no	—
72	C₂²×C₅₉	#334	1	2	yes	—
73	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	—	2	1	no	—
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	#334	1	2	yes	—
78	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
79	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
80	C₄×C₅₉	#504	1	2	yes	—
81	#2150	—	59	1	no	—
82	C₂×C₄⋊C₁₆	—	59	1	no	—
83	#1450	D₅₉	1	1	yes	characteristic
84	#1367	—	59	4	no	—
85	C₂×C₅₉	—	2	2	no	—
86	C₂×C₅₉	C₂×#334	1	1	yes	characteristic
87	C₂×C₅₉	#2151	1	1	yes	characteristic
88	C₂×C₅₉	#2153	1	1	yes	characteristic
89	C₂×M₃₂	—	118	1	no	—
90	C₂²×C₁₆	—	118	1	no	—
91	#431	Dic₅₉	1	1	yes	characteristic
92	#431	Dic₅₉	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
94	C₄⋊C₁₆	—	59	2	no	—
95	C₄⋊C₁₆	—	59	2	no	—
96	C₄⋊C₁₆	—	118	1	no	—
97	C₄⋊C₁₆	—	118	1	no	—
98	#322	Dic₅₉	1	2	yes	—
99	C₈²	C₂×D₅₉	1	2	yes	—
100	C₅₉	#11997	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
102	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
104	M₃₂	—	236	2	no	—
105	C₂×C₁₆	—	118	2	no	—
106	C₂×C₁₆	—	118	2	no	—
107	C₂×C₁₆	—	236	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₅₉⋊C₈	1	2	yes	—
110	C₂²⋊C₈	—	2	1	no	—
111	C₂²⋊C₈	C₅₉⋊C₈	1	2	yes	—
112	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
113	C₄×C₈	D₂₃₆	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
115	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
116	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	#11745664	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	#11745660	1	2	yes	—
124	C₂×C₈	#11745660	1	2	yes	—
125	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
126	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
129	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
130	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
131	C₄²	#11745664	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
133	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
134	C₁₆	—	236	2	no	—
135	C₁₆	—	236	2	no	—
136	C₂³	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
140	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
141	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
142	C₂×C₄	#14273659	1	2	yes	—
143	C₂×C₄	#14273663	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
146	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	4	no	—
150	C₈	—	4	2	no	—
151	C₈	#14273659	1	2	yes	—
152	C₈	C₂×#11745660	1	1	yes	characteristic
153	C₈	C₂×#11745660	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#16837153	1	2	yes	—
157	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#16837153	1	2	yes	—
162	C₄	C₂×#14273659	1	1	yes	characteristic
163	C₄	C₂×#14273663	1	1	yes	characteristic
164	C₄	#16837819	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#16837153	1	1	yes	characteristic
167	C₂	#21425855	1	1	yes	characteristic
168	C₂	#21425862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₄⋊C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#11997
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #21422160	C₂	semidirect
100. C₅₉	#11997	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21422160	2
C₅₉×#1450	2
#11997	59