Subgroups — #28500331

Order 15104.

242 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422855	C₂	1	2	yes	maximal
2	#21422168	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837426	—	2	2	no	—
5	#16837155	—	2	2	no	—
6	#16837155	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₂×#11745664	—	4	2	no	—
13	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
14	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	—	2	4	no	—
19	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₄²×C₅₉	D₈	1	1	yes	characteristic
36	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
37	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
38	#11745664	—	8	4	no	—
39	C₂×C₅₉⋊C₈	—	4	4	no	—
40	C₁₆×C₅₉	—	2	2	no	—
41	C₁₆×C₅₉	M₁₆	1	4	yes	complemented
42	C₂³×C₅₉	#107	1	1	yes	characteristic
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	1	no	—
46	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
47	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
50	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
51	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
54	C₈×C₅₉	—	2	1	no	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	#108	1	2	yes	—
57	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
58	C₅₉⋊C₈	—	8	4	no	—
59	#12022	—	59	1	no	maximal, Sylow
60	C₂²×C₅₉	—	2	1	no	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	#366	1	2	yes	—
63	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	#366	1	2	yes	—
67	C₄×C₅₉	#368	1	2	yes	—
68	C₄×C₅₉	#423	1	1	yes	characteristic
69	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
70	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
71	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
72	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
73	#1567	—	59	2	no	—
74	#1371	—	59	4	no	—
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	#1567	1	1	yes	characteristic
77	C₂×C₅₉	C₂×#366	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#368	1	1	yes	characteristic
79	C₂²×C₁₆	Dic₅₉	1	2	yes	—
80	#423	—	118	2	no	—
81	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
82	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
83	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
84	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
85	#335	—	59	4	no	—
86	#335	—	118	2	no	—
87	C₅₉	#12022	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₁₆	—	236	2	no	—
89	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
91	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
94	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
95	C₄⋊C₈	—	118	4	no	—
96	C₄⋊C₈	—	236	2	no	—
97	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
98	C₄×C₈	D₂₃₆	1	1	yes	characteristic
99	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
100	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
101	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
102	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
103	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
104	M₁₆	—	472	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	236	4	no	—
108	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
109	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
110	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
111	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
112	C₄²	D₄₇₂	1	1	yes	characteristic
113	C₄²	Dic₂₃₆	1	1	yes	characteristic
114	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
115	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	#11745664	1	4	yes	complemented
118	C₂³	#14273661	1	1	yes	characteristic
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	#14273661	1	1	yes	characteristic
123	C₂×C₄	#14273661	1	1	yes	characteristic
124	C₂×C₄	#14273661	1	1	yes	characteristic
125	C₂×C₄	D₉₄₄	1	2	yes	—
126	C₂×C₄	Dic₄₇₂	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	472	4	no	—
133	C₈	#14273663	1	2	yes	—
134	C₈	C₂×#11745664	1	2	yes	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	#16837228	1	2	yes	—
138	C₂²	C₂×#14273661	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	#16837228	1	2	yes	—
142	C₄	#16837232	1	2	yes	—
143	C₄	#16837426	1	1	yes	characteristic
144	C₄	C₂×#14273663	1	1	yes	characteristic
145	C₄	C₂×D₉₄₄	1	1	yes	characteristic
146	C₄	C₂×Dic₄₇₂	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#21422855	1	1	yes	characteristic
149	C₂	C₂×#16837228	1	1	yes	characteristic
150	C₂	C₂×#16837232	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	#107
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#12022
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21422168	C₂	semidirect
41. C₁₆×C₅₉	M₁₆	semidirect
87. C₅₉	#12022	semidirect
117. C₁₆	#11745664	semidirect

Maximal subgroups

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

subgroup	index
#21422855	2
#21422168	2
C₂×C₄×C₁₆×C₅₉	2
#12022	59