Subgroups — #28482370

Order 15104.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424268	C₂	1	1	yes	maximal, characteristic
2	#21424260	C₂	1	1	yes	maximal, characteristic
3	#21421955	C₂	1	4	yes	maximal
4	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273663	—	2	1	no	—
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273661	—	2	1	no	—
8	C₂×#14273659	—	2	1	no	—
9	#16837426	C₂²	1	2	yes	—
10	#16837268	C₂²	1	2	yes	—
11	C₅₉×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#322	C₄	1	2	yes	complemented
14	C₈²×C₅₉	C₄	1	2	yes	complemented
15	C₂×#11745664	—	2	2	no	—
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	#14273663	—	4	2	no	—
18	#14273663	—	4	2	no	—
19	#14273661	—	4	2	no	—
20	#14273659	—	4	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
22	C₅₉⋊#95	—	2	4	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₉	—	2	2	no	—
27	C₂²⋊C₈×C₅₉	—	2	2	no	—
28	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	#41	1	2	yes	—
35	C₂²×Dic₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	—	2	4	no	—
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	D₈	1	2	yes	—
42	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
44	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
45	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₉	#41	1	2	yes	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	#11745664	—	4	2	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂³×C₅₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	4	2	no	—
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#117	1	2	yes	—
57	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#125	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₉	—	8	4	no	—
64	C₂×Dic₅₉	—	8	2	no	—
65	C₈×C₅₉	—	4	4	no	—
66	C₈×C₅₉	#106	1	2	yes	—
67	C₈×C₅₉	#106	1	2	yes	—
68	C₈×C₅₉	#107	1	2	yes	—
69	C₈×C₅₉	#107	1	2	yes	—
70	C₅₉⋊C₈	—	8	2	no	—
71	#8790	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	#382	1	1	yes	characteristic
74	C₂²×C₅₉	#486	1	1	yes	characteristic
75	C₂²×C₅₉	#487	1	1	yes	characteristic
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	2	2	no	—
78	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
79	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
80	C₄×C₅₉	#486	1	1	yes	characteristic
81	C₄×C₅₉	#487	1	1	yes	characteristic
82	Dic₅₉	—	16	2	no	—
83	#1848	—	59	1	no	—
84	#1846	—	59	1	no	—
85	#1450	D₅₉	1	1	yes	characteristic
86	#1286	—	59	4	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	—	2	1	no	—
89	C₂×C₅₉	#1833	1	2	yes	—
90	C₂×C₅₉	#1834	1	1	yes	characteristic
91	#431	C₂×D₅₉	1	2	yes	—
92	C₂×#108	—	118	1	no	—
93	C₂×#108	—	118	1	no	—
94	C₂×#107	—	118	1	no	—
95	C₂×#106	—	118	1	no	—
96	#423	—	59	2	no	—
97	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	#382	—	59	2	no	—
99	#322	Dic₅₉	1	2	yes	complemented
100	C₈²	Dic₅₉	1	2	yes	complemented
101	C₅₉	#8790	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	118	2	no	—
103	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
104	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	118	2	no	—
106	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
107	#108	—	236	2	no	—
108	#108	—	236	2	no	—
109	#107	—	236	2	no	—
110	#106	—	236	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	118	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
115	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
117	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
119	#95	—	118	4	no	—
120	C₂²×C₄	—	236	2	no	—
121	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
122	C₂²×C₄	#11745685	1	2	yes	—
123	M₁₆	—	236	2	no	—
124	M₁₆	—	236	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	236	2	no	—
127	C₂×C₈	#11745660	1	2	yes	—
128	C₂×C₈	#11745660	1	2	yes	—
129	C₂×C₈	D₄₇₂	1	2	yes	—
130	C₂×C₈	Dic₂₃₆	1	2	yes	—
131	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
132	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
135	C₄⋊C₄	—	236	2	no	—
136	C₄⋊C₄	—	236	2	no	—
137	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
138	C₄²	#11745685	1	2	yes	—
139	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
140	C₂³	#14273715	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	472	4	no	—
143	C₂×C₄	—	472	2	no	—
144	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
145	C₂×C₄	#14273686	1	2	yes	—
146	C₂×C₄	#14273715	1	1	yes	characteristic
147	C₂×C₄	#14273719	1	2	yes	—
148	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
149	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
150	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	472	2	no	—
154	C₈	#14273659	1	2	yes	—
155	C₈	#14273659	1	2	yes	—
156	C₈	#14273661	1	2	yes	—
157	C₈	#14273661	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#16837268	1	1	yes	characteristic
160	C₂²	#16837749	1	1	yes	characteristic
161	C₂²	#16837753	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	944	2	no	—
165	C₄	C₂×#14273659	1	1	yes	characteristic
166	C₄	C₂×#14273661	1	1	yes	characteristic
167	C₄	#16837749	1	1	yes	characteristic
168	C₄	#16837753	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#21424220	1	2	yes	—
172	C₂	#21424224	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837268
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#8790
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. C₅₉×#322	C₄	semidirect
14. C₈²×C₅₉	C₄	semidirect
99. #322	Dic₅₉	semidirect
100. C₈²	Dic₅₉	semidirect
101. C₅₉	#8790	semidirect

Maximal subgroups

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

subgroup	index
#21424268	2
#21424260	2
#21421955	2
C₅₉×#1450	2
#8790	59