Subgroups — #28505065

Order 15104.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 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	#21422195	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273663	—	2	1	no	—
6	C₂×#14273663	C₂²	1	2	yes	—
7	C₂×#14273661	—	2	1	no	—
8	C₂×#14273659	C₂²	1	2	yes	—
9	#16837426	—	2	1	no	—
10	#16837268	—	2	1	no	—
11	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₅₉	C₄	1	2	yes	complemented
15	C₂×#11745664	—	2	2	no	—
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	#14273663	—	2	4	no	—
18	#14273663	—	4	2	no	—
19	#14273661	—	4	2	no	—
20	#14273659	—	2	4	no	—
21	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
22	C₅₉⋊#95	—	4	2	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
26	M₃₂×C₅₉	—	2	2	no	—
27	M₃₂×C₅₉	—	2	2	no	—
28	C₂×C₁₆×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	SD₁₆	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	D₈	1	2	yes	—
39	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₉	#41	1	2	yes	—
43	C₂×C₈×C₅₉	#41	1	2	yes	—
44	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₉	SD₁₆	1	2	yes	—
47	C₄⋊Dic₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	#11745664	—	4	2	no	—
50	#11745664	—	4	2	no	—
51	C₂×C₅₉⋊C₈	—	4	2	no	—
52	C₁₆×C₅₉	—	2	4	no	—
53	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	4	2	no	—
55	C₂×C₄×C₅₉	#106	1	2	yes	—
56	C₂×C₄×C₅₉	#106	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₈	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	#107	1	2	yes	—
66	C₈×C₅₉	#107	1	2	yes	—
67	C₈×C₅₉	#117	1	2	yes	—
68	C₈×C₅₉	#125	1	2	yes	—
69	C₅₉⋊C₈	—	8	2	no	—
70	#12887	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
73	C₂²×C₅₉	#486	1	1	yes	characteristic
74	C₂²×C₅₉	#487	1	1	yes	characteristic
75	C₄×C₅₉	—	2	2	no	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	#382	1	1	yes	characteristic
78	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
79	C₄×C₅₉	#486	1	1	yes	characteristic
80	C₄×C₅₉	#487	1	1	yes	characteristic
81	Dic₅₉	—	16	2	no	—
82	C₄×M₃₂	D₅₉	1	1	yes	characteristic
83	#1848	—	59	1	no	—
84	#1846	—	59	1	no	—
85	#1383	—	59	4	no	—
86	C₂×C₅₉	—	2	1	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	#1834	1	1	yes	characteristic
89	C₂×C₅₉	#2155	1	2	yes	—
90	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
91	C₂×#108	—	59	2	no	—
92	C₂×#108	—	118	1	no	—
93	C₂×#107	—	118	1	no	—
94	C₂×#106	—	59	2	no	—
95	#423	—	118	1	no	—
96	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
97	#382	—	118	1	no	—
98	#346	Dic₅₉	1	2	yes	complemented
99	C₄×C₁₆	Dic₅₉	1	2	yes	complemented
100	C₅₉	#12887	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	118	2	no	—
102	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	118	2	no	—
105	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
109	#108	—	118	4	no	—
110	#108	—	236	2	no	—
111	#107	—	236	2	no	—
112	#106	—	118	4	no	—
113	C₄⋊C₈	—	236	2	no	—
114	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
115	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
118	#95	—	236	2	no	—
119	C₂²×C₄	—	236	2	no	—
120	C₂²×C₄	#11745660	1	2	yes	—
121	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
122	M₁₆	—	236	2	no	—
123	M₁₆	—	236	2	no	—
124	C₂×C₈	—	236	2	no	—
125	C₂×C₈	D₄₇₂	1	2	yes	—
126	C₂×C₈	Dic₂₃₆	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
130	C₂×C₈	#11745685	1	2	yes	—
131	C₂×C₈	#11745685	1	2	yes	—
132	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
133	C₄⋊C₄	—	236	2	no	—
134	C₄⋊C₄	—	236	2	no	—
135	C₄²	#11745660	1	2	yes	—
136	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#11745660	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	472	4	no	—
142	C₂×C₄	—	472	2	no	—
143	C₂×C₄	#14273659	1	2	yes	—
144	C₂×C₄	#14273659	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
146	C₂×C₄	#14273715	1	1	yes	characteristic
147	C₂×C₄	#14273715	1	1	yes	characteristic
148	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
149	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
151	C₈	—	472	2	no	—
152	C₈	#14273661	1	2	yes	—
153	C₈	#14273661	1	2	yes	—
154	C₈	#14273686	1	2	yes	—
155	C₈	#14273719	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#14273659	1	1	yes	characteristic
158	C₂²	#16837749	1	1	yes	characteristic
159	C₂²	#16837753	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	944	2	no	—
163	C₄	#16837268	1	1	yes	characteristic
164	C₄	C₂×#14273661	1	1	yes	characteristic
165	C₄	#16837749	1	1	yes	characteristic
166	C₄	#16837753	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#21424224	1	1	yes	characteristic
170	C₂	#21425869	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14273659
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#12887
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₅₉×#346	C₄	semidirect
14. C₄×C₁₆×C₅₉	C₄	semidirect
98. #346	Dic₅₉	semidirect
99. C₄×C₁₆	Dic₅₉	semidirect
100. C₅₉	#12887	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
#21422195	2
C₄×M₃₂×C₅₉	2
#12887	59