Subgroups — #28475882

Order 15104.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
5	C₂×#14273663	—	2	1	no	—
6	C₂×#14273663	C₄	1	2	yes	—
7	#16837426	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	—
10	C₅₉×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#335	—	2	1	no	—
13	C₅₉×#334	C₄	1	2	yes	—
14	C₂×#11745664	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	—	2	2	no	—
16	C₂×C₅₉⋊C₁₆	—	4	1	no	—
17	C₂×C₅₉⋊C₁₆	—	4	1	no	—
18	#14273663	—	4	1	no	—
19	#14273663	—	4	1	no	—
20	#14273663	—	4	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
22	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₅₉	—	2	2	no	—
27	C₂²⋊C₈×C₅₉	—	4	1	no	—
28	C₄×C₈×C₅₉	—	4	1	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₉	D₈	1	1	yes	characteristic
50	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
51	#11745664	—	4	2	no	—
52	#11745664	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	C₂³×C₅₉	#103	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	4	1	no	—
58	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#101	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
68	C₈×C₅₉	—	2	1	no	—
69	C₈×C₅₉	—	2	1	no	—
70	C₈×C₅₉	—	8	2	no	—
71	C₈×C₅₉	#102	1	1	yes	characteristic
72	C₈×C₅₉	#103	1	1	yes	characteristic
73	C₈×C₅₉	#106	1	1	yes	characteristic
74	C₈×C₅₉	#107	1	1	yes	characteristic
75	C₅₉⋊C₈	—	8	2	no	—
76	#7276	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	1	no	—
79	C₂²×C₅₉	#332	1	1	yes	characteristic
80	C₂²×C₅₉	#333	1	1	yes	characteristic
81	C₂²×C₅₉	#340	1	1	yes	characteristic
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	#331	1	1	yes	characteristic
86	C₄×C₅₉	#332	1	1	yes	characteristic
87	C₄×C₅₉	#340	1	1	yes	characteristic
88	C₄×C₅₉	#343	1	1	yes	characteristic
89	C₄×C₅₉	#368	1	2	yes	—
90	C₂×C₄⋊C₁₆	—	59	1	no	—
91	#1848	—	59	1	no	—
92	#1564	D₅₉	1	1	yes	characteristic
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	—	2	1	no	—
95	C₂×C₅₉	#1296	1	1	yes	characteristic
96	C₂×C₅₉	#1381	1	1	yes	characteristic
97	C₂×C₅₉	#1389	1	1	yes	characteristic
98	C₂²×C₁₆	—	118	1	no	—
99	#431	—	2	1	no	—
100	C₂×#108	—	59	2	no	—
101	C₂×#108	—	118	1	no	—
102	#423	—	118	1	no	—
103	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
104	C₄⋊C₁₆	—	59	2	no	—
105	C₄⋊C₁₆	—	118	1	no	—
106	#335	—	2	1	no	—
107	#334	Dic₅₉	1	2	yes	—
108	C₅₉	#7276	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	118	2	no	—
110	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
111	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
113	C₂×C₁₆	—	118	2	no	—
114	C₂×C₁₆	—	236	1	no	—
115	C₂×C₁₆	—	236	1	no	—
116	#108	—	236	2	no	—
117	#108	—	236	1	no	—
118	#108	—	236	1	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	236	1	no	—
121	C₄⋊C₈	—	236	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
125	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
128	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	#11745670	1	1	yes	characteristic
130	C₂²×C₄	#11745672	1	1	yes	characteristic
131	M₁₆	—	236	2	no	—
132	M₁₆	—	236	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	236	2	no	—
137	C₂×C₈	#11745660	1	1	yes	characteristic
138	C₂×C₈	#11745660	1	1	yes	characteristic
139	C₂×C₈	D₄₇₂	1	1	yes	characteristic
140	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
141	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
142	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
144	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
145	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
146	C₄²	#11745669	1	1	yes	characteristic
147	C₄²	#11745671	1	1	yes	characteristic
148	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
149	C₁₆	—	236	2	no	—
150	C₂³	#14273650	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
154	C₂×C₄	#14273643	1	1	yes	characteristic
155	C₂×C₄	#14273645	1	1	yes	characteristic
156	C₂×C₄	#14273647	1	1	yes	characteristic
157	C₂×C₄	#14273648	1	1	yes	characteristic
158	C₂×C₄	#14273649	1	1	yes	characteristic
159	C₂×C₄	#14273651	1	1	yes	characteristic
160	C₂×C₄	#14273652	1	1	yes	characteristic
161	C₂×C₄	#14273658	1	1	yes	characteristic
162	C₂×C₄	#14273660	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	472	2	no	—
167	C₈	#14273649	1	1	yes	characteristic
168	C₈	#14273652	1	1	yes	characteristic
169	C₈	#14273659	1	1	yes	characteristic
170	C₈	#14273661	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#16837147	1	1	yes	characteristic
174	C₂²	#16837150	1	1	yes	characteristic
175	C₂²	#16837164	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#16837145	1	1	yes	characteristic
180	C₄	#16837149	1	1	yes	characteristic
181	C₄	#16837165	1	1	yes	characteristic
182	C₄	#16837170	1	1	yes	characteristic
183	C₄	#16837231	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#21421980	1	1	yes	characteristic
187	C₂	#21422189	1	1	yes	characteristic
188	C₂	#21422212	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273647
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#1564	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#103
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#7276
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
108. C₅₉	#7276	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21424268	2
C₅₉×#1564	2
#7276	59