Subgroups — #28475631

Order 15104.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837223	C₂	1	1	yes	maximal, characteristic
2	#21423975	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273667	—	2	1	no	—
5	C₂×#14273663	C₄	1	1	yes	characteristic
6	#16837440	C₄	1	1	yes	characteristic
7	#16837357	—	2	1	no	—
8	#16837223	—	2	1	no	—
9	#16837223	C₄	1	2	yes	complemented
10	#16837162	—	2	1	no	—
11	C₅₉×#431	—	2	1	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#322	—	2	1	no	—
14	C₈²×C₅₉	C₄	1	2	yes	complemented
15	C₂×#11745664	—	2	1	no	—
16	#14273686	—	2	1	no	—
17	#14273667	—	2	2	no	—
18	#14273667	—	4	1	no	—
19	C₂×C₅₉⋊C₁₆	—	4	1	no	—
20	#14273663	—	2	1	no	—
21	#14273663	C₈	1	2	yes	complemented
22	#14273661	—	2	1	no	—
23	#14273659	C₈	1	2	yes	complemented
24	#14273637	—	4	1	no	—
25	C₈×Dic₅₉	—	4	1	no	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	2	no	—
30	C₂²⋊C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	—	2	1	no	—
37	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	—	2	4	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	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
51	C₄²×C₅₉	—	2	1	no	—
52	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²⋊Dic₅₉	—	4	1	no	—
55	C₄⋊Dic₅₉	—	2	2	no	—
56	C₄×Dic₅₉	—	4	1	no	—
57	#11745664	—	2	2	no	—
58	#11745664	—	4	1	no	—
59	C₂×C₅₉⋊C₈	—	4	1	no	—
60	C₅₉⋊C₁₆	—	4	2	no	—
61	C₂³×C₅₉	#104	1	1	yes	characteristic
62	C₂×C₄×C₅₉	—	2	1	no	—
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#104	1	1	yes	characteristic
71	C₂×Dic₅₉	—	4	2	no	—
72	C₈×C₅₉	—	4	4	no	—
73	C₈×C₅₉	C₄×C₈	1	1	yes	characteristic
74	C₈×C₅₉	#97	1	1	yes	characteristic
75	C₈×C₅₉	#102	1	1	yes	characteristic
76	C₈×C₅₉	#103	1	1	yes	characteristic
77	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₅₉	#106	1	1	yes	characteristic
80	C₈×C₅₉	#107	1	1	yes	characteristic
81	C₅₉⋊C₈	—	4	2	no	—
82	#7180	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	4	1	no	—
84	C₂²×C₅₉	D₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₅₉	Q₈⋊C₈	1	1	yes	characteristic
86	C₂²×C₅₉	#339	1	1	yes	characteristic
87	C₄×C₅₉	—	2	2	no	—
88	C₄×C₅₉	—	4	1	no	—
89	C₄×C₅₉	D₄⋊C₈	1	1	yes	characteristic
90	C₄×C₅₉	Q₈⋊C₈	1	1	yes	characteristic
91	C₄×C₅₉	#336	1	1	yes	characteristic
92	C₄×C₅₉	#340	1	1	yes	characteristic
93	Dic₅₉	—	8	2	no	—
94	C₂×#364	—	59	1	no	—
95	#1772	—	59	1	no	—
96	#1450	D₅₉	1	1	yes	characteristic
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#1279	1	1	yes	characteristic
101	C₂×M₃₂	—	118	1	no	—
102	#431	—	2	1	no	—
103	C₂×#108	—	59	1	no	—
104	#427	—	59	1	no	—
105	#405	—	118	1	no	—
106	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	#364	—	59	2	no	—
108	#364	—	118	1	no	—
109	#339	—	118	1	no	—
110	#322	—	2	1	no	—
111	C₈²	Dic₅₉	1	2	yes	complemented
112	C₅₉	#7180	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	118	1	no	—
114	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
115	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
116	#117	—	118	1	no	—
117	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
118	M₃₂	—	118	2	no	—
119	M₃₂	—	236	1	no	—
120	C₂×C₁₆	—	236	1	no	—
121	#108	—	59	2	no	—
122	#108	—	118	1	no	—
123	#107	—	118	1	no	—
124	#106	—	59	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	#97	—	236	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	236	1	no	—
130	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
131	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
136	M₁₆	—	118	2	no	—
137	M₁₆	—	236	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	236	1	no	—
140	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
141	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
142	C₂×C₈	#11745659	1	1	yes	characteristic
143	C₂×C₈	#11745659	1	1	yes	characteristic
144	C₂×C₈	#11745660	1	1	yes	characteristic
145	C₂×C₈	#11745660	1	1	yes	characteristic
146	C₂×C₈	D₄₇₂	1	1	yes	characteristic
147	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
149	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
151	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
152	C₄⋊C₄	—	118	2	no	—
153	C₂²⋊C₄	—	236	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	236	1	no	—
156	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
158	C₁₆	—	236	2	no	—
159	C₂³	#14273655	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	236	2	no	—
164	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
165	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
166	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
167	C₂×C₄	#14273649	1	1	yes	characteristic
168	C₂×C₄	#14273652	1	1	yes	characteristic
169	C₂×C₄	#14273655	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	236	2	no	—
172	C₈	C₈×Dic₅₉	1	1	yes	characteristic
173	C₈	#14273637	1	1	yes	characteristic
174	C₈	#14273649	1	1	yes	characteristic
175	C₈	#14273652	1	1	yes	characteristic
176	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
177	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
178	C₈	#14273659	1	1	yes	characteristic
179	C₈	#14273661	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	#16837131	1	1	yes	characteristic
182	C₂²	#16837134	1	1	yes	characteristic
183	C₂²	#16837162	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	472	2	no	—
187	C₄	#16837131	1	1	yes	characteristic
188	C₄	#16837134	1	1	yes	characteristic
189	C₄	#16837156	1	1	yes	characteristic
190	C₄	#16837165	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#21421936	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273659
commutator	C₈×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂×C₅₉	#1279
prime radical	C₂³×C₅₉	#104
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#7180
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
9. #16837223	C₄	semidirect
14. C₈²×C₅₉	C₄	semidirect
21. #14273663	C₈	semidirect
23. #14273659	C₈	semidirect
111. C₈²	Dic₅₉	semidirect
112. C₅₉	#7180	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837223	2
#21423975	2
C₅₉×#1450	2
#7180	59