Subgroups — #28482352

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424213	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#21421930	C₂	1	4	yes	maximal
4	C₅₉×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₅₉⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
10	#16837157	—	2	2	no	—
11	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
12	C₄³×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#336	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
17	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
18	C₄×C₅₉⋊C₈	—	2	2	no	—
19	C₄×C₅₉⋊C₈	—	2	2	no	—
20	C₄×C₅₉⋊C₈	—	2	2	no	—
21	C₄×C₅₉⋊C₈	—	2	4	no	—
22	C₄×C₅₉⋊C₈	C₈	1	4	yes	complemented
23	C₂²×C₈×C₅₉	—	2	2	no	—
24	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	D₄	1	2	yes	—
27	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	4	no	—
30	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	—	4	4	no	—
37	C₄²×C₅₉	—	2	4	no	—
38	C₄²×C₅₉	—	2	2	no	—
39	C₄²×C₅₉	—	2	4	no	—
40	C₄²×C₅₉	C₂×C₈	1	2	yes	—
41	C₄²×C₅₉	M₁₆	1	2	yes	—
42	C₄²×C₅₉	#41	1	2	yes	—
43	C₄²×C₅₉	#41	1	2	yes	—
44	C₂×C₅₉⋊C₈	—	2	2	no	—
45	C₂×C₅₉⋊C₈	—	2	2	no	—
46	C₂×C₅₉⋊C₈	—	2	2	no	—
47	C₂×C₅₉⋊C₈	—	2	2	no	—
48	C₂×C₅₉⋊C₈	—	4	2	no	—
49	C₂×C₅₉⋊C₈	—	4	4	no	—
50	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
51	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
52	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₅₉	#108	1	4	yes	—
60	C₂×C₄×C₅₉	#117	1	2	yes	—
61	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#125	1	2	yes	—
64	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#135	1	4	yes	—
68	C₈×C₅₉	—	8	2	no	—
69	C₅₉⋊C₈	—	2	4	no	—
70	C₅₉⋊C₈	—	2	4	no	—
71	C₅₉⋊C₈	—	8	2	no	—
72	#8785	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	#382	1	1	yes	characteristic
74	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₂²×C₅₉	#423	1	1	yes	characteristic
76	C₂²×C₅₉	#427	1	2	yes	—
77	C₂²×C₅₉	C₂×#108	1	2	yes	—
78	C₄×C₅₉	—	2	4	no	—
79	C₄×C₅₉	—	2	4	no	—
80	C₄×C₅₉	—	2	2	no	—
81	C₄×C₅₉	#431	1	2	yes	—
82	C₄×C₅₉	#432	1	2	yes	—
83	C₄×C₅₉	#488	1	2	yes	—
84	C₄×C₅₉	#488	1	2	yes	—
85	#1831	—	59	1	no	—
86	#1831	D₅₉	1	1	yes	characteristic
87	C₄²×C₈	—	59	1	no	—
88	#1277	—	59	4	no	—
89	C₂×C₅₉	#1564	1	2	yes	—
90	C₂×C₅₉	#1568	1	2	yes	—
91	C₂×C₅₉	#1831	1	1	yes	characteristic
92	C₂×C₅₉	#1835	1	1	yes	characteristic
93	C₂×C₅₉	#1837	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	59	2	no	—
95	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
96	C₂×C₄×C₈	—	59	1	no	—
97	C₂×C₄×C₈	—	59	2	no	—
98	C₂×C₄×C₈	—	59	1	no	—
99	C₂×C₄×C₈	—	59	2	no	—
100	C₄³	C₂×D₅₉	1	1	yes	characteristic
101	#336	—	2	2	no	—
102	#336	—	118	2	no	—
103	C₅₉	#8785	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	59	2	no	—
106	C₂²×C₈	—	59	2	no	—
107	C₂²×C₈	—	118	2	no	—
108	C₂×C₄²	Dic₁₁₈	1	2	yes	—
109	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
111	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
112	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	118	4	no	—
115	C₄×C₈	—	59	4	no	—
116	C₄×C₈	—	118	4	no	—
117	C₄×C₈	—	118	2	no	—
118	C₄×C₈	—	118	2	no	—
119	C₄×C₈	—	118	2	no	—
120	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
121	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
122	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	59	4	no	—
128	C₂×C₈	—	59	4	no	—
129	C₂×C₈	—	118	2	no	—
130	C₂×C₈	—	118	2	no	—
131	C₂×C₈	—	118	2	no	—
132	C₂×C₈	—	118	2	no	—
133	C₂×C₈	—	236	2	no	—
134	C₂×C₈	—	236	4	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	4	no	—
138	C₄²	C₈×D₅₉	1	2	yes	—
139	C₄²	#11745659	1	2	yes	—
140	C₄²	#11745687	1	2	yes	—
141	C₄²	#11745689	1	2	yes	—
142	C₂³	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	Dic₅₉⋊C₈	1	4	yes	—
149	C₂×C₄	#14273662	1	4	yes	—
150	C₂×C₄	#14273685	1	2	yes	—
151	C₂×C₄	#14273714	1	1	yes	characteristic
152	C₂×C₄	#14273716	1	1	yes	characteristic
153	C₂×C₄	#14273718	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
155	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
156	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
157	C₂×C₄	#14273747	1	4	yes	—
158	C₈	—	8	2	no	—
159	C₈	—	118	4	no	—
160	C₈	—	118	4	no	—
161	C₈	—	472	2	no	—
162	C₂²	#16837267	1	1	yes	characteristic
163	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
164	C₂²	#16837425	1	1	yes	characteristic
165	C₂²	#16837438	1	2	yes	—
166	C₂²	C₂×#14273662	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	4	no	—
170	C₄	#16837454	1	2	yes	—
171	C₄	#16837458	1	2	yes	—
172	C₄	#16837756	1	2	yes	—
173	C₄	#16837759	1	2	yes	—
174	C₂	#21422842	1	2	yes	—
175	C₂	#21422857	1	2	yes	—
176	C₂	#21424212	1	1	yes	characteristic
177	C₂	#21424226	1	1	yes	characteristic
178	C₂	#21424233	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1831	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1831	D₅₉
hypocenter	C₅₉	#8785
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
22. C₄×C₅₉⋊C₈	C₈	semidirect
103. C₅₉	#8785	semidirect

Maximal subgroups

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

subgroup	index
#21424213	2
C₄²×C₅₉⋊C₈	2
#21421930	2
C₅₉×#1831	2
#8785	59