Subgroups — #28504006

Order 15104.

398 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837222	C₂	1	2	yes	maximal
2	M₃₂×Dic₅₉	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#21422178	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
13	#16837222	C₄	1	8	yes	complemented
14	#16837177	—	2	2	no	—
15	C₁₆×Dic₅₉	—	2	2	no	—
16	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₉	—	2	2	no	—
21	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
23	#14273667	—	2	2	no	—
24	#14273667	C₂×C₄	1	4	yes	—
25	C₂×C₅₉⋊C₁₆	—	2	2	no	—
26	C₄×C₅₉⋊C₈	C₈	1	4	yes	—
27	C₈×Dic₅₉	—	2	2	no	—
28	C₈×Dic₅₉	C₈	1	4	yes	—
29	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
30	C₈×Dic₅₉	D₄	1	1	yes	characteristic
31	C₈×Dic₅₉	D₄	1	1	yes	characteristic
32	C₈×Dic₅₉	Q₈	1	1	yes	characteristic
33	C₈×Dic₅₉	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₅₉	—	2	2	no	—
38	M₃₂×C₅₉	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₅₉	—	2	2	no	—
40	C₄×C₈×C₅₉	—	2	2	no	—
41	C₂²×C₄×C₅₉	—	2	1	no	—
42	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₉	—	2	1	no	—
44	C₂²×Dic₅₉	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₉	—	2	1	no	—
46	C₂×C₈×C₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₉	#41	1	2	yes	—
52	C₄²×C₅₉	—	2	1	no	—
53	C₄²×C₅₉	C₂×C₈	1	2	yes	—
54	C₄×Dic₅₉	—	2	8	no	—
55	C₄×Dic₅₉	—	2	1	no	—
56	C₄×Dic₅₉	C₄²	1	2	yes	—
57	C₄×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
60	C₂×C₅₉⋊C₈	C₄²	1	2	yes	—
61	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
64	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
65	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
66	C₂×C₅₉⋊C₈	M₁₆	1	2	yes	—
67	C₁₆×C₅₉	—	2	4	no	—
68	C₅₉⋊C₁₆	—	2	4	no	—
69	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	4	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₅₉	—	2	4	no	—
79	C₂×Dic₅₉	—	2	2	no	—
80	C₂×Dic₅₉	—	2	4	no	—
81	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
82	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	C₄×D₄	1	2	yes	—
85	C₈×C₅₉	C₄×Q₈	1	2	yes	—
86	C₅₉⋊C₈	C₄×C₈	1	4	yes	—
87	C₅₉⋊C₈	C₄⋊C₈	1	2	yes	—
88	C₅₉⋊C₈	C₄⋊C₈	1	2	yes	—
89	C₄×#364	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	—	2	2	no	—
91	C₂²×C₅₉	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₅₉	#431	1	1	yes	characteristic
94	C₄×C₅₉	—	2	4	no	—
95	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₅₉	#431	1	1	yes	characteristic
99	Dic₅₉	—	2	4	no	—
100	Dic₅₉	—	2	2	no	—
101	Dic₅₉	#364	1	4	yes	complemented
102	C₂×#364	—	59	2	no	—
103	C₄×M₃₂	—	59	2	no	—
104	C₄²×C₈	—	59	1	no	—
105	#1376	—	59	1	no	—
106	#1376	D₅₉	1	1	yes	characteristic
107	C₂×C₅₉	—	2	1	no	—
108	C₂×C₅₉	—	2	1	no	—
109	C₂×C₅₉	C₄×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₅₉	C₂×#364	1	2	yes	—
111	C₂×M₃₂	—	59	2	no	—
112	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
113	C₂×C₄×C₈	—	59	1	no	—
114	C₂×C₄×C₈	—	59	1	no	—
115	C₂×C₄×C₈	—	59	1	no	—
116	C₂×C₄×C₈	—	59	1	no	—
117	C₂×C₄×C₈	—	59	1	no	—
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	C₄³	—	59	1	no	—
120	#364	—	59	8	no	—
121	#346	—	118	2	no	—
122	C₄×C₁₆	—	118	2	no	—
123	C₅₉	C₄×#364	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	59	1	no	—
125	C₂²×C₈	—	59	1	no	—
126	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
128	C₂×C₄²	—	59	1	no	—
129	C₂×C₄²	—	59	1	no	—
130	C₂×C₄²	—	118	2	no	—
131	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	59	4	no	—
134	M₃₂	—	118	2	no	—
135	M₃₂	C₄×D₅₉	1	4	yes	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	118	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	59	4	no	—
140	C₄×C₈	—	59	4	no	—
141	C₄×C₈	—	59	1	no	—
142	C₄×C₈	—	59	1	no	—
143	C₄×C₈	—	59	1	no	—
144	C₄×C₈	—	59	1	no	—
145	C₄×C₈	—	59	4	no	—
146	C₄×C₈	—	118	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	59	2	no	—
149	C₂²×C₄	—	118	1	no	—
150	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	59	2	no	—
154	C₂×C₈	—	59	1	no	—
155	C₂×C₈	—	59	1	no	—
156	C₂×C₈	—	59	2	no	—
157	C₂×C₈	—	59	2	no	—
158	C₂×C₈	—	59	2	no	—
159	C₂×C₈	—	59	2	no	—
160	C₂×C₈	C₈×D₅₉	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
162	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
163	C₂×C₈	#11745687	1	2	yes	—
164	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
165	C₄²	—	2	1	no	—
166	C₄²	—	59	1	no	—
167	C₄²	—	59	1	no	—
168	C₄²	—	59	2	no	—
169	C₄²	—	59	2	no	—
170	C₄²	—	118	8	no	—
171	C₄²	—	118	1	no	—
172	C₄²	C₈×D₅₉	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	118	4	no	—
175	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	59	2	no	—
179	C₂×C₄	—	59	2	no	—
180	C₂×C₄	—	118	4	no	—
181	C₂×C₄	—	118	4	no	—
182	C₂×C₄	—	118	2	no	—
183	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
184	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
185	C₂×C₄	#14273683	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
188	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	59	2	no	—
191	C₈	—	59	2	no	—
192	C₈	—	59	4	no	—
193	C₈	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
194	C₈	#14273692	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	#16837354	1	1	yes	characteristic
197	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
198	C₂²	#16837452	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	59	4	no	—
201	C₄	—	118	4	no	—
202	C₄	—	118	2	no	—
203	C₄	#16837253	1	1	yes	characteristic
204	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
205	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
206	C₄	#16837452	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#21423944	1	1	yes	characteristic
210	C₂	#364×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1376	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1376	D₅₉
hypocenter	C₅₉	C₄×#364
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
13. #16837222	C₄	semidirect
101. Dic₅₉	#364	semidirect
123. C₅₉	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837222	2
M₃₂×Dic₅₉	2
C₄×C₈×Dic₅₉	2
#21422178	2
C₅₉×#1376	2
C₄×#364	59