Subgroups — C₄×#97×D₅₉ #28586010

Order 15104.

1790 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×D₅₉	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₅₉	C₂	1	2	yes	maximal
3	C₄³×D₅₉	C₂	1	1	yes	maximal, characteristic
4	C₄×#14273638	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#14273637	C₂	1	2	yes	maximal, complemented
6	#97×Dic₅₉	C₂	1	4	yes	maximal, complemented
7	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₉	C₂²	1	4	yes	—
9	C₂×C₄²×D₅₉	C₄	1	4	yes	—
10	C₂×C₄²×D₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×D₅₉	C₂²	1	2	yes	—
12	C₂×#14273638	C₂²	1	4	yes	—
13	C₂×#14273637	C₂²	1	8	yes	—
14	#97×D₅₉	C₄	1	32	yes	direct product factor
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₅₉	C₂²	1	4	yes	—
17	C₄×C₈×D₅₉	C₄	1	16	yes	complemented
18	C₄²×Dic₅₉	C₄	1	4	yes	—
19	C₄²×Dic₅₉	C₂²	1	2	yes	—
20	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₉	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
23	C₄³×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₅₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×D₅₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×D₅₉	C₂³	1	1	yes	characteristic
27	C₂²×C₅₉⋊C₈	C₂³	1	4	yes	—
28	C₂×C₈×D₅₉	C₂×C₄	1	16	yes	—
29	C₂×C₈×D₅₉	C₂×C₄	1	32	yes	—
30	C₂×C₄×Dic₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₅₉	C₂×C₄	1	8	yes	—
32	C₂×C₄×Dic₅₉	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
36	C₄²×D₅₉	—	2	16	no	—
37	C₄²×D₅₉	C₂×C₄	1	8	yes	—
38	C₄²×D₅₉	C₂×C₄	1	16	yes	—
39	#14273638	C₂×C₄	1	16	yes	complemented
40	#14273637	C₂×C₄	1	32	yes	complemented
41	C₄×C₅₉⋊C₈	C₂×C₄	1	8	yes	complemented
42	C₈×Dic₅₉	C₂×C₄	1	16	yes	complemented
43	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
44	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	#97×C₅₉	C₂×C₄	1	16	yes	complemented
48	C₄×C₈×C₅₉	C₂×C₄	1	8	yes	complemented
49	C₂³×D₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₅₉	C₂²×C₄	1	4	yes	—
54	C₂²×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₉	C₂²×C₄	1	2	yes	—
56	C₂×C₄×D₅₉	—	2	16	no	—
57	C₂×C₄×D₅₉	C₄²	1	16	yes	—
58	C₂×C₄×D₅₉	C₄²	1	8	yes	—
59	C₂×C₄×D₅₉	M₁₆	1	16	yes	—
60	C₂×C₄×D₅₉	C₂²×C₄	1	4	yes	—
61	C₂×C₄×D₅₉	C₂²×C₄	1	8	yes	—
62	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
63	C₂×C₈×C₅₉	C₂²×C₄	1	16	yes	—
64	C₄²×C₅₉	—	2	8	no	—
65	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
66	C₄²×C₅₉	C₂²×C₄	1	8	yes	—
67	C₄×Dic₅₉	—	2	8	no	—
68	C₄×Dic₅₉	—	2	16	no	—
69	C₄×Dic₅₉	C₄²	1	16	yes	—
70	C₄×Dic₅₉	C₄²	1	8	yes	—
71	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
72	C₄×Dic₅₉	C₂²×C₄	1	8	yes	—
73	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
74	C₂×C₅₉⋊C₈	C₂²×C₄	1	16	yes	—
75	C₈×D₅₉	C₄²	1	64	yes	complemented
76	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
77	C₂²×D₅₉	C₂×C₄²	1	2	yes	—
78	C₂²×D₅₉	C₂×C₄²	1	4	yes	—
79	C₂²×D₅₉	C₂×M₁₆	1	8	yes	—
80	C₂×C₄×C₅₉	—	2	8	no	—
81	C₂×C₄×C₅₉	C₂×C₄²	1	8	yes	—
82	C₂×C₄×C₅₉	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₅₉	C₂×M₁₆	1	8	yes	—
84	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
85	C₂×C₄×C₅₉	C₂³×C₄	1	4	yes	—
86	C₂×Dic₅₉	—	2	8	no	—
87	C₂×Dic₅₉	C₂×C₄²	1	8	yes	—
88	C₂×Dic₅₉	C₂×C₄²	1	2	yes	—
89	C₂×Dic₅₉	C₂×C₄²	1	4	yes	—
90	C₂×Dic₅₉	C₂×C₄²	1	4	yes	—
91	C₂×Dic₅₉	C₂×M₁₆	1	8	yes	—
92	C₄×D₅₉	—	2	32	no	—
93	C₄×D₅₉	#97	1	32	yes	direct product factor
94	C₄×D₅₉	C₂×C₄²	1	16	yes	—
95	C₈×C₅₉	C₂×C₄²	1	16	yes	complemented
96	C₅₉⋊C₈	C₂×C₄²	1	16	yes	complemented
97	C₂×C₄×#97	—	59	1	no	maximal, Sylow
98	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
99	C₂²×C₅₉	C₂²×C₄²	1	2	yes	—
100	C₂²×C₅₉	C₂²×M₁₆	1	4	yes	—
101	C₂×D₅₉	C₄³	1	4	yes	—
102	C₂×D₅₉	C₂×#97	1	8	yes	—
103	C₂×D₅₉	C₄×M₁₆	1	16	yes	—
104	C₄×C₅₉	—	2	8	no	—
105	C₄×C₅₉	C₂×#97	1	8	yes	complemented
106	C₄×C₅₉	C₂²×C₄²	1	4	yes	—
107	Dic₅₉	—	2	8	no	—
108	Dic₅₉	C₄³	1	4	yes	—
109	Dic₅₉	C₂×#97	1	8	yes	complemented
110	C₂²×#97	—	59	4	no	—
111	C₂²×C₄×C₈	—	59	2	no	—
112	C₂×C₄³	—	59	1	no	—
113	C₄×#97	—	59	4	no	—
114	C₄×#97	—	59	2	no	—
115	C₄×#97	—	59	1	no	—
116	C₄×#97	D₅₉	1	1	yes	characteristic, direct product factor
117	C₂×C₅₉	C₂×C₄³	1	1	yes	characteristic
118	C₂×C₅₉	C₂²×#97	1	2	yes	—
119	C₂×C₅₉	C₂×C₄×M₁₆	1	4	yes	—
120	D₅₉	C₄×#97	1	8	yes	direct product factor
121	C₂³×C₈	—	59	4	no	—
122	C₂²×C₄²	—	59	2	no	—
123	C₂²×C₄²	—	59	1	no	—
124	C₂²×C₄²	—	59	4	no	—
125	C₂×#97	—	59	8	no	—
126	C₂×#97	—	59	4	no	—
127	C₂×#97	—	59	32	no	—
128	C₂×#97	C₂×D₅₉	1	4	yes	—
129	C₂×C₄×C₈	—	59	4	no	—
130	C₂×C₄×C₈	—	59	2	no	—
131	C₂×C₄×C₈	—	59	16	no	—
132	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
133	C₄³	—	59	4	no	—
134	C₄³	—	59	2	no	—
135	C₄³	—	59	1	no	—
136	C₄³	C₂×D₅₉	1	1	yes	characteristic
137	C₅₉	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	—	59	2	no	—
139	C₂³×C₄	—	59	1	no	—
140	C₂³×C₄	—	59	4	no	—
141	C₂²×C₈	—	59	4	no	—
142	C₂²×C₈	—	59	16	no	—
143	C₂²×C₈	—	59	32	no	—
144	C₂²×C₈	C₂²×D₅₉	1	4	yes	—
145	C₂×C₄²	—	59	2	no	—
146	C₂×C₄²	—	59	4	no	—
147	C₂×C₄²	—	59	1	no	—
148	C₂×C₄²	—	59	2	no	—
149	C₂×C₄²	—	59	8	no	—
150	C₂×C₄²	—	59	16	no	—
151	C₂×C₄²	—	59	8	no	—
152	C₂×C₄²	—	59	4	no	—
153	C₂×C₄²	—	118	16	no	—
154	C₂×C₄²	C₄×D₅₉	1	4	yes	—
155	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
156	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
157	#97	—	59	32	no	—
158	#97	—	59	16	no	—
159	#97	C₄×D₅₉	1	16	yes	direct product factor
160	C₄×C₈	—	59	16	no	—
161	C₄×C₈	—	59	8	no	—
162	C₄×C₈	C₄×D₅₉	1	8	yes	complemented
163	C₂⁴	—	59	1	no	—
164	C₂²×C₄	—	59	2	no	—
165	C₂²×C₄	—	59	1	no	—
166	C₂²×C₄	—	59	8	no	—
167	C₂²×C₄	—	59	16	no	—
168	C₂²×C₄	—	59	4	no	—
169	C₂²×C₄	—	59	8	no	—
170	C₂²×C₄	—	59	4	no	—
171	C₂²×C₄	—	59	16	no	—
172	C₂²×C₄	—	118	16	no	—
173	C₂²×C₄	C₂×C₄×D₅₉	1	4	yes	—
174	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
175	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
176	C₂×C₈	—	59	8	no	—
177	C₂×C₈	—	59	16	no	—
178	C₂×C₈	—	59	64	no	—
179	C₂×C₈	C₂×C₄×D₅₉	1	8	yes	—
180	C₂×C₈	C₂×C₄×D₅₉	1	16	yes	—
181	C₄²	—	2	8	no	—
182	C₄²	—	59	8	no	—
183	C₄²	—	59	16	no	—
184	C₄²	—	59	4	no	—
185	C₄²	—	59	8	no	—
186	C₄²	—	118	16	no	—
187	C₄²	—	118	8	no	—
188	C₄²	C₂×C₄×D₅₉	1	4	yes	—
189	C₄²	C₂×C₄×D₅₉	1	8	yes	—
190	C₂³	—	59	8	no	—
191	C₂³	—	59	2	no	—
192	C₂³	—	59	4	no	—
193	C₂³	C₂²×C₄×D₅₉	1	1	yes	characteristic
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	59	8	no	—
196	C₂×C₄	—	59	4	no	—
197	C₂×C₄	—	59	2	no	—
198	C₂×C₄	—	59	4	no	—
199	C₂×C₄	—	59	16	no	—
200	C₂×C₄	—	59	32	no	—
201	C₂×C₄	—	59	8	no	—
202	C₂×C₄	—	118	32	no	—
203	C₂×C₄	—	118	8	no	—
204	C₂×C₄	C₄²×D₅₉	1	8	yes	—
205	C₂×C₄	C₄²×D₅₉	1	4	yes	—
206	C₂×C₄	M₁₆×D₅₉	1	8	yes	—
207	C₂×C₄	C₂²×C₄×D₅₉	1	2	yes	—
208	C₂×C₄	C₂²×C₄×D₅₉	1	4	yes	—
209	C₈	—	59	16	no	—
210	C₈	C₄²×D₅₉	1	16	yes	complemented
211	C₂²	—	59	8	no	—
212	C₂²	—	59	16	no	—
213	C₂²	—	59	4	no	—
214	C₂²	C₂×C₄²×D₅₉	1	1	yes	characteristic
215	C₂²	C₂×C₄²×D₅₉	1	2	yes	—
216	C₂²	C₂×M₁₆×D₅₉	1	4	yes	—
217	C₄	—	2	8	no	—
218	C₄	—	59	8	no	—
219	C₄	—	59	4	no	—
220	C₄	—	118	8	no	—
221	C₄	#97×D₅₉	1	8	yes	direct product factor
222	C₄	C₂×C₄²×D₅₉	1	4	yes	—
223	C₂	—	59	8	no	—
224	C₂	C₄³×D₅₉	1	1	yes	characteristic
225	C₂	C₂×#97×D₅₉	1	2	yes	—
226	C₂	C₄×M₁₆×D₅₉	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₉
commutator	C₂×C₅₉	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₄×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂³×C₄
prime radical	C₂³×D₅₉	C₂²×C₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₄×#97	D₅₉
hypocenter	C₅₉	C₂×C₄×#97
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
4. C₄×#14273638	C₂	semidirect
5. C₄×#14273637	C₂	semidirect
6. #97×Dic₅₉	C₂	semidirect
7. C₄×#97×C₅₉	C₂	semidirect
14. #97×D₅₉	C₄	direct (N × H)
17. C₄×C₈×D₅₉	C₄	semidirect
39. #14273638	C₂×C₄	semidirect
40. #14273637	C₂×C₄	semidirect
41. C₄×C₅₉⋊C₈	C₂×C₄	semidirect
42. C₈×Dic₅₉	C₂×C₄	semidirect
47. #97×C₅₉	C₂×C₄	semidirect
48. C₄×C₈×C₅₉	C₂×C₄	semidirect
75. C₈×D₅₉	C₄²	semidirect
93. C₄×D₅₉	#97	direct (N × H)
95. C₈×C₅₉	C₂×C₄²	semidirect
96. C₅₉⋊C₈	C₂×C₄²	semidirect
105. C₄×C₅₉	C₂×#97	semidirect
109. Dic₅₉	C₂×#97	semidirect
116. C₄×#97	D₅₉	direct (N × H)
120. D₅₉	C₄×#97	direct (N × H)
137. C₅₉	C₂×C₄×#97	semidirect
159. #97	C₄×D₅₉	direct (N × H)
162. C₄×C₈	C₄×D₅₉	semidirect
210. C₈	C₄²×D₅₉	semidirect
221. C₄	#97×D₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#97×D₅₉	2
C₂×C₄×C₈×D₅₉	2
C₄³×D₅₉	2
C₄×#14273638	2
C₄×#14273637	2
#97×Dic₅₉	2
C₄×#97×C₅₉	2
C₂×C₄×#97	59

Subgroups — C4×#97×D59 #28586010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#97×D₅₉ #28586010