Subgroups — #494×Dic₅₉ #28597127

Order 15104.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₅₉	C₂	1	4	yes	maximal
2	C₄⋊Q₈×Dic₅₉	C₂	1	2	yes	maximal
3	#21424958	C₂	1	4	yes	maximal
4	#21424899	C₂	1	2	yes	maximal
5	#21424256	C₂	1	1	yes	maximal, characteristic
6	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₉	C₂²	1	4	yes	—
9	Q₁₆×Dic₅₉	—	2	16	no	—
10	C₂×#14273661	C₂²	1	4	yes	—
11	C₂×#14273651	C₂²	1	8	yes	—
12	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
14	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
15	#16837309	C₂²	1	4	yes	—
16	#16837301	—	2	4	no	—
17	#16837272	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₅₉	—	2	4	no	—
19	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₅₉	C₂²	1	4	yes	—
21	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
22	C₅₉×#494	C₄	1	8	yes	complemented
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
25	Q₈×Dic₅₉	—	2	16	no	—
26	C₂×C₄⋊Dic₅₉	—	2	4	no	—
27	C₂×C₄⋊Dic₅₉	—	2	4	no	—
28	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
29	C₂×C₄×Dic₅₉	—	2	4	no	—
30	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
31	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
32	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
34	#14273661	—	2	8	no	—
35	#14273651	—	2	16	no	—
36	C₄×C₅₉⋊C₈	—	2	2	no	—
37	C₈×Dic₅₉	—	2	4	no	—
38	C₈×Dic₅₉	D₄	1	8	yes	—
39	C₅₉⋊#95	—	2	8	no	—
40	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
41	C₂×Q₁₆×C₅₉	—	2	8	no	—
42	C₂×Q₁₆×C₅₉	—	2	8	no	—
43	C₂×Q₁₆×C₅₉	C₂×C₄	1	16	yes	—
44	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₅₉	—	2	4	no	—
46	C₄⋊Q₈×C₅₉	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
48	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
50	C₂×Q₈×C₅₉	—	2	4	no	—
51	C₂×Q₈×C₅₉	—	2	8	no	—
52	C₂×Q₈×C₅₉	—	2	8	no	—
53	C₂×Q₈×C₅₉	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₅₉	—	2	4	no	—
55	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₅₉	—	2	4	no	—
58	C₂²×Dic₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₉	C₂×D₄	1	2	yes	—
60	C₂²×Dic₅₉	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₅₉	—	2	32	no	—
62	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₅₉	#41	1	4	yes	—
65	C₄⋊C₄×C₅₉	—	2	16	no	—
66	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₉	#41	1	2	yes	—
68	C₄⋊Dic₅₉	—	2	8	no	—
69	C₄⋊Dic₅₉	—	4	8	no	—
70	C₄⋊Dic₅₉	—	4	8	no	—
71	C₄×Dic₅₉	—	2	4	no	—
72	C₄×Dic₅₉	—	2	2	no	—
73	C₄×Dic₅₉	—	4	8	no	—
74	C₄×Dic₅₉	Q₁₆	1	8	yes	complemented
75	C₄×Dic₅₉	C₂×D₄	1	4	yes	—
76	C₂×C₅₉⋊C₈	—	2	2	no	—
77	C₂×C₅₉⋊C₈	—	2	2	no	—
78	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
79	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
80	Q₈×C₅₉	—	2	16	no	—
81	Q₈×C₅₉	—	4	8	no	—
82	C₂×C₄×C₅₉	—	2	8	no	—
83	C₂×C₄×C₅₉	—	4	4	no	—
84	C₂×C₄×C₅₉	—	4	4	no	—
85	C₂×C₄×C₅₉	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₅₉	#135	1	4	yes	—
88	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
91	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
92	C₂×Dic₅₉	—	2	2	no	—
93	C₂×Dic₅₉	—	2	2	no	—
94	C₂×Dic₅₉	—	2	2	no	—
95	C₂×Dic₅₉	—	4	4	no	—
96	C₂×Dic₅₉	—	4	8	no	—
97	C₂×Dic₅₉	#127	1	2	yes	—
98	C₂×Dic₅₉	#127	1	2	yes	—
99	C₂×Dic₅₉	C₂×Q₁₆	1	4	yes	—
100	C₂×Dic₅₉	C₂×Q₁₆	1	4	yes	—
101	C₈×C₅₉	C₄×D₄	1	8	yes	—
102	C₅₉⋊C₈	—	2	4	no	—
103	C₄×#494	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
106	C₂²×C₅₉	#532	1	1	yes	characteristic
107	C₂²×C₅₉	C₂²×Q₁₆	1	2	yes	—
108	C₂²×C₅₉	C₂×#135	1	2	yes	—
109	C₄×C₅₉	—	4	8	no	—
110	C₄×C₅₉	C₄×Q₁₆	1	8	yes	complemented
111	C₄×C₅₉	C₂×C₄×D₄	1	2	yes	—
112	C₄×C₅₉	#532	1	2	yes	—
113	Dic₅₉	—	2	2	no	—
114	Dic₅₉	—	2	4	no	—
115	Dic₅₉	—	8	4	no	—
116	Dic₅₉	#494	1	4	yes	direct product factor
117	C₂×#494	D₅₉	1	1	yes	characteristic
118	C₂×C₄×Q₁₆	—	59	4	no	—
119	C₄×C₄⋊Q₈	—	59	2	no	—
120	#1969	—	59	4	no	—
121	#1956	—	59	2	no	—
122	#1845	—	59	1	no	—
123	C₄²×C₈	—	59	1	no	—
124	C₂×C₅₉	C₄×#127	1	1	yes	characteristic
125	C₂×C₅₉	C₂×C₄×Q₁₆	1	2	yes	—
126	C₂×C₅₉	C₂×#494	1	1	yes	characteristic
127	C₂×C₅₉	#3147	1	1	yes	characteristic
128	C₂×C₅₉	#3170	1	2	yes	—
129	C₂²×Q₁₆	C₂×D₅₉	1	4	yes	—
130	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
131	C₂×C₄×Q₈	—	59	4	no	—
132	#494	Dic₅₉	1	8	yes	direct product factor
133	C₄×Q₁₆	—	118	16	no	—
134	C₂×#107	—	59	4	no	—
135	C₂×#103	—	59	8	no	—
136	C₂×C₄×C₈	—	59	2	no	—
137	C₂×C₄×C₈	—	59	2	no	—
138	C₂×C₄×C₈	—	59	1	no	—
139	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
140	#391	—	59	4	no	—
141	#389	—	118	4	no	—
142	#384	—	59	2	no	—
143	C₄×C₄⋊C₄	—	118	4	no	—
144	C₄³	—	59	1	no	—
145	C₅₉	C₄×#494	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	C₂²×D₅₉	1	4	yes	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	2	8	no	—
149	C₂×Q₁₆	C₂×Dic₅₉	1	16	yes	—
150	C₂²×C₈	—	59	2	no	—
151	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₅₉	1	8	yes	—
154	C₄×Q₈	—	118	16	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	59	4	no	—
157	C₂×C₄⋊C₄	—	118	4	no	—
158	C₂×C₄⋊C₄	—	118	4	no	—
159	C₂×C₄²	—	59	2	no	—
160	C₂×C₄²	—	59	1	no	—
161	C₂×C₄²	—	59	2	no	—
162	C₂×C₄²	—	59	1	no	—
163	C₂×C₄²	—	118	4	no	—
164	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
165	#107	—	118	8	no	—
166	#103	—	118	16	no	—
167	C₄×C₈	—	59	8	no	—
168	C₄×C₈	—	118	4	no	—
169	C₄×C₈	—	118	2	no	—
170	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
171	#95	—	118	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	C₂²×Dic₅₉	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	59	1	no	—
178	C₂²×C₄	—	59	2	no	—
179	C₂²×C₄	—	59	1	no	—
180	C₂²×C₄	—	118	4	no	—
181	C₂²×C₄	D₄×D₅₉	1	2	yes	—
182	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
183	Q₁₆	—	2	32	no	—
184	C₂×C₈	—	59	4	no	—
185	C₂×C₈	—	118	2	no	—
186	C₂×C₈	—	118	2	no	—
187	C₂×C₈	D₄×D₅₉	1	4	yes	—
188	C₂×C₈	#11745687	1	4	yes	—
189	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	118	8	no	—
192	C₄⋊C₄	—	236	8	no	—
193	C₄⋊C₄	—	236	8	no	—
194	C₄²	—	59	8	no	—
195	C₄²	—	59	4	no	—
196	C₄²	—	118	4	no	—
197	C₄²	—	118	2	no	—
198	C₄²	—	236	8	no	—
199	C₄²	#11745687	1	2	yes	—
200	C₄²	C₂²×Dic₅₉	1	2	yes	—
201	C₂³	C₂×D₄×D₅₉	1	1	yes	characteristic
202	Q₈	—	2	16	no	—
203	Q₈	—	4	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	59	4	no	—
208	C₂×C₄	—	59	2	no	—
209	C₂×C₄	—	59	4	no	—
210	C₂×C₄	—	59	2	no	—
211	C₂×C₄	—	118	2	no	—
212	C₂×C₄	—	118	2	no	—
213	C₂×C₄	—	118	2	no	—
214	C₂×C₄	—	236	8	no	—
215	C₂×C₄	—	236	4	no	—
216	C₂×C₄	D₄×Dic₅₉	1	4	yes	—
217	C₂×C₄	Q₁₆×D₅₉	1	4	yes	—
218	C₂×C₄	#14273750	1	4	yes	—
219	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₅₉	1	2	yes	—
221	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
222	C₂×C₄	C₂×#11745687	1	2	yes	—
223	C₈	—	118	4	no	—
224	C₈	D₄×Dic₅₉	1	8	yes	—
225	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
226	C₂²	#127×D₅₉	1	1	yes	characteristic
227	C₂²	#16837950	1	1	yes	characteristic
228	C₂²	C₂×Q₁₆×D₅₉	1	2	yes	—
229	C₂²	C₂×#14273750	1	2	yes	—
230	C₄	—	4	8	no	—
231	C₄	—	59	4	no	—
232	C₄	—	118	4	no	—
233	C₄	—	118	2	no	—
234	C₄	—	472	4	no	—
235	C₄	Q₁₆×Dic₅₉	1	8	yes	complemented
236	C₄	C₂×D₄×Dic₅₉	1	2	yes	—
237	C₄	#16837950	1	2	yes	—
238	C₂	#127×Dic₅₉	1	1	yes	characteristic
239	C₂	C₂×Q₁₆×Dic₅₉	1	2	yes	—
240	C₂	#494×D₅₉	1	1	yes	characteristic
241	C₂	#21434525	1	1	yes	characteristic
242	C₂	#21434680	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle, prime radical	C₂³×C₅₉	C₂²×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	C₄×#494
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
22. C₅₉×#494	C₄	semidirect
74. C₄×Dic₅₉	Q₁₆	semidirect
110. C₄×C₅₉	C₄×Q₁₆	semidirect
116. Dic₅₉	#494	direct (N × H)
132. #494	Dic₅₉	direct (N × H)
145. C₅₉	C₄×#494	semidirect
235. C₄	Q₁₆×Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₅₉	2
C₄⋊Q₈×Dic₅₉	2
#21424958	2
#21424899	2
#21424256	2
C₄×C₈×Dic₅₉	2
C₂×C₅₉×#494	2
C₄×#494	59

Subgroups — #494×Dic59 #28597127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #494×Dic₅₉ #28597127