Subgroups — #28503733

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837184	C₂	1	2	yes	maximal
2	C₄×#14273667	C₂	1	2	yes	maximal, complemented
3	#21422180	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₉×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	4	yes	—
6	#16837184	C₄	1	8	yes	complemented
7	#16837178	—	2	2	no	—
8	C₄×C₅₉⋊C₁₆	—	2	2	no	—
9	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₉	C₄	1	1	yes	characteristic
11	C₂×#117×C₅₉	C₄	1	1	yes	characteristic
12	C₅₉×#405	—	2	2	no	—
13	C₅₉×#405	—	2	2	no	—
14	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
15	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	#14273667	—	2	4	no	—
19	#14273667	C₂×C₄	1	8	yes	—
20	C₂×C₅₉⋊C₁₆	—	2	4	no	—
21	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₅₉	—	2	2	no	—
23	C₂×M₁₆×C₅₉	—	2	2	no	—
24	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₉	—	2	1	no	—
26	C₂²×C₈×C₅₉	—	2	1	no	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	C₈	1	4	yes	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	#117×C₅₉	—	2	2	no	—
34	#117×C₅₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₉	C₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₉	C₈	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
39	#97×C₅₉	—	2	2	no	—
40	#97×C₅₉	—	2	2	no	—
41	C₄×C₈×C₅₉	—	2	1	no	—
42	C₄×C₈×C₅₉	—	2	1	no	—
43	C₄×C₈×C₅₉	D₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	—	2	1	no	—
47	C₂²×C₄×C₅₉	—	2	1	no	—
48	C₂²×C₄×C₅₉	—	2	1	no	—
49	C₂²×C₄×C₅₉	—	2	1	no	—
50	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₅₉	—	2	8	no	—
57	C₂×C₈×C₅₉	—	2	1	no	—
58	C₂×C₈×C₅₉	—	2	1	no	—
59	C₂×C₈×C₅₉	—	2	1	no	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	4	no	—
62	C₂×C₈×C₅₉	—	2	4	no	—
63	C₂×C₈×C₅₉	C₄²	1	2	yes	—
64	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
69	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₉	#41	1	2	yes	—
72	C₄⋊C₄×C₅₉	—	2	4	no	—
73	C₂²⋊C₄×C₅₉	—	2	4	no	—
74	C₄²×C₅₉	—	2	1	no	—
75	C₄²×C₅₉	—	2	1	no	—
76	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₉	M₁₆	1	2	yes	—
78	C₅₉⋊C₁₆	—	2	8	no	—
79	C₂³×C₅₉	—	2	1	no	—
80	C₂³×C₅₉	—	2	1	no	—
81	C₂³×C₅₉	—	4	2	no	—
82	C₂³×C₅₉	C₄×C₈	1	2	yes	—
83	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₉	—	2	1	no	—
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₅₉	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₅₉	—	2	2	no	—
102	C₈×C₅₉	—	2	2	no	—
103	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
104	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
105	C₈×C₅₉	C₄×D₄	1	4	yes	—
106	#12602	—	59	1	no	maximal, Sylow
107	C₂²×C₅₉	—	2	2	no	—
108	C₂²×C₅₉	—	2	2	no	—
109	C₂²×C₅₉	—	4	2	no	—
110	C₂²×C₅₉	—	4	2	no	—
111	C₂²×C₅₉	—	4	2	no	—
112	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₅₉	#431	1	1	yes	characteristic
115	C₄×C₅₉	—	2	4	no	—
116	C₄×C₅₉	—	2	2	no	—
117	C₄×C₅₉	—	4	2	no	—
118	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
120	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₅₉	#431	1	1	yes	characteristic
122	C₂×#405	D₅₉	1	1	yes	characteristic
123	C₂×#349	—	59	2	no	—
124	C₄×M₃₂	—	59	2	no	—
125	#1377	—	59	2	no	—
126	C₂×C₅₉	—	2	1	no	—
127	C₂×C₅₉	—	2	1	no	—
128	C₂×C₅₉	—	4	2	no	—
129	C₂×C₅₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₅₉	#2115	1	2	yes	—
131	C₂²×M₁₆	C₂×D₅₉	1	1	yes	characteristic
132	C₂³×C₈	Dic₅₉	1	1	yes	characteristic
133	C₂×#117	Dic₅₉	1	1	yes	characteristic
134	C₂×M₃₂	—	59	4	no	—
135	#405	—	2	2	no	—
136	#405	—	2	2	no	—
137	C₂×#97	Dic₅₉	1	1	yes	characteristic
138	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
139	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
141	#349	—	59	8	no	—
142	#346	—	118	2	no	—
143	C₄×C₁₆	—	118	2	no	—
144	C₅₉	#12602	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	C₄×D₅₉	1	4	yes	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₅₉⋊C₈	1	4	yes	—
153	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	C₂×C₄⋊C₄	C₅₉⋊C₈	1	2	yes	—
160	C₂×C₂²⋊C₄	C₅₉⋊C₈	1	2	yes	—
161	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
163	M₃₂	—	59	8	no	—
164	M₃₂	—	118	4	no	—
165	C₂×C₁₆	—	118	4	no	—
166	#97	—	2	2	no	—
167	#97	—	2	2	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
171	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
172	C₂⁴	C₂×C₅₉⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
179	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
180	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
181	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
183	M₁₆	—	2	8	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
191	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
192	C₂×C₈	#11745664	1	2	yes	—
193	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
194	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
197	C₂×C₈	#11745685	1	2	yes	—
198	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	#11745664	1	2	yes	—
204	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
205	C₁₆	—	118	8	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₄×C₅₉⋊C₈	1	2	yes	—
210	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
221	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	4	yes	—
222	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
224	C₂×C₄	#14273686	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
227	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
231	C₈	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
232	C₈	C₄×C₂²⋊D₅₉	1	4	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
239	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
240	C₂²	#16837455	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
245	C₄	C₄×#11745664	1	1	yes	characteristic
246	C₄	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
247	C₄	#16837455	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
252	C₂	#21425717	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#405	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#405	D₅₉
hypocenter	C₅₉	#12602
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₄×#14273667	C₂	semidirect
3. #21422180	C₂	semidirect
6. #16837184	C₄	semidirect
144. C₅₉	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837184	2
C₄×#14273667	2
#21422180	2
C₂×C₅₉×#405	2
#12602	59