Subgroups — #28599123

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432729	C₂	1	2	yes	maximal
2	#21424872	C₂	1	8	yes	maximal, complemented
3	#21424138	C₂	1	4	yes	maximal
4	C₂×C₅₉×#423	C₂	1	1	yes	maximal, characteristic
5	#16837459	—	2	2	no	—
6	#16837459	—	2	2	no	—
7	#16837455	—	2	2	no	—
8	#16837455	—	2	2	no	—
9	C₂×#14273663	C₂²	1	8	yes	—
10	#16837426	C₂²	1	4	yes	—
11	#16837366	C₂²	1	4	yes	—
12	C₄×#11745664	C₂²	1	4	yes	—
13	#16837167	C₂²	1	8	yes	—
14	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#423	C₄	1	2	yes	—
17	C₅₉×#423	C₄	1	4	yes	—
18	C₅₉×#423	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
20	C₂×#11745664	C₂³	1	8	yes	—
21	#14273663	—	4	8	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
26	#14273638	—	2	4	no	—
27	C₄×C₅₉⋊C₈	—	2	4	no	—
28	C₂³×C₄×C₅₉	Q₈	1	2	yes	—
29	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₅₉	—	4	4	no	—
31	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
34	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	2	yes	—
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
43	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
45	C₂⁴×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	—	2	2	no	—
47	C₂²×C₄×C₅₉	—	2	2	no	—
48	C₂²×C₄×C₅₉	—	4	4	no	—
49	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₅₉	—	4	4	no	—
62	M₁₆×C₅₉	—	4	4	no	—
63	C₂×C₈×C₅₉	—	2	8	no	—
64	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
65	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₉	C₂×D₄	1	2	yes	—
74	C₄²×C₅₉	C₂×D₄	1	2	yes	—
75	#11745664	—	2	8	no	—
76	#11745664	—	2	8	no	—
77	C₂×C₅₉⋊C₈	—	2	8	no	—
78	C₂³×C₅₉	—	2	2	no	—
79	C₂³×C₅₉	—	4	2	no	—
80	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₅₉	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	2	1	no	—
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	4	no	—
91	C₂×C₄×C₅₉	—	2	4	no	—
92	C₂×C₄×C₅₉	—	2	4	no	—
93	C₂×C₄×C₅₉	—	4	2	no	—
94	C₂×C₄×C₅₉	—	4	2	no	—
95	C₂×C₄×C₅₉	—	4	2	no	—
96	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
105	C₂×C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
106	C₈×C₅₉	—	2	8	no	—
107	C₅₉⋊C₈	—	4	8	no	—
108	#21970	—	59	1	no	maximal, Sylow
109	C₂²×C₅₉	—	2	2	no	—
110	C₂²×C₅₉	—	2	1	no	—
111	C₂²×C₅₉	—	2	1	no	—
112	C₂²×C₅₉	—	2	2	no	—
113	C₂²×C₅₉	—	2	2	no	—
114	C₂²×C₅₉	—	4	4	no	—
115	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₅₉	#544	1	2	yes	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	2	no	—
120	C₄×C₅₉	—	2	2	no	—
121	C₄×C₅₉	—	2	2	no	—
122	C₄×C₅₉	#430	1	4	yes	—
123	C₄×C₅₉	#518	1	1	yes	characteristic
124	C₄×C₅₉	#519	1	1	yes	characteristic
125	C₄×C₅₉	#537	1	2	yes	—
126	#2920	—	59	2	no	—
127	C₂×#423	D₅₉	1	1	yes	characteristic
128	#1950	—	59	8	no	—
129	#1811	—	59	4	no	—
130	C₂×C₅₉	—	2	2	no	—
131	C₂×C₅₉	—	4	1	no	—
132	C₂×C₅₉	—	4	1	no	—
133	C₂×C₅₉	#2316	1	1	yes	characteristic
134	C₂×C₅₉	C₂×#430	1	2	yes	—
135	C₂²×M₁₆	C₂×D₅₉	1	2	yes	—
136	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
137	#432	—	118	2	no	—
138	#432	—	118	2	no	—
139	#431	—	118	2	no	—
140	#431	—	118	2	no	—
141	C₂×#108	—	59	8	no	—
142	#423	—	59	4	no	—
143	#423	Dic₅₉	1	2	yes	—
144	#423	Dic₅₉	1	4	yes	—
145	#423	Dic₅₉	1	2	yes	—
146	C₂×C₄⋊C₈	C₂×D₅₉	1	4	yes	—
147	#407	—	59	4	no	—
148	C₄×M₁₆	—	59	4	no	—
149	#341	—	59	8	no	—
150	C₅₉	#21970	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	Dic₁₁₈	1	2	yes	—
152	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
153	C₂×M₁₆	—	4	4	no	—
154	C₂×M₁₆	—	59	8	no	—
155	C₂×M₁₆	C₂×Dic₅₉	1	4	yes	—
156	C₂×M₁₆	C₂×Dic₅₉	1	4	yes	—
157	C₂²×C₈	C₂²×D₅₉	1	4	yes	—
158	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
159	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
160	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
161	C₂×C₄²	D₂₃₆	1	2	yes	—
162	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
165	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
167	#108	—	236	8	no	—
168	C₄⋊C₈	—	118	4	no	—
169	C₄⋊C₈	—	118	4	no	—
170	C₄⋊C₈	C₂×Dic₅₉	1	8	yes	—
171	C₄⋊C₈	C₂×Dic₅₉	1	8	yes	—
172	C₂²⋊C₈	—	118	4	no	—
173	C₂²⋊C₈	—	118	4	no	—
174	#97	—	118	4	no	—
175	C₄×C₈	—	118	4	no	—
176	C₂⁴	C₂×Dic₁₁₈	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
181	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
182	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
183	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
184	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
186	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
187	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
192	M₁₆	—	4	4	no	—
193	M₁₆	—	4	4	no	—
194	M₁₆	—	118	8	no	—
195	M₁₆	—	118	8	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	118	8	no	—
198	C₂×C₈	C₂²×Dic₅₉	1	8	yes	—
199	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
200	C₄²	C₄⋊Dic₅₉	1	2	yes	—
201	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
202	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
203	C₄²	C₄⋊Dic₅₉	1	2	yes	—
204	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
205	C₄²	C₂×D₂₃₆	1	2	yes	—
206	C₄²	C₂×D₂₃₆	1	2	yes	—
207	C₄²	C₂²×Dic₅₉	1	2	yes	—
208	C₄²	C₂²×Dic₅₉	1	2	yes	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
212	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
213	C₂³	C₂²×Dic₁₁₈	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
233	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
235	C₂×C₄	#14273772	1	2	yes	—
236	C₂×C₄	#14273775	1	2	yes	—
237	C₈	—	2	8	no	—
238	C₈	—	236	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	4	no	—
245	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
246	C₂²	#16838033	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	#16837451	1	4	yes	—
253	C₄	#16837877	1	1	yes	characteristic
254	C₄	#16837880	1	1	yes	characteristic
255	C₄	#16837983	1	2	yes	—
256	C₂	—	2	2	no	—
257	C₂	—	4	1	no	—
258	C₂	—	4	1	no	—
259	C₂	#21426714	1	1	yes	characteristic
260	C₂	C₂×#16837451	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₅₉×#423	C₂
socle	C₂²×C₅₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₉	C₂×Q₈
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×#423	D₅₉
hypocenter	C₅₉	#21970
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #21424872	C₂	semidirect
150. C₅₉	#21970	semidirect

Maximal subgroups

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

subgroup	index
#21432729	2
#21424872	2
#21424138	2
C₂×C₅₉×#423	2
#21970	59