Subgroups — C₂²×C₅₉×#408 #28472170

Order 15104.

5326 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₉×#408	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂²⋊C₈×C₅₉	C₂	1	4	yes	maximal, complemented
6	C₂²×#41×C₅₉	—	2	8	no	—
7	C₂²×#41×C₅₉	—	2	8	no	—
8	C₂²×#41×C₅₉	C₄	1	12	yes	—
9	C₂²×#41×C₅₉	C₂²	1	12	yes	—
10	C₂³×Q₈×C₅₉	C₄	1	1	yes	characteristic
11	C₂³×D₄×C₅₉	C₄	1	2	yes	—
12	C₂³×D₄×C₅₉	C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₅₉	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₅₉	—	2	8	no	—
16	C₂²×M₁₆×C₅₉	C₂²	1	12	yes	complemented
17	C₂³×C₈×C₅₉	—	2	8	no	—
18	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
19	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
20	C₂³×C₈×C₅₉	C₂²	1	12	yes	complemented
21	C₅₉×#408	C₂²	1	64	yes	direct product factor
22	C₂×C₂²⋊C₈×C₅₉	C₂²	1	48	yes	complemented
23	C₂⁵×C₅₉	C₂×C₄	1	2	yes	—
24	C₂⁵×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×#41×C₅₉	—	2	96	no	—
26	C₂×#41×C₅₉	—	2	48	no	—
27	C₂×#41×C₅₉	—	2	48	no	—
28	C₂×#41×C₅₉	C₂×C₄	1	48	yes	—
29	C₂×#41×C₅₉	C₂³	1	16	yes	—
30	C₂²×Q₈×C₅₉	—	2	8	no	—
31	C₂²×Q₈×C₅₉	C₂×C₄	1	12	yes	—
32	C₂²×D₄×C₅₉	—	2	16	no	—
33	C₂²×D₄×C₅₉	—	2	8	no	—
34	C₂²×D₄×C₅₉	C₂×C₄	1	24	yes	—
35	C₂²×D₄×C₅₉	C₂×C₄	1	12	yes	—
36	C₂³×C₄×C₅₉	—	2	8	no	—
37	C₂³×C₄×C₅₉	—	2	8	no	—
38	C₂³×C₄×C₅₉	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₅₉	C₂×C₄	1	12	yes	—
40	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₉	C₂×C₄	1	6	yes	—
42	C₂³×C₄×C₅₉	D₄	1	8	yes	—
43	C₂³×C₄×C₅₉	D₄	1	8	yes	—
44	C₂³×C₄×C₅₉	C₂³	1	12	yes	—
45	C₂³×C₄×C₅₉	C₂³	1	6	yes	—
46	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₅₉	—	2	48	no	—
48	C₂×M₁₆×C₅₉	C₂³	1	16	yes	complemented
49	C₂²×C₈×C₅₉	—	2	8	no	—
50	C₂²×C₈×C₅₉	—	2	48	no	—
51	C₂²×C₈×C₅₉	—	2	8	no	—
52	C₂²×C₈×C₅₉	C₂³	1	12	yes	—
53	C₂²×C₈×C₅₉	C₂³	1	12	yes	—
54	C₂²×C₈×C₅₉	C₂³	1	16	yes	complemented
55	C₂²⋊C₈×C₅₉	C₂³	1	64	yes	complemented
56	C₂⁴×C₅₉	—	2	8	no	—
57	C₂⁴×C₅₉	—	2	8	no	—
58	C₂⁴×C₅₉	—	2	16	no	—
59	C₂⁴×C₅₉	C₂²⋊C₄	1	8	yes	—
60	C₂⁴×C₅₉	C₂²×C₄	1	12	yes	—
61	C₂⁴×C₅₉	C₂²×C₄	1	6	yes	—
62	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	#41×C₅₉	—	2	192	no	—
64	#41×C₅₉	—	2	32	no	—
65	#41×C₅₉	—	2	32	no	—
66	C₂×Q₈×C₅₉	—	2	48	no	—
67	C₂×Q₈×C₅₉	C₂²×C₄	1	16	yes	—
68	C₂×D₄×C₅₉	—	2	96	no	—
69	C₂×D₄×C₅₉	—	2	48	no	—
70	C₂×D₄×C₅₉	C₂²×C₄	1	32	yes	—
71	C₂×D₄×C₅₉	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₅₉	—	2	8	no	—
73	C₂²×C₄×C₅₉	—	2	48	no	—
74	C₂²×C₄×C₅₉	—	2	24	no	—
75	C₂²×C₄×C₅₉	—	2	24	no	—
76	C₂²×C₄×C₅₉	C₂²⋊C₄	1	8	yes	—
77	C₂²×C₄×C₅₉	C₂²⋊C₄	1	48	yes	—
78	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
79	C₂²×C₄×C₅₉	C₂²×C₄	1	24	yes	—
80	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
81	C₂²×C₄×C₅₉	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
83	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₅₉	C₂×D₄	1	24	yes	—
86	C₂²×C₄×C₅₉	C₂×D₄	1	24	yes	—
87	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
88	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
89	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
90	M₁₆×C₅₉	—	2	32	no	—
91	C₂×C₈×C₅₉	—	2	24	no	—
92	C₂×C₈×C₅₉	—	2	32	no	—
93	C₂×C₈×C₅₉	—	2	24	no	—
94	C₂×C₈×C₅₉	C₂⁴	1	8	yes	—
95	C₂×C₈×C₅₉	C₂⁴	1	8	yes	—
96	C₂³×C₅₉	—	2	24	no	—
97	C₂³×C₅₉	—	2	32	no	—
98	C₂³×C₅₉	—	2	24	no	—
99	C₂³×C₅₉	—	2	48	no	—
100	C₂³×C₅₉	—	4	32	no	—
101	C₂³×C₅₉	C₂×C₂²⋊C₄	1	24	yes	—
102	C₂³×C₅₉	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂³×C₅₉	#131	1	8	yes	—
104	C₂³×C₅₉	C₂³×C₄	1	8	yes	—
105	C₂³×C₅₉	C₂³×C₄	1	4	yes	—
106	C₂³×C₅₉	C₂³×C₄	1	3	yes	—
107	Q₈×C₅₉	—	2	32	no	—
108	D₄×C₅₉	—	2	64	no	—
109	D₄×C₅₉	—	2	32	no	—
110	C₂×C₄×C₅₉	—	2	24	no	—
111	C₂×C₄×C₅₉	—	2	48	no	—
112	C₂×C₄×C₅₉	—	2	8	no	—
113	C₂×C₄×C₅₉	—	2	8	no	—
114	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	12	yes	—
115	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	24	yes	—
116	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	48	yes	—
117	C₂×C₄×C₅₉	C₂³×C₄	1	8	yes	—
118	C₂×C₄×C₅₉	C₂³×C₄	1	3	yes	—
119	C₂×C₄×C₅₉	C₂³×C₄	1	4	yes	—
120	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
121	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
122	C₂×C₄×C₅₉	C₂²×D₄	1	8	yes	—
123	C₂×C₄×C₅₉	C₂²×D₄	1	8	yes	—
124	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
125	C₈×C₅₉	—	2	8	no	—
126	C₈×C₅₉	—	2	8	no	—
127	C₂²×#408	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₉	—	2	48	no	—
129	C₂²×C₅₉	—	2	8	no	—
130	C₂²×C₅₉	—	2	8	no	—
131	C₂²×C₅₉	—	2	16	no	—
132	C₂²×C₅₉	—	4	48	no	—
133	C₂²×C₅₉	#408	1	16	yes	direct product factor
134	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	8	yes	—
135	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	6	yes	—
136	C₂²×C₅₉	C₂×#131	1	12	yes	—
137	C₂²×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
138	C₄×C₅₉	—	2	8	no	—
139	C₄×C₅₉	C₂²×C₂²⋊C₄	1	6	yes	—
140	C₄×C₅₉	C₂²×C₂²⋊C₄	1	8	yes	—
141	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
143	C₂³×#41	C₂×C₅₉	1	1	yes	characteristic
144	C₂³×M₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
145	C₂⁴×C₈	C₂×C₅₉	1	1	yes	characteristic, complemented
146	C₂×#408	C₂×C₅₉	1	24	yes	direct product factor
147	C₂²×C₂²⋊C₈	C₂×C₅₉	1	4	yes	complemented
148	C₂×C₅₉	—	2	8	no	—
149	C₂×C₅₉	—	4	8	no	—
150	C₂×C₅₉	C₂×#408	1	12	yes	direct product factor
151	C₂×C₅₉	C₂³×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₅₉	C₂²×#131	1	2	yes	—
153	C₂²×#41	—	2	8	no	—
154	C₂²×#41	—	2	8	no	—
155	C₂²×#41	C₄×C₅₉	1	12	yes	—
156	C₂²×#41	C₂²×C₅₉	1	12	yes	—
157	C₂³×Q₈	C₄×C₅₉	1	1	yes	characteristic
158	C₂³×D₄	C₄×C₅₉	1	2	yes	—
159	C₂³×D₄	C₄×C₅₉	1	1	yes	characteristic
160	C₂⁴×C₄	C₂²×C₅₉	1	2	yes	—
161	C₂⁴×C₄	C₂²×C₅₉	1	1	yes	characteristic
162	C₂²×M₁₆	—	2	8	no	—
163	C₂²×M₁₆	C₂²×C₅₉	1	12	yes	complemented
164	C₂³×C₈	—	2	8	no	—
165	C₂³×C₈	C₂²×C₅₉	1	2	yes	—
166	C₂³×C₈	C₂²×C₅₉	1	2	yes	—
167	C₂³×C₈	C₂²×C₅₉	1	12	yes	complemented
168	#408	C₂²×C₅₉	1	64	yes	direct product factor
169	C₂×C₂²⋊C₈	C₂²×C₅₉	1	48	yes	complemented
170	C₅₉	C₂²×#408	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁵	C₂×C₄×C₅₉	1	2	yes	—
172	C₂⁵	C₂×C₄×C₅₉	1	1	yes	characteristic
173	C₂×#41	—	2	96	no	—
174	C₂×#41	—	2	48	no	—
175	C₂×#41	—	2	48	no	—
176	C₂×#41	C₂×C₄×C₅₉	1	48	yes	—
177	C₂×#41	C₂³×C₅₉	1	16	yes	—
178	C₂²×Q₈	—	2	8	no	—
179	C₂²×Q₈	C₂×C₄×C₅₉	1	12	yes	—
180	C₂²×D₄	—	2	16	no	—
181	C₂²×D₄	—	2	8	no	—
182	C₂²×D₄	C₂×C₄×C₅₉	1	24	yes	—
183	C₂²×D₄	C₂×C₄×C₅₉	1	12	yes	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	C₂×C₄×C₅₉	1	2	yes	—
187	C₂³×C₄	C₂×C₄×C₅₉	1	12	yes	—
188	C₂³×C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₅₉	1	6	yes	—
190	C₂³×C₄	D₄×C₅₉	1	8	yes	—
191	C₂³×C₄	D₄×C₅₉	1	8	yes	—
192	C₂³×C₄	C₂³×C₅₉	1	12	yes	—
193	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
194	C₂³×C₄	C₂³×C₅₉	1	6	yes	—
195	C₂×M₁₆	—	2	48	no	—
196	C₂×M₁₆	C₂³×C₅₉	1	16	yes	complemented
197	C₂²×C₈	—	2	8	no	—
198	C₂²×C₈	—	2	48	no	—
199	C₂²×C₈	—	2	8	no	—
200	C₂²×C₈	C₂³×C₅₉	1	12	yes	—
201	C₂²×C₈	C₂³×C₅₉	1	12	yes	—
202	C₂²×C₈	C₂³×C₅₉	1	16	yes	complemented
203	C₂²⋊C₈	C₂³×C₅₉	1	64	yes	complemented
204	C₂⁴	—	2	8	no	—
205	C₂⁴	—	2	8	no	—
206	C₂⁴	—	2	16	no	—
207	C₂⁴	C₂²⋊C₄×C₅₉	1	8	yes	—
208	C₂⁴	C₂²×C₄×C₅₉	1	12	yes	—
209	C₂⁴	C₂²×C₄×C₅₉	1	6	yes	—
210	C₂⁴	C₂²×C₄×C₅₉	1	1	yes	characteristic
211	#41	—	2	192	no	—
212	#41	—	2	32	no	—
213	#41	—	2	32	no	—
214	C₂×Q₈	—	2	48	no	—
215	C₂×Q₈	C₂²×C₄×C₅₉	1	16	yes	—
216	C₂×D₄	—	2	96	no	—
217	C₂×D₄	—	2	48	no	—
218	C₂×D₄	C₂²×C₄×C₅₉	1	32	yes	—
219	C₂×D₄	C₂²×C₄×C₅₉	1	16	yes	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	48	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	2	24	no	—
224	C₂²×C₄	C₂²⋊C₄×C₅₉	1	8	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₅₉	1	48	yes	—
226	C₂²×C₄	C₂²×C₄×C₅₉	1	12	yes	—
227	C₂²×C₄	C₂²×C₄×C₅₉	1	24	yes	—
228	C₂²×C₄	C₂²×C₄×C₅₉	1	3	yes	—
229	C₂²×C₄	C₂²×C₄×C₅₉	1	6	yes	—
230	C₂²×C₄	C₂²×C₄×C₅₉	1	12	yes	—
231	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
232	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
233	C₂²×C₄	C₂×D₄×C₅₉	1	24	yes	—
234	C₂²×C₄	C₂×D₄×C₅₉	1	24	yes	—
235	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
236	C₂²×C₄	C₂⁴×C₅₉	1	3	yes	—
237	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
238	M₁₆	—	2	32	no	—
239	C₂×C₈	—	2	24	no	—
240	C₂×C₈	—	2	32	no	—
241	C₂×C₈	—	2	24	no	—
242	C₂×C₈	C₂⁴×C₅₉	1	8	yes	—
243	C₂×C₈	C₂⁴×C₅₉	1	8	yes	—
244	C₂³	—	2	32	no	—
245	C₂³	—	2	24	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	2	48	no	—
248	C₂³	—	4	32	no	—
249	C₂³	C₂×C₂²⋊C₄×C₅₉	1	24	yes	—
250	C₂³	C₂×C₂²⋊C₄×C₅₉	1	4	yes	—
251	C₂³	#131×C₅₉	1	8	yes	—
252	C₂³	C₂³×C₄×C₅₉	1	8	yes	—
253	C₂³	C₂³×C₄×C₅₉	1	4	yes	—
254	C₂³	C₂³×C₄×C₅₉	1	3	yes	—
255	Q₈	—	2	32	no	—
256	D₄	—	2	64	no	—
257	D₄	—	2	32	no	—
258	C₂×C₄	—	2	24	no	—
259	C₂×C₄	—	2	48	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	12	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	24	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	48	yes	—
265	C₂×C₄	C₂³×C₄×C₅₉	1	8	yes	—
266	C₂×C₄	C₂³×C₄×C₅₉	1	3	yes	—
267	C₂×C₄	C₂³×C₄×C₅₉	1	4	yes	—
268	C₂×C₄	C₂²×D₄×C₅₉	1	6	yes	—
269	C₂×C₄	C₂²×D₄×C₅₉	1	6	yes	—
270	C₂×C₄	C₂²×D₄×C₅₉	1	8	yes	—
271	C₂×C₄	C₂²×D₄×C₅₉	1	8	yes	—
272	C₂×C₄	C₂⁵×C₅₉	1	1	yes	characteristic
273	C₈	—	2	8	no	—
274	C₈	—	2	8	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	2	48	no	—
277	C₂²	—	2	16	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	4	48	no	—
280	C₂²	C₅₉×#408	1	16	yes	direct product factor
281	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	8	yes	—
282	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	6	yes	—
283	C₂²	C₂×#131×C₅₉	1	12	yes	—
284	C₂²	C₂⁴×C₄×C₅₉	1	1	yes	characteristic
285	C₄	—	2	8	no	—
286	C₄	C₂²×C₂²⋊C₄×C₅₉	1	6	yes	—
287	C₄	C₂²×C₂²⋊C₄×C₅₉	1	8	yes	—
288	C₄	C₂³×D₄×C₅₉	1	1	yes	characteristic
289	C₄	C₂³×D₄×C₅₉	1	1	yes	characteristic
290	C₂	—	2	8	no	—
291	C₂	—	4	8	no	—
292	C₂	C₂×C₅₉×#408	1	12	yes	direct product factor
293	C₂	C₂³×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
294	C₂	C₂²×#131×C₅₉	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₉	C₂³
commutator	C₂²	C₂⁴×C₄×C₅₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₅₉
socle	C₂⁴×C₅₉	C₂²×C₄
prime radical	C₂³×#41×C₅₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₅₉	C₂	semidirect
3. C₂⁴×C₈×C₅₉	C₂	semidirect
4. C₂×C₅₉×#408	C₂	direct (N × H)
5. C₂²×C₂²⋊C₈×C₅₉	C₂	semidirect
16. C₂²×M₁₆×C₅₉	C₂²	semidirect
20. C₂³×C₈×C₅₉	C₂²	semidirect
21. C₅₉×#408	C₂²	direct (N × H)
22. C₂×C₂²⋊C₈×C₅₉	C₂²	semidirect
48. C₂×M₁₆×C₅₉	C₂³	semidirect
54. C₂²×C₈×C₅₉	C₂³	semidirect
55. C₂²⋊C₈×C₅₉	C₂³	semidirect
127. C₂²×#408	C₅₉	direct (N × H)
133. C₂²×C₅₉	#408	direct (N × H)
144. C₂³×M₁₆	C₂×C₅₉	semidirect
145. C₂⁴×C₈	C₂×C₅₉	semidirect
146. C₂×#408	C₂×C₅₉	direct (N × H)
147. C₂²×C₂²⋊C₈	C₂×C₅₉	semidirect
150. C₂×C₅₉	C₂×#408	direct (N × H)
163. C₂²×M₁₆	C₂²×C₅₉	semidirect
167. C₂³×C₈	C₂²×C₅₉	semidirect
168. #408	C₂²×C₅₉	direct (N × H)
169. C₂×C₂²⋊C₈	C₂²×C₅₉	semidirect
170. C₅₉	C₂²×#408	direct (N × H)
196. C₂×M₁₆	C₂³×C₅₉	semidirect
202. C₂²×C₈	C₂³×C₅₉	semidirect
203. C₂²⋊C₈	C₂³×C₅₉	semidirect
280. C₂²	C₅₉×#408	direct (N × H)
292. C₂	C₂×C₅₉×#408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₅₉	2
C₂³×M₁₆×C₅₉	2
C₂⁴×C₈×C₅₉	2
C₂×C₅₉×#408	2
C₂²×C₂²⋊C₈×C₅₉	2
C₂²×#408	59

Subgroups — C22×C59×#408 #28472170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₉×#408 #28472170