Subgroups — C₂×#21422184 #28503557

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273667	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
3	#21422184	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11745664	C₄	1	1	yes	characteristic
6	C₂×#14273686	C₄	1	1	yes	characteristic
7	C₂×#14273667	—	2	1	no	—
8	C₂×#14273667	—	2	2	no	—
9	C₂×#14273667	C₄	1	4	yes	complemented
10	C₂×#14273667	C₂²	1	2	yes	—
11	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
12	#16837357	—	2	1	no	—
13	#16837357	—	2	1	no	—
14	#16837357	C₄	1	2	yes	—
15	#16837357	C₂²	1	2	yes	—
16	C₂×#14273637	—	2	1	no	—
17	C₂×C₈×Dic₅₉	—	2	1	no	—
18	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₅₉	—	2	2	no	—
20	C₂²×C₁₆×C₅₉	—	2	1	no	—
21	C₂²×C₁₆×C₅₉	—	2	1	no	—
22	C₂²×C₁₆×C₅₉	C₄	1	4	yes	complemented
23	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
24	C₂×#11745664	—	2	1	no	—
25	C₂×#11745664	—	2	2	no	—
26	C₂×#11745664	C₈	1	4	yes	—
27	C₂×#11745664	C₂×C₄	1	2	yes	—
28	C₂²×C₅₉⋊C₈	—	2	1	no	—
29	#14273686	—	2	1	no	—
30	#14273686	—	2	1	no	—
31	#14273686	C₈	1	2	yes	—
32	#14273686	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₅₉	C₈	1	2	yes	—
34	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
35	C₂×C₄×Dic₅₉	—	2	1	no	—
36	#14273667	—	2	4	no	—
37	#14273667	—	2	2	no	—
38	#14273667	C₂×C₄	1	4	yes	complemented
39	C₂×C₅₉⋊C₁₆	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	—	2	2	no	—
41	#14273637	—	2	2	no	—
42	#14273637	—	2	2	no	—
43	C₈×Dic₅₉	—	2	2	no	—
44	C₈×Dic₅₉	—	2	2	no	—
45	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	D₄	1	2	yes	—
52	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
55	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₅₉	—	2	4	no	—
57	C₂×C₁₆×C₅₉	—	2	4	no	—
58	C₂×C₁₆×C₅₉	—	2	2	no	—
59	C₂×C₁₆×C₅₉	—	2	2	no	—
60	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
61	C₂⁴×C₅₉	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
71	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₅₉	—	2	2	no	—
73	C₂×C₈×C₅₉	C₄²	1	2	yes	—
74	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
91	C₂²⋊Dic₅₉	—	2	2	no	—
92	C₂²⋊Dic₅₉	—	2	2	no	—
93	C₄⋊Dic₅₉	—	2	2	no	—
94	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
95	C₄×Dic₅₉	—	2	2	no	—
96	C₄×Dic₅₉	—	2	2	no	—
97	#11745664	—	2	4	no	—
98	#11745664	—	2	2	no	—
99	#11745664	C₂×C₈	1	4	yes	—
100	C₂×C₅₉⋊C₈	—	2	4	no	—
101	C₂×C₅₉⋊C₈	—	2	2	no	—
102	C₁₆×C₅₉	—	2	4	no	—
103	C₅₉⋊C₁₆	—	2	4	no	—
104	C₂³×C₅₉	—	2	4	no	—
105	C₂³×C₅₉	#97	1	2	yes	—
106	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₅₉	C₄⋊C₈	1	2	yes	—
109	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#95	1	2	yes	—
111	C₂×C₄×C₅₉	#95	1	2	yes	—
112	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₅₉	#97	1	2	yes	—
115	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
118	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
128	C₂×Dic₅₉	—	2	4	no	—
129	C₂×Dic₅₉	—	4	4	no	—
130	C₈×C₅₉	#95	1	2	yes	—
131	C₈×C₅₉	#95	1	2	yes	—
132	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₅₉⋊C₈	—	2	4	no	—
137	C₂×#1379	—	59	1	no	maximal, Sylow
138	C₂²×C₅₉	—	2	2	no	—
139	C₂²×C₅₉	—	2	2	no	—
140	C₂²×C₅₉	—	2	4	no	—
141	C₂²×C₅₉	—	2	4	no	—
142	C₂²×C₅₉	#336	1	2	yes	—
143	C₂²×C₅₉	#336	1	2	yes	—
144	C₂²×C₅₉	#350	1	4	yes	—
145	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
146	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₅₉	#336	1	2	yes	—
149	C₄×C₅₉	#336	1	2	yes	—
150	C₄×C₅₉	C₂×#95	1	1	yes	characteristic
151	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
154	Dic₅₉	—	4	4	no	—
155	C₂²×M₃₂	—	59	1	no	—
156	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
157	C₂×#405	—	59	1	no	—
158	#1379	—	59	4	no	—
159	C₂×C₅₉	—	2	2	no	—
160	C₂×C₅₉	—	2	1	no	—
161	C₂×C₅₉	—	2	1	no	—
162	C₂×C₅₉	#1379	1	4	yes	complemented
163	C₂×C₅₉	C₂×#336	1	1	yes	characteristic
164	C₂×C₅₉	C₂×#350	1	2	yes	—
165	C₂²×M₁₆	—	59	1	no	—
166	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
167	C₂×#117	—	59	1	no	—
168	C₂×M₃₂	—	59	4	no	—
169	C₂×M₃₂	—	59	2	no	—
170	C₂×M₃₂	—	118	1	no	—
171	C₂×M₃₂	—	118	2	no	—
172	C₂²×C₁₆	—	2	1	no	—
173	C₂²×C₁₆	—	2	1	no	—
174	C₂²×C₁₆	—	2	2	no	—
175	C₂²×C₁₆	—	118	1	no	—
176	C₂²×C₁₆	Dic₅₉	1	4	yes	complemented
177	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
178	#405	—	59	2	no	—
179	#405	—	59	2	no	—
180	#405	—	118	1	no	—
181	#405	—	118	1	no	—
182	C₂×#97	—	118	1	no	—
183	C₂×C₄×C₈	—	118	1	no	—
184	C₅₉	C₂×#1379	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
186	C₂×M₁₆	—	59	4	no	—
187	C₂×M₁₆	—	59	2	no	—
188	C₂×M₁₆	—	118	2	no	—
189	C₂×M₁₆	—	118	1	no	—
190	C₂²×C₈	—	118	1	no	—
191	C₂²×C₈	Dic₁₁₈	1	2	yes	—
192	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
193	C₂²×C₈	C₄×D₅₉	1	2	yes	—
194	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
195	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
196	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
197	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
200	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
201	#117	—	59	2	no	—
202	#117	—	59	2	no	—
203	#117	—	118	1	no	—
204	#117	—	118	1	no	—
205	C₂×C₄⋊C₄	—	59	2	no	—
206	C₂×C₂²⋊C₄	—	118	1	no	—
207	C₂×C₄²	—	118	1	no	—
208	M₃₂	—	59	4	no	—
209	M₃₂	—	118	4	no	—
210	M₃₂	—	118	2	no	—
211	C₂×C₁₆	—	2	4	no	—
212	C₂×C₁₆	—	2	4	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	—	2	2	no	—
215	C₂×C₁₆	—	118	4	no	—
216	C₂×C₁₆	—	118	2	no	—
217	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	complemented
218	#97	—	118	2	no	—
219	#97	—	118	2	no	—
220	C₄×C₈	—	118	2	no	—
221	C₄×C₈	—	118	2	no	—
222	C₂⁴	#11745659	1	1	yes	characteristic
223	C₂²×C₄	—	118	2	no	—
224	C₂²×C₄	C₈×D₅₉	1	2	yes	—
225	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
226	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
227	C₂²×C₄	#11745659	1	1	yes	characteristic
228	C₂²×C₄	#11745659	1	2	yes	—
229	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
230	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
231	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
234	M₁₆	—	59	4	no	—
235	M₁₆	—	118	4	no	—
236	M₁₆	—	118	2	no	—
237	C₂×C₈	—	118	4	no	—
238	C₂×C₈	—	118	2	no	—
239	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
240	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
241	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
242	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
243	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
244	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
245	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
246	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
247	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
248	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
249	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
250	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
251	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
253	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
254	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	59	4	no	—
258	C₄⋊C₄	—	118	2	no	—
259	C₂²⋊C₄	—	118	2	no	—
260	C₂²⋊C₄	—	118	2	no	—
261	C₄²	—	118	2	no	—
262	C₄²	—	118	2	no	—
263	C₁₆	—	2	4	no	—
264	C₁₆	—	118	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	#14273637	1	2	yes	—
267	C₂³	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
268	C₂³	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
269	C₂³	Dic₅₉⋊C₈	1	2	yes	—
270	C₂³	C₂×#11745659	1	1	yes	characteristic
271	C₂×C₄	—	118	4	no	—
272	C₂×C₄	—	236	4	no	—
273	C₂×C₄	C₅₉⋊#95	1	2	yes	—
274	C₂×C₄	C₅₉⋊#95	1	2	yes	—
275	C₂×C₄	C₈×Dic₅₉	1	2	yes	—
276	C₂×C₄	C₈×Dic₅₉	1	2	yes	—
277	C₂×C₄	#14273637	1	2	yes	—
278	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
279	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
280	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
281	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
282	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
283	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
284	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
286	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
290	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
291	C₈	—	118	4	no	—
292	C₈	C₅₉⋊#95	1	2	yes	—
293	C₈	C₅₉⋊#95	1	2	yes	—
294	C₈	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
295	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
296	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
297	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#16837156	1	2	yes	—
303	C₂²	#16837156	1	2	yes	—
304	C₂²	#16837185	1	4	yes	—
305	C₂²	C₂×#14273637	1	1	yes	characteristic
306	C₂²	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
307	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
308	C₄	—	236	4	no	—
309	C₄	#16837156	1	2	yes	—
310	C₄	#16837156	1	2	yes	—
311	C₄	C₂×C₅₉⋊#95	1	1	yes	characteristic
312	C₄	C₂×C₈×Dic₅₉	1	1	yes	characteristic
313	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
314	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	#21422184	1	4	yes	direct product factor
319	C₂	C₂×#16837156	1	1	yes	characteristic
320	C₂	C₂×#16837185	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₂₃₆
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂²⋊C₈
prime radical	C₂⁴×C₅₉	M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂³×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#1379
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21422184	C₂	direct (N × H)
9. C₂×#14273667	C₄	semidirect
22. C₂²×C₁₆×C₅₉	C₄	semidirect
38. #14273667	C₂×C₄	semidirect
60. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
162. C₂×C₅₉	#1379	semidirect
176. C₂²×C₁₆	Dic₅₉	semidirect
184. C₅₉	C₂×#1379	semidirect
217. C₂×C₁₆	C₂×Dic₅₉	semidirect
318. C₂	#21422184	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273667	2
C₂×#16837357	2
#21422184	2
C₂³×C₁₆×C₅₉	2
C₂×#1379	59

Subgroups — C2×#21422184 #28503557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422184 #28503557