Subgroups — C₂×#21423977 #28586697

Order 15104.

1102 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273663	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837358	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#21423977	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16837161	C₂	1	2	yes	maximal
6	C₂×C₅₉×#427	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11745664	C₂²	1	2	yes	—
8	C₂³×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×#14273663	C₄	1	8	yes	complemented
11	C₂×#14273663	C₂²	1	4	yes	—
12	#16837358	—	2	4	no	—
13	#16837358	C₂²	1	8	yes	—
14	C₂×#14273638	—	2	2	no	—
15	C₂×C₄×C₅₉⋊C₈	—	2	2	no	—
16	C₂×C₈×Dic₅₉	C₄	1	8	yes	complemented
17	C₂×C₈×Dic₅₉	C₂²	1	4	yes	—
18	#16837161	C₂²	1	8	yes	—
19	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#117×C₅₉	C₂²	1	2	yes	—
21	C₅₉×#427	C₄	1	4	yes	complemented
22	C₅₉×#427	C₂²	1	4	yes	—
23	C₂×#107×C₅₉	C₄	1	2	yes	complemented
24	C₂×#106×C₅₉	C₄	1	2	yes	complemented
25	C₂³×Dic₅₉	—	2	1	no	—
26	C₂×#11745664	—	2	2	no	—
27	C₂×#11745664	—	2	4	no	—
28	C₂×#11745664	C₂×C₄	1	8	yes	—
29	C₂×#11745664	C₂³	1	4	yes	—
30	C₂²×C₅₉⋊C₈	—	2	2	no	—
31	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
35	C₂²×C₅₉⋊C₈	Q₈	1	4	yes	—
36	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
37	C₂×C₄×Dic₅₉	—	2	1	no	—
38	C₂×C₄×Dic₅₉	—	2	2	no	—
39	C₂×C₄×Dic₅₉	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
41	#14273663	C₂×C₄	1	16	yes	complemented
42	#14273638	—	2	8	no	—
43	C₄×C₅₉⋊C₈	—	2	8	no	—
44	C₈×Dic₅₉	C₂×C₄	1	16	yes	complemented
45	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
52	#117×C₅₉	—	2	2	no	—
53	#117×C₅₉	—	2	2	no	—
54	#117×C₅₉	C₂×C₄	1	4	yes	—
55	#117×C₅₉	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
57	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
58	C₂×C₄²×C₅₉	—	2	2	no	—
59	#107×C₅₉	C₂×C₄	1	8	yes	complemented
60	#106×C₅₉	C₂×C₄	1	8	yes	complemented
61	C₂⁴×C₅₉	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	—	2	4	no	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	#41	1	2	yes	—
72	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₉	—	2	8	no	—
74	C₂²×Dic₅₉	—	2	4	no	—
75	C₂²×Dic₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	C₄²	1	4	yes	—
77	C₂×C₈×C₅₉	C₄²	1	4	yes	—
78	C₂×C₈×C₅₉	C₄²	1	8	yes	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₅₉	—	2	4	no	—
84	C₄⋊C₄×C₅₉	C₂²×C₄	1	8	yes	—
85	C₂²⋊C₄×C₅₉	—	2	4	no	—
86	C₂²⋊C₄×C₅₉	—	2	4	no	—
87	C₄²×C₅₉	—	2	4	no	—
88	C₄²×C₅₉	—	2	4	no	—
89	C₄×Dic₅₉	—	2	4	no	—
90	C₄×Dic₅₉	—	2	2	no	—
91	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
92	#11745664	—	2	8	no	—
93	#11745664	—	2	4	no	—
94	#11745664	C₂²×C₄	1	8	yes	—
95	C₂×C₅₉⋊C₈	—	2	8	no	—
96	C₂×C₅₉⋊C₈	—	2	4	no	—
97	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
98	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
99	C₂×C₅₉⋊C₈	C₄⋊C₄	1	8	yes	—
100	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₅₉⋊C₈	C₂×Q₈	1	4	yes	—
104	C₂³×C₅₉	—	2	4	no	—
105	C₂³×C₅₉	#117	1	2	yes	—
106	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₅₉	C₄×Q₈	1	2	yes	—
109	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₅₉	—	2	8	no	—
111	C₂×C₄×C₅₉	—	4	8	no	—
112	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₉	#117	1	2	yes	—
117	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₅₉	—	2	8	no	—
131	C₂×Dic₅₉	—	2	16	no	—
132	C₂×Dic₅₉	—	2	4	no	—
133	C₈×C₅₉	C₂×C₄²	1	4	yes	—
134	C₈×C₅₉	C₂×C₄²	1	4	yes	—
135	C₅₉⋊C₈	—	2	8	no	—
136	C₅₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
137	C₅₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
138	C₂×#1772	—	59	1	no	maximal, Sylow
139	C₂²×C₅₉	—	2	2	no	—
140	C₂²×C₅₉	—	2	2	no	—
141	C₂²×C₅₉	—	2	4	no	—
142	C₂²×C₅₉	—	2	4	no	—
143	C₂²×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
144	C₂²×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
145	C₂²×C₅₉	#443	1	4	yes	—
146	C₂²×C₅₉	C₂×#117	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₅₉	—	4	8	no	—
150	C₄×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
151	C₄×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
153	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
156	Dic₅₉	—	2	8	no	—
157	C₂²×#108	—	59	1	no	—
158	C₂×#427	D₅₉	1	1	yes	characteristic
159	C₂×#405	—	59	2	no	—
160	C₂²×C₄×C₈	—	59	1	no	—
161	#1772	—	59	8	no	—
162	C₂×#339	—	59	2	no	—
163	C₂×C₅₉	—	2	1	no	—
164	C₂×C₅₉	—	2	1	no	—
165	C₂×C₅₉	—	2	2	no	—
166	C₂×C₅₉	#1772	1	4	yes	complemented
167	C₂×C₅₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#443	1	2	yes	—
169	C₂²×M₁₆	—	59	2	no	—
170	C₂³×C₈	—	59	1	no	—
171	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
172	C₂×#117	C₂×D₅₉	1	2	yes	—
173	C₂²×C₄²	—	59	1	no	—
174	C₂×#108	—	59	8	no	—
175	C₂×#108	—	59	4	no	—
176	#427	Dic₅₉	1	4	yes	complemented
177	#427	C₂×D₅₉	1	4	yes	—
178	C₂×#107	Dic₅₉	1	2	yes	complemented
179	C₂×#106	Dic₅₉	1	2	yes	complemented
180	#405	—	59	8	no	—
181	#405	—	118	4	no	—
182	C₂×#97	—	118	2	no	—
183	C₂×C₄×C₈	—	59	8	no	—
184	C₂×C₄×C₈	—	59	4	no	—
185	C₂×C₄×C₈	—	118	2	no	—
186	#339	—	59	8	no	—
187	C₅₉	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	118	1	no	—
189	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
190	C₂×M₁₆	—	59	8	no	—
191	C₂×M₁₆	—	59	4	no	—
192	C₂×M₁₆	—	118	2	no	—
193	C₂×M₁₆	—	118	4	no	—
194	C₂²×C₈	—	59	4	no	—
195	C₂²×C₈	—	59	2	no	—
196	C₂²×C₈	—	59	1	no	—
197	C₂²×C₈	—	59	1	no	—
198	C₂²×C₈	—	59	4	no	—
199	C₂²×C₈	—	59	2	no	—
200	C₂²×C₈	—	118	2	no	—
201	C₂²×C₈	C₄×D₅₉	1	4	yes	—
202	C₂²×C₈	C₄×D₅₉	1	4	yes	—
203	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
204	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
205	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
206	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	C₂×Dic₅₉	1	4	yes	—
210	#117	C₂²×D₅₉	1	4	yes	—
211	C₂×C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	59	4	no	—
215	C₂×C₄²	—	59	2	no	—
216	C₂×C₄²	—	118	1	no	—
217	C₂×C₄²	—	118	2	no	—
218	#108	—	59	16	no	—
219	#107	C₂×Dic₅₉	1	8	yes	complemented
220	#106	C₂×Dic₅₉	1	8	yes	complemented
221	#97	—	118	8	no	—
222	C₄×C₈	—	59	16	no	—
223	C₄×C₈	—	118	8	no	—
224	C₂⁴	#11745687	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	118	8	no	—
227	C₂²×C₄	—	118	4	no	—
228	C₂²×C₄	—	118	2	no	—
229	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
231	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
233	C₂²×C₄	#11745687	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₅₉	1	2	yes	—
235	C₂²×C₄	#11745689	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
238	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
239	M₁₆	—	59	8	no	—
240	M₁₆	—	118	8	no	—
241	M₁₆	—	118	4	no	—
242	C₂×C₈	—	59	8	no	—
243	C₂×C₈	—	59	4	no	—
244	C₂×C₈	—	59	4	no	—
245	C₂×C₈	—	59	4	no	—
246	C₂×C₈	—	59	2	no	—
247	C₂×C₈	—	59	2	no	—
248	C₂×C₈	—	59	4	no	—
249	C₂×C₈	—	118	8	no	—
250	C₂×C₈	—	118	4	no	—
251	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
252	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
253	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
254	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
255	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
256	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
257	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	C₂²×Dic₅₉	1	8	yes	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	59	4	no	—
265	C₄²	—	118	4	no	—
266	C₄²	—	118	2	no	—
267	C₂³	—	2	4	no	—
268	C₂³	#14273685	1	2	yes	—
269	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
270	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
271	C₂³	#14273692	1	2	yes	—
272	C₂³	C₂×#11745687	1	1	yes	characteristic
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	—	118	16	no	—
276	C₂×C₄	—	118	8	no	—
277	C₂×C₄	—	118	4	no	—
278	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
280	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
281	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
282	C₂×C₄	#14273685	1	2	yes	—
283	C₂×C₄	#14273689	1	2	yes	—
284	C₂×C₄	#14273689	1	2	yes	—
285	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
287	C₂×C₄	#14273692	1	2	yes	—
288	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
289	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
295	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
296	C₈	—	59	4	no	—
297	C₈	—	59	4	no	—
298	C₈	—	118	8	no	—
299	C₈	C₂×C₄×Dic₅₉	1	4	yes	—
300	C₈	C₂×C₄×Dic₅₉	1	4	yes	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₄⋊C₄×Dic₅₉	1	2	yes	—
306	C₂²	C₄⋊C₄×Dic₅₉	1	2	yes	—
307	C₂²	#16837519	1	4	yes	—
308	C₂²	C₂×#14273685	1	1	yes	characteristic
309	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
310	C₂²	C₂×#14273692	1	1	yes	characteristic
311	C₄	—	4	8	no	—
312	C₄	—	118	8	no	—
313	C₄	C₄⋊C₄×Dic₅₉	1	2	yes	—
314	C₄	C₄⋊C₄×Dic₅₉	1	2	yes	—
315	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
316	C₄	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
317	C₄	C₂×#14273689	1	1	yes	characteristic
318	C₄	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	#21423977	1	4	yes	direct product factor
323	C₂	C₂×C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
324	C₂	C₂×#16837519	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₅₉×#427	C₂
socle	C₂³×C₅₉	C₄×D₄
prime radical	C₂⁴×C₅₉	#41
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×#427	D₅₉
hypocenter	C₅₉	C₂×#1772
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #21423977	C₂	direct (N × H)
10. C₂×#14273663	C₄	semidirect
16. C₂×C₈×Dic₅₉	C₄	semidirect
21. C₅₉×#427	C₄	semidirect
23. C₂×#107×C₅₉	C₄	semidirect
24. C₂×#106×C₅₉	C₄	semidirect
41. #14273663	C₂×C₄	semidirect
44. C₈×Dic₅₉	C₂×C₄	semidirect
59. #107×C₅₉	C₂×C₄	semidirect
60. #106×C₅₉	C₂×C₄	semidirect
166. C₂×C₅₉	#1772	semidirect
176. #427	Dic₅₉	semidirect
178. C₂×#107	Dic₅₉	semidirect
179. C₂×#106	Dic₅₉	semidirect
187. C₅₉	C₂×#1772	semidirect
219. #107	C₂×Dic₅₉	semidirect
220. #106	C₂×Dic₅₉	semidirect
322. C₂	#21423977	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×#14273663	2
C₂×#16837358	2
C₂²×C₈×Dic₅₉	2
#21423977	2
C₂×#16837161	2
C₂×C₅₉×#427	2
C₂×#1772	59

Subgroups — C2×#21423977 #28586697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423977 #28586697