Subgroups — C₂×C₅₉×#2078 #28435753

Order 15104.

950 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#417	C₂	1	2	yes	maximal
3	C₂×C₅₉×#408	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2078	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₉×#343	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₅₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
12	C₂×#107×C₅₉	—	2	2	no	—
13	C₂×#107×C₅₉	—	2	2	no	—
14	C₂×#107×C₅₉	C₂²	1	4	yes	complemented
15	C₅₉×#417	—	2	4	no	—
16	C₅₉×#417	C₂²	1	8	yes	—
17	C₂×#103×C₅₉	—	2	2	no	—
18	C₂×#102×C₅₉	—	2	2	no	—
19	C₅₉×#408	—	2	2	no	—
20	C₅₉×#408	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₈×C₅₉	—	2	2	no	—
22	C₅₉×#343	C₂²	1	8	yes	complemented
23	C₅₉×#341	C₂²	1	4	yes	complemented
24	C₂×#41×C₅₉	—	2	1	no	—
25	C₂×#41×C₅₉	—	2	1	no	—
26	C₂×#41×C₅₉	—	2	4	no	—
27	C₂×#41×C₅₉	—	2	4	no	—
28	C₂×#41×C₅₉	D₄	1	2	yes	—
29	C₂×#41×C₅₉	C₂³	1	2	yes	—
30	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₅₉	—	2	1	no	—
32	C₂²×D₄×C₅₉	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₉	—	2	1	no	—
34	C₂³×C₄×C₅₉	—	2	2	no	—
35	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₅₉	—	2	1	no	—
37	C₂×M₁₆×C₅₉	—	2	2	no	—
38	C₂×M₁₆×C₅₉	—	2	4	no	—
39	C₂×M₁₆×C₅₉	—	2	4	no	—
40	C₂×M₁₆×C₅₉	—	2	4	no	—
41	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
42	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₉	—	2	4	no	—
44	C₂²×C₈×C₅₉	—	2	1	no	—
45	C₂²×C₈×C₅₉	—	2	2	no	—
46	C₂²×C₈×C₅₉	D₄	1	2	yes	—
47	C₂²×C₈×C₅₉	D₄	1	2	yes	—
48	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
55	#107×C₅₉	—	2	8	no	—
56	#103×C₅₉	—	2	8	no	—
57	#102×C₅₉	—	2	8	no	—
58	C₂²⋊C₈×C₅₉	—	2	8	no	—
59	C₂⁴×C₅₉	—	2	1	no	—
60	C₂⁴×C₅₉	#41	1	1	yes	characteristic
61	#41×C₅₉	—	2	8	no	—
62	#41×C₅₉	—	2	4	no	—
63	#41×C₅₉	—	2	4	no	—
64	#41×C₅₉	—	4	2	no	—
65	#41×C₅₉	—	4	2	no	—
66	#41×C₅₉	—	4	4	no	—
67	C₂×Q₈×C₅₉	—	2	1	no	—
68	C₂×Q₈×C₅₉	—	2	4	no	—
69	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×D₄×C₅₉	—	2	2	no	—
71	C₂×D₄×C₅₉	—	2	2	no	—
72	C₂×D₄×C₅₉	—	2	1	no	—
73	C₂×D₄×C₅₉	—	2	4	no	—
74	C₂×D₄×C₅₉	—	4	4	no	—
75	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₉	—	2	1	no	—
77	C₂²×C₄×C₅₉	—	2	2	no	—
78	C₂²×C₄×C₅₉	—	2	2	no	—
79	C₂²×C₄×C₅₉	—	2	4	no	—
80	C₂²×C₄×C₅₉	—	4	2	no	—
81	C₂²×C₄×C₅₉	—	4	2	no	—
82	C₂²×C₄×C₅₉	—	4	2	no	—
83	C₂²×C₄×C₅₉	—	4	2	no	—
84	C₂²×C₄×C₅₉	—	8	4	no	—
85	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
90	C₂²×C₄×C₅₉	#41	1	4	yes	—
91	C₂²×C₄×C₅₉	#41	1	4	yes	—
92	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₅₉	—	2	4	no	—
94	M₁₆×C₅₉	—	2	4	no	—
95	M₁₆×C₅₉	—	2	4	no	—
96	M₁₆×C₅₉	—	4	4	no	—
97	M₁₆×C₅₉	—	4	2	no	—
98	C₂×C₈×C₅₉	—	2	4	no	—
99	C₂×C₈×C₅₉	—	2	4	no	—
100	C₂×C₈×C₅₉	—	2	4	no	—
101	C₂×C₈×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	—	2	4	no	—
103	C₂×C₈×C₅₉	—	4	4	no	—
104	C₂×C₈×C₅₉	—	4	2	no	—
105	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₅₉	—	2	4	no	—
107	C₄⋊C₄×C₅₉	—	2	4	no	—
108	C₄⋊C₄×C₅₉	—	4	2	no	—
109	C₄⋊C₄×C₅₉	—	4	2	no	—
110	C₂³×C₅₉	—	2	2	no	—
111	C₂³×C₅₉	—	2	2	no	—
112	C₂³×C₅₉	—	2	2	no	—
113	C₂³×C₅₉	—	4	2	no	—
114	C₂³×C₅₉	—	4	2	no	—
115	C₂³×C₅₉	—	4	2	no	—
116	C₂³×C₅₉	#121	1	1	yes	characteristic
117	C₂³×C₅₉	#121	1	1	yes	characteristic
118	C₂³×C₅₉	#126	1	4	yes	—
119	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
120	Q₈×C₅₉	—	2	4	no	—
121	Q₈×C₅₉	—	4	2	no	—
122	D₄×C₅₉	—	2	4	no	—
123	D₄×C₅₉	—	4	4	no	—
124	D₄×C₅₉	—	4	4	no	—
125	D₄×C₅₉	—	4	2	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	4	2	no	—
128	C₂×C₄×C₅₉	—	4	2	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	4	4	no	—
131	C₂×C₄×C₅₉	—	4	4	no	—
132	C₂×C₄×C₅₉	—	8	4	no	—
133	C₂×C₄×C₅₉	—	8	4	no	—
134	C₂×C₄×C₅₉	—	8	4	no	—
135	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₅₉	#123	1	2	yes	—
138	C₂×C₄×C₅₉	#123	1	2	yes	—
139	C₂×C₄×C₅₉	#123	1	4	yes	—
140	C₂×C₄×C₅₉	#123	1	4	yes	—
141	C₂×C₄×C₅₉	#124	1	2	yes	—
142	C₂×C₄×C₅₉	#124	1	2	yes	—
143	C₂×C₄×C₅₉	#126	1	4	yes	—
144	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
147	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
148	C₈×C₅₉	—	2	4	no	—
149	C₈×C₅₉	—	4	4	no	—
150	C₈×C₅₉	—	4	2	no	—
151	C₂×#2078	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₅₉	—	2	8	no	—
153	C₂²×C₅₉	—	2	1	no	—
154	C₂²×C₅₉	—	2	1	no	—
155	C₂²×C₅₉	—	2	1	no	—
156	C₂²×C₅₉	—	2	1	no	—
157	C₂²×C₅₉	—	4	2	no	—
158	C₂²×C₅₉	—	4	2	no	—
159	C₂²×C₅₉	—	4	2	no	—
160	C₂²×C₅₉	—	8	4	no	—
161	C₂²×C₅₉	#397	1	2	yes	—
162	C₂²×C₅₉	#397	1	2	yes	—
163	C₂²×C₅₉	#472	1	2	yes	—
164	C₂²×C₅₉	#473	1	2	yes	—
165	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
166	C₂²×C₅₉	C₂×#126	1	2	yes	—
167	C₄×C₅₉	—	4	2	no	—
168	C₄×C₅₉	—	8	4	no	—
169	C₄×C₅₉	#397	1	2	yes	—
170	C₄×C₅₉	#397	1	2	yes	—
171	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
172	C₄×C₅₉	C₂×#123	1	2	yes	—
173	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
174	C₂²×#107	C₂×C₅₉	1	1	yes	characteristic, complemented
175	C₂×#417	C₂×C₅₉	1	2	yes	—
176	C₂×#408	C₂×C₅₉	1	1	yes	characteristic
177	#2078	C₂×C₅₉	1	8	yes	direct product factor
178	C₂×#343	C₂×C₅₉	1	2	yes	complemented
179	C₂×#341	C₂×C₅₉	1	1	yes	characteristic, complemented
180	C₂×C₅₉	—	2	2	no	—
181	C₂×C₅₉	—	2	2	no	—
182	C₂×C₅₉	—	8	2	no	—
183	C₂×C₅₉	#2078	1	4	yes	direct product factor
184	C₂×C₅₉	C₂×#397	1	1	yes	characteristic
185	C₂×C₅₉	C₂×#472	1	1	yes	characteristic
186	C₂×C₅₉	C₂×#473	1	1	yes	characteristic
187	C₂²×#41	C₂²×C₅₉	1	1	yes	characteristic
188	C₂²×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
189	C₂²×M₁₆	C₂²×C₅₉	1	2	yes	—
190	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
191	C₂²×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
192	C₂×#107	—	2	2	no	—
193	C₂×#107	—	2	2	no	—
194	C₂×#107	C₂²×C₅₉	1	4	yes	complemented
195	#417	—	2	4	no	—
196	#417	C₂²×C₅₉	1	8	yes	—
197	C₂×#103	—	2	2	no	—
198	C₂×#102	—	2	2	no	—
199	#408	—	2	2	no	—
200	#408	C₂²×C₅₉	1	4	yes	—
201	C₂×C₂²⋊C₈	—	2	2	no	—
202	#343	C₂²×C₅₉	1	8	yes	complemented
203	#341	C₂²×C₅₉	1	4	yes	complemented
204	C₅₉	C₂×#2078	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×#41	—	2	1	no	—
206	C₂×#41	—	2	1	no	—
207	C₂×#41	—	2	4	no	—
208	C₂×#41	—	2	4	no	—
209	C₂×#41	D₄×C₅₉	1	2	yes	—
210	C₂×#41	C₂³×C₅₉	1	2	yes	—
211	C₂²×Q₈	D₄×C₅₉	1	1	yes	characteristic
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	D₄×C₅₉	1	1	yes	characteristic
214	C₂³×C₄	—	2	1	no	—
215	C₂³×C₄	—	2	2	no	—
216	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	—	2	4	no	—
221	C₂×M₁₆	—	2	4	no	—
222	C₂×M₁₆	C₂³×C₅₉	1	4	yes	—
223	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
224	C₂²×C₈	—	2	4	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	2	no	—
227	C₂²×C₈	D₄×C₅₉	1	2	yes	—
228	C₂²×C₈	D₄×C₅₉	1	2	yes	—
229	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
236	#107	—	2	8	no	—
237	#103	—	2	8	no	—
238	#102	—	2	8	no	—
239	C₂²⋊C₈	—	2	8	no	—
240	C₂⁴	—	2	1	no	—
241	C₂⁴	#41×C₅₉	1	1	yes	characteristic
242	#41	—	2	8	no	—
243	#41	—	2	4	no	—
244	#41	—	2	4	no	—
245	#41	—	4	2	no	—
246	#41	—	4	2	no	—
247	#41	—	4	4	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	4	4	no	—
256	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	8	4	no	—
266	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
270	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
271	C₂²×C₄	#41×C₅₉	1	4	yes	—
272	C₂²×C₄	#41×C₅₉	1	4	yes	—
273	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
274	M₁₆	—	2	4	no	—
275	M₁₆	—	2	4	no	—
276	M₁₆	—	2	4	no	—
277	M₁₆	—	4	4	no	—
278	M₁₆	—	4	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	#121×C₅₉	1	1	yes	characteristic
298	C₂³	#121×C₅₉	1	1	yes	characteristic
299	C₂³	#126×C₅₉	1	4	yes	—
300	C₂³	C₂×#41×C₅₉	1	1	yes	characteristic
301	Q₈	—	2	4	no	—
302	Q₈	—	4	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
317	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
318	C₂×C₄	#123×C₅₉	1	2	yes	—
319	C₂×C₄	#123×C₅₉	1	2	yes	—
320	C₂×C₄	#123×C₅₉	1	4	yes	—
321	C₂×C₄	#123×C₅₉	1	4	yes	—
322	C₂×C₄	#124×C₅₉	1	2	yes	—
323	C₂×C₄	#124×C₅₉	1	2	yes	—
324	C₂×C₄	#126×C₅₉	1	4	yes	—
325	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
327	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
328	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
329	C₈	—	2	4	no	—
330	C₈	—	4	4	no	—
331	C₈	—	4	2	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	8	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	4	no	—
341	C₂²	C₅₉×#397	1	2	yes	—
342	C₂²	C₅₉×#397	1	2	yes	—
343	C₂²	C₅₉×#472	1	2	yes	—
344	C₂²	C₅₉×#473	1	2	yes	—
345	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
346	C₂²	C₂×#126×C₅₉	1	2	yes	—
347	C₄	—	4	2	no	—
348	C₄	—	8	4	no	—
349	C₄	C₅₉×#397	1	2	yes	—
350	C₄	C₅₉×#397	1	2	yes	—
351	C₄	C₂×#123×C₅₉	1	1	yes	characteristic
352	C₄	C₂×#123×C₅₉	1	2	yes	—
353	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	—	8	2	no	—
357	C₂	C₅₉×#2078	1	4	yes	direct product factor
358	C₂	C₂×C₅₉×#397	1	1	yes	characteristic
359	C₂	C₂×C₅₉×#472	1	1	yes	characteristic
360	C₂	C₂×C₅₉×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×#41×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#107×C₅₉	C₂	semidirect
4. C₅₉×#2078	C₂	direct (N × H)
5. C₂×C₅₉×#343	C₂	semidirect
6. C₂×C₅₉×#341	C₂	semidirect
14. C₂×#107×C₅₉	C₂²	semidirect
22. C₅₉×#343	C₂²	semidirect
23. C₅₉×#341	C₂²	semidirect
151. C₂×#2078	C₅₉	direct (N × H)
174. C₂²×#107	C₂×C₅₉	semidirect
177. #2078	C₂×C₅₉	direct (N × H)
178. C₂×#343	C₂×C₅₉	semidirect
179. C₂×#341	C₂×C₅₉	semidirect
183. C₂×C₅₉	#2078	direct (N × H)
194. C₂×#107	C₂²×C₅₉	semidirect
202. #343	C₂²×C₅₉	semidirect
203. #341	C₂²×C₅₉	semidirect
204. C₅₉	C₂×#2078	direct (N × H)
357. C₂	C₅₉×#2078	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×#107×C₅₉	2
C₂×C₅₉×#417	2
C₂×C₅₉×#408	2
C₅₉×#2078	2
C₂×C₅₉×#343	2
C₂×C₅₉×#341	2
C₂×#2078	59

Subgroups — C2×C59×#2078 #28435753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#2078 #28435753