Subgroups — C₂×C₅₉×#2237 #28445751

Order 15104.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#427	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2237	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₉×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₉×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₅₉	—	2	1	no	—
12	C₂×#117×C₅₉	—	2	1	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#467	—	2	2	no	—
16	C₅₉×#467	C₂²	1	4	yes	—
17	C₅₉×#427	—	2	2	no	—
18	C₅₉×#427	C₂²	1	4	yes	—
19	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#106×C₅₉	—	2	1	no	—
23	C₂×#103×C₅₉	—	2	1	no	—
24	C₅₉×#367	C₂²	1	4	yes	complemented
25	C₅₉×#366	C₂²	1	4	yes	complemented
26	C₅₉×#358	C₂²	1	4	yes	complemented
27	C₅₉×#358	C₂²	1	4	yes	complemented
28	C₅₉×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₂²×Q₈×C₅₉	—	2	1	no	—
30	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂×Q₁₆×C₅₉	—	2	1	no	—
32	C₂×Q₁₆×C₅₉	—	2	2	no	—
33	C₂×Q₁₆×C₅₉	—	2	2	no	—
34	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₉	—	2	2	no	—
36	C₂²×C₈×C₅₉	—	2	2	no	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	D₄	1	2	yes	—
39	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²⋊Q₈×C₅₉	—	2	4	no	—
42	C₂²⋊Q₈×C₅₉	—	4	2	no	—
43	#117×C₅₉	—	2	4	no	—
44	#117×C₅₉	—	4	2	no	—
45	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
49	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
50	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
51	C₂×C₄²×C₅₉	—	4	1	no	—
52	C₂×C₁₆×C₅₉	—	2	2	no	—
53	C₂×C₁₆×C₅₉	—	2	2	no	—
54	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
55	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
56	#107×C₅₉	—	2	1	no	—
57	#107×C₅₉	—	2	1	no	—
58	#107×C₅₉	—	2	1	no	—
59	#107×C₅₉	C₂³	1	2	yes	—
60	#107×C₅₉	C₂³	1	2	yes	—
61	#107×C₅₉	C₂³	1	2	yes	—
62	#106×C₅₉	—	2	4	no	—
63	#103×C₅₉	—	2	4	no	—
64	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	—	4	1	no	—
67	C₂×Q₈×C₅₉	—	4	2	no	—
68	C₂×Q₈×C₅₉	—	4	2	no	—
69	C₂²×C₄×C₅₉	—	2	2	no	—
70	C₂²×C₄×C₅₉	—	2	2	no	—
71	C₂²×C₄×C₅₉	—	4	1	no	—
72	C₂²×C₄×C₅₉	—	4	1	no	—
73	C₂²×C₄×C₅₉	—	4	1	no	—
74	C₂²×C₄×C₅₉	—	4	1	no	—
75	C₂²×C₄×C₅₉	D₈	1	2	yes	—
76	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₅₉	—	2	2	no	—
80	Q₁₆×C₅₉	—	2	2	no	—
81	Q₁₆×C₅₉	—	4	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	4	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	#41	1	2	yes	—
89	C₂×C₈×C₅₉	#41	1	2	yes	—
90	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₄⋊C₄×C₅₉	—	2	2	no	—
93	C₄⋊C₄×C₅₉	—	2	2	no	—
94	C₄⋊C₄×C₅₉	—	4	1	no	—
95	C₄⋊C₄×C₅₉	—	4	1	no	—
96	C₄⋊C₄×C₅₉	—	4	1	no	—
97	C₄⋊C₄×C₅₉	—	4	4	no	—
98	C₂²⋊C₄×C₅₉	—	4	4	no	—
99	C₂²⋊C₄×C₅₉	—	4	4	no	—
100	C₄²×C₅₉	—	4	4	no	—
101	C₁₆×C₅₉	—	2	2	no	—
102	C₁₆×C₅₉	—	2	2	no	—
103	C₂³×C₅₉	—	2	2	no	—
104	C₂³×C₅₉	—	2	2	no	—
105	C₂³×C₅₉	—	2	2	no	—
106	C₂³×C₅₉	C₂×D₈	1	2	yes	—
107	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₅₉	—	4	2	no	—
109	Q₈×C₅₉	—	4	2	no	—
110	Q₈×C₅₉	—	8	2	no	—
111	C₂×C₄×C₅₉	—	2	2	no	—
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	4	no	—
115	C₂×C₄×C₅₉	—	4	2	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	—	8	2	no	—
120	C₂×C₄×C₅₉	—	8	2	no	—
121	C₂×C₄×C₅₉	—	8	2	no	—
122	C₂×C₄×C₅₉	—	8	2	no	—
123	C₂×C₄×C₅₉	#123	1	2	yes	—
124	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#137	1	2	yes	—
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₈×C₅₉	—	2	2	no	—
129	C₈×C₅₉	#123	1	2	yes	—
130	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
131	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
132	C₂×#2237	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₉	—	2	2	no	—
134	C₂²×C₅₉	—	2	2	no	—
135	C₂²×C₅₉	—	2	2	no	—
136	C₂²×C₅₉	—	4	4	no	—
137	C₂²×C₅₉	#480	1	2	yes	—
138	C₂²×C₅₉	#508	1	2	yes	—
139	C₂²×C₅₉	#510	1	2	yes	—
140	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	8	2	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	—	8	2	no	—
145	C₄×C₅₉	—	8	2	no	—
146	C₄×C₅₉	#480	1	2	yes	—
147	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
148	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
149	C₂×#467	C₂×C₅₉	1	1	yes	characteristic
150	C₂×#427	C₂×C₅₉	1	1	yes	characteristic
151	#2237	C₂×C₅₉	1	8	yes	direct product factor
152	C₂×#367	C₂×C₅₉	1	1	yes	characteristic, complemented
153	C₂×#366	C₂×C₅₉	1	1	yes	characteristic, complemented
154	C₂×#358	C₂×C₅₉	1	1	yes	characteristic, complemented
155	C₂×#358	C₂×C₅₉	1	1	yes	characteristic, complemented
156	C₂×C₂²⋊C₁₆	C₂×C₅₉	1	1	yes	characteristic
157	C₂×C₅₉	—	4	2	no	—
158	C₂×C₅₉	#2237	1	4	yes	direct product factor
159	C₂×C₅₉	C₂×#480	1	1	yes	characteristic
160	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
164	C₂×C₂²⋊Q₈	—	2	1	no	—
165	C₂×#117	—	2	1	no	—
166	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
167	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
168	#467	—	2	2	no	—
169	#467	C₂²×C₅₉	1	4	yes	—
170	#427	—	2	2	no	—
171	#427	C₂²×C₅₉	1	4	yes	—
172	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
173	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
174	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
175	C₂×#106	—	2	1	no	—
176	C₂×#103	—	2	1	no	—
177	#367	C₂²×C₅₉	1	4	yes	complemented
178	#366	C₂²×C₅₉	1	4	yes	complemented
179	#358	C₂²×C₅₉	1	4	yes	complemented
180	#358	C₂²×C₅₉	1	4	yes	complemented
181	C₂²⋊C₁₆	C₂²×C₅₉	1	4	yes	—
182	C₅₉	C₂×#2237	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	—	2	1	no	—
184	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₅₉	1	2	yes	—
193	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
194	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
195	C₂²⋊Q₈	—	2	4	no	—
196	C₂²⋊Q₈	—	4	2	no	—
197	#117	—	2	4	no	—
198	#117	—	4	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	4	1	no	—
203	C₂×C₂²⋊C₄	—	4	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	C₂×C₄²	—	4	1	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
209	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
210	#107	—	2	1	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	C₂³×C₅₉	1	2	yes	—
214	#107	C₂³×C₅₉	1	2	yes	—
215	#107	C₂³×C₅₉	1	2	yes	—
216	#106	—	2	4	no	—
217	#103	—	2	4	no	—
218	C₂⁴	D₈×C₅₉	1	1	yes	characteristic
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	D₈×C₅₉	1	1	yes	characteristic
230	C₂²×C₄	D₈×C₅₉	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
242	C₂×C₈	#41×C₅₉	1	2	yes	—
243	C₂×C₈	#41×C₅₉	1	2	yes	—
244	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₄²	—	4	4	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×D₈×C₅₉	1	2	yes	—
261	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	#123×C₅₉	1	2	yes	—
278	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
280	C₂×C₄	#137×C₅₉	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#123×C₅₉	1	2	yes	—
284	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
285	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₅₉×#480	1	2	yes	—
291	C₂²	C₅₉×#508	1	2	yes	—
292	C₂²	C₅₉×#510	1	2	yes	—
293	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	C₅₉×#480	1	2	yes	—
300	C₄	C₂×#123×C₅₉	1	1	yes	characteristic
301	C₄	C₂×#137×C₅₉	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	C₅₉×#2237	1	4	yes	direct product factor
304	C₂	C₂×C₅₉×#480	1	1	yes	characteristic
305	C₂	C₂×C₅₉×#508	1	1	yes	characteristic
306	C₂	C₂×C₅₉×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂⁴×C₅₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ C₂³×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₉×#2237	C₂	direct (N × H)
4. C₂×C₅₉×#367	C₂	semidirect
5. C₂×C₅₉×#366	C₂	semidirect
6. C₂×C₅₉×#358	C₂	semidirect
7. C₂×C₅₉×#358	C₂	semidirect
24. C₅₉×#367	C₂²	semidirect
25. C₅₉×#366	C₂²	semidirect
26. C₅₉×#358	C₂²	semidirect
27. C₅₉×#358	C₂²	semidirect
132. C₂×#2237	C₅₉	direct (N × H)
151. #2237	C₂×C₅₉	direct (N × H)
152. C₂×#367	C₂×C₅₉	semidirect
153. C₂×#366	C₂×C₅₉	semidirect
154. C₂×#358	C₂×C₅₉	semidirect
155. C₂×#358	C₂×C₅₉	semidirect
158. C₂×C₅₉	#2237	direct (N × H)
177. #367	C₂²×C₅₉	semidirect
178. #366	C₂²×C₅₉	semidirect
179. #358	C₂²×C₅₉	semidirect
180. #358	C₂²×C₅₉	semidirect
182. C₅₉	C₂×#2237	direct (N × H)
303. C₂	C₅₉×#2237	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#467	2
C₂×C₅₉×#427	2
C₅₉×#2237	2
C₂×C₅₉×#367	2
C₂×C₅₉×#366	2
C₂×C₅₉×#358	2
C₂×C₅₉×C₂²⋊C₁₆	2
C₂×#2237	59

Subgroups — C2×C59×#2237 #28445751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#2237 #28445751