Subgroups — C₂×#21424121 #28588246

Order 15104.

1790 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273659	C₂	1	2	yes	maximal, complemented
2	C₂×#16837262	C₂	1	2	yes	maximal
3	#21424121	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837165	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₅₉	C₂²	1	4	yes	—
7	C₂²×C₂²⋊Dic₅₉	—	2	2	no	—
8	C₂×#14273659	—	2	8	no	—
9	C₂×#14273659	C₂²	1	8	yes	complemented
10	#16837262	—	2	4	no	—
11	#16837262	C₂²	1	8	yes	—
12	C₂×C₅₉⋊#95	—	2	2	no	—
13	#16837165	C₂²	1	8	yes	complemented
14	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₉	—	2	8	no	—
16	C₂³×C₈×C₅₉	C₄	1	8	yes	complemented
17	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₉	C₂²	1	4	yes	—
19	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂³×Dic₅₉	—	2	4	no	—
21	C₂×C₄⋊Dic₅₉	—	2	8	no	—
22	C₂×C₄⋊Dic₅₉	—	2	8	no	—
23	C₂×C₄⋊Dic₅₉	—	2	4	no	—
24	C₂×C₄⋊Dic₅₉	C₂³	1	8	yes	—
25	C₂×C₂²⋊Dic₅₉	—	2	8	no	—
26	C₂×C₂²⋊Dic₅₉	—	4	8	no	—
27	#14273659	—	2	16	no	—
28	C₅₉⋊#95	—	2	8	no	—
29	C₂⁵×C₅₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₉	—	2	4	no	—
31	C₂³×C₄×C₅₉	—	2	4	no	—
32	C₂³×C₄×C₅₉	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₉	D₄	1	2	yes	—
35	C₂³×C₄×C₅₉	Q₈	1	4	yes	—
36	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	—	2	4	no	—
39	C₂²×C₈×C₅₉	—	2	32	no	—
40	C₂²×C₈×C₅₉	—	2	16	no	—
41	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₉	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₅₉	D₄	1	8	yes	—
45	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
47	C₂⁴×C₅₉	—	2	4	no	—
48	C₂⁴×C₅₉	—	2	4	no	—
49	C₂⁴×C₅₉	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₅₉	SD₁₆	1	8	yes	—
51	C₂⁴×C₅₉	C₂×D₄	1	2	yes	—
52	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	—	2	4	no	—
54	C₂²×C₄×C₅₉	—	2	8	no	—
55	C₂²×C₄×C₅₉	—	2	8	no	—
56	C₂²×C₄×C₅₉	—	2	8	no	—
57	C₂²×C₄×C₅₉	—	2	16	no	—
58	C₂²×C₄×C₅₉	—	2	4	no	—
59	C₂²×C₄×C₅₉	—	2	4	no	—
60	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₅₉	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₉	#41	1	4	yes	—
70	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₅₉	—	2	8	no	—
72	C₂²×Dic₅₉	—	4	8	no	—
73	C₂²×Dic₅₉	—	4	8	no	—
74	C₂²×Dic₅₉	—	4	8	no	—
75	C₂×C₈×C₅₉	—	2	8	no	—
76	C₂×C₈×C₅₉	—	2	4	no	—
77	C₂×C₈×C₅₉	—	2	32	no	—
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₅₉	C₂×D₄	1	8	yes	—
82	C₂²⋊Dic₅₉	—	4	16	no	—
83	C₄⋊Dic₅₉	—	2	8	no	—
84	C₄⋊Dic₅₉	—	2	8	no	—
85	C₄⋊Dic₅₉	—	4	8	no	—
86	C₂³×C₅₉	—	2	4	no	—
87	C₂³×C₅₉	—	2	4	no	—
88	C₂³×C₅₉	—	2	8	no	—
89	C₂³×C₅₉	—	2	8	no	—
90	C₂³×C₅₉	—	2	32	no	—
91	C₂³×C₅₉	#106	1	16	yes	complemented
92	C₂³×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
93	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₅₉	#121	1	4	yes	—
95	C₂³×C₅₉	C₂×SD₁₆	1	8	yes	—
96	C₂³×C₅₉	C₂×SD₁₆	1	4	yes	—
97	C₂³×C₅₉	#135	1	4	yes	—
98	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	—	2	4	no	—
100	C₂×C₄×C₅₉	—	2	8	no	—
101	C₂×C₄×C₅₉	—	2	4	no	—
102	C₂×C₄×C₅₉	—	2	4	no	—
103	C₂×C₄×C₅₉	—	2	8	no	—
104	C₂×C₄×C₅₉	—	2	16	no	—
105	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅₉	#117	1	4	yes	—
109	C₂×C₄×C₅₉	#121	1	4	yes	—
110	C₂×C₄×C₅₉	C₂²⋊Q₈	1	8	yes	—
111	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
115	C₂×Dic₅₉	—	4	8	no	—
116	C₂×Dic₅₉	—	4	8	no	—
117	C₂×Dic₅₉	—	4	8	no	—
118	C₂×Dic₅₉	—	8	16	no	—
119	C₈×C₅₉	—	2	4	no	—
120	C₈×C₅₉	C₂×C₂²⋊C₄	1	8	yes	complemented
121	C₂×#1807	—	59	1	no	maximal, Sylow
122	C₂²×C₅₉	—	2	32	no	—
123	C₂²×C₅₉	—	2	4	no	—
124	C₂²×C₅₉	—	2	4	no	—
125	C₂²×C₅₉	—	2	16	no	—
126	C₂²×C₅₉	#380	1	4	yes	—
127	C₂²×C₅₉	C₂×#106	1	4	yes	—
128	C₂²×C₅₉	C₂×#106	1	8	yes	complemented
129	C₂²×C₅₉	#427	1	4	yes	—
130	C₂²×C₅₉	#465	1	16	yes	—
131	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₉	C₂×#121	1	2	yes	—
133	C₂²×C₅₉	C₂²×SD₁₆	1	2	yes	—
134	C₂²×C₅₉	C₂×#135	1	2	yes	—
135	C₄×C₅₉	—	2	4	no	—
136	C₄×C₅₉	#380	1	4	yes	—
137	C₄×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
139	C₄×C₅₉	C₂×C₂²⋊Q₈	1	2	yes	—
140	Dic₅₉	—	8	8	no	—
141	C₂⁴×C₈	D₅₉	1	1	yes	characteristic
142	C₂²×#106	—	59	2	no	—
143	C₂×#380	—	59	2	no	—
144	#1807	—	59	8	no	—
145	C₂×#340	—	59	2	no	—
146	C₂×C₅₉	—	2	8	no	—
147	C₂×C₅₉	#1807	1	8	yes	complemented
148	C₂×C₅₉	C₂×#380	1	1	yes	characteristic
149	C₂×C₅₉	C₂²×#106	1	1	yes	characteristic
150	C₂×C₅₉	C₂×#427	1	1	yes	characteristic
151	C₂×C₅₉	C₂×#465	1	4	yes	—
152	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
153	C₂³×C₈	—	2	8	no	—
154	C₂³×C₈	Dic₅₉	1	8	yes	complemented
155	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₅₉	1	4	yes	—
157	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	59	4	no	—
159	C₂²×C₂²⋊C₄	—	118	2	no	—
160	C₂×#106	—	59	8	no	—
161	C₂×#106	—	118	8	no	—
162	#380	—	59	8	no	—
163	#380	—	118	4	no	—
164	C₂×#95	—	118	2	no	—
165	#340	—	59	8	no	—
166	C₅₉	C₂×#1807	1	1	yes	characteristic, complemented, Sylow
167	C₂⁵	D₂₃₆	1	1	yes	characteristic
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	118	4	no	—
171	C₂³×C₄	Dic₁₁₈	1	4	yes	—
172	C₂³×C₄	D₂₃₆	1	2	yes	—
173	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₅₉	1	4	yes	—
175	C₂³×C₄	C₂²×D₅₉	1	2	yes	—
176	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	32	no	—
179	C₂²×C₈	—	2	16	no	—
180	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
181	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
182	C₂²×C₈	C₂×Dic₅₉	1	16	yes	complemented
183	C₂²×C₈	C₂²⋊D₅₉	1	8	yes	—
184	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
185	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
186	C₂×C₄⋊C₄	—	59	8	no	—
187	C₂×C₄⋊C₄	—	118	8	no	—
188	C₂×C₄⋊C₄	—	118	8	no	—
189	C₂×C₄⋊C₄	—	118	4	no	—
190	C₂×C₂²⋊C₄	—	118	8	no	—
191	C₂×C₂²⋊C₄	—	236	8	no	—
192	#106	—	118	16	no	—
193	#95	—	118	8	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	#11745660	1	8	yes	—
197	C₂⁴	C₄⋊Dic₅₉	1	4	yes	—
198	C₂⁴	C₂×D₂₃₆	1	2	yes	—
199	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	118	8	no	—
208	C₂²×C₄	—	236	8	no	—
209	C₂²×C₄	—	236	8	no	—
210	C₂²×C₄	—	236	8	no	—
211	C₂²×C₄	C₄⋊Dic₅₉	1	4	yes	—
212	C₂²×C₄	C₄⋊Dic₅₉	1	8	yes	—
213	C₂²×C₄	C₂×Dic₁₁₈	1	4	yes	—
214	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
215	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
216	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
217	C₂²×C₄	#11745685	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₅₉	1	4	yes	—
219	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₅₉	1	4	yes	—
221	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	32	no	—
225	C₂×C₈	C₂²⋊Dic₅₉	1	16	yes	complemented
226	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
227	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
228	C₂×C₈	C₂×C₂²⋊D₅₉	1	8	yes	—
229	C₄⋊C₄	—	118	8	no	—
230	C₄⋊C₄	—	118	8	no	—
231	C₄⋊C₄	—	236	8	no	—
232	C₂²⋊C₄	—	236	16	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	32	no	—
238	C₂³	#14273659	1	16	yes	complemented
239	C₂³	C₂×C₄⋊Dic₅₉	1	4	yes	—
240	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
241	C₂³	#14273701	1	4	yes	—
242	C₂³	C₂×#11745660	1	8	yes	—
243	C₂³	C₂×#11745660	1	4	yes	—
244	C₂³	#14273749	1	4	yes	—
245	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	472	16	no	—
253	C₂×C₄	—	236	8	no	—
254	C₂×C₄	—	236	8	no	—
255	C₂×C₄	—	236	8	no	—
256	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
259	C₂×C₄	#14273686	1	4	yes	—
260	C₂×C₄	#14273701	1	4	yes	—
261	C₂×C₄	#14273706	1	8	yes	—
262	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₂²⋊D₅₉	1	2	yes	—
264	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
265	C₂×C₄	C₂×#11745685	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	C₂×C₂²⋊Dic₅₉	1	8	yes	complemented
268	C₂²	—	2	4	no	—
269	C₂²	—	2	16	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	32	no	—
272	C₂²	#16837262	1	4	yes	—
273	C₂²	C₂×#14273659	1	4	yes	—
274	C₂²	C₂×#14273659	1	8	yes	complemented
275	C₂²	#16837440	1	4	yes	—
276	C₂²	#16837631	1	16	yes	—
277	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
278	C₂²	C₂×#14273701	1	2	yes	—
279	C₂²	C₂²×#11745660	1	2	yes	—
280	C₂²	C₂×#14273749	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	472	8	no	—
283	C₄	#16837262	1	4	yes	—
284	C₄	C₂²×C₂²⋊Dic₅₉	1	1	yes	characteristic
285	C₄	C₂×#14273686	1	1	yes	characteristic
286	C₄	C₂×#14273706	1	2	yes	—
287	C₂	—	2	8	no	—
288	C₂	#21424121	1	8	yes	direct product factor
289	C₂	C₂×#16837262	1	1	yes	characteristic
290	C₂	C₂²×#14273659	1	1	yes	characteristic
291	C₂	C₂×#16837440	1	1	yes	characteristic
292	C₂	C₂×#16837631	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂⁴×C₈×C₅₉	C₂
socle	C₂⁴×C₅₉	C₂×D₄
prime radical	C₂⁵×C₅₉	D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂⁴×C₈	D₅₉
hypocenter	C₅₉	C₂×#1807
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273659	C₂	semidirect
3. #21424121	C₂	direct (N × H)
4. C₂×#16837165	C₂	semidirect
9. C₂×#14273659	C₂²	semidirect
13. #16837165	C₂²	semidirect
16. C₂³×C₈×C₅₉	C₄	semidirect
43. C₂²×C₈×C₅₉	C₂×C₄	semidirect
78. C₂×C₈×C₅₉	C₂²⋊C₄	semidirect
91. C₂³×C₅₉	#106	semidirect
120. C₈×C₅₉	C₂×C₂²⋊C₄	semidirect
128. C₂²×C₅₉	C₂×#106	semidirect
147. C₂×C₅₉	#1807	semidirect
154. C₂³×C₈	Dic₅₉	semidirect
166. C₅₉	C₂×#1807	semidirect
182. C₂²×C₈	C₂×Dic₅₉	semidirect
225. C₂×C₈	C₂²⋊Dic₅₉	semidirect
238. C₂³	#14273659	semidirect
267. C₈	C₂×C₂²⋊Dic₅₉	semidirect
274. C₂²	C₂×#14273659	semidirect
288. C₂	#21424121	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×#14273659	2
C₂×#16837262	2
#21424121	2
C₂×#16837165	2
C₂⁴×C₈×C₅₉	2
C₂×#1807	59

Subgroups — C2×#21424121 #28588246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21424121 #28588246