Subgroups — C₂×C₅₉×#2017 #28435542

Order 15104.

1938 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#575	C₂	1	3	yes	maximal, complemented
2	C₂×C₅₉×#408	C₂	1	3	yes	maximal, complemented
3	C₅₉×#2017	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₉×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#41×C₅₉	C₂²	1	3	yes	—
6	C₅₉×#575	—	2	12	no	—
7	C₅₉×#575	—	2	6	no	—
8	C₅₉×#575	C₂²	1	12	yes	complemented
9	C₂×#137×C₅₉	—	2	6	no	—
10	C₂×AGL(1,ℤ/8)×C₅₉	—	2	6	no	—
11	C₂×#135×C₅₉	—	2	6	no	—
12	C₂²×M₁₆×C₅₉	C₂²	1	3	yes	—
13	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#408	—	2	6	no	—
15	C₅₉×#408	C₂²	1	12	yes	complemented
16	C₂×C₂²⋊C₈×C₅₉	—	2	6	no	—
17	C₅₉×#338	C₂²	1	4	yes	complemented
18	C₂×#41×C₅₉	—	2	3	no	—
19	C₂×#41×C₅₉	—	2	3	no	—
20	C₂×#41×C₅₉	—	2	6	no	—
21	C₂×#41×C₅₉	—	2	6	no	—
22	C₂×#41×C₅₉	—	2	6	no	—
23	C₂×#41×C₅₉	—	2	6	no	—
24	C₂×#41×C₅₉	D₄	1	6	yes	—
25	C₂×#41×C₅₉	C₂³	1	6	yes	—
26	C₂²×Q₈×C₅₉	D₄	1	3	yes	—
27	C₂²×D₄×C₅₉	—	2	3	no	—
28	C₂²×D₄×C₅₉	D₄	1	3	yes	—
29	C₂³×C₄×C₅₉	—	2	3	no	—
30	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	#137×C₅₉	—	2	24	no	—
32	#137×C₅₉	—	2	24	no	—
33	AGL(1,ℤ/8)×C₅₉	—	2	24	no	—
34	AGL(1,ℤ/8)×C₅₉	—	2	24	no	—
35	#135×C₅₉	—	2	24	no	—
36	#135×C₅₉	—	2	24	no	—
37	C₂×Q₁₆×C₅₉	—	2	6	no	—
38	C₂×SD₁₆×C₅₉	—	2	12	no	—
39	C₂×D₈×C₅₉	—	2	6	no	—
40	C₂×M₁₆×C₅₉	—	2	6	no	—
41	C₂×M₁₆×C₅₉	—	2	6	no	—
42	C₂×M₁₆×C₅₉	D₄	1	6	yes	—
43	C₂×M₁₆×C₅₉	C₂³	1	6	yes	—
44	C₂²×C₈×C₅₉	—	2	3	no	—
45	C₂²×C₈×C₅₉	—	2	3	no	—
46	C₂²×C₈×C₅₉	D₄	1	6	yes	—
47	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
48	C₂²⋊C₈×C₅₉	—	2	24	no	—
49	C₂⁴×C₅₉	—	2	3	no	—
50	C₂⁴×C₅₉	#41	1	1	yes	characteristic
51	#41×C₅₉	—	2	12	no	—
52	#41×C₅₉	—	2	12	no	—
53	#41×C₅₉	—	2	12	no	—
54	#41×C₅₉	—	2	12	no	—
55	#41×C₅₉	—	4	6	no	—
56	#41×C₅₉	—	4	6	no	—
57	#41×C₅₉	—	4	6	no	—
58	#41×C₅₉	—	4	6	no	—
59	C₂×Q₈×C₅₉	—	2	3	no	—
60	C₂×Q₈×C₅₉	—	2	6	no	—
61	C₂×Q₈×C₅₉	—	2	6	no	—
62	C₂×Q₈×C₅₉	C₂×D₄	1	6	yes	—
63	C₂×D₄×C₅₉	—	2	6	no	—
64	C₂×D₄×C₅₉	—	2	6	no	—
65	C₂×D₄×C₅₉	—	2	6	no	—
66	C₂×D₄×C₅₉	—	2	6	no	—
67	C₂×D₄×C₅₉	—	2	3	no	—
68	C₂×D₄×C₅₉	—	4	6	no	—
69	C₂×D₄×C₅₉	—	4	6	no	—
70	C₂×D₄×C₅₉	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₅₉	—	2	3	no	—
72	C₂²×C₄×C₅₉	—	2	6	no	—
73	C₂²×C₄×C₅₉	—	2	6	no	—
74	C₂²×C₄×C₅₉	—	4	3	no	—
75	C₂²×C₄×C₅₉	—	4	3	no	—
76	C₂²×C₄×C₅₉	—	4	3	no	—
77	C₂²×C₄×C₅₉	—	4	3	no	—
78	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
83	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₅₉	—	2	24	no	—
85	SD₁₆×C₅₉	—	2	48	no	—
86	D₈×C₅₉	—	2	24	no	—
87	M₁₆×C₅₉	—	2	12	no	—
88	M₁₆×C₅₉	—	2	12	no	—
89	C₂×C₈×C₅₉	—	2	6	no	—
90	C₂×C₈×C₅₉	—	2	6	no	—
91	C₂×C₈×C₅₉	—	2	6	no	—
92	C₂×C₈×C₅₉	—	4	3	no	—
93	C₂×C₈×C₅₉	—	4	1	no	—
94	C₂×C₈×C₅₉	C₂×D₄	1	12	yes	—
95	C₂³×C₅₉	—	2	6	no	—
96	C₂³×C₅₉	—	4	2	no	—
97	C₂³×C₅₉	—	4	12	no	—
98	C₂³×C₅₉	—	4	3	no	—
99	C₂³×C₅₉	—	4	3	no	—
100	C₂³×C₅₉	#121	1	3	yes	—
101	C₂³×C₅₉	#121	1	3	yes	—
102	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
103	Q₈×C₅₉	—	2	12	no	—
104	Q₈×C₅₉	—	4	6	no	—
105	D₄×C₅₉	—	2	12	no	—
106	D₄×C₅₉	—	4	12	no	—
107	D₄×C₅₉	—	4	12	no	—
108	D₄×C₅₉	—	4	6	no	—
109	C₂×C₄×C₅₉	—	2	6	no	—
110	C₂×C₄×C₅₉	—	4	3	no	—
111	C₂×C₄×C₅₉	—	4	3	no	—
112	C₂×C₄×C₅₉	—	4	6	no	—
113	C₂×C₄×C₅₉	—	4	6	no	—
114	C₂×C₄×C₅₉	—	4	6	no	—
115	C₂×C₄×C₅₉	—	4	6	no	—
116	C₂×C₄×C₅₉	#120	1	3	yes	—
117	C₂×C₄×C₅₉	#120	1	3	yes	—
118	C₂×C₄×C₅₉	#120	1	6	yes	—
119	C₂×C₄×C₅₉	#121	1	3	yes	—
120	C₂×C₄×C₅₉	#121	1	3	yes	—
121	C₂×C₄×C₅₉	#127	1	3	yes	—
122	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
124	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
125	C₈×C₅₉	—	2	12	no	—
126	C₈×C₅₉	—	4	2	no	—
127	C₂×#2017	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₉	—	2	6	no	—
129	C₂²×C₅₉	—	4	3	no	—
130	C₂²×C₅₉	—	4	1	no	—
131	C₂²×C₅₉	—	4	12	no	—
132	C₂²×C₅₉	—	4	6	no	—
133	C₂²×C₅₉	—	8	6	no	—
134	C₂²×C₅₉	—	8	6	no	—
135	C₂²×C₅₉	#392	1	3	yes	—
136	C₂²×C₅₉	#392	1	1	yes	characteristic
137	C₂²×C₅₉	C₂×#121	1	3	yes	—
138	C₄×C₅₉	—	4	6	no	—
139	C₄×C₅₉	#392	1	1	yes	characteristic
140	C₄×C₅₉	#392	1	3	yes	—
141	C₄×C₅₉	C₂×#120	1	3	yes	—
142	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
143	C₂×#575	C₂×C₅₉	1	3	yes	complemented
144	C₂×#408	C₂×C₅₉	1	3	yes	complemented
145	#2017	C₂×C₅₉	1	8	yes	direct product factor
146	C₂×#338	C₂×C₅₉	1	1	yes	characteristic, complemented
147	C₂×C₅₉	—	2	3	no	—
148	C₂×C₅₉	—	2	3	no	—
149	C₂×C₅₉	—	8	6	no	—
150	C₂×C₅₉	#2017	1	2	yes	direct product factor
151	C₂×C₅₉	C₂×#392	1	1	yes	characteristic
152	C₂²×#41	C₂²×C₅₉	1	3	yes	—
153	#575	—	2	12	no	—
154	#575	—	2	6	no	—
155	#575	C₂²×C₅₉	1	12	yes	complemented
156	C₂×#137	—	2	6	no	—
157	C₂×AGL(1,ℤ/8)	—	2	6	no	—
158	C₂×#135	—	2	6	no	—
159	C₂²×M₁₆	C₂²×C₅₉	1	3	yes	—
160	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
161	#408	—	2	6	no	—
162	#408	C₂²×C₅₉	1	12	yes	complemented
163	C₂×C₂²⋊C₈	—	2	6	no	—
164	#338	C₂²×C₅₉	1	4	yes	complemented
165	C₅₉	C₂×#2017	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#41	—	2	3	no	—
167	C₂×#41	—	2	3	no	—
168	C₂×#41	—	2	6	no	—
169	C₂×#41	—	2	6	no	—
170	C₂×#41	—	2	6	no	—
171	C₂×#41	—	2	6	no	—
172	C₂×#41	D₄×C₅₉	1	6	yes	—
173	C₂×#41	C₂³×C₅₉	1	6	yes	—
174	C₂²×Q₈	D₄×C₅₉	1	3	yes	—
175	C₂²×D₄	—	2	3	no	—
176	C₂²×D₄	D₄×C₅₉	1	3	yes	—
177	C₂³×C₄	—	2	3	no	—
178	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
179	#137	—	2	24	no	—
180	#137	—	2	24	no	—
181	AGL(1,ℤ/8)	—	2	24	no	—
182	AGL(1,ℤ/8)	—	2	24	no	—
183	#135	—	2	24	no	—
184	#135	—	2	24	no	—
185	C₂×Q₁₆	—	2	6	no	—
186	C₂×SD₁₆	—	2	12	no	—
187	C₂×D₈	—	2	6	no	—
188	C₂×M₁₆	—	2	6	no	—
189	C₂×M₁₆	—	2	6	no	—
190	C₂×M₁₆	D₄×C₅₉	1	6	yes	—
191	C₂×M₁₆	C₂³×C₅₉	1	6	yes	—
192	C₂²×C₈	—	2	3	no	—
193	C₂²×C₈	—	2	3	no	—
194	C₂²×C₈	D₄×C₅₉	1	6	yes	—
195	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
196	C₂²⋊C₈	—	2	24	no	—
197	C₂⁴	—	2	3	no	—
198	C₂⁴	#41×C₅₉	1	1	yes	characteristic
199	#41	—	2	12	no	—
200	#41	—	2	12	no	—
201	#41	—	2	12	no	—
202	#41	—	2	12	no	—
203	#41	—	4	6	no	—
204	#41	—	4	6	no	—
205	#41	—	4	6	no	—
206	#41	—	4	6	no	—
207	C₂×Q₈	—	2	3	no	—
208	C₂×Q₈	—	2	6	no	—
209	C₂×Q₈	—	2	6	no	—
210	C₂×Q₈	C₂×D₄×C₅₉	1	6	yes	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	—	2	6	no	—
214	C₂×D₄	—	2	6	no	—
215	C₂×D₄	—	2	3	no	—
216	C₂×D₄	—	4	6	no	—
217	C₂×D₄	—	4	6	no	—
218	C₂×D₄	C₂×D₄×C₅₉	1	6	yes	—
219	C₂²×C₄	—	2	3	no	—
220	C₂²×C₄	—	2	6	no	—
221	C₂²×C₄	—	2	6	no	—
222	C₂²×C₄	—	4	3	no	—
223	C₂²×C₄	—	4	3	no	—
224	C₂²×C₄	—	4	3	no	—
225	C₂²×C₄	—	4	3	no	—
226	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
227	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
228	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
229	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
230	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
232	Q₁₆	—	2	24	no	—
233	SD₁₆	—	2	48	no	—
234	D₈	—	2	24	no	—
235	M₁₆	—	2	12	no	—
236	M₁₆	—	2	12	no	—
237	C₂×C₈	—	2	6	no	—
238	C₂×C₈	—	2	6	no	—
239	C₂×C₈	—	2	6	no	—
240	C₂×C₈	—	4	3	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	C₂×D₄×C₅₉	1	12	yes	—
243	C₂³	—	2	6	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	12	no	—
246	C₂³	—	4	3	no	—
247	C₂³	—	4	3	no	—
248	C₂³	#121×C₅₉	1	3	yes	—
249	C₂³	#121×C₅₉	1	3	yes	—
250	C₂³	C₂×#41×C₅₉	1	1	yes	characteristic
251	Q₈	—	2	12	no	—
252	Q₈	—	4	6	no	—
253	D₄	—	2	12	no	—
254	D₄	—	4	12	no	—
255	D₄	—	4	12	no	—
256	D₄	—	4	6	no	—
257	C₂×C₄	—	2	6	no	—
258	C₂×C₄	—	4	3	no	—
259	C₂×C₄	—	4	3	no	—
260	C₂×C₄	—	4	6	no	—
261	C₂×C₄	—	4	6	no	—
262	C₂×C₄	—	4	6	no	—
263	C₂×C₄	—	4	6	no	—
264	C₂×C₄	#120×C₅₉	1	3	yes	—
265	C₂×C₄	#120×C₅₉	1	3	yes	—
266	C₂×C₄	#120×C₅₉	1	6	yes	—
267	C₂×C₄	#121×C₅₉	1	3	yes	—
268	C₂×C₄	#121×C₅₉	1	3	yes	—
269	C₂×C₄	#127×C₅₉	1	3	yes	—
270	C₂×C₄	#127×C₅₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₅₉	1	3	yes	—
272	C₂×C₄	C₂²×D₄×C₅₉	1	3	yes	—
273	C₈	—	2	12	no	—
274	C₈	—	4	2	no	—
275	C₂²	—	2	6	no	—
276	C₂²	—	4	3	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	12	no	—
279	C₂²	—	4	6	no	—
280	C₂²	—	8	6	no	—
281	C₂²	—	8	6	no	—
282	C₂²	C₅₉×#392	1	3	yes	—
283	C₂²	C₅₉×#392	1	1	yes	characteristic
284	C₂²	C₂×#121×C₅₉	1	3	yes	—
285	C₄	—	4	6	no	—
286	C₄	C₅₉×#392	1	1	yes	characteristic
287	C₄	C₅₉×#392	1	3	yes	—
288	C₄	C₂×#120×C₅₉	1	3	yes	—
289	C₄	C₂×#127×C₅₉	1	1	yes	characteristic
290	C₂	—	2	3	no	—
291	C₂	—	2	3	no	—
292	C₂	—	8	6	no	—
293	C₂	C₅₉×#2017	1	2	yes	direct product factor
294	C₂	C₂×C₅₉×#392	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
socle	C₂²×C₅₉	#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#575	C₂	semidirect
2. C₂×C₅₉×#408	C₂	semidirect
3. C₅₉×#2017	C₂	direct (N × H)
4. C₂×C₅₉×#338	C₂	semidirect
8. C₅₉×#575	C₂²	semidirect
15. C₅₉×#408	C₂²	semidirect
17. C₅₉×#338	C₂²	semidirect
127. C₂×#2017	C₅₉	direct (N × H)
143. C₂×#575	C₂×C₅₉	semidirect
144. C₂×#408	C₂×C₅₉	semidirect
145. #2017	C₂×C₅₉	direct (N × H)
146. C₂×#338	C₂×C₅₉	semidirect
150. C₂×C₅₉	#2017	direct (N × H)
155. #575	C₂²×C₅₉	semidirect
162. #408	C₂²×C₅₉	semidirect
164. #338	C₂²×C₅₉	semidirect
165. C₅₉	C₂×#2017	direct (N × H)
293. C₂	C₅₉×#2017	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×#575	2
C₂×C₅₉×#408	2
C₅₉×#2017	2
C₂×C₅₉×#338	2
C₂×#2017	59

Subgroups — C2×C59×#2017 #28435542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#2017 #28435542