Subgroups — C₅₉×#23640 #28435749

Order 15104.

1014 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#408	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2311	C₂	1	2	yes	maximal
4	C₅₉×#2077	C₂	1	8	yes	maximal, complemented
5	C₅₉×#1736	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1735	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₅₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₉	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
12	C₂×#106×C₅₉	—	2	4	no	—
13	C₂×#106×C₅₉	C₂²	1	4	yes	—
14	C₅₉×#408	—	2	4	no	—
15	C₂×C₂²⋊C₈×C₅₉	C₂²	1	4	yes	—
16	C₅₉×#391	—	2	2	no	—
17	C₅₉×#390	—	2	2	no	—
18	C₅₉×#386	—	2	4	no	—
19	C₅₉×#380	C₂²	1	8	yes	—
20	C₅₉×#340	C₂²	1	4	yes	—
21	C₅₉×#340	C₂²	1	8	yes	—
22	C₂×#41×C₅₉	—	2	2	no	—
23	C₂×#41×C₅₉	—	2	2	no	—
24	C₂×#41×C₅₉	—	2	4	no	—
25	C₂×#41×C₅₉	—	2	4	no	—
26	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₅₉	D₄	1	2	yes	—
28	C₂²×D₄×C₅₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₉	—	2	2	no	—
30	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
31	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₅₉	—	2	2	no	—
33	C₂×M₁₆×C₅₉	—	2	4	no	—
34	C₂²×C₈×C₅₉	—	2	4	no	—
35	C₂²×C₈×C₅₉	D₄	1	4	yes	—
36	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
38	#117×C₅₉	—	2	4	no	—
39	#117×C₅₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
45	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₅₉	—	2	8	no	—
47	C₂×C₄²×C₅₉	—	2	2	no	—
48	#106×C₅₉	—	2	8	no	—
49	C₂²⋊C₈×C₅₉	—	2	8	no	—
50	#95×C₅₉	—	2	8	no	—
51	C₂⁴×C₅₉	C₂×D₄	1	2	yes	—
52	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
53	#41×C₅₉	—	4	4	no	—
54	#41×C₅₉	—	4	4	no	—
55	#41×C₅₉	—	4	8	no	—
56	C₂×Q₈×C₅₉	—	2	2	no	—
57	C₂×Q₈×C₅₉	—	2	4	no	—
58	C₂×D₄×C₅₉	—	2	4	no	—
59	C₂×D₄×C₅₉	—	2	8	no	—
60	C₂×D₄×C₅₉	—	2	2	no	—
61	C₂×D₄×C₅₉	—	2	4	no	—
62	C₂²×C₄×C₅₉	—	2	2	no	—
63	C₂²×C₄×C₅₉	—	2	2	no	—
64	C₂²×C₄×C₅₉	—	2	4	no	—
65	C₂²×C₄×C₅₉	—	2	4	no	—
66	C₂²×C₄×C₅₉	—	2	4	no	—
67	C₂²×C₄×C₅₉	—	2	4	no	—
68	C₂²×C₄×C₅₉	—	4	4	no	—
69	C₂²×C₄×C₅₉	—	8	4	no	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅₉	#41	1	2	yes	—
74	C₂²×C₄×C₅₉	#41	1	4	yes	—
75	C₂²×C₄×C₅₉	#41	1	4	yes	—
76	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₅₉	—	4	4	no	—
78	C₂×C₈×C₅₉	—	2	8	no	—
79	C₂×C₈×C₅₉	—	2	4	no	—
80	C₂×C₈×C₅₉	—	2	2	no	—
81	C₂×C₈×C₅₉	—	2	4	no	—
82	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
83	C₄⋊C₄×C₅₉	—	2	4	no	—
84	C₄⋊C₄×C₅₉	—	2	4	no	—
85	C₄⋊C₄×C₅₉	—	2	4	no	—
86	C₄⋊C₄×C₅₉	—	2	4	no	—
87	C₄⋊C₄×C₅₉	—	4	4	no	—
88	C₂²⋊C₄×C₅₉	—	4	4	no	—
89	C₂²⋊C₄×C₅₉	—	8	8	no	—
90	C₄²×C₅₉	—	4	4	no	—
91	C₂³×C₅₉	—	2	4	no	—
92	C₂³×C₅₉	—	2	2	no	—
93	C₂³×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	—	2	8	no	—
95	C₂³×C₅₉	#121	1	2	yes	—
96	C₂³×C₅₉	#123	1	4	yes	—
97	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₅₉	—	4	4	no	—
99	D₄×C₅₉	—	4	8	no	—
100	D₄×C₅₉	—	4	4	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	2	4	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	—	4	4	no	—
106	C₂×C₄×C₅₉	—	4	4	no	—
107	C₂×C₄×C₅₉	—	4	4	no	—
108	C₂×C₄×C₅₉	—	4	4	no	—
109	C₂×C₄×C₅₉	—	4	8	no	—
110	C₂×C₄×C₅₉	—	4	4	no	—
111	C₂×C₄×C₅₉	—	8	8	no	—
112	C₂×C₄×C₅₉	#121	1	2	yes	—
113	C₂×C₄×C₅₉	#123	1	4	yes	—
114	C₂×C₄×C₅₉	#123	1	4	yes	—
115	C₂×C₄×C₅₉	#124	1	4	yes	—
116	C₂×C₄×C₅₉	#126	1	8	yes	—
117	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
119	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
120	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
121	C₈×C₅₉	—	2	4	no	—
122	C₈×C₅₉	—	4	2	no	—
123	#23640	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₉	—	2	8	no	—
125	C₂²×C₅₉	—	2	4	no	—
126	C₂²×C₅₉	—	4	8	no	—
127	C₂²×C₅₉	—	4	2	no	—
128	C₂²×C₅₉	—	4	4	no	—
129	C₂²×C₅₉	#397	1	4	yes	—
130	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
131	C₂²×C₅₉	C₂×#123	1	2	yes	—
132	C₂²×C₅₉	#577	1	4	yes	—
133	C₄×C₅₉	—	4	2	no	—
134	C₄×C₅₉	—	8	4	no	—
135	C₄×C₅₉	—	8	4	no	—
136	C₄×C₅₉	#397	1	4	yes	—
137	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
138	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
139	C₄×C₅₉	C₂×#126	1	2	yes	—
140	C₂²×#106	C₂×C₅₉	1	1	yes	characteristic, complemented
141	C₂×#408	C₂×C₅₉	1	1	yes	characteristic
142	#2311	C₂×C₅₉	1	2	yes	—
143	#2077	C₂×C₅₉	1	8	yes	complemented
144	#1736	C₂×C₅₉	1	2	yes	complemented
145	#1735	C₂×C₅₉	1	1	yes	characteristic, complemented
146	C₂×C₅₉	—	2	4	no	—
147	C₂×C₅₉	—	8	2	no	—
148	C₂×C₅₉	C₂×#397	1	1	yes	characteristic
149	C₂×C₅₉	#3057	1	2	yes	—
150	C₂×C₅₉	#3099	1	4	yes	—
151	C₂²×#41	C₂²×C₅₉	1	1	yes	characteristic
152	C₂²×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
154	C₂×#117	C₂²×C₅₉	1	2	yes	—
155	C₂²×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
156	C₂×#106	—	2	4	no	—
157	C₂×#106	C₂²×C₅₉	1	4	yes	—
158	#408	—	2	4	no	—
159	C₂×C₂²⋊C₈	C₂²×C₅₉	1	4	yes	—
160	#391	—	2	2	no	—
161	#390	—	2	2	no	—
162	#386	—	2	4	no	—
163	#380	C₂²×C₅₉	1	8	yes	—
164	#340	C₂²×C₅₉	1	4	yes	—
165	#340	C₂²×C₅₉	1	8	yes	—
166	C₅₉	#23640	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#41	—	2	2	no	—
168	C₂×#41	—	2	2	no	—
169	C₂×#41	—	2	4	no	—
170	C₂×#41	—	2	4	no	—
171	C₂²×Q₈	D₄×C₅₉	1	1	yes	characteristic
172	C₂²×D₄	D₄×C₅₉	1	2	yes	—
173	C₂²×D₄	D₄×C₅₉	1	1	yes	characteristic
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
176	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	D₄×C₅₉	1	4	yes	—
181	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
182	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	8	no	—
192	C₂×C₄²	—	2	2	no	—
193	#106	—	2	8	no	—
194	C₂²⋊C₈	—	2	8	no	—
195	#95	—	2	8	no	—
196	C₂⁴	C₂×D₄×C₅₉	1	2	yes	—
197	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	#41	—	4	8	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	8	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
218	C₂²×C₄	#41×C₅₉	1	2	yes	—
219	C₂²×C₄	#41×C₅₉	1	4	yes	—
220	C₂²×C₄	#41×C₅₉	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
222	M₁₆	—	4	4	no	—
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	8	8	no	—
235	C₄²	—	4	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	8	no	—
240	C₂³	#121×C₅₉	1	2	yes	—
241	C₂³	#123×C₅₉	1	4	yes	—
242	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	D₄	—	4	8	no	—
245	D₄	—	4	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	#121×C₅₉	1	2	yes	—
258	C₂×C₄	#123×C₅₉	1	4	yes	—
259	C₂×C₄	#123×C₅₉	1	4	yes	—
260	C₂×C₄	#124×C₅₉	1	4	yes	—
261	C₂×C₄	#126×C₅₉	1	8	yes	—
262	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
264	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
265	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	4	8	no	—
272	C₂²	—	4	2	no	—
273	C₂²	C₅₉×#397	1	4	yes	—
274	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
275	C₂²	C₂×#123×C₅₉	1	2	yes	—
276	C₂²	C₅₉×#577	1	4	yes	—
277	C₄	—	4	2	no	—
278	C₄	—	8	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₅₉×#397	1	4	yes	—
281	C₄	C₂×#123×C₅₉	1	1	yes	characteristic
282	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
283	C₄	C₂×#126×C₅₉	1	2	yes	—
284	C₂	—	2	4	no	—
285	C₂	—	8	2	no	—
286	C₂	C₂×C₅₉×#397	1	1	yes	characteristic
287	C₂	C₅₉×#3057	1	2	yes	—
288	C₂	C₅₉×#3099	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×#41×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂²×#41×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#106×C₅₉	C₂	semidirect
4. C₅₉×#2077	C₂	semidirect
5. C₅₉×#1736	C₂	semidirect
6. C₅₉×#1735	C₂	semidirect
123. #23640	C₅₉	direct (N × H)
140. C₂²×#106	C₂×C₅₉	semidirect
143. #2077	C₂×C₅₉	semidirect
144. #1736	C₂×C₅₉	semidirect
145. #1735	C₂×C₅₉	semidirect
166. C₅₉	#23640	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×#106×C₅₉	2
C₂×C₅₉×#408	2
C₅₉×#2311	2
C₅₉×#2077	2
C₅₉×#1736	2
C₅₉×#1735	2
#23640	59

Subgroups — C59×#23640 #28435749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#23640 #28435749