Subgroups — C₅₉×#10603 #28422712

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#1965	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1545	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₉×#331	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#431	C₂²	1	1	yes	characteristic
10	C₂×#108×C₅₉	—	2	1	no	—
11	C₅₉×#423	C₂²	1	1	yes	characteristic
12	C₅₉×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#100×C₅₉	—	2	1	no	—
16	C₂×#100×C₅₉	—	2	1	no	—
17	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#341	—	2	1	no	—
19	C₅₉×#335	—	2	1	no	—
20	C₅₉×#335	—	2	2	no	—
21	C₅₉×#334	—	2	1	no	—
22	C₅₉×#331	—	2	2	no	—
23	C₅₉×#331	—	2	1	no	—
24	C₅₉×#331	—	2	1	no	—
25	C₂²×D₄×C₅₉	—	2	1	no	—
26	C₂²×D₄×C₅₉	—	2	1	no	—
27	C₂²×D₄×C₅₉	Q₈	1	2	yes	—
28	C₂×M₁₆×C₅₉	—	2	1	no	—
29	C₂×M₁₆×C₅₉	—	2	1	no	—
30	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
31	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	—	2	1	no	—
33	C₂²×C₈×C₅₉	—	2	1	no	—
34	#127×C₅₉	—	2	1	no	—
35	#127×C₅₉	—	2	1	no	—
36	#127×C₅₉	—	2	1	no	—
37	#127×C₅₉	—	2	1	no	—
38	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	#108×C₅₉	—	4	2	no	—
40	C₄⋊C₈×C₅₉	—	2	1	no	—
41	C₄⋊C₈×C₅₉	—	2	1	no	—
42	C₄⋊C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₈×C₅₉	—	2	1	no	—
44	C₄⋊C₈×C₅₉	—	2	1	no	—
45	C₄⋊C₈×C₅₉	—	2	1	no	—
46	C₄⋊C₈×C₅₉	—	2	1	no	—
47	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
49	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
50	#100×C₅₉	—	4	2	no	—
51	#100×C₅₉	—	4	2	no	—
52	C₂²⋊C₈×C₅₉	—	2	1	no	—
53	#97×C₅₉	—	2	1	no	—
54	C₄×C₈×C₅₉	—	2	1	no	—
55	C₄×C₈×C₅₉	—	2	1	no	—
56	C₂⁴×C₅₉	—	2	2	no	—
57	C₂×D₄×C₅₉	—	2	2	no	—
58	C₂×D₄×C₅₉	—	2	2	no	—
59	C₂×D₄×C₅₉	—	4	1	no	—
60	C₂×D₄×C₅₉	—	4	1	no	—
61	C₂×D₄×C₅₉	—	4	1	no	—
62	C₂×D₄×C₅₉	—	4	1	no	—
63	C₂×D₄×C₅₉	—	4	1	no	—
64	C₂×D₄×C₅₉	—	4	1	no	—
65	C₂×D₄×C₅₉	—	4	1	no	—
66	C₂×D₄×C₅₉	—	4	1	no	—
67	C₂×D₄×C₅₉	—	8	1	no	—
68	C₂×D₄×C₅₉	—	8	1	no	—
69	C₂×D₄×C₅₉	—	8	2	no	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
73	M₁₆×C₅₉	—	2	1	no	—
74	M₁₆×C₅₉	—	2	1	no	—
75	M₁₆×C₅₉	—	2	1	no	—
76	M₁₆×C₅₉	—	4	1	no	—
77	M₁₆×C₅₉	—	4	1	no	—
78	M₁₆×C₅₉	—	4	1	no	—
79	M₁₆×C₅₉	—	4	1	no	—
80	M₁₆×C₅₉	D₈	1	2	yes	—
81	C₂×C₈×C₅₉	—	2	1	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	—	4	1	no	—
88	C₂×C₈×C₅₉	—	4	1	no	—
89	C₂×C₈×C₅₉	—	4	1	no	—
90	C₂×C₈×C₅₉	—	4	1	no	—
91	C₂×C₈×C₅₉	—	8	1	no	—
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₉	#41	1	1	yes	characteristic
95	C₄²×C₅₉	#41	1	1	yes	characteristic
96	C₂³×C₅₉	—	4	2	no	—
97	C₂³×C₅₉	—	4	2	no	—
98	C₂³×C₅₉	—	4	2	no	—
99	C₂³×C₅₉	—	8	2	no	—
100	C₂³×C₅₉	—	8	2	no	—
101	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
102	D₄×C₅₉	—	8	2	no	—
103	D₄×C₅₉	—	8	2	no	—
104	D₄×C₅₉	—	8	2	no	—
105	D₄×C₅₉	—	8	1	no	—
106	D₄×C₅₉	—	8	1	no	—
107	D₄×C₅₉	—	8	1	no	—
108	D₄×C₅₉	—	8	1	no	—
109	D₄×C₅₉	—	8	1	no	—
110	D₄×C₅₉	—	8	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	4	1	no	—
114	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	#126	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
124	C₈×C₅₉	—	2	2	no	—
125	C₈×C₅₉	—	4	2	no	—
126	C₈×C₅₉	—	4	1	no	—
127	C₈×C₅₉	—	4	1	no	—
128	C₈×C₅₉	—	4	1	no	—
129	C₈×C₅₉	—	8	1	no	—
130	C₈×C₅₉	—	8	1	no	—
131	C₈×C₅₉	—	8	1	no	—
132	#10603	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₉	—	2	1	no	—
134	C₂²×C₅₉	—	2	1	no	—
135	C₂²×C₅₉	—	8	2	no	—
136	C₂²×C₅₉	—	8	2	no	—
137	C₂²×C₅₉	—	8	2	no	—
138	C₂²×C₅₉	—	16	2	no	—
139	C₂²×C₅₉	—	16	2	no	—
140	C₂²×C₅₉	#396	1	1	yes	characteristic
141	C₂²×C₅₉	#474	1	1	yes	characteristic
142	C₂²×C₅₉	#478	1	1	yes	characteristic
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	#459	1	1	yes	characteristic
147	C₄×C₅₉	#464	1	1	yes	characteristic
148	C₄×C₅₉	#466	1	1	yes	characteristic
149	C₄×C₅₉	#468	1	1	yes	characteristic
150	C₄×C₅₉	#471	1	1	yes	characteristic
151	C₄×C₅₉	#472	1	1	yes	characteristic
152	#1965	C₂×C₅₉	1	1	yes	characteristic
153	#1950	C₂×C₅₉	1	1	yes	characteristic, complemented
154	#1571	C₂×C₅₉	1	1	yes	characteristic, complemented
155	#1565	C₂×C₅₉	1	1	yes	characteristic, complemented
156	#1545	C₂×C₅₉	1	1	yes	characteristic
157	C₂×#331	C₂×C₅₉	1	1	yes	characteristic
158	#1292	C₂×C₅₉	1	1	yes	characteristic, complemented
159	C₂×C₅₉	—	2	1	no	—
160	C₂×C₅₉	—	2	1	no	—
161	C₂×C₅₉	—	16	2	no	—
162	C₂×C₅₉	#2063	1	1	yes	characteristic
163	C₂×C₅₉	#2067	1	1	yes	characteristic
164	C₂×C₅₉	#2069	1	1	yes	characteristic
165	C₂×#127	C₂²×C₅₉	1	1	yes	characteristic
166	#431	C₂²×C₅₉	1	1	yes	characteristic
167	C₂×#108	—	2	1	no	—
168	#423	C₂²×C₅₉	1	1	yes	characteristic
169	#423	C₂²×C₅₉	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
172	C₂×#100	—	2	1	no	—
173	C₂×#100	—	2	1	no	—
174	C₄×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
175	#341	—	2	1	no	—
176	#335	—	2	1	no	—
177	#335	—	2	2	no	—
178	#334	—	2	1	no	—
179	#331	—	2	2	no	—
180	#331	—	2	1	no	—
181	#331	—	2	1	no	—
182	C₅₉	#10603	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	Q₈×C₅₉	1	2	yes	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	D₄×C₅₉	1	1	yes	characteristic
189	C₂×M₁₆	D₄×C₅₉	1	1	yes	characteristic
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	#127	—	2	1	no	—
193	#127	—	2	1	no	—
194	#127	—	2	1	no	—
195	#127	—	2	1	no	—
196	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
197	#108	—	4	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
206	C₄⋊C₈	D₄×C₅₉	1	2	yes	—
207	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
208	#100	—	4	2	no	—
209	#100	—	4	2	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	8	1	no	—
226	C₂×D₄	—	8	1	no	—
227	C₂×D₄	—	8	2	no	—
228	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
230	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	M₁₆	—	4	1	no	—
238	M₁₆	D₈×C₅₉	1	2	yes	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	8	1	no	—
250	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
252	C₄²	#41×C₅₉	1	1	yes	characteristic
253	C₄²	#41×C₅₉	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
260	D₄	—	8	2	no	—
261	D₄	—	8	2	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	1	no	—
264	D₄	—	8	1	no	—
265	D₄	—	8	1	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	D₄	—	8	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
273	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
274	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
275	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
276	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
277	C₂×C₄	#126×C₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
279	C₂×C₄	#135×C₅₉	1	1	yes	characteristic
280	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
281	C₂×C₄	#137×C₅₉	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	8	1	no	—
288	C₈	—	8	1	no	—
289	C₈	—	8	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	2	no	—
296	C₂²	—	16	2	no	—
297	C₂²	C₅₉×#396	1	1	yes	characteristic
298	C₂²	C₅₉×#474	1	1	yes	characteristic
299	C₂²	C₅₉×#478	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	C₅₉×#459	1	1	yes	characteristic
304	C₄	C₅₉×#464	1	1	yes	characteristic
305	C₄	C₅₉×#466	1	1	yes	characteristic
306	C₄	C₅₉×#468	1	1	yes	characteristic
307	C₄	C₅₉×#471	1	1	yes	characteristic
308	C₄	C₅₉×#472	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	16	2	no	—
312	C₂	C₅₉×#2063	1	1	yes	characteristic
313	C₂	C₅₉×#2067	1	1	yes	characteristic
314	C₂	C₅₉×#2069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
prime radical	C₂×#127×C₅₉	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#1950	C₂	semidirect
3. C₅₉×#1571	C₂	semidirect
4. C₅₉×#1565	C₂	semidirect
7. C₅₉×#1292	C₂	semidirect
132. #10603	C₅₉	direct (N × H)
153. #1950	C₂×C₅₉	semidirect
154. #1571	C₂×C₅₉	semidirect
155. #1565	C₂×C₅₉	semidirect
158. #1292	C₂×C₅₉	semidirect
182. C₅₉	#10603	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#1965	2
C₅₉×#1950	2
C₅₉×#1571	2
C₅₉×#1565	2
C₅₉×#1545	2
C₂×C₅₉×#331	2
C₅₉×#1292	2
#10603	59

Subgroups — C59×#10603 #28422712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#10603 #28422712