Subgroups — C₅₉×#29210 #28441319

Order 15104.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2059	C₂	1	4	yes	maximal
4	C₅₉×#1991	C₂	1	2	yes	maximal
5	C₅₉×#1948	C₂	1	2	yes	maximal
6	C₅₉×#1947	C₂	1	2	yes	maximal
7	C₅₉×#1846	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×#125×C₅₉	C₂²	1	2	yes	—
11	C₂×#107×C₅₉	C₂²	1	4	yes	—
12	C₂×#106×C₅₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
16	C₅₉×#389	—	2	2	no	—
17	C₅₉×#389	—	2	2	no	—
18	C₅₉×#389	C₂²	1	2	yes	—
19	C₅₉×#387	—	2	2	no	—
20	C₅₉×#384	C₂²	1	2	yes	—
21	C₅₉×#382	C₂²	1	2	yes	—
22	C₅₉×#382	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
24	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
25	C₄³×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#340	C₂²	1	4	yes	—
27	C₅₉×#340	C₂²	1	4	yes	—
28	C₅₉×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
31	#125×C₅₉	—	2	2	no	—
32	#125×C₅₉	—	4	2	no	—
33	#125×C₅₉	—	4	2	no	—
34	#125×C₅₉	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₅₉	—	2	2	no	—
44	C₂×C₄²×C₅₉	D₄	1	2	yes	—
45	C₂×C₄²×C₅₉	D₄	1	2	yes	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
49	#107×C₅₉	—	4	4	no	—
50	#106×C₅₉	—	4	4	no	—
51	C₄⋊C₈×C₅₉	Q₈	1	4	yes	—
52	C₄⋊C₈×C₅₉	Q₈	1	4	yes	—
53	C₄×C₈×C₅₉	—	2	4	no	—
54	#95×C₅₉	—	2	2	no	—
55	#95×C₅₉	—	2	2	no	—
56	#95×C₅₉	—	2	2	no	—
57	#95×C₅₉	—	2	4	no	—
58	C₂²×C₄×C₅₉	—	2	2	no	—
59	C₂²×C₄×C₅₉	—	2	2	no	—
60	C₂²×C₄×C₅₉	—	2	4	no	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	—	2	2	no	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
72	C₄⋊C₄×C₅₉	—	2	8	no	—
73	C₄⋊C₄×C₅₉	—	2	2	no	—
74	C₄⋊C₄×C₅₉	—	2	2	no	—
75	C₄⋊C₄×C₅₉	—	2	4	no	—
76	C₄⋊C₄×C₅₉	—	2	2	no	—
77	C₄⋊C₄×C₅₉	—	2	4	no	—
78	C₄⋊C₄×C₅₉	—	4	4	no	—
79	C₄⋊C₄×C₅₉	—	4	4	no	—
80	C₄⋊C₄×C₅₉	—	4	8	no	—
81	C₄⋊C₄×C₅₉	—	4	4	no	—
82	C₄⋊C₄×C₅₉	C₂×Q₈	1	4	yes	—
83	C₄²×C₅₉	—	2	4	no	—
84	C₄²×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	4	2	no	—
87	C₄²×C₅₉	—	4	4	no	—
88	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
89	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
90	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₅₉	#41	1	2	yes	—
92	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	4	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	—	4	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	2	no	—
107	C₂×C₄×C₅₉	—	4	4	no	—
108	C₂×C₄×C₅₉	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₅₉	#135	1	4	yes	—
112	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
118	C₈×C₅₉	—	2	4	no	—
119	C₈×C₅₉	—	2	4	no	—
120	#29210	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₅₉	#535	1	1	yes	characteristic
124	C₂²×C₅₉	C₂×#135	1	2	yes	—
125	C₂²×C₅₉	#575	1	2	yes	—
126	C₄×C₅₉	—	2	4	no	—
127	C₄×C₅₉	—	2	2	no	—
128	C₄×C₅₉	—	4	4	no	—
129	C₄×C₅₉	—	4	2	no	—
130	C₄×C₅₉	—	8	4	no	—
131	C₄×C₅₉	—	8	2	no	—
132	C₄×C₅₉	#499	1	4	yes	—
133	C₄×C₅₉	#554	1	1	yes	characteristic
134	C₄×C₅₉	#554	1	1	yes	characteristic
135	C₄×C₅₉	Q₈²	1	1	yes	characteristic
136	C₄×C₅₉	Q₈²	1	1	yes	characteristic
137	#2859	C₂×C₅₉	1	1	yes	characteristic
138	C₄×#125	C₂×C₅₉	1	1	yes	characteristic
139	#2059	C₂×C₅₉	1	4	yes	—
140	#1991	C₂×C₅₉	1	2	yes	—
141	#1948	C₂×C₅₉	1	2	yes	—
142	#1947	C₂×C₅₉	1	2	yes	—
143	#1846	C₂×C₅₉	1	2	yes	—
144	C₄×C₄⋊C₈	C₂×C₅₉	1	1	yes	characteristic
145	C₂×C₅₉	#2567	1	1	yes	characteristic
146	C₂×C₅₉	#3076	1	2	yes	—
147	C₂×C₅₉	C₂×#499	1	1	yes	characteristic
148	C₂×C₅₉	#3166	1	1	yes	characteristic
149	C₂×C₅₉	#3212	1	2	yes	—
150	C₂×#125	C₂²×C₅₉	1	1	yes	characteristic
151	C₂×#125	C₂²×C₅₉	1	2	yes	—
152	C₂×#107	C₂²×C₅₉	1	4	yes	—
153	C₂×#106	C₂²×C₅₉	1	4	yes	—
154	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
157	#389	—	2	2	no	—
158	#389	—	2	2	no	—
159	#389	C₂²×C₅₉	1	2	yes	—
160	#387	—	2	2	no	—
161	#384	C₂²×C₅₉	1	2	yes	—
162	#382	C₂²×C₅₉	1	1	yes	characteristic
163	#382	C₂²×C₅₉	1	2	yes	—
164	C₄×C₄⋊C₄	—	2	2	no	—
165	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
166	C₄³	C₂²×C₅₉	1	1	yes	characteristic
167	#340	C₂²×C₅₉	1	4	yes	—
168	#340	C₂²×C₅₉	1	4	yes	—
169	#336	C₂²×C₅₉	1	2	yes	—
170	C₅₉	#29210	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
172	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
173	#125	—	2	2	no	—
174	#125	—	4	2	no	—
175	#125	—	4	2	no	—
176	#125	Q₈×C₅₉	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	D₄×C₅₉	1	2	yes	—
187	C₂×C₄²	D₄×C₅₉	1	2	yes	—
188	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
191	#107	—	4	4	no	—
192	#106	—	4	4	no	—
193	C₄⋊C₈	Q₈×C₅₉	1	4	yes	—
194	C₄⋊C₈	Q₈×C₅₉	1	4	yes	—
195	C₄×C₈	—	2	4	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
213	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	8	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	C₂×Q₈×C₅₉	1	4	yes	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	C₂×Q₈×C₅₉	1	2	yes	—
231	C₄²	C₂×Q₈×C₅₉	1	2	yes	—
232	C₄²	C₂×Q₈×C₅₉	1	2	yes	—
233	C₄²	#41×C₅₉	1	2	yes	—
234	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₂²⋊Q₈×C₅₉	1	4	yes	—
251	C₂×C₄	C₄⋊Q₈×C₅₉	1	2	yes	—
252	C₂×C₄	C₄⋊Q₈×C₅₉	1	2	yes	—
253	C₂×C₄	#135×C₅₉	1	4	yes	—
254	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
258	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	2	4	no	—
262	C₂²	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
264	C₂²	C₅₉×#535	1	1	yes	characteristic
265	C₂²	C₂×#135×C₅₉	1	2	yes	—
266	C₂²	C₅₉×#575	1	2	yes	—
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	4	no	—
272	C₄	—	8	2	no	—
273	C₄	C₅₉×#499	1	4	yes	—
274	C₄	C₅₉×#554	1	1	yes	characteristic
275	C₄	C₅₉×#554	1	1	yes	characteristic
276	C₄	Q₈²×C₅₉	1	1	yes	characteristic
277	C₄	Q₈²×C₅₉	1	1	yes	characteristic
278	C₂	C₅₉×#2567	1	1	yes	characteristic
279	C₂	C₅₉×#3076	1	2	yes	—
280	C₂	C₂×C₅₉×#499	1	1	yes	characteristic
281	C₂	C₅₉×#3166	1	1	yes	characteristic
282	C₂	C₅₉×#3212	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
120. #29210	C₅₉	direct (N × H)
170. C₅₉	#29210	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×#2859	2
C₄×#125×C₅₉	2
C₅₉×#2059	2
C₅₉×#1991	2
C₅₉×#1948	2
C₅₉×#1947	2
C₅₉×#1846	2
C₄×C₄⋊C₈×C₅₉	2
#29210	59

Subgroups — C59×#29210 #28441319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#29210 #28441319