Subgroups — C₅₉×#2431 #21420749

Order 7552.

952 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#127×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#121×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#392	C₂	1	4	yes	maximal, complemented
6	C₅₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#386	C₂	1	4	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₅₉	—	2	4	no	—
11	C₂×#41×C₅₉	—	2	4	no	—
12	C₂×#41×C₅₉	—	2	4	no	—
13	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₅₉	C₂²	1	2	yes	—
15	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₅₉	C₂²	1	4	yes	—
17	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₅₉	C₂²	1	2	yes	—
19	C₂³×C₄×C₅₉	C₂²	1	2	yes	—
20	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
21	#127×C₅₉	—	2	4	no	—
22	#124×C₅₉	—	2	4	no	—
23	#121×C₅₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
28	C₂×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
29	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
31	#95×C₅₉	C₂²	1	2	yes	—
32	#95×C₅₉	C₂²	1	4	yes	—
33	C₂⁴×C₅₉	C₂³	1	2	yes	—
34	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
35	C₂⁴×C₅₉	C₂³	1	2	yes	—
36	#41×C₅₉	—	2	32	no	—
37	C₂×Q₈×C₅₉	—	2	2	no	—
38	C₂×Q₈×C₅₉	—	2	2	no	—
39	C₂×Q₈×C₅₉	D₄	1	4	yes	complemented
40	C₂×D₄×C₅₉	—	2	4	no	—
41	C₂×D₄×C₅₉	—	2	4	no	—
42	C₂×D₄×C₅₉	—	2	4	no	—
43	C₂×D₄×C₅₉	—	2	8	no	—
44	C₂×D₄×C₅₉	—	2	4	no	—
45	C₂×D₄×C₅₉	—	2	2	no	—
46	C₂×D₄×C₅₉	—	2	2	no	—
47	C₂×D₄×C₅₉	—	2	2	no	—
48	C₂×D₄×C₅₉	—	2	2	no	—
49	C₂×D₄×C₅₉	—	2	2	no	—
50	C₂×D₄×C₅₉	—	2	8	no	—
51	C₂×D₄×C₅₉	—	4	4	no	—
52	C₂×D₄×C₅₉	—	4	4	no	—
53	C₂×D₄×C₅₉	D₄	1	4	yes	complemented
54	C₂²×C₄×C₅₉	—	2	4	no	—
55	C₂²×C₄×C₅₉	—	2	4	no	—
56	C₂²×C₄×C₅₉	—	2	4	no	—
57	C₂²×C₄×C₅₉	—	2	2	no	—
58	C₂²×C₄×C₅₉	—	2	2	no	—
59	C₂²×C₄×C₅₉	D₄	1	4	yes	complemented
60	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₅₉	—	2	2	no	—
67	C₂²⋊C₄×C₅₉	—	2	4	no	—
68	C₂²⋊C₄×C₅₉	—	2	8	no	—
69	C₂²⋊C₄×C₅₉	—	2	4	no	—
70	C₂²⋊C₄×C₅₉	—	4	4	no	—
71	C₄²×C₅₉	—	2	2	no	—
72	C₄²×C₅₉	—	2	2	no	—
73	C₂³×C₅₉	—	2	2	no	—
74	C₂³×C₅₉	—	2	1	no	—
75	C₂³×C₅₉	—	2	2	no	—
76	C₂³×C₅₉	—	2	1	no	—
77	C₂³×C₅₉	—	2	2	no	—
78	C₂³×C₅₉	—	2	4	no	—
79	C₂³×C₅₉	—	2	4	no	—
80	C₂³×C₅₉	—	2	2	no	—
81	C₂³×C₅₉	—	2	2	no	—
82	C₂³×C₅₉	—	2	2	no	—
83	C₂³×C₅₉	—	2	2	no	—
84	C₂³×C₅₉	—	2	4	no	—
85	C₂³×C₅₉	—	2	4	no	—
86	C₂³×C₅₉	C₂×D₄	1	4	yes	—
87	C₂³×C₅₉	C₂×D₄	1	2	yes	—
88	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
89	Q₈×C₅₉	—	2	8	no	—
90	D₄×C₅₉	—	2	16	no	—
91	D₄×C₅₉	—	2	8	no	—
92	D₄×C₅₉	—	4	16	no	—
93	D₄×C₅₉	—	4	8	no	—
94	D₄×C₅₉	—	4	4	no	—
95	C₂×C₄×C₅₉	—	2	16	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	2	2	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	2	8	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₅₉	#41	1	4	yes	complemented
114	C₂²×C₅₉	—	2	4	no	—
115	C₂²×C₅₉	—	2	2	no	—
116	C₂²×C₅₉	—	2	2	no	—
117	C₂²×C₅₉	—	2	2	no	—
118	C₂²×C₅₉	—	2	8	no	—
119	C₂²×C₅₉	—	4	2	no	—
120	C₂²×C₅₉	—	4	2	no	—
121	C₂²×C₅₉	—	4	2	no	—
122	C₂²×C₅₉	—	4	2	no	—
123	C₂²×C₅₉	—	4	4	no	—
124	C₂²×C₅₉	—	4	2	no	—
125	C₂²×C₅₉	—	4	2	no	—
126	C₂²×C₅₉	—	4	2	no	—
127	C₂²×C₅₉	—	4	4	no	—
128	C₂²×C₅₉	—	4	4	no	—
129	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₅₉	C₂×#41	1	2	yes	—
133	C₂²×C₅₉	H(5,𝔽₂)	1	2	yes	—
134	C₄×C₅₉	—	2	2	no	—
135	C₄×C₅₉	—	2	4	no	—
136	C₄×C₅₉	—	2	4	no	—
137	C₄×C₅₉	—	4	2	no	—
138	C₄×C₅₉	—	4	2	no	—
139	C₄×C₅₉	#120	1	4	yes	—
140	#2431	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₅₉	—	4	2	no	—
142	C₂×C₅₉	—	4	4	no	—
143	C₂×C₅₉	—	8	2	no	—
144	C₂×C₅₉	C₂×#120	1	1	yes	characteristic
145	C₂×C₅₉	#532	1	1	yes	characteristic
146	C₂×C₅₉	#535	1	1	yes	characteristic
147	C₂×C₅₉	#546	1	2	yes	—
148	C₂×C₅₉	#550	1	2	yes	—
149	C₂²×#41	C₂×C₅₉	1	1	yes	characteristic, complemented
150	C₂×#127	C₂×C₅₉	1	1	yes	characteristic, complemented
151	C₂×#124	C₂×C₅₉	1	1	yes	characteristic, complemented
152	C₂×#121	C₂×C₅₉	1	1	yes	characteristic, complemented
153	#392	C₂×C₅₉	1	4	yes	complemented
154	#391	C₂×C₅₉	1	1	yes	characteristic, complemented
155	#390	C₂×C₅₉	1	1	yes	characteristic, complemented
156	#386	C₂×C₅₉	1	4	yes	complemented
157	C₄×C₂²⋊C₄	C₂×C₅₉	1	1	yes	characteristic, complemented
158	C₅₉	#2431	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#41	—	2	4	no	—
160	C₂×#41	—	2	4	no	—
161	C₂×#41	—	2	4	no	—
162	C₂²×Q₈	C₂²×C₅₉	1	1	yes	characteristic
163	C₂²×D₄	C₂²×C₅₉	1	1	yes	characteristic
164	C₂²×D₄	C₂²×C₅₉	1	2	yes	—
165	C₂²×D₄	C₂²×C₅₉	1	4	yes	—
166	C₂²×D₄	C₂²×C₅₉	1	1	yes	characteristic
167	C₂²×D₄	C₂²×C₅₉	1	2	yes	—
168	C₂³×C₄	C₂²×C₅₉	1	2	yes	—
169	C₂³×C₄	C₂²×C₅₉	1	1	yes	characteristic
170	#127	—	2	4	no	—
171	#124	—	2	4	no	—
172	#121	—	2	4	no	—
173	C₂×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₂²×C₅₉	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₅₉	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
178	C₂×C₄²	C₂²×C₅₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²×C₅₉	1	1	yes	characteristic
180	#95	C₂²×C₅₉	1	2	yes	—
181	#95	C₂²×C₅₉	1	4	yes	—
182	C₂⁴	C₂³×C₅₉	1	2	yes	—
183	C₂⁴	C₂³×C₅₉	1	1	yes	characteristic
184	C₂⁴	C₂³×C₅₉	1	2	yes	—
185	#41	—	2	32	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	D₄×C₅₉	1	4	yes	complemented
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	D₄×C₅₉	1	4	yes	complemented
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	D₄×C₅₉	1	4	yes	complemented
209	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
210	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
211	C₂²×C₄	C₂³×C₅₉	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
213	C₂²×C₄	C₂³×C₅₉	1	2	yes	—
214	C₂²×C₄	C₂³×C₅₉	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	C₂×D₄×C₅₉	1	4	yes	—
236	C₂³	C₂×D₄×C₅₉	1	2	yes	—
237	C₂³	C₂⁴×C₅₉	1	1	yes	characteristic
238	Q₈	—	2	8	no	—
239	D₄	—	2	16	no	—
240	D₄	—	2	8	no	—
241	D₄	—	4	16	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	16	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
258	C₂×C₄	C₂×D₄×C₅₉	1	4	yes	—
259	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
262	C₂×C₄	#41×C₅₉	1	4	yes	complemented
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
279	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
280	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
281	C₂²	C₂×#41×C₅₉	1	2	yes	—
282	C₂²	H(5,𝔽₂)×C₅₉	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	#120×C₅₉	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	—	8	2	no	—
292	C₂	C₂×#120×C₅₉	1	1	yes	characteristic
293	C₂	C₅₉×#532	1	1	yes	characteristic
294	C₂	C₅₉×#535	1	1	yes	characteristic
295	C₂	C₅₉×#546	1	2	yes	—
296	C₂	C₅₉×#550	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₅₉	C₂	semidirect
2. C₂×#127×C₅₉	C₂	semidirect
3. C₂×#124×C₅₉	C₂	semidirect
4. C₂×#121×C₅₉	C₂	semidirect
5. C₅₉×#392	C₂	semidirect
6. C₅₉×#391	C₂	semidirect
7. C₅₉×#390	C₂	semidirect
8. C₅₉×#386	C₂	semidirect
9. C₄×C₂²⋊C₄×C₅₉	C₂	semidirect
39. C₂×Q₈×C₅₉	D₄	semidirect
53. C₂×D₄×C₅₉	D₄	semidirect
59. C₂²×C₄×C₅₉	D₄	semidirect
113. C₂×C₄×C₅₉	#41	semidirect
140. #2431	C₅₉	direct (N × H)
149. C₂²×#41	C₂×C₅₉	semidirect
150. C₂×#127	C₂×C₅₉	semidirect
151. C₂×#124	C₂×C₅₉	semidirect
152. C₂×#121	C₂×C₅₉	semidirect
153. #392	C₂×C₅₉	semidirect
154. #391	C₂×C₅₉	semidirect
155. #390	C₂×C₅₉	semidirect
156. #386	C₂×C₅₉	semidirect
157. C₄×C₂²⋊C₄	C₂×C₅₉	semidirect
158. C₅₉	#2431	direct (N × H)
188. C₂×Q₈	D₄×C₅₉	semidirect
202. C₂×D₄	D₄×C₅₉	semidirect
208. C₂²×C₄	D₄×C₅₉	semidirect
262. C₂×C₄	#41×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₅₉	2
C₂×#127×C₅₉	2
C₂×#124×C₅₉	2
C₂×#121×C₅₉	2
C₅₉×#392	2
C₅₉×#391	2
C₅₉×#390	2
C₅₉×#386	2
C₄×C₂²⋊C₄×C₅₉	2
#2431	59

Subgroups — C59×#2431 #21420749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#2431 #21420749