Subgroups — C₅₉×#3220 #21421538

Order 7552.

420 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₉×#535	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#463	C₂	1	2	yes	maximal, complemented
8	C₅₉×#459	C₂	1	2	yes	maximal, complemented
9	C₅₉×#449	C₂	1	2	yes	maximal, complemented
10	C₅₉×#447	C₂	1	2	yes	maximal, complemented
11	C₅₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₅₉	C₂²	1	2	yes	complemented
15	C₂×D₈×C₅₉	C₂²	1	2	yes	—
16	C₂×D₈×C₅₉	C₂²	1	2	yes	—
17	C₂×D₈×C₅₉	C₂²	1	2	yes	complemented
18	#127×C₅₉	C₂²	1	1	yes	characteristic
19	#127×C₅₉	C₂²	1	1	yes	characteristic
20	#127×C₅₉	C₂²	1	1	yes	characteristic, complemented
21	#125×C₅₉	C₂²	1	1	yes	characteristic, complemented
22	#124×C₅₉	C₂²	1	1	yes	characteristic
23	#124×C₅₉	C₂²	1	1	yes	characteristic
24	#123×C₅₉	—	2	2	no	—
25	#121×C₅₉	—	2	2	no	—
26	#121×C₅₉	—	2	2	no	—
27	#121×C₅₉	C₂²	1	2	yes	—
28	#120×C₅₉	—	2	2	no	—
29	C₄×D₄×C₅₉	C₂²	1	2	yes	complemented
30	#117×C₅₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	C₂²	1	2	yes	complemented
32	#103×C₅₉	C₂²	1	2	yes	complemented
33	#102×C₅₉	C₂²	1	2	yes	—
34	#102×C₅₉	C₂²	1	2	yes	—
35	#102×C₅₉	C₂²	1	2	yes	complemented
36	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
37	#97×C₅₉	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₅₉	—	2	1	no	—
40	#41×C₅₉	—	4	2	no	—
41	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₅₉	—	2	1	no	—
43	C₂×D₄×C₅₉	—	2	2	no	—
44	C₂×D₄×C₅₉	—	2	1	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	—	4	2	no	—
49	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₅₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₉	—	2	1	no	—
54	C₂²×C₄×C₅₉	—	2	2	no	—
55	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
56	SD₁₆×C₅₉	—	4	2	no	—
57	D₈×C₅₉	—	4	2	no	—
58	D₈×C₅₉	—	4	2	no	—
59	D₈×C₅₉	—	4	2	no	—
60	C₂×C₈×C₅₉	C₂³	1	2	yes	—
61	C₂×C₈×C₅₉	C₂³	1	2	yes	—
62	C₄⋊C₄×C₅₉	—	2	2	no	—
63	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
64	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₅₉	—	2	2	no	—
66	C₂²⋊C₄×C₅₉	—	2	2	no	—
67	C₂²⋊C₄×C₅₉	—	2	2	no	—
68	C₂²⋊C₄×C₅₉	—	2	2	no	—
69	C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
70	C₄²×C₅₉	C₂³	1	1	yes	characteristic
71	C₄²×C₅₉	C₂³	1	1	yes	characteristic
72	C₂³×C₅₉	—	2	2	no	—
73	C₂³×C₅₉	—	2	1	no	—
74	C₂³×C₅₉	—	2	1	no	—
75	C₂³×C₅₉	—	2	1	no	—
76	C₂³×C₅₉	—	4	2	no	—
77	C₂³×C₅₉	—	8	2	no	—
78	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
79	Q₈×C₅₉	—	2	2	no	—
80	D₄×C₅₉	—	2	2	no	—
81	D₄×C₅₉	—	2	2	no	—
82	D₄×C₅₉	—	2	2	no	—
83	D₄×C₅₉	—	2	2	no	—
84	D₄×C₅₉	—	2	2	no	—
85	D₄×C₅₉	—	4	2	no	—
86	D₄×C₅₉	—	4	2	no	—
87	D₄×C₅₉	—	8	2	no	—
88	D₄×C₅₉	—	8	2	no	—
89	D₄×C₅₉	—	8	2	no	—
90	D₄×C₅₉	—	8	2	no	—
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	4	2	no	—
95	C₂×C₄×C₅₉	—	8	2	no	—
96	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₅₉	—	2	2	no	—
101	C₈×C₅₉	—	2	2	no	—
102	C₂²×C₅₉	—	2	1	no	—
103	C₂²×C₅₉	—	2	2	no	—
104	C₂²×C₅₉	—	4	1	no	—
105	C₂²×C₅₉	—	4	1	no	—
106	C₂²×C₅₉	—	4	1	no	—
107	C₂²×C₅₉	—	4	2	no	—
108	C₂²×C₅₉	—	4	2	no	—
109	C₂²×C₅₉	—	4	2	no	—
110	C₂²×C₅₉	—	4	2	no	—
111	C₂²×C₅₉	—	4	2	no	—
112	C₂²×C₅₉	—	4	2	no	—
113	C₂²×C₅₉	—	8	4	no	—
114	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₅₉	—	2	1	no	—
116	C₄×C₅₉	—	2	2	no	—
117	C₄×C₅₉	—	2	2	no	—
118	C₄×C₅₉	—	4	1	no	—
119	C₄×C₅₉	—	4	2	no	—
120	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
121	#3220	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₅₉	—	4	1	no	—
123	C₂×C₅₉	—	8	1	no	—
124	C₂×C₅₉	—	8	2	no	—
125	C₂×C₅₉	—	8	1	no	—
126	C₂×C₅₉	—	8	1	no	—
127	C₂×C₅₉	#535	1	1	yes	characteristic
128	C₂×C₅₉	#576	1	2	yes	—
129	#540	C₂×C₅₉	1	1	yes	characteristic, complemented
130	#535	C₂×C₅₉	1	1	yes	characteristic, complemented
131	#496	C₂×C₅₉	1	1	yes	characteristic, complemented
132	#493	C₂×C₅₉	1	1	yes	characteristic, complemented
133	#490	C₂×C₅₉	1	1	yes	characteristic, complemented
134	#488	C₂×C₅₉	1	1	yes	characteristic, complemented
135	#463	C₂×C₅₉	1	2	yes	complemented
136	#459	C₂×C₅₉	1	2	yes	complemented
137	#449	C₂×C₅₉	1	2	yes	complemented
138	#447	C₂×C₅₉	1	2	yes	complemented
139	#433	C₂×C₅₉	1	1	yes	characteristic, complemented
140	C₅₉	#3220	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#41	C₂²×C₅₉	1	1	yes	characteristic
142	C₂²×D₄	C₂²×C₅₉	1	1	yes	characteristic
143	C₂×SD₁₆	C₂²×C₅₉	1	2	yes	complemented
144	C₂×D₈	C₂²×C₅₉	1	2	yes	—
145	C₂×D₈	C₂²×C₅₉	1	2	yes	—
146	C₂×D₈	C₂²×C₅₉	1	2	yes	complemented
147	#127	C₂²×C₅₉	1	1	yes	characteristic
148	#127	C₂²×C₅₉	1	1	yes	characteristic
149	#127	C₂²×C₅₉	1	1	yes	characteristic, complemented
150	#125	C₂²×C₅₉	1	1	yes	characteristic, complemented
151	#124	C₂²×C₅₉	1	1	yes	characteristic
152	#124	C₂²×C₅₉	1	1	yes	characteristic
153	#123	—	2	2	no	—
154	#121	—	2	2	no	—
155	#121	—	2	2	no	—
156	#121	C₂²×C₅₉	1	2	yes	—
157	#120	—	2	2	no	—
158	C₄×D₄	C₂²×C₅₉	1	2	yes	complemented
159	#117	C₂²×C₅₉	1	1	yes	characteristic
160	C₄⋊C₈	C₂²×C₅₉	1	2	yes	complemented
161	#103	C₂²×C₅₉	1	2	yes	complemented
162	#102	C₂²×C₅₉	1	2	yes	—
163	#102	C₂²×C₅₉	1	2	yes	—
164	#102	C₂²×C₅₉	1	2	yes	complemented
165	C₂²⋊C₈	C₂²×C₅₉	1	2	yes	—
166	#97	C₂²×C₅₉	1	1	yes	characteristic, complemented
167	C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic, complemented
168	C₂⁴	—	2	1	no	—
169	#41	—	4	2	no	—
170	C₂×Q₈	C₂³×C₅₉	1	1	yes	characteristic
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	C₂³×C₅₉	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₅₉	1	1	yes	characteristic
180	C₂×D₄	C₂³×C₅₉	1	1	yes	characteristic
181	C₂×D₄	C₂³×C₅₉	1	2	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂³×C₅₉	1	1	yes	characteristic
185	SD₁₆	—	4	2	no	—
186	D₈	—	4	2	no	—
187	D₈	—	4	2	no	—
188	D₈	—	4	2	no	—
189	C₂×C₈	C₂³×C₅₉	1	2	yes	—
190	C₂×C₈	C₂³×C₅₉	1	2	yes	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	D₄×C₅₉	1	2	yes	—
193	C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	D₄×C₅₉	1	2	yes	—
199	C₄²	C₂³×C₅₉	1	1	yes	characteristic
200	C₄²	C₂³×C₅₉	1	1	yes	characteristic
201	C₂³	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	8	2	no	—
207	C₂³	C₂×D₄×C₅₉	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	D₄	—	2	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	2	no	—
212	D₄	—	2	2	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	D₄	—	8	2	no	—
219	D₄	—	8	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₅₉	1	2	yes	—
227	C₂×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	8	4	no	—
243	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	H(5,𝔽₂)×C₅₉	1	2	yes	—
250	C₂	—	4	1	no	—
251	C₂	—	8	1	no	—
252	C₂	—	8	1	no	—
253	C₂	—	8	1	no	—
254	C₂	—	8	2	no	—
255	C₂	C₅₉×#535	1	1	yes	characteristic
256	C₂	C₅₉×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ #117×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#540	C₂	semidirect
2. C₅₉×#535	C₂	semidirect
3. C₅₉×#496	C₂	semidirect
4. C₅₉×#493	C₂	semidirect
5. C₅₉×#490	C₂	semidirect
6. C₅₉×#488	C₂	semidirect
7. C₅₉×#463	C₂	semidirect
8. C₅₉×#459	C₂	semidirect
9. C₅₉×#449	C₂	semidirect
10. C₅₉×#447	C₂	semidirect
11. C₅₉×#433	C₂	semidirect
14. C₂×SD₁₆×C₅₉	C₂²	semidirect
17. C₂×D₈×C₅₉	C₂²	semidirect
20. #127×C₅₉	C₂²	semidirect
21. #125×C₅₉	C₂²	semidirect
29. C₄×D₄×C₅₉	C₂²	semidirect
31. C₄⋊C₈×C₅₉	C₂²	semidirect
32. #103×C₅₉	C₂²	semidirect
35. #102×C₅₉	C₂²	semidirect
37. #97×C₅₉	C₂²	semidirect
38. C₄×C₈×C₅₉	C₂²	semidirect
121. #3220	C₅₉	direct (N × H)
129. #540	C₂×C₅₉	semidirect
130. #535	C₂×C₅₉	semidirect
131. #496	C₂×C₅₉	semidirect
132. #493	C₂×C₅₉	semidirect
133. #490	C₂×C₅₉	semidirect
134. #488	C₂×C₅₉	semidirect
135. #463	C₂×C₅₉	semidirect
136. #459	C₂×C₅₉	semidirect
137. #449	C₂×C₅₉	semidirect
138. #447	C₂×C₅₉	semidirect
139. #433	C₂×C₅₉	semidirect
140. C₅₉	#3220	direct (N × H)
143. C₂×SD₁₆	C₂²×C₅₉	semidirect
146. C₂×D₈	C₂²×C₅₉	semidirect
149. #127	C₂²×C₅₉	semidirect
150. #125	C₂²×C₅₉	semidirect
158. C₄×D₄	C₂²×C₅₉	semidirect
160. C₄⋊C₈	C₂²×C₅₉	semidirect
161. #103	C₂²×C₅₉	semidirect
164. #102	C₂²×C₅₉	semidirect
166. #97	C₂²×C₅₉	semidirect
167. C₄×C₈	C₂²×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#540	2
C₅₉×#535	2
C₅₉×#496	2
C₅₉×#493	2
C₅₉×#490	2
C₅₉×#488	2
C₅₉×#463	2
C₅₉×#459	2
C₅₉×#449	2
C₅₉×#447	2
C₅₉×#433	2
#3220	59

Subgroups — C59×#3220 #21421538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#3220 #21421538