Subgroups — C₅₉×#33919 #28446028

Order 15104.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3336	C₂	1	2	yes	maximal
2	C₅₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2237	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2236	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2229	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2219	C₂	1	2	yes	maximal, complemented
7	C₄×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2169	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2159	C₂	1	2	yes	maximal, complemented
10	C₅₉×#552	—	2	2	no	—
11	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#499	C₂²	1	2	yes	—
14	C₅₉×#485	—	2	2	no	—
15	C₅₉×#483	—	2	2	no	—
16	C₅₉×#479	—	2	2	no	—
17	C₅₉×#467	C₂²	1	2	yes	—
18	C₅₉×#467	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
20	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#427	C₂²	1	2	yes	—
22	C₂×#107×C₅₉	C₂²	1	2	yes	—
23	C₅₉×#367	C₂²	1	2	yes	—
24	C₅₉×#367	C₂²	1	2	yes	—
25	C₅₉×#366	C₂²	1	2	yes	—
26	C₅₉×#366	C₂²	1	1	yes	characteristic
27	C₅₉×#366	C₂²	1	1	yes	characteristic
28	C₅₉×#366	C₂²	1	1	yes	characteristic
29	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₅₉×#358	C₂²	1	2	yes	—
31	C₅₉×#358	C₂²	1	2	yes	—
32	C₅₉×#358	C₂²	1	2	yes	—
33	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
37	#126×C₅₉	—	4	2	no	—
38	#125×C₅₉	—	2	2	no	—
39	#125×C₅₉	—	4	2	no	—
40	#123×C₅₉	—	4	2	no	—
41	C₂²⋊Q₈×C₅₉	—	2	2	no	—
42	C₂²⋊Q₈×C₅₉	—	2	2	no	—
43	C₄×Q₈×C₅₉	—	2	2	no	—
44	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
45	#117×C₅₉	—	2	2	no	—
46	#117×C₅₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
48	Q₃₂×C₅₉	—	2	2	no	—
49	M₃₂×C₅₉	—	2	4	no	—
50	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
52	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
53	#107×C₅₉	—	2	2	no	—
54	#107×C₅₉	C₂³	1	2	yes	—
55	#107×C₅₉	C₂³	1	2	yes	—
56	#107×C₅₉	C₂³	1	2	yes	—
57	#106×C₅₉	—	2	2	no	—
58	#106×C₅₉	—	2	2	no	—
59	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
60	#103×C₅₉	—	2	2	no	—
61	#103×C₅₉	—	2	2	no	—
62	#103×C₅₉	—	2	2	no	—
63	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
64	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	—	4	2	no	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
69	Q₁₆×C₅₉	—	2	2	no	—
70	Q₁₆×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	—	2	2	no	—
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	—	2	2	no	—
77	C₄⋊C₄×C₅₉	—	2	2	no	—
78	C₄⋊C₄×C₅₉	—	2	2	no	—
79	C₄⋊C₄×C₅₉	—	4	2	no	—
80	C₄⋊C₄×C₅₉	—	4	2	no	—
81	C₄⋊C₄×C₅₉	—	4	2	no	—
82	C₄⋊C₄×C₅₉	—	4	2	no	—
83	C₄⋊C₄×C₅₉	—	4	2	no	—
84	C₄⋊C₄×C₅₉	—	4	2	no	—
85	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₅₉	—	4	2	no	—
87	C₂²⋊C₄×C₅₉	—	4	2	no	—
88	C₂²⋊C₄×C₅₉	—	4	2	no	—
89	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
90	C₄²×C₅₉	—	4	2	no	—
91	C₄²×C₅₉	—	4	2	no	—
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₅₉	—	2	2	no	—
94	C₁₆×C₅₉	—	2	1	no	—
95	C₁₆×C₅₉	#41	1	2	yes	—
96	C₂³×C₅₉	C₂×D₈	1	2	yes	—
97	Q₈×C₅₉	—	4	2	no	—
98	Q₈×C₅₉	—	4	2	no	—
99	D₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	—	2	2	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	8	4	no	—
107	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₅₉	—	2	2	no	—
112	C₈×C₅₉	—	2	1	no	—
113	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
115	#33919	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₅₉	—	2	2	no	—
117	C₂²×C₅₉	—	2	2	no	—
118	C₂²×C₅₉	—	2	2	no	—
119	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₅₉	—	2	2	no	—
121	C₄×C₅₉	—	2	1	no	—
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	8	2	no	—
124	C₄×C₅₉	—	8	2	no	—
125	C₄×C₅₉	—	8	2	no	—
126	C₄×C₅₉	—	8	2	no	—
127	C₄×C₅₉	#547	1	1	yes	characteristic
128	C₄×C₅₉	#578	1	1	yes	characteristic
129	#3336	C₂×C₅₉	1	2	yes	—
130	#2253	C₂×C₅₉	1	1	yes	characteristic, complemented
131	#2237	C₂×C₅₉	1	2	yes	complemented
132	#2236	C₂×C₅₉	1	2	yes	complemented
133	#2229	C₂×C₅₉	1	2	yes	complemented
134	#2219	C₂×C₅₉	1	2	yes	complemented
135	C₄×Q₃₂	C₂×C₅₉	1	1	yes	characteristic, complemented
136	#2169	C₂×C₅₉	1	1	yes	characteristic
137	#2159	C₂×C₅₉	1	2	yes	complemented
138	C₂×C₅₉	—	4	2	no	—
139	C₂×C₅₉	#3334	1	1	yes	characteristic
140	C₂×C₅₉	#3414	1	1	yes	characteristic
141	C₂×C₅₉	#3417	1	1	yes	characteristic
142	#552	—	2	2	no	—
143	C₂×Q₃₂	C₂²×C₅₉	1	1	yes	characteristic
144	C₂×M₃₂	C₂²×C₅₉	1	2	yes	—
145	#499	C₂²×C₅₉	1	2	yes	—
146	#485	—	2	2	no	—
147	#483	—	2	2	no	—
148	#479	—	2	2	no	—
149	#467	C₂²×C₅₉	1	2	yes	—
150	#467	C₂²×C₅₉	1	2	yes	—
151	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
152	C₈×D₄	C₂²×C₅₉	1	1	yes	characteristic
153	#427	C₂²×C₅₉	1	2	yes	—
154	C₂×#107	C₂²×C₅₉	1	2	yes	—
155	#367	C₂²×C₅₉	1	2	yes	—
156	#367	C₂²×C₅₉	1	2	yes	—
157	#366	C₂²×C₅₉	1	1	yes	characteristic
158	#366	C₂²×C₅₉	1	2	yes	—
159	#366	C₂²×C₅₉	1	1	yes	characteristic
160	#366	C₂²×C₅₉	1	1	yes	characteristic
161	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
162	#358	C₂²×C₅₉	1	2	yes	—
163	#358	C₂²×C₅₉	1	2	yes	—
164	#358	C₂²×C₅₉	1	2	yes	—
165	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
166	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
167	C₅₉	#33919	1	1	yes	characteristic, direct product factor, Sylow
168	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
169	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
170	#126	—	4	2	no	—
171	#125	—	2	2	no	—
172	#125	—	4	2	no	—
173	#123	—	4	2	no	—
174	C₂²⋊Q₈	—	2	2	no	—
175	C₂²⋊Q₈	—	2	2	no	—
176	C₄×Q₈	—	2	2	no	—
177	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
178	#117	—	2	2	no	—
179	#117	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	Q₃₂	—	2	2	no	—
182	M₃₂	—	2	4	no	—
183	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
184	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
185	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
186	#107	—	2	2	no	—
187	#107	C₂³×C₅₉	1	2	yes	—
188	#107	C₂³×C₅₉	1	2	yes	—
189	#107	C₂³×C₅₉	1	2	yes	—
190	#106	—	2	2	no	—
191	#106	—	2	2	no	—
192	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
193	#103	—	2	2	no	—
194	#103	—	2	2	no	—
195	#103	—	2	2	no	—
196	C₂²⋊C₈	D₄×C₅₉	1	2	yes	—
197	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	D₈×C₅₉	1	1	yes	characteristic
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
202	Q₁₆	—	2	2	no	—
203	Q₁₆	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
208	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	D₈×C₅₉	1	2	yes	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
226	C₁₆	—	2	2	no	—
227	C₁₆	—	2	1	no	—
228	C₁₆	#41×C₅₉	1	2	yes	—
229	C₂³	C₂×D₈×C₅₉	1	2	yes	—
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	D₄	—	4	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
241	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
247	C₈	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	2	no	—
259	C₄	C₅₉×#547	1	1	yes	characteristic
260	C₄	C₅₉×#578	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	C₅₉×#3334	1	1	yes	characteristic
263	C₂	C₅₉×#3414	1	1	yes	characteristic
264	C₂	C₅₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂×D₄×C₅₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×D₄×C₅₉ ◁ C₈×D₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#2253	C₂	semidirect
3. C₅₉×#2237	C₂	semidirect
4. C₅₉×#2236	C₂	semidirect
5. C₅₉×#2229	C₂	semidirect
6. C₅₉×#2219	C₂	semidirect
7. C₄×Q₃₂×C₅₉	C₂	semidirect
9. C₅₉×#2159	C₂	semidirect
115. #33919	C₅₉	direct (N × H)
130. #2253	C₂×C₅₉	semidirect
131. #2237	C₂×C₅₉	semidirect
132. #2236	C₂×C₅₉	semidirect
133. #2229	C₂×C₅₉	semidirect
134. #2219	C₂×C₅₉	semidirect
135. C₄×Q₃₂	C₂×C₅₉	semidirect
137. #2159	C₂×C₅₉	semidirect
167. C₅₉	#33919	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×#3336	2
C₅₉×#2253	2
C₅₉×#2237	2
C₅₉×#2236	2
C₅₉×#2229	2
C₅₉×#2219	2
C₄×Q₃₂×C₅₉	2
C₅₉×#2169	2
C₅₉×#2159	2
#33919	59

Subgroups — C59×#33919 #28446028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33919 #28446028