Subgroups — C₅₉×#33701 #28445810

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2245	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2244	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2243	C₂	1	4	yes	maximal, complemented
6	C₅₉×#2138	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₉	—	2	1	no	—
9	C₂×#135×C₅₉	C₂²	1	2	yes	—
10	C₅₉×#533	—	2	1	no	—
11	C₅₉×#532	—	2	1	no	—
12	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#500	C₂²	1	2	yes	—
14	C₅₉×#499	—	2	1	no	—
15	C₅₉×#499	C₂²	1	2	yes	—
16	C₅₉×#498	—	2	1	no	—
17	C₅₉×#495	—	2	1	no	—
18	C₅₉×#495	C₂²	1	2	yes	—
19	C₅₉×#494	C₂²	1	1	yes	characteristic
20	C₅₉×#493	C₂²	1	1	yes	characteristic
21	C₅₉×#492	—	2	1	no	—
22	C₅₉×#427	—	2	1	no	—
23	C₅₉×#427	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#358	C₂²	1	8	yes	—
26	C₅₉×#357	C₂²	1	8	yes	—
27	C₅₉×#346	C₂²	1	2	yes	—
28	C₅₉×#346	C₂²	1	2	yes	—
29	C₂×#41×C₅₉	—	2	2	no	—
30	#135×C₅₉	—	2	8	no	—
31	#135×C₅₉	—	4	4	no	—
32	C₂×Q₁₆×C₅₉	—	2	1	no	—
33	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₅₉	—	2	2	no	—
35	C₂×SD₁₆×C₅₉	—	2	2	no	—
36	C₂×D₈×C₅₉	—	2	1	no	—
37	C₂×D₈×C₅₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₉	—	2	2	no	—
41	C₄⋊Q₈×C₅₉	—	2	1	no	—
42	#127×C₅₉	—	2	1	no	—
43	#125×C₅₉	—	2	2	no	—
44	#124×C₅₉	—	2	2	no	—
45	C₂²⋊Q₈×C₅₉	—	4	2	no	—
46	#121×C₅₉	—	4	2	no	—
47	C₄×Q₈×C₅₉	—	4	2	no	—
48	C₄×D₄×C₅₉	—	4	2	no	—
49	#117×C₅₉	—	2	2	no	—
50	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₉	—	2	4	no	—
52	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
53	#107×C₅₉	—	2	2	no	—
54	#107×C₅₉	C₂³	1	4	yes	—
55	#106×C₅₉	—	2	2	no	—
56	#106×C₅₉	—	2	2	no	—
57	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	#41×C₅₉	—	4	4	no	—
62	#41×C₅₉	—	4	4	no	—
63	C₂×Q₈×C₅₉	—	2	2	no	—
64	C₂×Q₈×C₅₉	—	4	2	no	—
65	C₂×D₄×C₅₉	—	2	2	no	—
66	C₂×D₄×C₅₉	—	4	2	no	—
67	C₂×D₄×C₅₉	—	4	2	no	—
68	C₂²×C₄×C₅₉	—	4	2	no	—
69	C₂²×C₄×C₅₉	D₈	1	2	yes	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	Q₁₆×C₅₉	—	2	4	no	—
72	Q₁₆×C₅₉	—	4	2	no	—
73	SD₁₆×C₅₉	—	4	4	no	—
74	SD₁₆×C₅₉	—	4	4	no	—
75	D₈×C₅₉	—	2	4	no	—
76	D₈×C₅₉	—	4	2	no	—
77	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	4	yes	—
82	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	—	2	4	no	—
84	C₄⋊C₄×C₅₉	—	4	2	no	—
85	C₄⋊C₄×C₅₉	—	4	4	no	—
86	C₄⋊C₄×C₅₉	—	4	2	no	—
87	C₂²⋊C₄×C₅₉	—	4	2	no	—
88	C₂²⋊C₄×C₅₉	—	4	4	no	—
89	C₄²×C₅₉	—	4	2	no	—
90	C₄²×C₅₉	D₈	1	2	yes	—
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	—	4	2	no	—
95	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
96	Q₈×C₅₉	—	4	2	no	—
97	Q₈×C₅₉	—	4	2	no	—
98	Q₈×C₅₉	—	8	4	no	—
99	D₄×C₅₉	—	4	2	no	—
100	D₄×C₅₉	—	4	2	no	—
101	D₄×C₅₉	—	8	4	no	—
102	D₄×C₅₉	—	8	4	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	4	4	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	8	4	no	—
107	C₂×C₄×C₅₉	#124	1	2	yes	—
108	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
112	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₅₉	#124	1	2	yes	—
114	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
115	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
116	C₈×C₅₉	C₂×#41	1	2	yes	—
117	#33701	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₅₉	—	2	2	no	—
119	C₂²×C₅₉	—	8	2	no	—
120	C₂²×C₅₉	—	8	4	no	—
121	C₂²×C₅₉	#486	1	2	yes	—
122	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₅₉	—	2	2	no	—
124	C₄×C₅₉	—	2	2	no	—
125	C₄×C₅₉	—	8	2	no	—
126	C₄×C₅₉	—	8	4	no	—
127	C₄×C₅₉	#486	1	2	yes	—
128	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
129	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
130	#3160	C₂×C₅₉	1	1	yes	characteristic, complemented
131	#3147	C₂×C₅₉	1	1	yes	characteristic
132	#2245	C₂×C₅₉	1	2	yes	complemented
133	#2244	C₂×C₅₉	1	2	yes	complemented
134	#2243	C₂×C₅₉	1	4	yes	complemented
135	#2138	C₂×C₅₉	1	4	yes	complemented
136	C₂×#346	C₂×C₅₉	1	1	yes	characteristic, complemented
137	C₂×C₅₉	—	2	2	no	—
138	C₂×C₅₉	—	16	2	no	—
139	C₂×C₅₉	C₂×#486	1	1	yes	characteristic
140	C₂×C₅₉	#3414	1	2	yes	—
141	C₂×#135	—	2	1	no	—
142	C₂×#135	C₂²×C₅₉	1	2	yes	—
143	#533	—	2	1	no	—
144	#532	—	2	1	no	—
145	C₂²×C₁₆	C₂²×C₅₉	1	2	yes	—
146	#500	C₂²×C₅₉	1	2	yes	—
147	#499	—	2	1	no	—
148	#499	C₂²×C₅₉	1	2	yes	—
149	#498	—	2	1	no	—
150	#495	—	2	1	no	—
151	#495	C₂²×C₅₉	1	2	yes	—
152	#494	C₂²×C₅₉	1	1	yes	characteristic
153	#493	C₂²×C₅₉	1	1	yes	characteristic
154	#492	—	2	1	no	—
155	#427	—	2	1	no	—
156	#427	C₂²×C₅₉	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
158	#358	C₂²×C₅₉	1	8	yes	—
159	#357	C₂²×C₅₉	1	8	yes	—
160	#346	C₂²×C₅₉	1	2	yes	—
161	#346	C₂²×C₅₉	1	2	yes	—
162	C₅₉	#33701	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#41	—	2	2	no	—
164	#135	—	2	8	no	—
165	#135	—	4	4	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	C₂³×C₅₉	1	2	yes	—
172	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	1	no	—
176	#127	—	2	1	no	—
177	#125	—	2	2	no	—
178	#124	—	2	2	no	—
179	C₂²⋊Q₈	—	4	2	no	—
180	#121	—	4	2	no	—
181	C₄×Q₈	—	4	2	no	—
182	C₄×D₄	—	4	2	no	—
183	#117	—	2	2	no	—
184	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	C₂³×C₅₉	1	4	yes	—
187	#107	—	2	2	no	—
188	#107	C₂³×C₅₉	1	4	yes	—
189	#106	—	2	2	no	—
190	#106	—	2	2	no	—
191	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
192	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	D₈×C₅₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	4	no	—
208	SD₁₆	—	4	4	no	—
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	SD₁₆×C₅₉	1	4	yes	—
212	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
215	C₂×C₈	#41×C₅₉	1	4	yes	—
216	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₄²	—	4	2	no	—
224	C₄²	D₈×C₅₉	1	2	yes	—
225	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
226	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
227	C₁₆	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	Q₈	—	8	4	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	4	no	—
236	D₄	—	8	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	#124×C₅₉	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
243	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
245	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
247	C₈	#124×C₅₉	1	2	yes	—
248	C₈	C₂×SD₁₆×C₅₉	1	2	yes	—
249	C₈	C₂×SD₁₆×C₅₉	1	2	yes	—
250	C₈	C₂×#41×C₅₉	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	C₅₉×#486	1	2	yes	—
255	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	4	no	—
260	C₄	C₅₉×#486	1	2	yes	—
261	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
262	C₄	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	16	2	no	—
265	C₂	C₂×C₅₉×#486	1	1	yes	characteristic
266	C₂	C₅₉×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₅₉×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ C₂×C₄×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3160	C₂	semidirect
3. C₅₉×#2245	C₂	semidirect
4. C₅₉×#2244	C₂	semidirect
5. C₅₉×#2243	C₂	semidirect
6. C₅₉×#2138	C₂	semidirect
7. C₂×C₅₉×#346	C₂	semidirect
117. #33701	C₅₉	direct (N × H)
130. #3160	C₂×C₅₉	semidirect
132. #2245	C₂×C₅₉	semidirect
133. #2244	C₂×C₅₉	semidirect
134. #2243	C₂×C₅₉	semidirect
135. #2138	C₂×C₅₉	semidirect
136. C₂×#346	C₂×C₅₉	semidirect
162. C₅₉	#33701	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×#3160	2
C₅₉×#3147	2
C₅₉×#2245	2
C₅₉×#2244	2
C₅₉×#2243	2
C₅₉×#2138	2
C₂×C₅₉×#346	2
#33701	59

Subgroups — C59×#33701 #28445810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33701 #28445810