Subgroups — C₅₉×#33705 #28445814

Order 15104.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 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
2	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2245	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2241	C₂	1	4	yes	maximal, complemented
5	C₅₉×#2141	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₅₉	—	2	1	no	—
8	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
9	C₅₉×#533	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
11	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
12	C₅₉×#500	C₂²	1	2	yes	—
13	C₅₉×#499	—	2	1	no	—
14	C₅₉×#499	C₂²	1	2	yes	—
15	C₅₉×#498	—	2	1	no	—
16	C₅₉×#494	—	2	1	no	—
17	C₅₉×#494	—	2	1	no	—
18	C₅₉×#494	C₂²	1	2	yes	—
19	C₅₉×#494	C₂²	1	2	yes	—
20	C₅₉×#427	—	2	1	no	—
21	C₅₉×#427	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#358	C₂²	1	16	yes	—
24	C₅₉×#346	C₂²	1	2	yes	—
25	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
26	C₂²×Q₈×C₅₉	—	2	2	no	—
27	C₂×Q₁₆×C₅₉	—	2	2	no	—
28	C₂×Q₁₆×C₅₉	—	2	2	no	—
29	C₂×Q₁₆×C₅₉	—	2	2	no	—
30	C₂×Q₁₆×C₅₉	—	2	8	no	—
31	C₂×Q₁₆×C₅₉	—	4	4	no	—
32	C₂×Q₁₆×C₅₉	C₂³	1	4	yes	—
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₉	—	2	2	no	—
36	C₄⋊Q₈×C₅₉	—	2	2	no	—
37	C₄⋊Q₈×C₅₉	—	2	2	no	—
38	C₄⋊Q₈×C₅₉	—	4	2	no	—
39	#125×C₅₉	—	2	2	no	—
40	C₂²⋊Q₈×C₅₉	—	4	2	no	—
41	C₄×Q₈×C₅₉	—	4	2	no	—
42	#117×C₅₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
44	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
45	M₃₂×C₅₉	—	2	4	no	—
46	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
47	#107×C₅₉	—	2	2	no	—
48	#107×C₅₉	C₂³	1	4	yes	—
49	#106×C₅₉	—	2	2	no	—
50	#106×C₅₉	—	2	2	no	—
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×Q₈×C₅₉	—	2	4	no	—
56	C₂×Q₈×C₅₉	—	4	2	no	—
57	C₂×Q₈×C₅₉	—	4	2	no	—
58	C₂×Q₈×C₅₉	—	4	4	no	—
59	C₂×Q₈×C₅₉	—	4	4	no	—
60	C₂²×C₄×C₅₉	—	4	2	no	—
61	C₂²×C₄×C₅₉	D₈	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₅₉	—	2	8	no	—
64	Q₁₆×C₅₉	—	4	4	no	—
65	Q₁₆×C₅₉	—	4	4	no	—
66	Q₁₆×C₅₉	—	4	4	no	—
67	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
68	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	#41	1	4	yes	—
72	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	—	2	4	no	—
74	C₄⋊C₄×C₅₉	—	4	4	no	—
75	C₄⋊C₄×C₅₉	—	4	4	no	—
76	C₄⋊C₄×C₅₉	—	4	4	no	—
77	C₄⋊C₄×C₅₉	—	4	2	no	—
78	C₂²⋊C₄×C₅₉	—	4	2	no	—
79	C₄²×C₅₉	—	4	2	no	—
80	C₄²×C₅₉	D₈	1	2	yes	—
81	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₁₆×C₅₉	—	2	4	no	—
84	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₅₉	—	4	4	no	—
86	Q₈×C₅₉	—	4	4	no	—
87	Q₈×C₅₉	—	8	4	no	—
88	Q₈×C₅₉	—	8	4	no	—
89	C₂×C₄×C₅₉	—	2	4	no	—
90	C₂×C₄×C₅₉	—	4	4	no	—
91	C₂×C₄×C₅₉	—	4	4	no	—
92	C₂×C₄×C₅₉	—	8	4	no	—
93	C₂×C₄×C₅₉	#124	1	2	yes	—
94	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₅₉	#124	1	2	yes	—
100	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
101	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
102	C₈×C₅₉	C₂×#41	1	2	yes	—
103	#33705	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₅₉	—	2	2	no	—
105	C₂²×C₅₉	#486	1	2	yes	—
106	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₅₉	—	2	2	no	—
108	C₄×C₅₉	—	8	4	no	—
109	C₄×C₅₉	—	8	4	no	—
110	C₄×C₅₉	#486	1	2	yes	—
111	C₄×C₅₉	#510	1	4	yes	—
112	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
113	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
114	#3160	C₂×C₅₉	1	1	yes	characteristic
115	C₂×#494	C₂×C₅₉	1	1	yes	characteristic
116	#2245	C₂×C₅₉	1	4	yes	complemented
117	#2241	C₂×C₅₉	1	4	yes	complemented
118	#2141	C₂×C₅₉	1	4	yes	—
119	C₄×M₃₂	C₂×C₅₉	1	1	yes	characteristic
120	C₂×C₅₉	—	2	2	no	—
121	C₂×C₅₉	C₂×#486	1	1	yes	characteristic
122	C₂×C₅₉	C₂×#510	1	2	yes	—
123	C₂²×Q₁₆	—	2	1	no	—
124	C₂²×Q₁₆	C₂²×C₅₉	1	2	yes	—
125	#533	—	2	1	no	—
126	C₂×C₄⋊Q₈	—	2	1	no	—
127	C₂×M₃₂	C₂²×C₅₉	1	2	yes	—
128	#500	C₂²×C₅₉	1	2	yes	—
129	#499	—	2	1	no	—
130	#499	C₂²×C₅₉	1	2	yes	—
131	#498	—	2	1	no	—
132	#494	—	2	1	no	—
133	#494	—	2	1	no	—
134	#494	C₂²×C₅₉	1	2	yes	—
135	#494	C₂²×C₅₉	1	2	yes	—
136	#427	—	2	1	no	—
137	#427	C₂²×C₅₉	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
139	#358	C₂²×C₅₉	1	16	yes	—
140	#346	C₂²×C₅₉	1	2	yes	—
141	C₄×C₁₆	C₂²×C₅₉	1	2	yes	—
142	C₅₉	#33705	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×Q₈	—	2	2	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	4	4	no	—
149	C₂×Q₁₆	C₂³×C₅₉	1	4	yes	—
150	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	#125	—	2	2	no	—
157	C₂²⋊Q₈	—	4	2	no	—
158	C₄×Q₈	—	4	2	no	—
159	#117	—	2	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
162	M₃₂	—	2	4	no	—
163	C₂×C₁₆	C₂³×C₅₉	1	4	yes	—
164	#107	—	2	2	no	—
165	#107	C₂³×C₅₉	1	4	yes	—
166	#106	—	2	2	no	—
167	#106	—	2	2	no	—
168	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
169	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
171	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	D₈×C₅₉	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	Q₁₆	—	4	4	no	—
184	C₂×C₈	SD₁₆×C₅₉	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
188	C₂×C₈	#41×C₅₉	1	4	yes	—
189	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	D₈×C₅₉	1	2	yes	—
198	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
200	C₁₆	—	2	4	no	—
201	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	8	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	8	4	no	—
210	C₂×C₄	#124×C₅₉	1	2	yes	—
211	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
212	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
213	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
214	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
215	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
216	C₈	#124×C₅₉	1	2	yes	—
217	C₈	C₂×SD₁₆×C₅₉	1	2	yes	—
218	C₈	C₂×SD₁₆×C₅₉	1	2	yes	—
219	C₈	C₂×#41×C₅₉	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	C₅₉×#486	1	2	yes	—
222	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	8	4	no	—
226	C₄	C₅₉×#486	1	2	yes	—
227	C₄	C₅₉×#510	1	4	yes	—
228	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
229	C₄	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×C₅₉×#486	1	1	yes	characteristic
232	C₂	C₂×C₅₉×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂³×C₅₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
3. C₅₉×#2245	C₂	semidirect
4. C₅₉×#2241	C₂	semidirect
103. #33705	C₅₉	direct (N × H)
116. #2245	C₂×C₅₉	semidirect
117. #2241	C₂×C₅₉	semidirect
142. C₅₉	#33705	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
C₂×C₅₉×#494	2
C₅₉×#2245	2
C₅₉×#2241	2
C₅₉×#2141	2
C₄×M₃₂×C₅₉	2
#33705	59

Subgroups — C59×#33705 #28445814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33705 #28445814