Subgroups — C₅₉×#17619 #28429728

Order 15104.

938 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2982	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2351	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1518	C₂	1	6	yes	maximal
5	C₅₉×#1503	C₂	1	6	yes	maximal
6	Q₈²×C₅₉	C₄	1	4	yes	complemented
7	C₅₉×#557	—	2	4	no	—
8	C₅₉×#557	C₄	1	4	yes	complemented
9	C₅₉×#544	—	2	3	no	—
10	C₅₉×#533	C₂²	1	6	yes	—
11	C₅₉×#519	—	2	1	no	—
12	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₉	—	2	4	no	—
15	C₅₉×#431	C₂²	1	6	yes	—
16	C₅₉×#389	—	2	3	no	—
17	C₅₉×#382	C₂²	1	3	yes	—
18	C₄×C₄⋊C₄×C₅₉	C₂²	1	3	yes	—
19	C₅₉×#328	C₂²	1	4	yes	—
20	C₅₉×#328	C₂²	1	12	yes	—
21	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₅₉	—	2	12	no	—
23	C₄⋊Q₈×C₅₉	C₂×C₄	1	6	yes	—
24	C₄⋊Q₈×C₅₉	C₂×C₄	1	6	yes	—
25	#125×C₅₉	—	2	12	no	—
26	#125×C₅₉	—	2	6	no	—
27	C₂²⋊Q₈×C₅₉	—	2	6	no	—
28	C₂²⋊Q₈×C₅₉	C₂³	1	4	yes	—
29	C₄×Q₈×C₅₉	—	2	4	no	—
30	C₄×Q₈×C₅₉	—	2	4	no	—
31	C₄×Q₈×C₅₉	—	2	6	no	—
32	C₄×Q₈×C₅₉	—	4	4	no	—
33	C₄×Q₈×C₅₉	C₂×C₄	1	12	yes	—
34	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
35	C₄×Q₈×C₅₉	D₄	1	4	yes	—
36	#117×C₅₉	—	2	6	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	3	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	3	no	—
39	C₂×C₄⋊C₄×C₅₉	D₄	1	3	yes	—
40	C₂×C₄²×C₅₉	—	2	3	no	—
41	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
42	C₄⋊C₈×C₅₉	—	2	6	no	—
43	C₂²⋊C₈×C₅₉	C₂³	1	4	yes	—
44	C₄×C₈×C₅₉	—	2	6	no	—
45	#95×C₅₉	—	2	3	no	—
46	#95×C₅₉	C₂³	1	1	yes	characteristic
47	#95×C₅₉	C₂³	1	3	yes	—
48	C₂×Q₈×C₅₉	—	2	6	no	—
49	C₂×Q₈×C₅₉	C₂²⋊C₄	1	4	yes	—
50	C₂×Q₈×C₅₉	SD₁₆	1	4	yes	—
51	C₂×Q₈×C₅₉	Q₁₆	1	4	yes	—
52	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₅₉	—	2	1	no	—
54	C₂²×C₄×C₅₉	—	2	3	no	—
55	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₅₉	—	2	4	no	—
59	C₄⋊C₄×C₅₉	—	2	6	no	—
60	C₄⋊C₄×C₅₉	—	2	6	no	—
61	C₄⋊C₄×C₅₉	—	2	6	no	—
62	C₄⋊C₄×C₅₉	—	2	12	no	—
63	C₄⋊C₄×C₅₉	—	2	4	no	—
64	C₄⋊C₄×C₅₉	—	4	6	no	—
65	C₄⋊C₄×C₅₉	—	4	6	no	—
66	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	12	yes	—
67	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
68	C₂²⋊C₄×C₅₉	—	2	4	no	—
69	C₄²×C₅₉	—	2	12	no	—
70	C₄²×C₅₉	—	4	6	no	—
71	C₄²×C₅₉	C₂²×C₄	1	6	yes	—
72	C₄²×C₅₉	C₂×D₄	1	6	yes	—
73	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
74	Q₈×C₅₉	—	4	4	no	—
75	Q₈×C₅₉	—	8	6	no	—
76	Q₈×C₅₉	#103	1	16	yes	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	4	no	—
79	C₂×C₄×C₅₉	—	2	4	no	—
80	C₂×C₄×C₅₉	—	4	3	no	—
81	C₂×C₄×C₅₉	—	4	1	no	—
82	C₂×C₄×C₅₉	—	4	1	no	—
83	C₂×C₄×C₅₉	—	4	1	no	—
84	C₂×C₄×C₅₉	—	4	3	no	—
85	C₂×C₄×C₅₉	—	4	3	no	—
86	C₂×C₄×C₅₉	—	4	3	no	—
87	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	6	yes	—
88	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₅₉	C₂×SD₁₆	1	6	yes	—
90	C₂×C₄×C₅₉	C₂×Q₁₆	1	6	yes	—
91	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₅₉	—	4	4	no	—
94	#17619	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₅₉	—	2	1	no	—
96	C₂²×C₅₉	—	2	1	no	—
97	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
99	C₂²×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
100	C₄×C₅₉	—	2	1	no	—
101	C₄×C₅₉	—	2	1	no	—
102	C₄×C₅₉	—	2	6	no	—
103	C₄×C₅₉	—	4	2	no	—
104	C₄×C₅₉	—	4	4	no	—
105	C₄×C₅₉	—	4	4	no	—
106	C₄×C₅₉	—	8	3	no	—
107	C₄×C₅₉	C₂×#103	1	12	yes	—
108	#3502	C₂×C₅₉	1	1	yes	characteristic
109	#2982	C₂×C₅₉	1	1	yes	characteristic
110	#2351	C₂×C₅₉	1	1	yes	characteristic
111	#1518	C₂×C₅₉	1	6	yes	—
112	#1503	C₂×C₅₉	1	6	yes	—
113	C₂×C₅₉	—	2	1	no	—
114	C₂×C₅₉	—	2	1	no	—
115	C₂×C₅₉	#2884	1	1	yes	characteristic
116	C₂×C₅₉	C₂²×#103	1	1	yes	characteristic
117	C₂×C₅₉	#2901	1	1	yes	characteristic
118	Q₈²	C₄×C₅₉	1	4	yes	complemented
119	#557	—	2	4	no	—
120	#557	C₄×C₅₉	1	4	yes	complemented
121	#544	—	2	3	no	—
122	#533	C₂²×C₅₉	1	6	yes	—
123	#519	—	2	1	no	—
124	C₂×C₄×Q₈	—	2	1	no	—
125	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
126	C₈×Q₈	—	2	4	no	—
127	#431	C₂²×C₅₉	1	6	yes	—
128	#389	—	2	3	no	—
129	#382	C₂²×C₅₉	1	3	yes	—
130	C₄×C₄⋊C₄	C₂²×C₅₉	1	3	yes	—
131	#328	C₂²×C₅₉	1	4	yes	—
132	#328	C₂²×C₅₉	1	12	yes	—
133	C₅₉	#17619	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×Q₈	D₄×C₅₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	12	no	—
136	C₄⋊Q₈	C₂×C₄×C₅₉	1	6	yes	—
137	C₄⋊Q₈	C₂×C₄×C₅₉	1	6	yes	—
138	#125	—	2	12	no	—
139	#125	—	2	6	no	—
140	C₂²⋊Q₈	—	2	6	no	—
141	C₂²⋊Q₈	C₂³×C₅₉	1	4	yes	—
142	C₄×Q₈	—	2	4	no	—
143	C₄×Q₈	—	2	4	no	—
144	C₄×Q₈	—	2	6	no	—
145	C₄×Q₈	—	4	4	no	—
146	C₄×Q₈	C₂×C₄×C₅₉	1	12	yes	—
147	C₄×Q₈	C₂×C₄×C₅₉	1	4	yes	—
148	C₄×Q₈	D₄×C₅₉	1	4	yes	—
149	#117	—	2	6	no	—
150	C₂×C₄⋊C₄	—	2	3	no	—
151	C₂×C₄⋊C₄	—	2	3	no	—
152	C₂×C₄⋊C₄	D₄×C₅₉	1	3	yes	—
153	C₂×C₄²	—	2	3	no	—
154	C₂×C₄²	C₂³×C₅₉	1	3	yes	—
155	C₄⋊C₈	—	2	6	no	—
156	C₂²⋊C₈	C₂³×C₅₉	1	4	yes	—
157	C₄×C₈	—	2	6	no	—
158	#95	—	2	3	no	—
159	#95	C₂³×C₅₉	1	1	yes	characteristic
160	#95	C₂³×C₅₉	1	3	yes	—
161	C₂×Q₈	—	2	6	no	—
162	C₂×Q₈	C₂²⋊C₄×C₅₉	1	4	yes	—
163	C₂×Q₈	SD₁₆×C₅₉	1	4	yes	—
164	C₂×Q₈	Q₁₆×C₅₉	1	4	yes	—
165	C₂×Q₈	C₂²×C₄×C₅₉	1	4	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
169	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
171	C₂×C₈	—	2	4	no	—
172	C₄⋊C₄	—	2	6	no	—
173	C₄⋊C₄	—	2	6	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	—	2	12	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	6	no	—
178	C₄⋊C₄	—	4	6	no	—
179	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	12	yes	—
180	C₄⋊C₄	C₂²×C₄×C₅₉	1	4	yes	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	12	no	—
183	C₄²	—	4	6	no	—
184	C₄²	C₂²×C₄×C₅₉	1	6	yes	—
185	C₄²	C₂×D₄×C₅₉	1	6	yes	—
186	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	8	6	no	—
189	Q₈	#103×C₅₉	1	16	yes	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	3	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	3	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	6	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	6	yes	—
202	C₂×C₄	C₂×SD₁₆×C₅₉	1	6	yes	—
203	C₂×C₄	C₂×Q₁₆×C₅₉	1	6	yes	—
204	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
210	C₂²	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
211	C₂²	C₂²×Q₁₆×C₅₉	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	6	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	3	no	—
219	C₄	C₂×#103×C₅₉	1	12	yes	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₅₉×#2884	1	1	yes	characteristic
223	C₂	C₂²×#103×C₅₉	1	1	yes	characteristic
224	C₂	C₅₉×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂³×C₅₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄×Q₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. Q₈²×C₅₉	C₄	semidirect
8. C₅₉×#557	C₄	semidirect
94. #17619	C₅₉	direct (N × H)
118. Q₈²	C₄×C₅₉	semidirect
120. #557	C₄×C₅₉	semidirect
133. C₅₉	#17619	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×#3502	2
C₅₉×#2982	2
C₅₉×#2351	2
C₅₉×#1518	2
C₅₉×#1503	2
#17619	59

Subgroups — C59×#17619 #28429728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#17619 #28429728