Subgroups — C₅₉×#33303 #28445412

Order 15104.

1526 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#495	C₂	1	8	yes	maximal, complemented
2	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#492	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2858	C₂	1	2	yes	maximal, complemented
6	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₅₉	C₂²	1	4	yes	—
8	C₂²×SD₁₆×C₅₉	C₂²	1	8	yes	—
9	C₂²×D₈×C₅₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
11	C₂×#127×C₅₉	C₂²	1	2	yes	—
12	C₂×#124×C₅₉	C₂²	1	8	yes	—
13	C₅₉×#495	—	2	32	no	—
14	C₅₉×#494	—	2	4	no	—
15	C₅₉×#493	—	2	4	no	—
16	C₅₉×#492	—	2	8	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₉	C₂²	1	4	yes	—
20	C₅₉×#390	—	2	8	no	—
21	C₄³×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
23	C₂²×D₄×C₅₉	—	2	4	no	—
24	C₂²×D₄×C₅₉	C₂³	1	4	yes	—
25	C₂×Q₁₆×C₅₉	—	2	8	no	—
26	C₂×Q₁₆×C₅₉	—	2	16	no	—
27	C₂×SD₁₆×C₅₉	—	2	16	no	—
28	C₂×SD₁₆×C₅₉	—	2	32	no	—
29	C₂×D₈×C₅₉	—	2	8	no	—
30	C₂×D₈×C₅₉	—	2	16	no	—
31	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
32	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
33	C₄⋊Q₈×C₅₉	—	2	4	no	—
34	C₄⋊Q₈×C₅₉	—	2	4	no	—
35	#127×C₅₉	—	2	4	no	—
36	#127×C₅₉	—	2	4	no	—
37	#124×C₅₉	—	2	32	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₅₉	—	2	16	no	—
40	C₂×C₄²×C₅₉	D₄	1	4	yes	—
41	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₅₉	D₄	1	4	yes	—
44	C₄×C₈×C₅₉	D₄	1	4	yes	—
45	C₄×C₈×C₅₉	D₄	1	16	yes	complemented
46	C₂⁴×C₅₉	—	2	4	no	—
47	C₂×Q₈×C₅₉	—	2	16	no	—
48	C₂×Q₈×C₅₉	—	2	4	no	—
49	C₂×Q₈×C₅₉	—	2	4	no	—
50	C₂×D₄×C₅₉	—	2	16	no	—
51	C₂×D₄×C₅₉	—	2	4	no	—
52	C₂×D₄×C₅₉	—	2	4	no	—
53	C₂×D₄×C₅₉	—	4	8	no	—
54	C₂×D₄×C₅₉	—	4	16	no	—
55	C₂²×C₄×C₅₉	—	2	4	no	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₅₉	—	4	16	no	—
60	SD₁₆×C₅₉	—	4	32	no	—
61	D₈×C₅₉	—	4	16	no	—
62	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₅₉	C₂×D₄	1	8	yes	—
65	C₂×C₈×C₅₉	C₂×D₄	1	8	yes	—
66	C₄⋊C₄×C₅₉	—	4	8	no	—
67	C₂²⋊C₄×C₅₉	—	4	32	no	—
68	C₄²×C₅₉	—	2	8	no	—
69	C₄²×C₅₉	C₂×D₄	1	2	yes	—
70	C₄²×C₅₉	C₂×D₄	1	2	yes	—
71	C₄²×C₅₉	C₂×D₄	1	8	yes	—
72	C₂³×C₅₉	—	4	8	no	—
73	C₂³×C₅₉	—	4	4	no	—
74	C₂³×C₅₉	—	4	16	no	—
75	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
76	Q₈×C₅₉	—	4	8	no	—
77	Q₈×C₅₉	—	4	8	no	—
78	D₄×C₅₉	—	4	8	no	—
79	D₄×C₅₉	—	4	8	no	—
80	D₄×C₅₉	—	8	16	no	—
81	C₂×C₄×C₅₉	—	2	8	no	—
82	C₂×C₄×C₅₉	—	4	8	no	—
83	C₂×C₄×C₅₉	—	4	4	no	—
84	C₂×C₄×C₅₉	#127	1	8	yes	—
85	C₂×C₄×C₅₉	#127	1	4	yes	—
86	C₂×C₄×C₅₉	#135	1	8	yes	—
87	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
90	C₈×C₅₉	#127	1	8	yes	—
91	C₈×C₅₉	#127	1	8	yes	—
92	#33303	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₉	—	8	4	no	—
94	C₂²×C₅₉	—	8	8	no	—
95	C₂²×C₅₉	—	8	16	no	—
96	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
97	C₂²×C₅₉	C₂×#127	1	2	yes	—
98	C₂²×C₅₉	C₂×#135	1	4	yes	—
99	C₄×C₅₉	—	2	8	no	—
100	C₄×C₅₉	—	8	4	no	—
101	C₄×C₅₉	#495	1	8	yes	complemented
102	C₄×C₅₉	C₂×#127	1	2	yes	—
103	C₄×C₅₉	C₂×#127	1	2	yes	—
104	C₂×#495	C₂×C₅₉	1	8	yes	complemented
105	C₂×#494	C₂×C₅₉	1	1	yes	characteristic, complemented
106	C₂×#493	C₂×C₅₉	1	1	yes	characteristic
107	C₂×#492	C₂×C₅₉	1	2	yes	complemented
108	#2858	C₂×C₅₉	1	2	yes	complemented
109	C₄²×C₈	C₂×C₅₉	1	1	yes	characteristic, complemented
110	C₂×C₅₉	—	16	4	no	—
111	C₂×C₅₉	#2867	1	1	yes	characteristic
112	C₂×C₅₉	C₂×#495	1	2	yes	—
113	C₂×C₅₉	#3147	1	4	yes	—
114	C₂²×Q₁₆	C₂²×C₅₉	1	4	yes	—
115	C₂²×SD₁₆	C₂²×C₅₉	1	8	yes	—
116	C₂²×D₈	C₂²×C₅₉	1	4	yes	—
117	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
118	C₂×#127	C₂²×C₅₉	1	2	yes	—
119	C₂×#124	C₂²×C₅₉	1	8	yes	—
120	#495	—	2	32	no	—
121	#494	—	2	4	no	—
122	#493	—	2	4	no	—
123	#492	—	2	8	no	—
124	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₅₉	1	4	yes	—
127	#390	—	2	8	no	—
128	C₄³	C₂²×C₅₉	1	1	yes	characteristic
129	C₅₉	#33303	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
131	C₂²×D₄	—	2	4	no	—
132	C₂²×D₄	C₂³×C₅₉	1	4	yes	—
133	C₂×Q₁₆	—	2	8	no	—
134	C₂×Q₁₆	—	2	16	no	—
135	C₂×SD₁₆	—	2	16	no	—
136	C₂×SD₁₆	—	2	32	no	—
137	C₂×D₈	—	2	8	no	—
138	C₂×D₈	—	2	16	no	—
139	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
140	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	—	2	4	no	—
143	#127	—	2	4	no	—
144	#127	—	2	4	no	—
145	#124	—	2	32	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₂²⋊C₄	—	2	16	no	—
148	C₂×C₄²	D₄×C₅₉	1	4	yes	—
149	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
150	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
151	C₄×C₈	D₄×C₅₉	1	4	yes	—
152	C₄×C₈	D₄×C₅₉	1	4	yes	—
153	C₄×C₈	D₄×C₅₉	1	16	yes	complemented
154	C₂⁴	—	2	4	no	—
155	C₂×Q₈	—	2	16	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×D₄	—	2	16	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	8	no	—
162	C₂×D₄	—	4	16	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
165	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
166	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
167	Q₁₆	—	4	16	no	—
168	SD₁₆	—	4	32	no	—
169	D₈	—	4	16	no	—
170	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
171	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
172	C₂×C₈	C₂×D₄×C₅₉	1	8	yes	—
173	C₂×C₈	C₂×D₄×C₅₉	1	8	yes	—
174	C₄⋊C₄	—	4	8	no	—
175	C₂²⋊C₄	—	4	32	no	—
176	C₄²	—	2	8	no	—
177	C₄²	C₂×D₄×C₅₉	1	2	yes	—
178	C₄²	C₂×D₄×C₅₉	1	2	yes	—
179	C₄²	C₂×D₄×C₅₉	1	8	yes	—
180	C₂³	—	4	8	no	—
181	C₂³	—	4	4	no	—
182	C₂³	—	4	16	no	—
183	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
184	Q₈	—	4	8	no	—
185	Q₈	—	4	8	no	—
186	D₄	—	4	8	no	—
187	D₄	—	4	8	no	—
188	D₄	—	8	16	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	#127×C₅₉	1	8	yes	—
193	C₂×C₄	#127×C₅₉	1	4	yes	—
194	C₂×C₄	#135×C₅₉	1	8	yes	—
195	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
198	C₈	#127×C₅₉	1	8	yes	—
199	C₈	#127×C₅₉	1	8	yes	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	8	no	—
202	C₂²	—	8	16	no	—
203	C₂²	C₂×#127×C₅₉	1	1	yes	characteristic
204	C₂²	C₂×#127×C₅₉	1	2	yes	—
205	C₂²	C₂×#135×C₅₉	1	4	yes	—
206	C₄	—	2	8	no	—
207	C₄	—	8	4	no	—
208	C₄	C₅₉×#495	1	8	yes	complemented
209	C₄	C₂×#127×C₅₉	1	2	yes	—
210	C₄	C₂×#127×C₅₉	1	2	yes	—
211	C₂	—	16	4	no	—
212	C₂	C₅₉×#2867	1	1	yes	characteristic
213	C₂	C₂×C₅₉×#495	1	2	yes	—
214	C₂	C₅₉×#3147	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂×C₅₉×#493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#495	C₂	semidirect
2. C₂×C₅₉×#494	C₂	semidirect
4. C₂×C₅₉×#492	C₂	semidirect
5. C₅₉×#2858	C₂	semidirect
6. C₄²×C₈×C₅₉	C₂	semidirect
45. C₄×C₈×C₅₉	D₄	semidirect
92. #33303	C₅₉	direct (N × H)
101. C₄×C₅₉	#495	semidirect
104. C₂×#495	C₂×C₅₉	semidirect
105. C₂×#494	C₂×C₅₉	semidirect
107. C₂×#492	C₂×C₅₉	semidirect
108. #2858	C₂×C₅₉	semidirect
109. C₄²×C₈	C₂×C₅₉	semidirect
129. C₅₉	#33303	direct (N × H)
153. C₄×C₈	D₄×C₅₉	semidirect
208. C₄	C₅₉×#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#495	2
C₂×C₅₉×#494	2
C₂×C₅₉×#493	2
C₂×C₅₉×#492	2
C₅₉×#2858	2
C₄²×C₈×C₅₉	2
#33303	59

Subgroups — C59×#33303 #28445412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33303 #28445412