Subgroups — C₅₉×#33290 #28445399

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2862	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2859	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1991	C₂	1	4	yes	maximal
4	C₅₉×#1990	C₂	1	4	yes	maximal
5	C₅₉×#1835	C₂	1	4	yes	maximal
6	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
8	C₂×#125×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
12	C₅₉×#391	—	2	4	no	—
13	C₅₉×#389	—	2	4	no	—
14	C₅₉×#384	C₂²	1	2	yes	—
15	C₅₉×#382	C₂²	1	2	yes	—
16	C₅₉×#382	C₂²	1	2	yes	—
17	C₅₉×#382	C₂²	1	2	yes	—
18	C₄³×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#340	C₂²	1	16	yes	—
20	C₂²×Q₈×C₅₉	—	2	2	no	—
21	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
22	C₄⋊Q₈×C₅₉	—	2	8	no	—
23	#125×C₅₉	—	2	8	no	—
24	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
25	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
28	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
29	C₂×C₄²×C₅₉	D₄	1	4	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₉	—	2	4	no	—
34	C₄×C₈×C₅₉	—	2	4	no	—
35	C₄×C₈×C₅₉	Q₈	1	8	yes	complemented
36	#95×C₅₉	—	2	4	no	—
37	#95×C₅₉	—	2	4	no	—
38	#95×C₅₉	—	2	4	no	—
39	C₂×Q₈×C₅₉	—	4	4	no	—
40	C₂×Q₈×C₅₉	—	4	8	no	—
41	C₂²×C₄×C₅₉	—	2	4	no	—
42	C₂²×C₄×C₅₉	—	2	4	no	—
43	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₅₉	—	2	4	no	—
48	C₂×C₈×C₅₉	—	2	4	no	—
49	C₂×C₈×C₅₉	C₂×Q₈	1	8	yes	—
50	C₄⋊C₄×C₅₉	—	2	8	no	—
51	C₄⋊C₄×C₅₉	—	2	8	no	—
52	C₄⋊C₄×C₅₉	—	4	8	no	—
53	C₄⋊C₄×C₅₉	—	4	4	no	—
54	C₄⋊C₄×C₅₉	—	4	8	no	—
55	C₄²×C₅₉	—	2	8	no	—
56	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
57	C₄²×C₅₉	#41	1	4	yes	—
58	C₄²×C₅₉	#41	1	4	yes	—
59	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
60	Q₈×C₅₉	—	8	8	no	—
61	C₂×C₄×C₅₉	—	2	4	no	—
62	C₂×C₄×C₅₉	—	2	4	no	—
63	C₂×C₄×C₅₉	—	4	4	no	—
64	C₂×C₄×C₅₉	—	4	4	no	—
65	C₂×C₄×C₅₉	—	4	8	no	—
66	C₂×C₄×C₅₉	—	4	8	no	—
67	C₂×C₄×C₅₉	#124	1	4	yes	—
68	C₂×C₄×C₅₉	#124	1	4	yes	—
69	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
70	C₂×C₄×C₅₉	#135	1	8	yes	—
71	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
73	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
74	C₈×C₅₉	—	2	8	no	—
75	#33290	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₅₉	C₂×#124	1	1	yes	characteristic
77	C₂²×C₅₉	C₂×#124	1	1	yes	characteristic
78	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₅₉	C₂×#135	1	4	yes	—
80	C₄×C₅₉	—	2	4	no	—
81	C₄×C₅₉	—	2	4	no	—
82	C₄×C₅₉	—	8	4	no	—
83	C₄×C₅₉	—	8	4	no	—
84	C₄×C₅₉	#488	1	8	yes	—
85	C₄×C₅₉	#533	1	4	yes	—
86	#2862	C₂×C₅₉	1	1	yes	characteristic
87	#2859	C₂×C₅₉	1	1	yes	characteristic
88	#1991	C₂×C₅₉	1	4	yes	—
89	#1990	C₂×C₅₉	1	4	yes	—
90	#1835	C₂×C₅₉	1	4	yes	—
91	C₄²×C₈	C₂×C₅₉	1	1	yes	characteristic
92	C₂×C₅₉	#2862	1	1	yes	characteristic
93	C₂×C₅₉	C₂×#488	1	2	yes	—
94	C₂×C₅₉	#3131	1	2	yes	—
95	C₂×C₅₉	#3160	1	2	yes	—
96	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
97	C₂×#125	C₂²×C₅₉	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
101	#391	—	2	4	no	—
102	#389	—	2	4	no	—
103	#384	C₂²×C₅₉	1	2	yes	—
104	#382	C₂²×C₅₉	1	2	yes	—
105	#382	C₂²×C₅₉	1	2	yes	—
106	#382	C₂²×C₅₉	1	2	yes	—
107	C₄³	C₂²×C₅₉	1	1	yes	characteristic
108	#340	C₂²×C₅₉	1	16	yes	—
109	C₅₉	#33290	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×Q₈	—	2	2	no	—
111	C₂²×C₈	C₂³×C₅₉	1	4	yes	—
112	C₄⋊Q₈	—	2	8	no	—
113	#125	—	2	8	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
118	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
119	C₂×C₄²	D₄×C₅₉	1	4	yes	—
120	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
121	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
123	C₄×C₈	—	2	4	no	—
124	C₄×C₈	—	2	4	no	—
125	C₄×C₈	Q₈×C₅₉	1	8	yes	complemented
126	#95	—	2	4	no	—
127	#95	—	2	4	no	—
128	#95	—	2	4	no	—
129	C₂×Q₈	—	4	4	no	—
130	C₂×Q₈	—	4	8	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
136	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂×Q₈×C₅₉	1	8	yes	—
140	C₄⋊C₄	—	2	8	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄²	—	2	8	no	—
146	C₄²	C₂×Q₈×C₅₉	1	4	yes	—
147	C₄²	#41×C₅₉	1	4	yes	—
148	C₄²	#41×C₅₉	1	4	yes	—
149	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
150	Q₈	—	8	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	#124×C₅₉	1	4	yes	—
158	C₂×C₄	#124×C₅₉	1	4	yes	—
159	C₂×C₄	C₄⋊Q₈×C₅₉	1	4	yes	—
160	C₂×C₄	#135×C₅₉	1	8	yes	—
161	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
162	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
163	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
164	C₈	—	2	8	no	—
165	C₂²	C₂×#124×C₅₉	1	1	yes	characteristic
166	C₂²	C₂×#124×C₅₉	1	1	yes	characteristic
167	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
168	C₂²	C₂×#135×C₅₉	1	4	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	2	4	no	—
171	C₄	—	8	4	no	—
172	C₄	—	8	4	no	—
173	C₄	C₅₉×#488	1	8	yes	—
174	C₄	C₅₉×#533	1	4	yes	—
175	C₂	C₅₉×#2862	1	1	yes	characteristic
176	C₂	C₂×C₅₉×#488	1	2	yes	—
177	C₂	C₅₉×#3131	1	2	yes	—
178	C₂	C₅₉×#3160	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
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
35. C₄×C₈×C₅₉	Q₈	semidirect
75. #33290	C₅₉	direct (N × H)
109. C₅₉	#33290	direct (N × H)
125. C₄×C₈	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2862	2
C₅₉×#2859	2
C₅₉×#1991	2
C₅₉×#1990	2
C₅₉×#1835	2
C₄²×C₈×C₅₉	2
#33290	59

Subgroups — C59×#33290 #28445399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33290 #28445399