Subgroups — C₅₉×#33311 #28445420

Order 15104.

830 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#499	C₂	1	4	yes	maximal
2	C₅₉×#2859	C₂	1	2	yes	maximal
3	C₅₉×#1846	C₂	1	4	yes	maximal
4	C₅₉×#1845	C₂	1	2	yes	maximal
5	C₅₉×#1844	C₂	1	2	yes	maximal
6	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₅₉	C₂²	1	4	yes	—
8	C₅₉×#499	—	2	16	no	—
9	C₂×#107×C₅₉	C₂²	1	8	yes	—
10	C₂×#106×C₅₉	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#389	—	2	8	no	—
16	C₅₉×#384	C₂²	1	4	yes	—
17	C₅₉×#382	C₂²	1	4	yes	—
18	C₄³×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
20	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
21	#125×C₅₉	—	2	16	no	—
22	C₂×C₄⋊C₄×C₅₉	—	2	8	no	—
23	C₂×C₄⋊C₄×C₅₉	—	2	8	no	—
24	C₂×C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
25	C₂×C₄²×C₅₉	D₄	1	4	yes	—
26	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	#107×C₅₉	—	2	16	no	—
30	#106×C₅₉	—	2	16	no	—
31	C₄×C₈×C₅₉	D₄	1	8	yes	—
32	C₄×C₈×C₅₉	Q₈	1	4	yes	—
33	C₄×C₈×C₅₉	Q₈	1	4	yes	—
34	C₄×C₈×C₅₉	Q₈	1	8	yes	—
35	#95×C₅₉	—	2	8	no	—
36	C₂²×C₄×C₅₉	—	2	8	no	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
42	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
43	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
44	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
46	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
47	C₄⋊C₄×C₅₉	—	2	16	no	—
48	C₄⋊C₄×C₅₉	—	4	16	no	—
49	C₄⋊C₄×C₅₉	—	4	16	no	—
50	C₄²×C₅₉	—	2	8	no	—
51	C₄²×C₅₉	C₂×D₄	1	4	yes	—
52	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
53	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
54	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
55	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	4	no	—
57	C₂×C₄×C₅₉	—	2	4	no	—
58	C₂×C₄×C₅₉	—	4	8	no	—
59	C₂×C₄×C₅₉	—	4	16	no	—
60	C₂×C₄×C₅₉	#127	1	4	yes	—
61	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
62	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
63	C₂×C₄×C₅₉	#135	1	8	yes	—
64	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
65	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
68	C₈×C₅₉	C₄⋊Q₈	1	8	yes	—
69	C₈×C₅₉	C₄⋊Q₈	1	8	yes	—
70	#33311	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
72	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₉	C₂×#135	1	4	yes	—
75	C₄×C₅₉	—	2	4	no	—
76	C₄×C₅₉	—	2	4	no	—
77	C₄×C₅₉	—	8	8	no	—
78	C₄×C₅₉	#499	1	8	yes	complemented
79	C₄×C₅₉	C₂×C₄⋊Q₈	1	2	yes	—
80	C₄×C₅₉	C₂×C₄⋊Q₈	1	2	yes	—
81	C₂×#499	C₂×C₅₉	1	4	yes	—
82	#2859	C₂×C₅₉	1	2	yes	—
83	#1846	C₂×C₅₉	1	4	yes	—
84	#1845	C₂×C₅₉	1	2	yes	—
85	#1844	C₂×C₅₉	1	2	yes	—
86	C₄²×C₈	C₂×C₅₉	1	1	yes	characteristic
87	C₂×C₅₉	#2868	1	1	yes	characteristic
88	C₂×C₅₉	#3147	1	2	yes	—
89	C₂×C₅₉	C₂×#499	1	2	yes	—
90	C₂×C₅₉	#3160	1	2	yes	—
91	C₂×#125	C₂²×C₅₉	1	4	yes	—
92	#499	—	2	16	no	—
93	C₂×#107	C₂²×C₅₉	1	8	yes	—
94	C₂×#106	C₂²×C₅₉	1	8	yes	—
95	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
96	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
99	#389	—	2	8	no	—
100	#384	C₂²×C₅₉	1	4	yes	—
101	#382	C₂²×C₅₉	1	4	yes	—
102	C₄³	C₂²×C₅₉	1	1	yes	characteristic
103	C₅₉	#33311	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
105	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
106	#125	—	2	16	no	—
107	C₂×C₄⋊C₄	—	2	8	no	—
108	C₂×C₄⋊C₄	—	2	8	no	—
109	C₂×C₄⋊C₄	C₂³×C₅₉	1	8	yes	—
110	C₂×C₄²	D₄×C₅₉	1	4	yes	—
111	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
112	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
113	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
114	#107	—	2	16	no	—
115	#106	—	2	16	no	—
116	C₄×C₈	D₄×C₅₉	1	8	yes	—
117	C₄×C₈	Q₈×C₅₉	1	4	yes	—
118	C₄×C₈	Q₈×C₅₉	1	4	yes	—
119	C₄×C₈	Q₈×C₅₉	1	8	yes	—
120	#95	—	2	8	no	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
123	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
125	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
126	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
127	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
128	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
129	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
130	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
131	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
132	C₄⋊C₄	—	2	16	no	—
133	C₄⋊C₄	—	4	16	no	—
134	C₄⋊C₄	—	4	16	no	—
135	C₄²	—	2	8	no	—
136	C₄²	C₂×D₄×C₅₉	1	4	yes	—
137	C₄²	C₂×Q₈×C₅₉	1	4	yes	—
138	C₄²	C₂×Q₈×C₅₉	1	2	yes	—
139	C₄²	C₂×Q₈×C₅₉	1	2	yes	—
140	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	4	8	no	—
144	C₂×C₄	—	4	16	no	—
145	C₂×C₄	#127×C₅₉	1	4	yes	—
146	C₂×C₄	C₄⋊Q₈×C₅₉	1	4	yes	—
147	C₂×C₄	C₄⋊Q₈×C₅₉	1	4	yes	—
148	C₂×C₄	#135×C₅₉	1	8	yes	—
149	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
150	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
151	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
152	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
153	C₈	C₄⋊Q₈×C₅₉	1	8	yes	—
154	C₈	C₄⋊Q₈×C₅₉	1	8	yes	—
155	C₂²	C₂×#127×C₅₉	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
157	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
158	C₂²	C₂×#135×C₅₉	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	—	8	8	no	—
162	C₄	C₅₉×#499	1	8	yes	complemented
163	C₄	C₂×C₄⋊Q₈×C₅₉	1	2	yes	—
164	C₄	C₂×C₄⋊Q₈×C₅₉	1	2	yes	—
165	C₂	C₅₉×#2868	1	1	yes	characteristic
166	C₂	C₅₉×#3147	1	2	yes	—
167	C₂	C₂×C₅₉×#499	1	2	yes	—
168	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
70. #33311	C₅₉	direct (N × H)
78. C₄×C₅₉	#499	semidirect
103. C₅₉	#33311	direct (N × H)
162. C₄	C₅₉×#499	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#499	2
C₅₉×#2859	2
C₅₉×#1846	2
C₅₉×#1845	2
C₅₉×#1844	2
C₄²×C₈×C₅₉	2
#33311	59

Subgroups — C59×#33311 #28445420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33311 #28445420