Subgroups — C₅₉×#33317 #28445426

Order 15104.

822 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#501	C₂	1	4	yes	maximal
2	C₅₉×#2862	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2859	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1847	C₂	1	4	yes	maximal
5	C₅₉×#1846	C₂	1	4	yes	maximal
6	C₄×#97×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	C₅₉×#501	—	2	16	no	—
10	C₂×#107×C₅₉	C₂²	1	8	yes	—
11	C₂×#106×C₅₉	C₂²	1	8	yes	—
12	C₂×#97×C₅₉	C₂²	1	2	yes	—
13	C₂×#97×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#391	—	2	4	no	—
16	C₅₉×#389	—	2	4	no	—
17	C₅₉×#384	C₂²	1	2	yes	—
18	C₅₉×#382	C₂²	1	2	yes	—
19	C₅₉×#382	C₂²	1	2	yes	—
20	C₅₉×#382	C₂²	1	2	yes	—
21	C₄³×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₅₉	—	2	2	no	—
23	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
24	C₄⋊Q₈×C₅₉	—	2	8	no	—
25	#125×C₅₉	—	2	8	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
31	C₂×C₄²×C₅₉	D₄	1	4	yes	—
32	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	#107×C₅₉	—	2	16	no	—
36	#106×C₅₉	—	2	16	no	—
37	#97×C₅₉	D₄	1	8	yes	—
38	#97×C₅₉	Q₈	1	8	yes	complemented
39	C₄×C₈×C₅₉	Q₈	1	8	yes	complemented
40	#95×C₅₉	—	2	4	no	—
41	#95×C₅₉	—	2	4	no	—
42	#95×C₅₉	—	2	4	no	—
43	C₂×Q₈×C₅₉	—	4	8	no	—
44	C₂×Q₈×C₅₉	—	4	4	no	—
45	C₂²×C₄×C₅₉	—	2	4	no	—
46	C₂²×C₄×C₅₉	—	2	4	no	—
47	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×D₄	1	8	yes	—
52	C₂×C₈×C₅₉	C₂×Q₈	1	8	yes	—
53	C₂×C₈×C₅₉	C₂×Q₈	1	8	yes	—
54	C₄⋊C₄×C₅₉	—	2	4	no	—
55	C₄⋊C₄×C₅₉	—	2	8	no	—
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₄⋊C₄×C₅₉	—	4	8	no	—
58	C₄⋊C₄×C₅₉	—	4	8	no	—
59	C₄⋊C₄×C₅₉	—	4	4	no	—
60	C₄²×C₅₉	—	4	4	no	—
61	C₄²×C₅₉	C₂×D₄	1	4	yes	—
62	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
63	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
64	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
65	Q₈×C₅₉	—	8	8	no	—
66	C₂×C₄×C₅₉	—	2	4	no	—
67	C₂×C₄×C₅₉	—	2	4	no	—
68	C₂×C₄×C₅₉	—	2	4	no	—
69	C₂×C₄×C₅₉	—	4	4	no	—
70	C₂×C₄×C₅₉	—	4	4	no	—
71	C₂×C₄×C₅₉	—	4	8	no	—
72	C₂×C₄×C₅₉	—	4	8	no	—
73	C₂×C₄×C₅₉	#127	1	4	yes	—
74	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
75	C₂×C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
76	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
77	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
79	C₈×C₅₉	C₄⋊Q₈	1	16	yes	—
80	#33317	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
82	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₅₉	#575	1	4	yes	—
85	C₄×C₅₉	—	2	4	no	—
86	C₄×C₅₉	—	2	4	no	—
87	C₄×C₅₉	—	2	4	no	—
88	C₄×C₅₉	—	8	4	no	—
89	C₄×C₅₉	—	8	4	no	—
90	C₄×C₅₉	C₂×C₄⋊Q₈	1	4	yes	—
91	C₂×#501	C₂×C₅₉	1	4	yes	—
92	#2862	C₂×C₅₉	1	1	yes	characteristic
93	#2859	C₂×C₅₉	1	1	yes	characteristic
94	#1847	C₂×C₅₉	1	4	yes	—
95	#1846	C₂×C₅₉	1	4	yes	—
96	C₄×#97	C₂×C₅₉	1	1	yes	characteristic
97	C₂×C₅₉	#2868	1	1	yes	characteristic
98	C₂×C₅₉	#3153	1	2	yes	—
99	C₂×C₅₉	#3162	1	2	yes	—
100	C₂×C₅₉	#3166	1	2	yes	—
101	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
102	C₂×#125	C₂²×C₅₉	1	2	yes	—
103	#501	—	2	16	no	—
104	C₂×#107	C₂²×C₅₉	1	8	yes	—
105	C₂×#106	C₂²×C₅₉	1	8	yes	—
106	C₂×#97	C₂²×C₅₉	1	2	yes	—
107	C₂×#97	C₂²×C₅₉	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
109	#391	—	2	4	no	—
110	#389	—	2	4	no	—
111	#384	C₂²×C₅₉	1	2	yes	—
112	#382	C₂²×C₅₉	1	2	yes	—
113	#382	C₂²×C₅₉	1	2	yes	—
114	#382	C₂²×C₅₉	1	2	yes	—
115	C₄³	C₂²×C₅₉	1	1	yes	characteristic
116	C₅₉	#33317	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×Q₈	—	2	2	no	—
118	C₂²×C₈	C₂³×C₅₉	1	4	yes	—
119	C₄⋊Q₈	—	2	8	no	—
120	#125	—	2	8	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
125	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
126	C₂×C₄²	D₄×C₅₉	1	4	yes	—
127	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
128	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
129	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
130	#107	—	2	16	no	—
131	#106	—	2	16	no	—
132	#97	D₄×C₅₉	1	8	yes	—
133	#97	Q₈×C₅₉	1	8	yes	complemented
134	C₄×C₈	Q₈×C₅₉	1	8	yes	complemented
135	#95	—	2	4	no	—
136	#95	—	2	4	no	—
137	#95	—	2	4	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
145	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₅₉	1	8	yes	—
147	C₂×C₈	C₂×Q₈×C₅₉	1	8	yes	—
148	C₂×C₈	C₂×Q₈×C₅₉	1	8	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₂×D₄×C₅₉	1	4	yes	—
157	C₄²	C₂×Q₈×C₅₉	1	4	yes	—
158	C₄²	C₂×Q₈×C₅₉	1	4	yes	—
159	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
160	Q₈	—	8	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	#127×C₅₉	1	4	yes	—
169	C₂×C₄	C₄⋊Q₈×C₅₉	1	4	yes	—
170	C₂×C₄	C₄⋊Q₈×C₅₉	1	4	yes	—
171	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
172	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
173	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
174	C₈	C₄⋊Q₈×C₅₉	1	16	yes	—
175	C₂²	C₂×#127×C₅₉	1	1	yes	characteristic
176	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
177	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
178	C₂²	C₅₉×#575	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	8	4	no	—
184	C₄	C₂×C₄⋊Q₈×C₅₉	1	4	yes	—
185	C₂	C₅₉×#2868	1	1	yes	characteristic
186	C₂	C₅₉×#3153	1	2	yes	—
187	C₂	C₅₉×#3162	1	2	yes	—
188	C₂	C₅₉×#3166	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
38. #97×C₅₉	Q₈	semidirect
39. C₄×C₈×C₅₉	Q₈	semidirect
80. #33317	C₅₉	direct (N × H)
116. C₅₉	#33317	direct (N × H)
133. #97	Q₈×C₅₉	semidirect
134. 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
C₂×C₅₉×#501	2
C₅₉×#2862	2
C₅₉×#2859	2
C₅₉×#1847	2
C₅₉×#1846	2
C₄×#97×C₅₉	2
#33317	59

Subgroups — C59×#33317 #28445426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33317 #28445426