Subgroups — #28500233

Order 15104.

270 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#21422159	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	—	2	1	no	—
6	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
10	C₈×C₅₉⋊C₈	C₄	1	2	yes	—
11	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
12	C₂×M₃₂×C₅₉	—	2	1	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×C₄⋊C₁₆	—	2	1	no	—
15	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₅₉⋊C₈	—	2	1	no	—
17	C₂²×C₅₉⋊C₈	C₈	1	2	yes	—
18	#14273667	—	4	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₄×C₅₉⋊C₈	—	2	1	no	—
21	C₄×C₅₉⋊C₈	C₈	1	1	yes	characteristic
22	C₄×C₅₉⋊C₈	C₈	1	1	yes	characteristic
23	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₅₉	—	4	2	no	—
29	C₂×C₁₆×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
47	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅₉⋊C₈	—	2	4	no	—
49	C₂×C₅₉⋊C₈	—	2	2	no	—
50	C₂×C₅₉⋊C₈	—	2	4	no	—
51	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
53	C₁₆×C₅₉	—	4	2	no	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#106	1	2	yes	—
62	C₂×C₄×C₅₉	#108	1	2	yes	—
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₅₉	—	2	1	no	—
67	C₈×C₅₉	—	2	1	no	—
68	C₈×C₅₉	#106	1	2	yes	—
69	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₅₉⋊C₈	—	2	4	no	—
72	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	M₃₂	1	4	yes	complemented
74	#11996	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	—	2	1	no	—
77	C₂²×C₅₉	#334	1	2	yes	—
78	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₅₉	—	2	1	no	—
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	#334	1	2	yes	—
82	C₄×C₅₉	#364	1	2	yes	—
83	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
84	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
85	C₄×C₅₉	C₂×M₃₂	1	2	yes	—
86	#2150	—	59	1	no	—
87	#2150	D₅₉	1	1	yes	characteristic
88	C₂×C₈²	—	59	1	no	—
89	#1367	—	59	4	no	—
90	C₂×C₅₉	—	2	2	no	—
91	C₂×C₅₉	C₂×#334	1	1	yes	characteristic
92	C₂×C₅₉	#2150	1	1	yes	characteristic
93	C₂×C₅₉	C₂×#364	1	1	yes	characteristic
94	C₂×M₃₂	—	2	1	no	—
95	C₂×M₃₂	—	118	1	no	—
96	C₂×C₄×C₈	—	59	1	no	—
97	C₂×C₄×C₈	—	59	1	no	—
98	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	1	no	—
100	C₄⋊C₁₆	—	59	2	no	—
101	C₄⋊C₁₆	—	118	1	no	—
102	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
103	C₈²	—	59	2	no	—
104	C₈²	—	59	2	no	—
105	C₅₉	#11996	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	59	2	no	—
107	C₂²×C₈	—	118	1	no	—
108	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
111	M₃₂	—	4	2	no	—
112	M₃₂	—	236	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	118	2	no	—
115	C₄×C₈	—	59	1	no	—
116	C₄×C₈	—	59	1	no	—
117	C₄×C₈	—	59	2	no	—
118	C₄×C₈	—	59	2	no	—
119	C₄×C₈	—	118	1	no	—
120	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
121	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
125	C₂²×C₄	#11745659	1	1	yes	characteristic
126	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	59	2	no	—
129	C₂×C₈	—	59	2	no	—
130	C₂×C₈	—	118	4	no	—
131	C₂×C₈	—	118	4	no	—
132	C₂×C₈	—	118	2	no	—
133	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
135	C₂×C₈	#11745670	1	2	yes	—
136	C₂×C₈	#11745671	1	2	yes	—
137	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₄²	C₈×D₅₉	1	1	yes	characteristic
140	C₄²	#11745659	1	1	yes	characteristic
141	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₁₆	—	236	2	no	—
145	C₂³	Dic₅₉⋊C₈	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
149	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
150	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
151	C₂×C₄	#14273658	1	2	yes	—
152	C₂×C₄	#14273662	1	2	yes	—
153	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
155	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	59	4	no	—
159	C₈	—	118	4	no	—
160	C₈	—	118	2	no	—
161	C₈	#14273658	1	2	yes	—
162	C₈	C₂×#11745671	1	1	yes	characteristic
163	C₈	C₂×#11745670	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16837152	1	2	yes	—
167	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#16837152	1	2	yes	—
171	C₄	#16837222	1	2	yes	—
172	C₄	C₂×#14273658	1	1	yes	characteristic
173	C₄	C₂×#14273662	1	1	yes	characteristic
174	C₄	M₃₂×D₅₉	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	C₂×#16837152	1	1	yes	characteristic
177	C₂	#21425851	1	1	yes	characteristic
178	C₂	C₂×#16837222	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₄⋊C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#11996
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #21422159	C₂	semidirect
73. C₅₉⋊C₈	M₃₂	semidirect
105. C₅₉	#11996	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
C₂×C₈×C₅₉⋊C₈	2
#21422159	2
C₅₉×#2150	2
#11996	59