Subgroups — #28475627

Order 15104.

274 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423975	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273663	C₄	1	1	yes	characteristic
5	#16837440	C₄	1	1	yes	characteristic
6	#16837357	—	2	1	no	—
7	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
8	#16837162	—	2	1	no	—
9	C₈×C₅₉⋊C₈	—	2	1	no	—
10	C₈×C₅₉⋊C₈	C₄	1	2	yes	complemented
11	C₂×M₃₂×C₅₉	—	2	1	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#364	—	2	1	no	—
14	C₅₉×#364	C₄	1	2	yes	complemented
15	C₂×#11745664	—	2	1	no	—
16	C₂²×C₅₉⋊C₈	—	2	2	no	—
17	#14273686	—	2	1	no	—
18	#14273663	—	2	1	no	—
19	#14273663	C₈	1	2	yes	complemented
20	#14273661	C₈	1	2	yes	complemented
21	#14273659	—	2	1	no	—
22	#14273637	—	4	1	no	—
23	C₄×C₅₉⋊C₈	—	2	2	no	—
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₈×Dic₅₉	—	4	1	no	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
29	M₃₂×C₅₉	—	2	2	no	—
30	M₃₂×C₅₉	—	4	1	no	—
31	C₂×C₁₆×C₅₉	—	4	1	no	—
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	—	2	1	no	—
37	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
50	C₄²×C₅₉	—	2	1	no	—
51	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²⋊Dic₅₉	—	4	1	no	—
54	C₄⋊Dic₅₉	—	2	2	no	—
55	C₄×Dic₅₉	—	4	1	no	—
56	#11745664	—	2	2	no	—
57	#11745664	—	4	1	no	—
58	C₂×C₅₉⋊C₈	—	2	4	no	—
59	C₂×C₅₉⋊C₈	—	2	8	no	—
60	C₂×C₅₉⋊C₈	—	4	1	no	—
61	C₁₆×C₅₉	—	4	2	no	—
62	C₂³×C₅₉	#104	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	4	no	—
64	C₂×C₄×C₅₉	—	2	1	no	—
65	C₂×C₄×C₅₉	—	2	1	no	—
66	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#104	1	1	yes	characteristic
72	C₂×Dic₅₉	—	4	2	no	—
73	C₈×C₅₉	C₄×C₈	1	1	yes	characteristic
74	C₈×C₅₉	#97	1	1	yes	characteristic
75	C₈×C₅₉	#102	1	1	yes	characteristic
76	C₈×C₅₉	#103	1	1	yes	characteristic
77	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₅₉	#106	1	1	yes	characteristic
80	C₈×C₅₉	#107	1	1	yes	characteristic
81	C₅₉⋊C₈	—	2	8	no	—
82	C₅₉⋊C₈	—	4	2	no	—
83	#7179	—	59	1	no	maximal, Sylow
84	C₂²×C₅₉	—	2	2	no	—
85	C₂²×C₅₉	D₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₅₉	Q₈⋊C₈	1	1	yes	characteristic
87	C₂²×C₅₉	#339	1	1	yes	characteristic
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	—	2	2	no	—
90	C₄×C₅₉	D₄⋊C₈	1	1	yes	characteristic
91	C₄×C₅₉	Q₈⋊C₈	1	1	yes	characteristic
92	C₄×C₅₉	#336	1	1	yes	characteristic
93	C₄×C₅₉	#340	1	1	yes	characteristic
94	Dic₅₉	—	8	2	no	—
95	C₂×#364	D₅₉	1	1	yes	characteristic
96	#1772	—	59	1	no	—
97	C₂×C₈²	—	59	1	no	—
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#1279	1	1	yes	characteristic
101	C₂×C₅₉	#1335	1	2	yes	—
102	C₂×M₃₂	—	2	1	no	—
103	C₂×#108	—	59	1	no	—
104	#427	—	59	1	no	—
105	#405	—	118	1	no	—
106	C₂×C₄×C₈	—	118	1	no	—
107	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	#364	—	2	1	no	—
109	#364	Dic₅₉	1	2	yes	complemented
110	#339	—	118	1	no	—
111	C₈²	—	59	2	no	—
112	C₈²	—	118	1	no	—
113	C₅₉	#7179	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	118	1	no	—
115	C₂²×C₈	—	118	2	no	—
116	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
117	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
118	#117	—	118	1	no	—
119	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
120	M₃₂	—	2	2	no	—
121	M₃₂	—	4	1	no	—
122	C₂×C₁₆	—	4	1	no	—
123	#108	—	59	2	no	—
124	#108	—	118	1	no	—
125	#107	—	59	2	no	—
126	#106	—	118	1	no	—
127	#97	—	236	1	no	—
128	C₄×C₈	—	118	2	no	—
129	C₄×C₈	—	118	2	no	—
130	C₄×C₈	—	236	1	no	—
131	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
132	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
137	M₁₆	—	118	2	no	—
138	M₁₆	—	236	1	no	—
139	C₂×C₈	—	118	4	no	—
140	C₂×C₈	—	118	8	no	—
141	C₂×C₈	—	236	1	no	—
142	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
143	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
144	C₂×C₈	#11745659	1	1	yes	characteristic
145	C₂×C₈	#11745659	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
147	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
148	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
149	C₂×C₈	#11745669	1	1	yes	characteristic
150	C₂×C₈	#11745670	1	1	yes	characteristic
151	C₂×C₈	#11745671	1	1	yes	characteristic
152	C₂×C₈	#11745672	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
154	C₄⋊C₄	—	118	2	no	—
155	C₂²⋊C₄	—	236	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	236	1	no	—
158	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
159	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
160	C₁₆	—	4	2	no	—
161	C₂³	#14273653	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	236	2	no	—
166	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
167	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
168	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
169	C₂×C₄	#14273647	1	1	yes	characteristic
170	C₂×C₄	#14273650	1	1	yes	characteristic
171	C₂×C₄	#14273653	1	1	yes	characteristic
172	C₈	—	118	8	no	—
173	C₈	—	236	2	no	—
174	C₈	C₈×Dic₅₉	1	1	yes	characteristic
175	C₈	#14273637	1	1	yes	characteristic
176	C₈	#14273648	1	1	yes	characteristic
177	C₈	#14273651	1	1	yes	characteristic
178	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
179	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
180	C₈	#14273658	1	1	yes	characteristic
181	C₈	#14273660	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#16837129	1	1	yes	characteristic
184	C₂²	#16837132	1	1	yes	characteristic
185	C₂²	#16837163	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	472	2	no	—
189	C₄	#16837129	1	1	yes	characteristic
190	C₄	#16837132	1	1	yes	characteristic
191	C₄	#16837156	1	1	yes	characteristic
192	C₄	#16837164	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#21421935	1	1	yes	characteristic
196	C₂	#21422068	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11745669
commutator	C₈×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	D₄⋊C₈
prime radical	C₂³×C₅₉	#104
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#7179
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
10. C₈×C₅₉⋊C₈	C₄	semidirect
14. C₅₉×#364	C₄	semidirect
19. #14273663	C₈	semidirect
20. #14273661	C₈	semidirect
109. #364	Dic₅₉	semidirect
113. C₅₉	#7179	semidirect

Maximal subgroups

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

subgroup	index
#21423975	2
C₂×C₈×C₅₉⋊C₈	2
C₂×C₅₉×#364	2
#7179	59