Subgroups — #28506831

Order 15104.

310 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837225	C₂	1	2	yes	maximal
2	#21424253	C₂	1	1	yes	maximal, characteristic
3	#21423975	C₂	1	1	yes	maximal, characteristic
4	#21422202	C₂	1	2	yes	maximal
5	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	C₂²	1	1	yes	characteristic
7	#16837440	C₂²	1	1	yes	characteristic
8	C₂×#14273659	—	2	1	no	—
9	C₂×#14273659	C₂²	1	2	yes	—
10	#16837357	—	2	1	no	—
11	#16837272	—	2	1	no	—
12	#16837225	—	2	4	no	—
13	#16837162	—	2	1	no	—
14	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#364	C₄	1	4	yes	complemented
17	C₂×#11745664	—	2	1	no	—
18	#14273686	—	2	1	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	4	2	no	—
21	#14273663	—	2	1	no	—
22	#14273663	D₄	1	2	yes	—
23	#14273661	Q₈	1	2	yes	—
24	#14273659	—	2	1	no	—
25	#14273659	—	2	2	no	—
26	#14273659	—	2	2	no	—
27	#14273659	—	4	2	no	—
28	#14273637	—	4	1	no	—
29	C₈×Dic₅₉	—	4	1	no	—
30	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
33	M₃₂×C₅₉	—	2	2	no	—
34	M₃₂×C₅₉	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₅₉	—	2	2	no	—
36	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₉	—	2	1	no	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	4	2	no	—
41	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
51	C₄²×C₅₉	—	2	1	no	—
52	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂²⋊Dic₅₉	—	4	1	no	—
54	C₄⋊Dic₅₉	—	2	2	no	—
55	C₄⋊Dic₅₉	—	4	2	no	—
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄⋊Dic₅₉	—	8	4	no	—
58	C₄×Dic₅₉	—	4	1	no	—
59	#11745664	—	2	2	no	—
60	#11745664	—	4	1	no	—
61	C₂×C₅₉⋊C₈	—	4	1	no	—
62	C₁₆×C₅₉	—	2	4	no	—
63	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₅₉	—	2	4	no	—
65	C₂×C₄×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
72	C₂×Dic₅₉	—	4	2	no	—
73	C₂×Dic₅₉	—	8	4	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	C₈×C₅₉	#106	1	2	yes	—
76	C₈×C₅₉	#106	1	2	yes	—
77	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
78	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₅₉	#125	1	1	yes	characteristic
80	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₅₉⋊C₈	—	4	2	no	—
82	#13186	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	2	no	—
84	C₂²×C₅₉	#428	1	1	yes	characteristic
85	C₂²×C₅₉	#460	1	1	yes	characteristic
86	C₂²×C₅₉	#476	1	1	yes	characteristic
87	C₄×C₅₉	—	2	4	no	—
88	C₄×C₅₉	#389	1	1	yes	characteristic
89	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
90	C₄×C₅₉	#461	1	1	yes	characteristic
91	C₄×C₅₉	#475	1	1	yes	characteristic
92	Dic₅₉	—	8	2	no	—
93	Dic₅₉	—	16	2	no	—
94	C₂×#365	—	59	2	no	—
95	C₂×#364	D₅₉	1	1	yes	characteristic
96	#1844	—	59	1	no	—
97	#1772	—	59	1	no	—
98	#1386	—	59	2	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	—	2	1	no	—
101	C₂×C₅₉	#1948	1	1	yes	characteristic
102	C₂×C₅₉	#2230	1	2	yes	—
103	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
104	C₂×#108	—	59	1	no	—
105	#427	—	59	1	no	—
106	C₂×#106	—	59	2	no	—
107	C₂×#106	—	118	1	no	—
108	#405	—	118	1	no	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	#384	—	118	1	no	—
111	#365	—	118	4	no	—
112	#364	Dic₅₉	1	4	yes	complemented
113	#339	—	118	1	no	—
114	C₅₉	#13186	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	118	1	no	—
116	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	#117	—	118	1	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄⋊C₄	—	236	2	no	—
121	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	C₂×Dic₅₉	1	4	yes	—
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	59	2	no	—
126	#108	—	118	1	no	—
127	#107	—	59	2	no	—
128	#106	—	118	1	no	—
129	#106	—	118	2	no	—
130	#106	—	118	2	no	—
131	#106	—	236	2	no	—
132	#97	—	236	1	no	—
133	C₄×C₈	—	236	1	no	—
134	C₄×C₈	Dic₁₁₈	1	2	yes	—
135	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	236	2	no	—
138	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
139	M₁₆	—	118	2	no	—
140	M₁₆	—	236	1	no	—
141	C₂×C₈	—	236	1	no	—
142	C₂×C₈	#11745660	1	1	yes	characteristic
143	C₂×C₈	#11745660	1	1	yes	characteristic
144	C₂×C₈	#11745660	1	2	yes	—
145	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
146	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
147	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
148	C₂×C₈	#11745687	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
150	C₂×C₈	#11745689	1	1	yes	characteristic
151	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
152	C₄⋊C₄	—	118	2	no	—
153	C₄⋊C₄	—	472	4	no	—
154	C₄⋊C₄	—	236	2	no	—
155	C₄⋊C₄	—	236	2	no	—
156	C₂²⋊C₄	—	236	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	236	1	no	—
159	C₄²	C₄⋊Dic₅₉	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₂³	#14273755	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	472	4	no	—
165	C₂×C₄	—	472	2	no	—
166	C₂×C₄	—	236	2	no	—
167	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
168	C₂×C₄	#14273702	1	1	yes	characteristic
169	C₂×C₄	#14273707	1	1	yes	characteristic
170	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
171	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
172	C₂×C₄	#14273760	1	1	yes	characteristic
173	C₈	—	236	2	no	—
174	C₈	#14273659	1	2	yes	—
175	C₈	#14273659	1	2	yes	—
176	C₈	D₄×Dic₅₉	1	1	yes	characteristic
177	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
178	C₈	#14273718	1	1	yes	characteristic
179	C₈	#14273726	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	#16837444	1	1	yes	characteristic
182	C₂²	#16837603	1	1	yes	characteristic
183	C₂²	#16837693	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	944	2	no	—
186	C₄	—	472	2	no	—
187	C₄	#16837301	1	1	yes	characteristic
188	C₄	C₂×#14273659	1	1	yes	characteristic
189	C₄	#16837608	1	1	yes	characteristic
190	C₄	#16837688	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#21424860	1	1	yes	characteristic
194	C₂	#21426268	1	2	yes	—

	subgroup	G/H
center	C₂²	#16837693
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂³×C₅₉	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#13186
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
16. C₅₉×#364	C₄	semidirect
112. #364	Dic₅₉	semidirect
114. C₅₉	#13186	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837225	2
#21424253	2
#21423975	2
#21422202	2
C₂×C₅₉×#364	2
#13186	59