Subgroups — #28506841

Order 15104.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425869	C₂	1	2	yes	maximal
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	#21423975	C₂	1	1	yes	maximal, characteristic
4	#21422208	C₂	1	2	yes	maximal
5	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	C₂×#14273663	C₂²	1	1	yes	characteristic
9	#16837440	C₂²	1	1	yes	characteristic
10	#16837426	—	2	1	no	—
11	#16837357	—	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	2	yes	complemented
17	C₅₉×#364	C₄	1	2	yes	complemented
18	C₂×#11745664	—	2	1	no	—
19	C₂×#11745664	—	2	2	no	—
20	#14273686	—	2	1	no	—
21	#14273663	—	2	1	no	—
22	#14273663	—	2	2	no	—
23	#14273663	—	2	2	no	—
24	#14273663	—	4	1	no	—
25	#14273663	—	4	1	no	—
26	#14273663	Q₈	1	2	yes	—
27	#14273661	—	2	1	no	—
28	#14273659	D₄	1	2	yes	—
29	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
30	#14273637	—	4	1	no	—
31	C₈×Dic₅₉	—	4	1	no	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
35	M₃₂×C₅₉	—	2	2	no	—
36	M₃₂×C₅₉	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	—	2	1	no	—
43	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	SD₁₆	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	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
56	C₄²×C₅₉	—	2	1	no	—
57	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²⋊Dic₅₉	—	4	1	no	—
60	C₄⋊Dic₅₉	—	2	2	no	—
61	C₄×Dic₅₉	—	4	1	no	—
62	#11745664	—	2	2	no	—
63	#11745664	—	4	1	no	—
64	#11745664	—	4	2	no	—
65	#11745664	—	4	2	no	—
66	C₂×C₅₉⋊C₈	—	4	1	no	—
67	C₂×C₅₉⋊C₈	—	4	2	no	—
68	C₁₆×C₅₉	—	2	4	no	—
69	C₂³×C₅₉	#137	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	4	2	no	—
73	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₅₉	—	4	2	no	—
80	C₈×C₅₉	#106	1	1	yes	characteristic
81	C₈×C₅₉	#106	1	1	yes	characteristic
82	C₈×C₅₉	#106	1	1	yes	characteristic
83	C₈×C₅₉	#106	1	1	yes	characteristic
84	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
85	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₅₉	#125	1	1	yes	characteristic
87	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₅₉⋊C₈	—	4	2	no	—
89	C₅₉⋊C₈	—	8	2	no	—
90	#13188	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	4	1	no	—
92	C₂²×C₅₉	#428	1	1	yes	characteristic
93	C₂²×C₅₉	#461	1	1	yes	characteristic
94	C₂²×C₅₉	#475	1	1	yes	characteristic
95	C₄×C₅₉	—	2	2	no	—
96	C₄×C₅₉	—	4	1	no	—
97	C₄×C₅₉	#389	1	1	yes	characteristic
98	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
99	C₄×C₅₉	#460	1	1	yes	characteristic
100	C₄×C₅₉	#476	1	1	yes	characteristic
101	Dic₅₉	—	8	2	no	—
102	#2155	—	59	2	no	—
103	C₂×#364	D₅₉	1	1	yes	characteristic
104	#1848	—	59	1	no	—
105	#1772	—	59	1	no	—
106	#1388	—	59	2	no	—
107	C₂×C₅₉	—	2	1	no	—
108	C₂×C₅₉	—	2	1	no	—
109	C₂×C₅₉	—	2	1	no	—
110	C₂×C₅₉	#1948	1	1	yes	characteristic
111	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
112	C₂×#108	—	59	1	no	—
113	C₂×#108	—	59	2	no	—
114	C₂×#108	—	118	1	no	—
115	#427	—	59	1	no	—
116	#423	—	118	1	no	—
117	#405	—	118	1	no	—
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	#365	—	118	4	no	—
120	#364	Dic₅₉	1	2	yes	complemented
121	#364	Dic₅₉	1	2	yes	complemented
122	#339	—	118	1	no	—
123	C₅₉	#13188	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	—	118	1	no	—
125	C₂×M₁₆	—	118	2	no	—
126	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
128	#117	—	118	1	no	—
129	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	C₂×Dic₅₉	1	4	yes	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	—	59	2	no	—
134	#108	—	118	2	no	—
135	#108	—	118	1	no	—
136	#108	—	118	2	no	—
137	#108	—	236	1	no	—
138	#108	—	236	1	no	—
139	#107	—	118	1	no	—
140	#106	—	59	2	no	—
141	C₄⋊C₈	—	236	2	no	—
142	#97	—	236	1	no	—
143	C₄×C₈	—	236	1	no	—
144	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
145	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
150	M₁₆	—	118	2	no	—
151	M₁₆	—	236	2	no	—
152	M₁₆	—	236	2	no	—
153	M₁₆	—	236	1	no	—
154	C₂×C₈	—	236	2	no	—
155	C₂×C₈	—	236	1	no	—
156	C₂×C₈	#11745660	1	1	yes	characteristic
157	C₂×C₈	#11745660	1	1	yes	characteristic
158	C₂×C₈	#11745660	1	1	yes	characteristic
159	C₂×C₈	#11745660	1	1	yes	characteristic
160	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
162	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
163	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
164	C₂×C₈	#11745687	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
166	C₂×C₈	#11745689	1	1	yes	characteristic
167	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
168	C₄⋊C₄	—	118	2	no	—
169	C₂²⋊C₄	—	236	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	236	1	no	—
172	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
173	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	#14273760	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	236	2	no	—
180	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
181	C₂×C₄	#14273702	1	1	yes	characteristic
182	C₂×C₄	#14273707	1	1	yes	characteristic
183	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
184	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
185	C₂×C₄	#14273755	1	1	yes	characteristic
186	C₈	—	472	2	no	—
187	C₈	—	236	2	no	—
188	C₈	#14273659	1	1	yes	characteristic
189	C₈	#14273659	1	1	yes	characteristic
190	C₈	#14273659	1	1	yes	characteristic
191	C₈	#14273659	1	1	yes	characteristic
192	C₈	D₄×Dic₅₉	1	1	yes	characteristic
193	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
194	C₈	#14273718	1	1	yes	characteristic
195	C₈	#14273726	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#16837444	1	1	yes	characteristic
198	C₂²	#16837608	1	1	yes	characteristic
199	C₂²	#16837688	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	472	2	no	—
203	C₄	#16837301	1	1	yes	characteristic
204	C₄	C₂×#14273659	1	1	yes	characteristic
205	C₄	#16837603	1	1	yes	characteristic
206	C₄	#16837693	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#21424860	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837693
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂×C₅₉	#1948
prime radical	C₂³×C₅₉	#137
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#13188
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ 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
17. C₅₉×#364	C₄	semidirect
120. #364	Dic₅₉	semidirect
121. #364	Dic₅₉	semidirect
123. C₅₉	#13188	semidirect

Maximal subgroups

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

subgroup	index
#21425869	2
#21424268	2
#21423975	2
#21422208	2
C₂×C₅₉×#364	2
#13188	59