Subgroups — #28506826

Order 15104.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#21424268	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273659	C₂	1	1	yes	maximal, characteristic
4	#21422195	C₂	1	2	yes	maximal
5	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	C₂×#14273659	C₂²	1	1	yes	characteristic
9	C₂×#14273659	C₂²	1	1	yes	characteristic
10	#16837426	—	2	1	no	—
11	C₄×C₄⋊Dic₅₉	—	2	1	no	—
12	#16837225	—	2	4	no	—
13	#16837165	—	2	1	no	—
14	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×C₄⋊C₁₆	C₄	1	2	yes	complemented
17	C₅₉×C₄⋊C₁₆	C₄	1	2	yes	complemented
18	C₂×#11745664	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	1	no	—
20	C₂×C₄⋊Dic₅₉	—	2	1	no	—
21	C₂×C₄×Dic₅₉	—	4	1	no	—
22	#14273663	—	2	2	no	—
23	#14273663	—	2	2	no	—
24	#14273663	—	4	2	no	—
25	#14273659	—	2	1	no	—
26	#14273659	—	2	1	no	—
27	#14273659	D₄	1	2	yes	—
28	#14273659	Q₈	1	2	yes	—
29	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
30	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
31	C₅₉⋊#95	—	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	—	2	2	no	—
37	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
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	SD₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₉	—	4	1	no	—
46	C₂²×Dic₅₉	—	4	1	no	—
47	C₂×C₈×C₅₉	—	2	1	no	—
48	C₂×C₈×C₅₉	—	2	1	no	—
49	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
57	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
60	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
61	C₄⋊Dic₅₉	—	2	2	no	—
62	C₄⋊Dic₅₉	—	2	2	no	—
63	C₄⋊Dic₅₉	—	4	1	no	—
64	C₄⋊Dic₅₉	—	4	1	no	—
65	C₄×Dic₅₉	—	8	2	no	—
66	#11745664	—	4	2	no	—
67	#11745664	—	4	2	no	—
68	C₂×C₅₉⋊C₈	—	4	2	no	—
69	C₁₆×C₅₉	—	2	4	no	—
70	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	4	1	no	—
73	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
83	C₂×Dic₅₉	—	4	2	no	—
84	C₂×Dic₅₉	—	4	2	no	—
85	C₂×Dic₅₉	—	8	1	no	—
86	C₂×Dic₅₉	—	8	1	no	—
87	C₂×Dic₅₉	—	8	1	no	—
88	C₂×Dic₅₉	—	8	1	no	—
89	C₈×C₅₉	—	2	1	no	—
90	C₈×C₅₉	—	2	1	no	—
91	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
92	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
93	C₈×C₅₉	#125	1	1	yes	characteristic
94	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
95	C₅₉⋊C₈	—	8	2	no	—
96	#13185	—	59	1	no	maximal, Sylow
97	C₂²×C₅₉	—	2	1	no	—
98	C₂²×C₅₉	—	2	1	no	—
99	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
100	C₂²×C₅₉	#461	1	1	yes	characteristic
101	C₂²×C₅₉	#476	1	1	yes	characteristic
102	C₄×C₅₉	—	2	1	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	#365	1	2	yes	complemented
106	C₄×C₅₉	#389	1	1	yes	characteristic
107	C₄×C₅₉	#428	1	1	yes	characteristic
108	C₄×C₅₉	#460	1	1	yes	characteristic
109	C₄×C₅₉	#475	1	1	yes	characteristic
110	Dic₅₉	—	8	2	no	—
111	Dic₅₉	—	8	2	no	—
112	C₂×#365	—	59	2	no	—
113	#2150	D₅₉	1	1	yes	characteristic
114	#1848	—	59	1	no	—
115	C₄×#106	—	59	1	no	—
116	#1383	—	59	2	no	—
117	C₂×C₅₉	—	2	1	no	—
118	C₂×C₅₉	—	2	1	no	—
119	C₂×C₅₉	#1948	1	1	yes	characteristic
120	C₂×C₅₉	C₂×#365	1	1	yes	characteristic
121	C₂×C₅₉	#2155	1	1	yes	characteristic
122	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
123	C₂×#108	—	59	2	no	—
124	C₂×#108	—	118	1	no	—
125	C₂×#106	—	59	1	no	—
126	C₂×#106	—	59	1	no	—
127	#423	—	118	1	no	—
128	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	118	1	no	—
130	#365	—	118	4	no	—
131	C₄⋊C₁₆	Dic₅₉	1	2	yes	complemented
132	C₄⋊C₁₆	Dic₅₉	1	2	yes	complemented
133	#340	—	118	1	no	—
134	C₅₉	#13185	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	118	2	no	—
136	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	118	1	no	—
139	C₂×C₄⋊C₄	—	118	1	no	—
140	C₂×C₄²	—	236	1	no	—
141	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
142	M₃₂	—	2	2	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
145	#108	—	118	2	no	—
146	#108	—	118	2	no	—
147	#108	—	236	2	no	—
148	#106	—	59	2	no	—
149	#106	—	59	2	no	—
150	#106	—	118	1	no	—
151	#106	—	118	1	no	—
152	C₄⋊C₈	—	236	1	no	—
153	C₄⋊C₈	—	236	1	no	—
154	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
155	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
156	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
157	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
158	#95	—	236	1	no	—
159	C₂²×C₄	—	236	1	no	—
160	C₂²×C₄	—	236	1	no	—
161	C₂²×C₄	#11745660	1	1	yes	characteristic
162	C₂²×C₄	#11745660	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
164	M₁₆	—	236	2	no	—
165	M₁₆	—	236	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	236	2	no	—
169	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
170	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
172	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
173	C₂×C₈	#11745687	1	1	yes	characteristic
174	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
175	C₂×C₈	#11745689	1	1	yes	characteristic
176	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
177	C₄⋊C₄	—	118	2	no	—
178	C₄⋊C₄	—	118	2	no	—
179	C₄⋊C₄	—	236	1	no	—
180	C₄⋊C₄	—	236	1	no	—
181	C₄²	—	472	2	no	—
182	C₄²	#11745660	1	1	yes	characteristic
183	C₄²	#11745660	1	1	yes	characteristic
184	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
185	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₂³	C₂×#11745660	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	472	1	no	—
191	C₂×C₄	—	472	1	no	—
192	C₂×C₄	—	472	1	no	—
193	C₂×C₄	—	472	1	no	—
194	C₂×C₄	—	236	2	no	—
195	C₂×C₄	—	236	2	no	—
196	C₂×C₄	#14273659	1	1	yes	characteristic
197	C₂×C₄	#14273659	1	1	yes	characteristic
198	C₂×C₄	#14273659	1	1	yes	characteristic
199	C₂×C₄	#14273659	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
201	C₂×C₄	#14273702	1	1	yes	characteristic
202	C₂×C₄	#14273707	1	1	yes	characteristic
203	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
204	C₂×C₄	#14273755	1	1	yes	characteristic
205	C₂×C₄	#14273760	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	472	2	no	—
209	C₈	D₄×Dic₅₉	1	1	yes	characteristic
210	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
211	C₈	#14273718	1	1	yes	characteristic
212	C₈	#14273726	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#14273659	1	1	yes	characteristic
216	C₂²	#16837608	1	1	yes	characteristic
217	C₂²	#16837693	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	472	2	no	—
222	C₄	—	472	2	no	—
223	C₄	#16837225	1	2	yes	complemented
224	C₄	#16837301	1	1	yes	characteristic
225	C₄	#16837444	1	1	yes	characteristic
226	C₄	#16837603	1	1	yes	characteristic
227	C₄	#16837688	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#21424860	1	1	yes	characteristic
231	C₂	C₂×#16837225	1	1	yes	characteristic
232	C₂	#21425869	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273659
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#13185
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
16. C₅₉×C₄⋊C₁₆	C₄	semidirect
17. C₅₉×C₄⋊C₁₆	C₄	semidirect
105. C₄×C₅₉	#365	semidirect
131. C₄⋊C₁₆	Dic₅₉	semidirect
132. C₄⋊C₁₆	Dic₅₉	semidirect
134. C₅₉	#13185	semidirect
223. C₄	#16837225	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
#21424268	2
C₄×#14273659	2
#21422195	2
C₅₉×#2150	2
#13185	59