Subgroups — #28487306

Order 15104.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422848	C₂	1	2	yes	maximal
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
3	#21422379	C₂	1	1	yes	maximal, characteristic
4	#21421947	C₂	1	2	yes	maximal
5	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	1	yes	characteristic
7	#16837455	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	#16837155	—	2	2	no	—
10	#16837155	—	2	1	no	—
11	#16837155	—	2	1	no	—
12	#16837153	—	2	2	no	—
13	#16837123	—	2	1	no	—
14	C₈×C₅₉⋊C₈	—	2	1	no	—
15	C₂×#108×C₅₉	C₄	1	1	yes	characteristic
16	C₂×#108×C₅₉	C₄	1	1	yes	characteristic
17	C₄×M₁₆×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#341	C₄	1	2	yes	—
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
26	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
27	C₂²⋊(C₅₉⋊C₈)	Q₈	1	2	yes	—
28	C₄×C₅₉⋊C₈	—	2	1	no	—
29	C₄×C₅₉⋊C₈	—	2	1	no	—
30	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	#108×C₅₉	—	2	2	no	—
36	#108×C₅₉	C₈	1	4	yes	complemented
37	#97×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
46	M₁₆×C₅₉	—	2	2	no	—
47	M₁₆×C₅₉	—	2	2	no	—
48	M₁₆×C₅₉	—	2	2	no	—
49	M₁₆×C₅₉	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	—	2	1	no	—
52	C₂×C₈×C₅₉	—	2	1	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
60	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₉	#41	1	1	yes	characteristic
64	C₂×C₅₉⋊C₈	—	2	1	no	—
65	C₂×C₅₉⋊C₈	—	2	1	no	—
66	C₂×C₅₉⋊C₈	—	2	2	no	—
67	C₂×C₅₉⋊C₈	—	4	2	no	—
68	C₂×C₅₉⋊C₈	—	8	2	no	—
69	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	4	1	no	—
73	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
83	C₈×C₅₉	—	2	1	no	—
84	C₈×C₅₉	—	2	1	no	—
85	C₈×C₅₉	—	2	4	no	—
86	C₈×C₅₉	—	4	2	no	—
87	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
88	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
89	C₅₉⋊C₈	—	4	2	no	—
90	C₅₉⋊C₈	—	4	1	no	—
91	C₅₉⋊C₈	—	4	1	no	—
92	C₅₉⋊C₈	—	8	2	no	—
93	#9608	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	—	2	1	no	—
95	C₂²×C₅₉	—	2	1	no	—
96	C₂²×C₅₉	#389	1	1	yes	characteristic
97	C₂²×C₅₉	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₉	#446	1	1	yes	characteristic
99	C₄×C₅₉	—	2	1	no	—
100	C₄×C₅₉	—	2	1	no	—
101	C₄×C₅₉	—	2	1	no	—
102	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₅₉	#424	1	1	yes	characteristic
104	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
105	C₄×C₅₉	#436	1	1	yes	characteristic
106	C₄×C₅₉	#472	1	1	yes	characteristic
107	C₄×C₅₉	#473	1	1	yes	characteristic
108	C₄×#108	D₅₉	1	1	yes	characteristic
109	#1565	—	59	2	no	—
110	C₂×#335	—	59	1	no	—
111	#1450	—	59	1	no	—
112	#1283	—	59	2	no	—
113	C₂×C₅₉	—	2	1	no	—
114	C₂×C₅₉	—	2	1	no	—
115	C₂×C₅₉	#1939	1	1	yes	characteristic
116	C₂×C₅₉	#1952	1	1	yes	characteristic
117	C₂×C₅₉	#1953	1	1	yes	characteristic
118	#431	—	59	1	no	—
119	#431	—	59	1	no	—
120	C₂×#108	Dic₅₉	1	1	yes	characteristic
121	C₂×#108	Dic₅₉	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	59	2	no	—
123	C₄×M₁₆	C₂×D₅₉	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
125	#341	Dic₅₉	1	2	yes	—
126	#335	—	118	2	no	—
127	#335	—	118	1	no	—
128	#335	—	118	1	no	—
129	#334	—	118	2	no	—
130	#322	—	118	1	no	—
131	C₈²	—	118	1	no	—
132	C₅₉	#9608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
134	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
135	C₂²×C₈	—	118	2	no	—
136	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
139	#108	—	2	2	no	—
140	#108	C₅₉⋊C₈	1	4	yes	complemented
141	C₄⋊C₈	—	59	1	no	—
142	C₄⋊C₈	—	59	1	no	—
143	C₄⋊C₈	—	118	2	no	—
144	C₄⋊C₈	—	118	2	no	—
145	C₄⋊C₈	—	118	1	no	—
146	C₂²⋊C₈	—	59	2	no	—
147	C₂²⋊C₈	—	118	1	no	—
148	#97	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	118	1	no	—
151	C₄×C₈	—	118	1	no	—
152	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
153	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
154	C₄×C₈	D₂₃₆	1	1	yes	characteristic
155	C₄×C₈	D₂₃₆	1	1	yes	characteristic
156	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
157	C₂²×C₄	#11745689	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	—	2	2	no	—
162	M₁₆	C₂×C₅₉⋊C₈	1	4	yes	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	118	1	no	—
168	C₂×C₈	—	118	1	no	—
169	C₂×C₈	—	118	2	no	—
170	C₂×C₈	—	472	2	no	—
171	C₂×C₈	—	236	2	no	—
172	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
173	C₂×C₈	#11745664	1	2	yes	—
174	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
176	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
177	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
178	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
179	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
180	C₄²	D₄×D₅₉	1	1	yes	characteristic
181	C₄²	#11745687	1	1	yes	characteristic
182	C₂³	Q₈×Dic₅₉	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
188	C₂×C₄	#14273702	1	1	yes	characteristic
189	C₂×C₄	#14273707	1	1	yes	characteristic
190	C₂×C₄	#14273718	1	1	yes	characteristic
191	C₂×C₄	#14273726	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
194	C₂×C₄	#14273734	1	1	yes	characteristic
195	C₂×C₄	#14273734	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	472	2	no	—
201	C₈	—	236	2	no	—
202	C₈	—	236	1	no	—
203	C₈	—	236	1	no	—
204	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
205	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16837301	1	1	yes	characteristic
209	C₂²	Q₈×C₅₉⋊C₈	1	1	yes	characteristic
210	C₂²	#16837533	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
215	C₄	#16837429	1	1	yes	characteristic
216	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
217	C₄	#16837480	1	1	yes	characteristic
218	C₄	#16837673	1	1	yes	characteristic
219	C₄	#16837678	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#21424815	1	1	yes	characteristic
223	C₂	#21424882	1	1	yes	characteristic
224	C₂	#21424887	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273702
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	#389
prime radical	C₂³×C₅₉	C₄×Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	#9608
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
36. #108×C₅₉	C₈	semidirect
132. C₅₉	#9608	semidirect
140. #108	C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21422848	2
C₂×#16837155	2
#21422379	2
#21421947	2
C₄×#108×C₅₉	2
#9608	59