Subgroups — #28487719

Order 15104.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432494	C₂	1	1	yes	maximal, characteristic
2	#21424248	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837150	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273727	C₄	1	4	yes	—
6	C₂×#14273727	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	#16837272	—	2	2	no	—
10	#16837272	—	2	1	no	—
11	#16837272	—	2	1	no	—
12	#16837150	—	2	8	no	—
13	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	—	2	1	no	—
15	C₄³×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#336	—	2	1	no	—
17	C₂²×Dic₁₁₈	C₂×C₄	1	4	yes	—
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	#14273727	—	2	4	no	—
20	#14273727	—	2	4	no	—
21	#14273727	—	4	8	no	—
22	C₂×C₄⋊Dic₅₉	—	2	4	no	—
23	C₂×C₄⋊Dic₅₉	—	2	4	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
27	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
28	C₂²×C₈×C₅₉	—	2	2	no	—
29	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₉	D₄	1	2	yes	—
31	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₉	—	2	2	no	—
35	C₄⋊C₈×C₅₉	D₄	1	4	yes	—
36	C₄×C₈×C₅₉	—	4	2	no	—
37	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₉	—	2	4	no	—
43	C₂×Dic₁₁₈	—	2	8	no	—
44	C₂×Dic₁₁₈	—	4	8	no	—
45	C₂×C₈×C₅₉	—	2	4	no	—
46	C₂×C₈×C₅₉	—	4	2	no	—
47	C₂×C₈×C₅₉	—	4	2	no	—
48	C₄²×C₅₉	—	2	4	no	—
49	C₄²×C₅₉	—	2	2	no	—
50	C₄²×C₅₉	SD₁₆	1	4	yes	—
51	C₄²×C₅₉	SD₁₆	1	2	yes	—
52	C₄²×C₅₉	Q₁₆	1	4	yes	—
53	C₄²×C₅₉	Q₁₆	1	2	yes	—
54	C₄²×C₅₉	C₂×D₄	1	2	yes	—
55	C₄²×C₅₉	#41	1	2	yes	—
56	C₄⋊Dic₅₉	—	4	8	no	—
57	C₄⋊Dic₅₉	—	4	8	no	—
58	C₄⋊Dic₅₉	—	4	4	no	—
59	C₄⋊Dic₅₉	—	4	2	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂×C₅₉⋊C₈	—	4	4	no	—
63	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	#100	1	4	yes	—
68	C₂×C₄×C₅₉	#103	1	4	yes	—
69	C₂×C₄×C₅₉	#103	1	4	yes	—
70	C₂×C₄×C₅₉	#103	1	4	yes	—
71	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅₉	#121	1	2	yes	—
73	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₅₉	#137	1	2	yes	—
80	C₂×Dic₅₉	—	4	4	no	—
81	C₂×Dic₅₉	—	4	8	no	—
82	Dic₁₁₈	—	8	8	no	—
83	C₈×C₅₉	—	4	4	no	—
84	C₅₉⋊C₈	—	8	2	no	—
85	#9691	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	#390	1	1	yes	characteristic
87	C₂²×C₅₉	C₂×#100	1	2	yes	—
88	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
89	C₂²×C₅₉	C₂×#103	1	2	yes	—
90	C₂²×C₅₉	#419	1	1	yes	characteristic
91	C₄×C₅₉	—	2	4	no	—
92	C₄×C₅₉	—	2	2	no	—
93	C₄×C₅₉	#333	1	8	yes	complemented
94	C₄×C₅₉	#461	1	2	yes	—
95	C₄×C₅₉	#462	1	2	yes	—
96	C₄×C₅₉	#487	1	2	yes	—
97	C₄×C₅₉	#492	1	1	yes	characteristic
98	C₄×C₅₉	#494	1	1	yes	characteristic
99	Dic₅₉	—	8	4	no	—
100	#2868	—	59	1	no	—
101	#1842	—	59	1	no	—
102	C₄×C₄⋊C₈	D₅₉	1	1	yes	characteristic
103	C₂×#333	—	59	4	no	—
104	C₂×C₅₉	C₂×#333	1	2	yes	—
105	C₂×C₅₉	#1548	1	2	yes	—
106	C₂×C₅₉	#1956	1	1	yes	characteristic
107	C₂×C₅₉	#1965	1	1	yes	characteristic
108	C₂×C₅₉	#1981	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	—	59	4	no	—
110	C₂×C₄⋊Q₈	—	59	2	no	—
111	C₂×C₄⋊C₈	—	59	2	no	—
112	C₂×C₄⋊C₈	—	118	2	no	—
113	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
114	C₂×C₄×C₈	—	2	1	no	—
115	#384	—	118	2	no	—
116	#384	—	118	1	no	—
117	#384	—	118	1	no	—
118	C₄³	C₂×D₅₉	1	1	yes	characteristic
119	#336	—	2	1	no	—
120	#333	—	118	8	no	—
121	C₅₉	#9691	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₈	—	59	4	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	118	2	no	—
125	C₄⋊Q₈	—	118	4	no	—
126	C₄⋊Q₈	—	118	4	no	—
127	C₄⋊Q₈	—	236	8	no	—
128	C₂×C₄⋊C₄	—	118	4	no	—
129	C₂×C₄⋊C₄	—	118	4	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄⋊C₄	—	118	2	no	—
132	C₂×C₄²	C₄×D₅₉	1	2	yes	—
133	C₂×C₄²	D₂₃₆	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	118	4	no	—
139	C₄⋊C₈	—	236	4	no	—
140	C₄⋊C₈	D₂₃₆	1	4	yes	—
141	C₄×C₈	—	4	2	no	—
142	C₂×Q₈	—	118	8	no	—
143	C₂×Q₈	—	236	8	no	—
144	C₂²×C₄	—	118	4	no	—
145	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
146	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	236	2	no	—
154	C₂×C₈	—	236	4	no	—
155	C₄⋊C₄	—	236	8	no	—
156	C₄⋊C₄	—	236	8	no	—
157	C₄⋊C₄	—	236	4	no	—
158	C₄⋊C₄	—	236	2	no	—
159	C₄⋊C₄	—	236	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	#11745660	1	4	yes	—
163	C₄²	Dic₂₃₆	1	4	yes	—
164	C₄²	#11745670	1	2	yes	—
165	C₄²	#11745672	1	2	yes	—
166	C₄²	C₂×D₂₃₆	1	2	yes	—
167	C₄²	#11745685	1	2	yes	—
168	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
169	Q₈	—	472	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	236	8	no	—
174	C₂×C₄	—	236	4	no	—
175	C₂×C₄	#14273643	1	4	yes	—
176	C₂×C₄	#14273650	1	4	yes	—
177	C₂×C₄	#14273652	1	4	yes	—
178	C₂×C₄	#14273652	1	4	yes	—
179	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
180	C₂×C₄	#14273701	1	2	yes	—
181	C₂×C₄	#14273715	1	1	yes	characteristic
182	C₂×C₄	#14273724	1	1	yes	characteristic
183	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
184	C₂×C₄	C₂×#11745660	1	2	yes	—
185	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
187	C₂×C₄	#14273757	1	2	yes	—
188	C₈	—	4	4	no	—
189	C₈	—	472	2	no	—
190	C₂²	#16837304	1	1	yes	characteristic
191	C₂²	C₂×#14273643	1	2	yes	—
192	C₂²	C₂×#14273650	1	1	yes	characteristic
193	C₂²	C₂×#14273652	1	2	yes	—
194	C₂²	#16837410	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	472	4	no	—
198	C₄	#16837150	1	8	yes	complemented
199	C₄	#16837607	1	2	yes	—
200	C₄	#16837612	1	2	yes	—
201	C₄	#16837753	1	2	yes	—
202	C₄	#16837775	1	1	yes	characteristic
203	C₄	#16837782	1	1	yes	characteristic
204	C₂	C₂×#16837150	1	2	yes	—
205	C₂	#21422776	1	2	yes	—
206	C₂	#21424898	1	1	yes	characteristic
207	C₂	#21424936	1	1	yes	characteristic
208	C₂	#21425019	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#9691
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
93. C₄×C₅₉	#333	semidirect
121. C₅₉	#9691	semidirect
198. C₄	#16837150	semidirect

Maximal subgroups

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

subgroup	index
#21432494	2
#21424248	2
C₂×#16837150	2
C₄×C₄⋊C₈×C₅₉	2
#9691	59