Subgroups — #28504058

Order 15104.

318 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837179	C₂	1	2	yes	maximal
3	#21423963	C₂	1	1	yes	maximal, characteristic
4	#21422175	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
8	C₂×#14273637	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	complemented
11	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	complemented
12	#16837179	—	2	4	no	—
13	#16837156	C₄	1	2	yes	—
14	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#346	C₄	1	2	yes	complemented
18	C₄×C₁₆×C₅₉	C₄	1	2	yes	complemented
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₉	—	2	1	no	—
21	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
22	#14273667	—	2	2	no	—
23	#14273667	—	2	2	no	—
24	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
27	#14273637	—	2	2	no	—
28	#14273637	—	2	2	no	—
29	C₅₉⋊#95	—	2	1	no	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₅₉	—	2	1	no	—
34	M₃₂×C₅₉	—	2	1	no	—
35	M₃₂×C₅₉	D₄	1	2	yes	—
36	M₃₂×C₅₉	Q₈	1	2	yes	—
37	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	—	2	1	no	—
48	C₂×C₈×C₅₉	—	2	1	no	—
49	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×D₄	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	C₄²	1	1	yes	characteristic
58	C₄²×C₅₉	C₄²	1	1	yes	characteristic
59	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
60	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
61	C₄⋊Dic₅₉	—	4	2	no	—
62	C₄×Dic₅₉	—	2	2	no	—
63	C₄×Dic₅₉	—	2	2	no	—
64	C₂×C₅₉⋊C₈	—	2	2	no	—
65	C₂×C₅₉⋊C₈	—	2	2	no	—
66	C₂×C₅₉⋊C₈	—	2	2	no	—
67	C₁₆×C₅₉	—	2	2	no	—
68	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
69	C₅₉⋊C₁₆	—	2	4	no	—
70	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	4	1	no	—
73	C₂×C₄×C₅₉	#97	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#97	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#97	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#97	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₄×Q₈	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	C₂×Dic₅₉	—	4	2	no	—
84	C₂×Dic₅₉	—	4	2	no	—
85	C₂×Dic₅₉	—	4	2	no	—
86	C₈×C₅₉	—	2	1	no	—
87	C₈×C₅₉	—	2	1	no	—
88	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₈×C₅₉	#117	1	1	yes	characteristic
90	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
91	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
92	C₅₉⋊C₈	—	4	2	no	—
93	#12666	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	—	2	1	no	—
95	C₂²×C₅₉	—	2	1	no	—
96	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
97	C₂²×C₅₉	#436	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	#347	1	2	yes	complemented
103	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₅₉	#405	1	1	yes	characteristic
105	C₄×C₅₉	#436	1	1	yes	characteristic
106	C₄×C₅₉	#446	1	1	yes	characteristic
107	Dic₅₉	—	8	2	no	—
108	#2150	—	59	1	no	—
109	C₂×#347	—	59	2	no	—
110	C₄×M₃₂	D₅₉	1	1	yes	characteristic
111	#1770	—	59	1	no	—
112	#1375	—	59	2	no	—
113	C₂×C₅₉	—	2	1	no	—
114	C₂×C₅₉	—	2	1	no	—
115	C₂×C₅₉	#1770	1	1	yes	characteristic
116	C₂×C₅₉	C₂×#347	1	1	yes	characteristic
117	C₂×C₅₉	#2110	1	1	yes	characteristic
118	C₂×M₃₂	—	59	2	no	—
119	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
120	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
121	C₂×C₄⋊C₈	—	59	1	no	—
122	C₂×#97	—	59	2	no	—
123	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
124	C₄×C₄⋊C₄	—	59	1	no	—
125	C₄⋊C₁₆	—	59	2	no	—
126	C₄⋊C₁₆	—	59	2	no	—
127	#347	—	118	4	no	—
128	#346	Dic₅₉	1	2	yes	complemented
129	C₄×C₁₆	Dic₅₉	1	2	yes	complemented
130	#336	—	59	2	no	—
131	C₅₉	#12666	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	59	2	no	—
133	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	118	1	no	—
136	C₂×C₄²	—	59	2	no	—
137	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
138	M₃₂	—	2	1	no	—
139	M₃₂	—	2	1	no	—
140	M₃₂	—	118	2	no	—
141	M₃₂	—	118	2	no	—
142	M₃₂	Dic₁₁₈	1	2	yes	—
143	M₃₂	D₂₃₆	1	2	yes	—
144	C₂×C₁₆	—	59	4	no	—
145	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
146	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
147	C₄⋊C₈	—	118	1	no	—
148	C₄⋊C₈	—	118	1	no	—
149	#97	—	118	2	no	—
150	#97	—	118	2	no	—
151	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
152	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
154	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
155	#95	—	118	1	no	—
156	C₂²×C₄	—	118	2	no	—
157	C₂²×C₄	#11745659	1	1	yes	characteristic
158	C₂²×C₄	#11745659	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	118	2	no	—
163	C₂×C₈	—	118	2	no	—
164	C₂×C₈	—	118	2	no	—
165	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
166	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
167	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
170	C₂×C₈	#11745685	1	1	yes	characteristic
171	C₂×C₈	#11745685	1	1	yes	characteristic
172	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
173	C₄⋊C₄	—	236	2	no	—
174	C₄²	—	118	2	no	—
175	C₄²	—	118	2	no	—
176	C₄²	#11745659	1	1	yes	characteristic
177	C₄²	#11745659	1	1	yes	characteristic
178	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
179	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	118	4	no	—
182	C₁₆	C₄⋊Dic₅₉	1	4	yes	complemented
183	C₂³	C₂×#11745659	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	236	2	no	—
187	C₂×C₄	—	236	2	no	—
188	C₂×C₄	—	236	2	no	—
189	C₂×C₄	#14273637	1	1	yes	characteristic
190	C₂×C₄	#14273637	1	1	yes	characteristic
191	C₂×C₄	#14273637	1	1	yes	characteristic
192	C₂×C₄	#14273637	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
194	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
195	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
196	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
197	C₂×C₄	#14273733	1	1	yes	characteristic
198	C₂×C₄	#14273733	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	236	2	no	—
202	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
203	C₈	#14273686	1	1	yes	characteristic
204	C₈	C₄×D₂₃₆	1	1	yes	characteristic
205	C₈	C₄×Dic₁₁₈	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#14273637	1	1	yes	characteristic
209	C₂²	#16837481	1	1	yes	characteristic
210	C₂²	#16837532	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	472	2	no	—
215	C₄	#16837179	1	2	yes	complemented
216	C₄	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
217	C₄	#16837357	1	1	yes	characteristic
218	C₄	#16837481	1	1	yes	characteristic
219	C₄	#16837532	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#21423963	1	1	yes	characteristic
223	C₂	C₂×#16837179	1	1	yes	characteristic
224	C₂	#21425700	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#97
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#12666
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
11. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
17. C₅₉×#346	C₄	semidirect
18. C₄×C₁₆×C₅₉	C₄	semidirect
68. C₁₆×C₅₉	C₄⋊C₄	semidirect
102. C₄×C₅₉	#347	semidirect
128. #346	Dic₅₉	semidirect
129. C₄×C₁₆	Dic₅₉	semidirect
131. C₅₉	#12666	semidirect
182. C₁₆	C₄⋊Dic₅₉	semidirect
215. C₄	#16837179	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
C₂×#16837179	2
#21423963	2
#21422175	2
C₄×M₃₂×C₅₉	2
#12666	59