Subgroups — #28507216

Order 15104.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433030	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#21422186	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837959	C₄	1	1	yes	characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	#16837532	—	2	1	no	—
8	#16837532	—	2	1	no	—
9	C₈×Dic₁₁₈	—	2	1	no	—
10	C₈×Dic₁₁₈	—	2	1	no	—
11	#16837428	C₄	1	2	yes	—
12	#16837426	C₄	1	1	yes	characteristic
13	#16837357	C₂²	1	2	yes	—
14	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
16	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#346	—	2	1	no	—
19	C₄×C₁₆×C₅₉	—	2	1	no	—
20	C₂×#11745664	C₂×C₄	1	2	yes	—
21	#14273727	C₈	1	1	yes	characteristic
22	#14273727	C₈	1	1	yes	characteristic
23	#14273719	C₈	1	2	yes	—
24	#14273706	—	2	1	no	—
25	#14273706	—	2	1	no	—
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	—	2	1	no	—
28	#14273686	C₂×C₄	1	2	yes	—
29	C₂×C₅₉⋊C₁₆	—	2	2	no	—
30	C₂×C₅₉⋊C₁₆	—	2	4	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
32	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
33	Dic₅₉⋊C₈	—	2	2	no	—
34	Dic₅₉⋊C₈	—	2	2	no	—
35	#14273637	—	2	2	no	—
36	C₈×Dic₅₉	—	2	2	no	—
37	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₅₉	—	2	2	no	—
41	M₃₂×C₅₉	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₅₉	—	2	2	no	—
43	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₁₁₈	—	2	1	no	—
51	C₂×Dic₁₁₈	—	2	1	no	—
52	C₂×C₈×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₅₉	#41	1	2	yes	—
59	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊Dic₅₉	—	2	2	no	—
64	C₄⋊Dic₅₉	—	2	1	no	—
65	C₄⋊Dic₅₉	—	2	1	no	—
66	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
68	Dic₅₉⋊C₄	—	2	2	no	—
69	Dic₅₉⋊C₄	—	2	2	no	—
70	C₄×Dic₅₉	—	2	2	no	—
71	#11745664	—	2	4	no	—
72	C₂×C₅₉⋊C₈	—	2	2	no	—
73	C₁₆×C₅₉	—	2	4	no	—
74	C₅₉⋊C₁₆	—	4	2	no	—
75	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	4	1	no	—
78	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
84	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	#131	1	2	yes	—
86	C₂×Dic₅₉	—	2	2	no	—
87	C₂×Dic₅₉	—	2	2	no	—
88	Dic₁₁₈	—	4	2	no	—
89	Dic₁₁₈	—	4	2	no	—
90	C₈×C₅₉	—	2	1	no	—
91	C₈×C₅₉	—	2	1	no	—
92	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₅₉	#124	1	1	yes	characteristic
94	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
95	C₅₉⋊C₈	—	4	2	no	—
96	#13265	—	59	1	no	maximal, Sylow
97	C₂²×C₅₉	—	2	1	no	—
98	C₂²×C₅₉	—	2	1	no	—
99	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₅₉	#446	1	2	yes	—
101	C₄×C₅₉	—	2	1	no	—
102	C₄×C₅₉	—	2	1	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	#350	1	2	yes	—
105	C₄×C₅₉	#391	1	1	yes	characteristic
106	C₄×C₅₉	#408	1	1	yes	characteristic
107	C₄×C₅₉	C₈×Q₈	1	2	yes	—
108	Dic₅₉	—	4	2	no	—
109	Dic₅₉	—	4	2	no	—
110	#2960	—	59	1	no	—
111	C₂×C₄⋊C₁₆	—	59	1	no	—
112	C₄×M₃₂	D₅₉	1	1	yes	characteristic
113	#1379	—	59	4	no	—
114	C₂×C₅₉	—	2	2	no	—
115	C₂×C₅₉	#1984	1	1	yes	characteristic
116	C₂×C₅₉	C₂×#350	1	1	yes	characteristic
117	C₂×C₅₉	#2117	1	1	yes	characteristic
118	#533	—	59	1	no	—
119	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
120	C₂²×C₁₆	—	59	2	no	—
121	#446	—	118	1	no	—
122	#446	—	118	1	no	—
123	C₈×Q₈	—	118	1	no	—
124	C₈×Q₈	—	118	1	no	—
125	#424	—	59	2	no	—
126	#423	—	59	1	no	—
127	#405	—	59	2	no	—
128	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
129	C₄⋊C₁₆	—	118	1	no	—
130	C₄⋊C₁₆	—	118	1	no	—
131	#346	—	2	1	no	—
132	C₄×C₁₆	—	2	1	no	—
133	C₅₉	#13265	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	59	2	no	—
135	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
137	C₄⋊Q₈	—	59	1	no	—
138	C₄⋊Q₈	—	59	1	no	—
139	#125	—	59	2	no	—
140	C₂²⋊Q₈	—	118	1	no	—
141	C₂²⋊Q₈	—	118	1	no	—
142	C₄×Q₈	—	118	1	no	—
143	C₄×Q₈	—	118	1	no	—
144	#117	—	59	2	no	—
145	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
146	M₃₂	—	2	2	no	—
147	M₃₂	Dic₁₁₈	1	4	yes	complemented
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	118	2	no	—
150	C₂×C₁₆	—	118	4	no	—
151	C₄⋊C₈	—	118	2	no	—
152	C₄⋊C₈	—	118	2	no	—
153	C₄⋊C₈	—	118	1	no	—
154	C₄⋊C₈	—	118	1	no	—
155	#97	—	118	2	no	—
156	C₄×C₈	—	118	2	no	—
157	C₄×C₈	D₂₃₆	1	1	yes	characteristic
158	C₄×C₈	D₂₃₆	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
161	C₂×Q₈	—	118	1	no	—
162	C₂×Q₈	—	118	1	no	—
163	C₂²×C₄	#11745659	1	1	yes	characteristic
164	C₂²×C₄	#11745659	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
166	M₁₆	—	118	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	118	2	no	—
169	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
170	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
172	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
173	C₂×C₈	#11745685	1	2	yes	—
174	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
175	C₄⋊C₄	—	59	2	no	—
176	C₄⋊C₄	—	59	2	no	—
177	C₄⋊C₄	—	118	2	no	—
178	C₄⋊C₄	—	118	2	no	—
179	C₄⋊C₄	—	118	1	no	—
180	C₄⋊C₄	—	118	1	no	—
181	C₂²⋊C₄	—	118	2	no	—
182	C₄²	—	118	2	no	—
183	C₄²	C₈×D₅₉	1	1	yes	characteristic
184	C₄²	C₈×D₅₉	1	1	yes	characteristic
185	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
186	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
187	C₁₆	—	2	4	no	—
188	C₁₆	—	236	2	no	—
189	C₂³	C₂×#11745659	1	1	yes	characteristic
190	Q₈	—	236	2	no	—
191	Q₈	—	236	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	118	2	no	—
195	C₂×C₄	—	118	2	no	—
196	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
197	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
198	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
199	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
201	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
203	C₂×C₄	#14273733	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	236	2	no	—
207	C₈	#14273706	1	2	yes	—
208	C₈	#14273715	1	1	yes	characteristic
209	C₈	#14273727	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
213	C₂²	#16837532	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	236	2	no	—
218	C₄	—	236	2	no	—
219	C₄	#16837187	1	2	yes	—
220	C₄	#16837308	1	1	yes	characteristic
221	C₄	#16837368	1	1	yes	characteristic
222	C₄	C₈×Dic₁₁₈	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#21425032	1	1	yes	characteristic
225	C₂	C₂×#16837187	1	1	yes	characteristic
226	C₂	#21425725	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#13265
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
41. M₃₂×C₅₉	Q₈	semidirect
133. C₅₉	#13265	semidirect
147. M₃₂	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21433030	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21422186	2
C₄×M₃₂×C₅₉	2
#13265	59