Subgroups — #28741879

Order 15104.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837818	C₂	1	2	yes	maximal, complemented
2	#21432809	C₂	1	1	yes	maximal, characteristic
3	#21425929	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₂₃₆	C₂	1	4	yes	maximal, complemented
5	#21425852	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425709	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	2	yes	—
9	#16837952	C₄	1	1	yes	characteristic
10	#16837818	—	2	8	no	—
11	C₂×#14273667	C₂²	1	2	yes	—
12	C₂×#14273666	C₂²	1	4	yes	—
13	C₂×C₁₆×D₅₉	C₂²	1	4	yes	—
14	#16837481	C₄	1	4	yes	—
15	C₈×D₂₃₆	C₂²	1	4	yes	—
16	#16837426	C₄	1	1	yes	characteristic
17	#16837368	C₄	1	2	yes	—
18	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
19	(C₂×D₅₉)⋊C₁₆	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
24	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
25	C₂×#11745685	C₂×C₄	1	2	yes	—
26	#14273733	—	2	4	no	—
27	#14273733	—	2	4	no	—
28	C₂×#11745664	C₂×C₄	1	2	yes	—
29	C₂×#11745659	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₅₉	C₂³	1	4	yes	—
31	#14273727	C₈	1	1	yes	characteristic
32	#14273724	C₈	1	1	yes	characteristic
33	#14273715	C₈	1	2	yes	—
34	#14273701	C₈	1	4	yes	—
35	C₄×D₂₃₆	C₂×C₄	1	4	yes	—
36	#14273667	—	2	4	no	—
37	#14273666	—	2	8	no	—
38	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
39	C₁₆×D₅₉	—	2	8	no	—
40	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
41	(C₂×D₅₉)⋊C₈	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₉	—	2	2	no	—
46	C₂×C₁₆×C₅₉	D₄	1	4	yes	complemented
47	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
48	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₉	C₂×C₈	1	4	yes	—
55	#11745685	—	2	4	no	—
56	#11745685	—	2	4	no	—
57	C₂×D₂₃₆	C₂×C₈	1	4	yes	—
58	C₂×D₂₃₆	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₅₉	C₂²×C₄	1	4	yes	—
60	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₉	#41	1	2	yes	—
66	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
67	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
70	(C₂×D₅₉)⋊C₄	C₂×C₈	1	8	yes	—
71	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
72	#11745664	—	2	4	no	—
73	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
74	#11745659	—	2	8	no	—
75	C₈×D₅₉	—	2	4	no	—
76	C₈×D₅₉	—	2	4	no	—
77	C₁₆×C₅₉	—	2	2	no	—
78	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
79	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
80	C₅₉⋊C₁₆	—	2	4	no	—
81	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₅₉	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₅₉	—	2	4	no	—
84	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₉	#131	1	2	yes	—
89	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₅₉	—	2	8	no	—
91	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
92	D₂₃₆	—	2	4	no	—
93	D₂₃₆	—	2	8	no	—
94	C₄×D₅₉	—	2	4	no	—
95	C₄×D₅₉	—	2	4	no	—
96	Dic₁₁₈	—	2	4	no	—
97	C₈×C₅₉	—	2	2	no	—
98	C₈×C₅₉	C₄×D₄	1	2	yes	—
99	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
101	C₅₉⋊C₈	—	2	4	no	—
102	#33434	—	59	1	no	maximal, Sylow
103	C₂²×C₅₉	—	2	2	no	—
104	C₂²×C₅₉	C₈×D₄	1	2	yes	—
105	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
106	C₂×D₅₉	—	2	8	no	—
107	C₂×D₅₉	—	2	4	no	—
108	C₄×C₅₉	—	2	2	no	—
109	C₄×C₅₉	C₈×D₄	1	2	yes	—
110	C₄×C₅₉	#504	1	4	yes	complemented
111	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
113	Dic₅₉	—	2	4	no	—
114	C₂×#504	—	59	2	no	—
115	#2931	—	59	1	no	—
116	#2169	—	59	4	no	—
117	D₄×C₁₆	—	59	4	no	—
118	#2150	—	59	1	no	—
119	#2113	—	59	2	no	—
120	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
121	C₂×C₅₉	—	2	2	no	—
122	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₅₉	C₂×#504	1	2	yes	—
124	D₅₉	—	4	4	no	—
125	C₂×#131	—	59	2	no	—
126	#532	—	59	1	no	—
127	#504	—	118	8	no	—
128	C₂×M₃₂	—	59	2	no	—
129	C₂×M₃₂	—	59	4	no	—
130	C₂²×C₁₆	—	59	4	no	—
131	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
132	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
133	#436	—	59	4	no	—
134	C₈×D₄	—	59	4	no	—
135	#423	—	59	1	no	—
136	#408	—	59	2	no	—
137	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
138	C₄⋊C₁₆	—	59	4	no	—
139	C₂²⋊C₁₆	—	59	8	no	—
140	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
141	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
142	C₅₉	#33434	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	59	2	no	—
144	#131	—	118	4	no	—
145	#131	—	118	4	no	—
146	C₂×M₁₆	—	59	2	no	—
147	C₂×M₁₆	—	59	4	no	—
148	C₂²×C₈	—	59	4	no	—
149	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
151	C₄⋊Q₈	—	59	1	no	—
152	#127	—	59	1	no	—
153	#124	—	59	2	no	—
154	#121	—	59	4	no	—
155	C₄×D₄	—	59	4	no	—
156	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
157	M₃₂	—	118	4	no	—
158	M₃₂	—	118	8	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	59	4	no	—
161	C₂×C₁₆	—	118	8	no	—
162	C₂×C₁₆	D₂₃₆	1	4	yes	complemented
163	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
164	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
165	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
166	C₄⋊C₈	—	59	4	no	—
167	C₂²⋊C₈	—	59	8	no	—
168	C₄×C₈	C₄×D₅₉	1	2	yes	—
169	C₄×C₈	C₂²×D₅₉	1	2	yes	—
170	#41	—	118	4	no	—
171	#41	—	118	4	no	—
172	C₂×Q₈	—	59	2	no	—
173	C₂×D₄	—	59	4	no	—
174	C₂×D₄	—	59	4	no	—
175	C₂×D₄	—	59	2	no	—
176	C₂²×C₄	—	59	4	no	—
177	C₂²×C₄	C₈×D₅₉	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
179	M₁₆	—	118	4	no	—
180	M₁₆	—	118	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	59	4	no	—
183	C₂×C₈	—	118	4	no	—
184	C₂×C₈	—	118	4	no	—
185	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
188	C₂×C₈	#11745685	1	2	yes	—
189	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
190	C₄⋊C₄	—	59	4	no	—
191	C₂²⋊C₄	—	59	8	no	—
192	C₄²	C₈×D₅₉	1	1	yes	characteristic
193	C₄²	C₈×D₅₉	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₅₉	1	2	yes	—
195	C₁₆	—	2	2	no	—
196	C₁₆	—	118	4	no	—
197	C₁₆	C₂×D₂₃₆	1	2	yes	—
198	C₁₆	C₂×D₂₃₆	1	2	yes	—
199	C₂³	—	59	4	no	—
200	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
201	Q₈	—	118	4	no	—
202	D₄	—	118	8	no	—
203	D₄	—	118	4	no	—
204	D₄	—	118	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	59	4	no	—
207	C₂×C₄	—	118	4	no	—
208	C₂×C₄	—	118	4	no	—
209	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
213	C₂×C₄	#14273733	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	118	4	no	—
217	C₈	C₄×D₂₃₆	1	2	yes	—
218	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
219	C₈	C₂×#11745685	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	118	8	no	—
222	C₂²	—	118	4	no	—
223	C₂²	C₈×D₂₃₆	1	2	yes	—
224	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	118	4	no	—
227	C₄	C₈×D₂₃₆	1	2	yes	—
228	C₄	#16837818	1	4	yes	complemented
229	C₄	C₂×C₄×D₂₃₆	1	1	yes	characteristic
230	C₄	C₂×#14273733	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	236	4	no	—
233	C₂	C₂×C₈×D₂₃₆	1	1	yes	characteristic
234	C₂	C₂×#16837818	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
prime radical	#16837952	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#33434
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16837818	C₂	semidirect
3. #21425929	C₂	semidirect
4. C₁₆×D₂₃₆	C₂	semidirect
5. #21425852	C₂	semidirect
6. #21425709	C₂	semidirect
7. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
46. C₂×C₁₆×C₅₉	D₄	semidirect
110. C₄×C₅₉	#504	semidirect
120. C₂×C₄×C₁₆	D₅₉	semidirect
142. C₅₉	#33434	semidirect
162. C₂×C₁₆	D₂₃₆	semidirect
228. C₄	#16837818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837818	2
#21432809	2
#21425929	2
C₁₆×D₂₃₆	2
#21425852	2
#21425709	2
C₂×C₄×C₁₆×C₅₉	2
#33434	59