Subgroups — #28744095

Order 15104.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837834	C₂	1	4	yes	maximal, complemented
2	#21434524	C₂	1	2	yes	maximal, complemented
3	#21426357	C₂	1	4	yes	maximal, complemented
4	#21426352	C₂	1	2	yes	maximal, complemented
5	#21426349	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	—	2	2	no	—
9	C₂×#14273749	C₂²	1	4	yes	—
10	#16837952	—	2	2	no	—
11	#16837834	—	2	16	no	—
12	C₂×Dic₄₇₂	C₂²	1	4	yes	—
13	C₂×#14273672	C₂²	1	8	yes	—
14	C₂×D₉₄₄	C₂²	1	4	yes	—
15	#16837786	—	2	2	no	—
16	#16837786	C₂²	1	4	yes	—
17	#16837782	C₂²	1	2	yes	—
18	#16837779	C₂²	1	2	yes	—
19	#16837775	—	2	2	no	—
20	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
25	C₂×#11745685	—	2	4	no	—
26	#14273749	—	2	16	no	—
27	#14273749	—	4	8	no	—
28	C₂×Dic₂₃₆	—	2	2	no	—
29	C₂×Dic₂₃₆	C₂³	1	4	yes	—
30	C₂×#11745660	—	2	4	no	—
31	C₂×#11745660	—	2	4	no	—
32	C₂×D₄₇₂	—	2	2	no	—
33	C₂×D₄₇₂	C₂³	1	4	yes	—
34	#14273727	—	2	2	no	—
35	#14273724	—	2	2	no	—
36	#14273715	—	2	4	no	—
37	#14273701	—	4	4	no	—
38	C₄×D₂₃₆	—	4	4	no	—
39	Dic₄₇₂	—	2	8	no	—
40	#14273672	—	2	16	no	—
41	D₉₄₄	—	2	8	no	—
42	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₉	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
48	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	D₈	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₉	—	4	4	no	—
55	#11745685	—	4	8	no	—
56	#11745685	—	4	8	no	—
57	C₂×D₂₃₆	—	2	4	no	—
58	C₂×D₂₃₆	—	4	4	no	—
59	C₂×C₄×D₅₉	—	4	4	no	—
60	C₂×Dic₁₁₈	—	2	4	no	—
61	C₂×C₈×C₅₉	D₈	1	4	yes	—
62	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
67	C₄²×C₅₉	D₈	1	2	yes	—
68	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
70	(C₂×D₅₉)⋊C₄	—	4	8	no	—
71	C₄⋊Dic₅₉	—	4	4	no	—
72	Dic₂₃₆	—	2	8	no	—
73	Dic₂₃₆	—	4	4	no	—
74	D₄₇₂	—	2	8	no	—
75	D₄₇₂	—	4	4	no	—
76	#11745660	—	4	8	no	—
77	#11745660	—	4	8	no	—
78	C₁₆×C₅₉	C₂×D₄	1	4	yes	—
79	C₁₆×C₅₉	C₂×D₄	1	4	yes	—
80	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₅₉	—	4	4	no	—
82	C₂×C₄×C₅₉	—	2	4	no	—
83	C₂×C₄×C₅₉	#127	1	2	yes	—
84	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
89	C₂²⋊D₅₉	—	8	8	no	—
90	C₂×Dic₅₉	—	4	4	no	—
91	D₂₃₆	—	4	4	no	—
92	D₂₃₆	—	4	4	no	—
93	D₂₃₆	—	8	8	no	—
94	C₄×D₅₉	—	8	8	no	—
95	Dic₁₁₈	—	4	4	no	—
96	Dic₁₁₈	—	4	4	no	—
97	C₈×C₅₉	#127	1	2	yes	—
98	C₈×C₅₉	C₂×D₈	1	2	yes	—
99	C₈×C₅₉	C₂×D₈	1	2	yes	—
100	C₈×C₅₉	C₂²×D₄	1	2	yes	—
101	#33713	—	59	1	no	maximal, Sylow
102	C₂²×C₅₉	—	2	2	no	—
103	C₂²×C₅₉	#493	1	2	yes	—
104	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₅₉	—	8	4	no	—
106	C₂×D₅₉	—	8	8	no	—
107	C₄×C₅₉	—	2	2	no	—
108	C₄×C₅₉	#493	1	2	yes	—
109	C₄×C₅₉	#508	1	4	yes	complemented
110	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
111	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
112	Dic₅₉	—	8	4	no	—
113	C₂×#508	—	59	4	no	—
114	#3147	—	59	2	no	—
115	#2249	—	59	4	no	—
116	#2248	—	59	2	no	—
117	#2247	—	59	1	no	—
118	#2246	—	59	1	no	—
119	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
120	C₂×C₅₉	—	2	2	no	—
121	C₂×C₅₉	C₂×#493	1	1	yes	characteristic
122	C₂×C₅₉	C₂×#508	1	2	yes	—
123	D₅₉	—	16	4	no	—
124	C₂×#135	—	59	4	no	—
125	C₂×#135	—	118	2	no	—
126	#532	—	118	2	no	—
127	#508	—	118	16	no	—
128	C₂×Q₃₂	—	59	4	no	—
129	C₂×SD₃₂	—	59	8	no	—
130	C₂×D₁₆	—	59	4	no	—
131	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
132	#495	—	59	4	no	—
133	#495	—	118	2	no	—
134	#494	—	59	2	no	—
135	#493	—	59	2	no	—
136	#492	—	118	2	no	—
137	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
138	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
141	C₅₉	#33713	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	118	4	no	—
143	#135	—	118	16	no	—
144	#135	—	236	8	no	—
145	C₂×Q₁₆	—	59	4	no	—
146	C₂×Q₁₆	—	118	2	no	—
147	C₂×SD₁₆	—	118	4	no	—
148	C₂×SD₁₆	—	118	4	no	—
149	C₂×D₈	—	59	4	no	—
150	C₂×D₈	—	118	2	no	—
151	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	118	2	no	—
154	#127	—	118	2	no	—
155	#124	—	118	4	no	—
156	#121	—	236	4	no	—
157	C₄×D₄	—	236	4	no	—
158	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
159	Q₃₂	—	118	8	no	—
160	SD₃₂	—	118	16	no	—
161	D₁₆	—	118	8	no	—
162	C₂×C₁₆	D₂₃₆	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
164	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
165	C₄×C₈	D₂₃₆	1	1	yes	characteristic
166	C₄×C₈	D₂₃₆	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
169	#41	—	236	8	no	—
170	#41	—	236	8	no	—
171	C₂×Q₈	—	118	4	no	—
172	C₂×D₄	—	118	4	no	—
173	C₂×D₄	—	236	4	no	—
174	C₂×D₄	—	236	4	no	—
175	C₂²×C₄	—	236	4	no	—
176	C₂²×C₄	D₄₇₂	1	2	yes	—
177	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
178	Q₁₆	—	118	8	no	—
179	Q₁₆	—	236	4	no	—
180	SD₁₆	—	236	8	no	—
181	SD₁₆	—	236	8	no	—
182	D₈	—	118	8	no	—
183	D₈	—	236	4	no	—
184	C₂×C₈	D₄₇₂	1	4	yes	—
185	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
186	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
187	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
188	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
190	C₄⋊C₄	—	236	4	no	—
191	C₂²⋊C₄	—	236	8	no	—
192	C₄²	D₄₇₂	1	2	yes	—
193	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
194	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
195	C₁₆	C₂×D₂₃₆	1	4	yes	—
196	C₁₆	C₂×D₂₃₆	1	4	yes	—
197	C₂³	—	236	4	no	—
198	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
199	Q₈	—	236	4	no	—
200	Q₈	—	236	4	no	—
201	D₄	—	472	8	no	—
202	D₄	—	472	8	no	—
203	D₄	—	236	4	no	—
204	D₄	—	236	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	472	8	no	—
207	C₂×C₄	—	236	4	no	—
208	C₂×C₄	#14273724	1	2	yes	—
209	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
210	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
211	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
212	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
214	C₈	#14273724	1	2	yes	—
215	C₈	C₂×D₄₇₂	1	2	yes	—
216	C₈	C₂×D₄₇₂	1	2	yes	—
217	C₈	C₂²×D₂₃₆	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	472	8	no	—
220	C₂²	—	472	4	no	—
221	C₂²	#16837779	1	2	yes	—
222	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	472	4	no	—
225	C₄	#16837779	1	2	yes	—
226	C₄	#16837834	1	4	yes	complemented
227	C₄	C₂×#14273724	1	1	yes	characteristic
228	C₄	C₂²×D₄₇₂	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	944	4	no	—
231	C₂	C₂×#16837779	1	1	yes	characteristic
232	C₂	C₂×#16837834	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#33713
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16837834	C₂	semidirect
2. #21434524	C₂	semidirect
3. #21426357	C₂	semidirect
4. #21426352	C₂	semidirect
5. #21426349	C₂	semidirect
6. #21426346	C₂	semidirect
7. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
45. C₂×C₁₆×C₅₉	D₄	semidirect
109. C₄×C₅₉	#508	semidirect
119. C₂×C₄×C₁₆	D₅₉	semidirect
141. C₅₉	#33713	semidirect
162. C₂×C₁₆	D₂₃₆	semidirect
226. C₄	#16837834	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837834	2
#21434524	2
#21426357	2
#21426352	2
#21426349	2
#21426346	2
C₂×C₄×C₁₆×C₅₉	2
#33713	59