Subgroups — #28741931

Order 15104.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#21433000	C₂	1	1	yes	maximal, characteristic
3	#21425939	C₂	1	4	yes	maximal, complemented
4	#21425934	C₂	1	4	yes	maximal, complemented
5	#21425862	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425723	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#16838202	C₄	1	1	yes	characteristic
9	C₂×#14273733	C₂²	1	2	yes	—
10	#16837818	—	2	2	no	—
11	#16837818	—	2	2	no	—
12	#16837818	C₂²	1	4	yes	complemented
13	#16837818	C₂²	1	4	yes	complemented
14	C₂×#14273667	C₂²	1	2	yes	—
15	C₂×#14273666	—	2	2	no	—
16	C₂×C₁₆×D₅₉	—	2	2	no	—
17	#16837525	C₄	1	4	yes	—
18	#16837520	C₂²	1	4	yes	—
19	#16837451	C₄	1	1	yes	characteristic
20	#16837420	C₄	1	2	yes	—
21	#16837223	C₂²	1	4	yes	complemented
22	#16837185	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#405	C₂²	1	1	yes	characteristic
26	C₅₉×#346	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₅₉	C₂²	1	2	yes	complemented
28	C₂×#11745685	C₂×C₄	1	2	yes	—
29	#14273760	C₈	1	4	yes	—
30	#14273755	C₈	1	4	yes	—
31	#14273749	C₂×C₄	1	4	yes	—
32	#14273733	—	2	2	no	—
33	#14273733	—	2	2	no	—
34	#14273733	C₂×C₄	1	4	yes	—
35	#14273733	C₂³	1	4	yes	—
36	C₂×#11745664	C₂×C₄	1	2	yes	—
37	C₂×#11745659	—	2	2	no	—
38	C₂×C₈×D₅₉	—	2	2	no	—
39	#14273667	—	2	2	no	—
40	#14273667	C₂³	1	4	yes	—
41	#14273666	—	2	4	no	—
42	#14273666	—	2	4	no	—
43	C₂×C₅₉⋊C₁₆	—	2	2	no	—
44	C₁₆×D₅₉	—	2	4	no	—
45	C₁₆×D₅₉	—	2	4	no	—
46	#14273663	C₂×C₄	1	4	yes	—
47	#14273655	C₂×C₄	1	8	yes	—
48	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₉	—	2	2	no	—
52	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
53	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
54	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
55	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
57	#97×C₅₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₉	—	2	2	no	—
61	#11745685	—	2	2	no	—
62	#11745685	—	2	2	no	—
63	#11745685	C₂×C₈	1	4	yes	—
64	#11745685	C₂²×C₄	1	4	yes	—
65	C₂×D₂₃₆	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₅₉	—	2	2	no	—
67	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
68	M₁₆×C₅₉	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₉	#41	1	2	yes	—
73	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₂²⋊C₄×C₅₉	—	2	1	no	—
76	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
77	#11745664	—	2	2	no	—
78	#11745664	C₂²×C₄	1	4	yes	—
79	C₂×C₅₉⋊C₈	—	2	2	no	—
80	Dic₂₃₆	C₂×C₈	1	4	yes	—
81	D₄₇₂	C₂×C₈	1	4	yes	—
82	#11745660	C₂×C₈	1	8	yes	—
83	#11745659	—	2	4	no	—
84	#11745659	—	2	4	no	—
85	C₈×D₅₉	—	2	4	no	—
86	C₈×D₅₉	—	2	4	no	—
87	C₁₆×C₅₉	—	2	2	no	—
88	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
89	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₅₉⋊C₁₆	—	2	4	no	—
92	C₂³×C₅₉	#131	1	1	yes	characteristic
93	C₂²×D₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₅₉	—	2	4	no	—
100	C₂²⋊D₅₉	—	2	4	no	—
101	C₂×Dic₅₉	—	2	2	no	—
102	D₂₃₆	—	2	2	no	—
103	D₂₃₆	C₂²×C₈	1	4	yes	—
104	C₄×D₅₉	—	2	4	no	—
105	C₄×D₅₉	—	2	4	no	—
106	Dic₁₁₈	—	2	2	no	—
107	Dic₁₁₈	C₂²×C₈	1	4	yes	—
108	C₈×C₅₉	C₄×D₄	1	2	yes	—
109	C₈×C₅₉	C₂²×C₈	1	4	yes	—
110	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
112	C₅₉⋊C₈	—	2	4	no	—
113	#33441	—	59	1	no	maximal, Sylow
114	C₂²×C₅₉	—	4	1	no	—
115	C₂²×C₅₉	C₈×D₄	1	2	yes	—
116	C₂²×C₅₉	C₂×#131	1	1	yes	characteristic
117	C₂×D₅₉	—	2	4	no	—
118	C₂×D₅₉	—	4	4	no	—
119	C₄×C₅₉	—	4	2	no	—
120	C₄×C₅₉	C₈×D₄	1	2	yes	—
121	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
123	Dic₅₉	—	2	4	no	—
124	C₂×#504	—	59	2	no	—
125	#2955	—	59	1	no	—
126	#2171	—	59	4	no	—
127	#2170	—	59	4	no	—
128	#2153	—	59	1	no	—
129	#2117	—	59	2	no	—
130	#2108	D₅₉	1	1	yes	characteristic, complemented
131	C₂×C₅₉	—	2	1	no	—
132	C₂×C₅₉	—	2	2	no	—
133	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
134	D₅₉	—	4	4	no	—
135	#575	—	59	1	no	—
136	C₂×#131	—	59	2	no	—
137	#504	—	59	4	no	—
138	#504	—	59	4	no	—
139	#504	—	118	2	no	—
140	#504	—	118	2	no	—
141	C₂×M₃₂	—	59	2	no	—
142	C₂×M₃₂	—	118	2	no	—
143	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
144	C₂²×C₁₆	—	118	2	no	—
145	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
146	#444	—	59	4	no	—
147	#443	—	59	4	no	—
148	#430	—	59	1	no	—
149	#421	—	59	2	no	—
150	#405	C₂×D₅₉	1	1	yes	characteristic
151	#364	—	59	4	no	—
152	#350	—	59	8	no	—
153	#346	C₂×D₅₉	1	2	yes	complemented
154	C₄×C₁₆	C₂×D₅₉	1	2	yes	complemented
155	C₅₉	#33441	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	59	2	no	—
157	#137	—	59	4	no	—
158	AGL(1,ℤ/8)	—	59	4	no	—
159	#135	—	59	4	no	—
160	#131	—	59	4	no	—
161	#131	—	59	4	no	—
162	#131	—	118	2	no	—
163	#131	—	118	2	no	—
164	C₂×M₁₆	—	59	2	no	—
165	C₂×M₁₆	—	118	2	no	—
166	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
167	C₂²×C₈	—	118	2	no	—
168	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
169	#117	C₄×D₅₉	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	59	4	no	—
172	M₃₂	—	118	2	no	—
173	M₃₂	—	118	4	no	—
174	M₃₂	—	118	4	no	—
175	C₂×C₁₆	—	118	2	no	—
176	C₂×C₁₆	—	118	4	no	—
177	C₂×C₁₆	—	118	4	no	—
178	C₂×C₁₆	D₂₃₆	1	2	yes	—
179	C₂×C₁₆	D₂₃₆	1	2	yes	—
180	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
181	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
182	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
183	#108	—	59	4	no	—
184	#104	—	59	8	no	—
185	#97	C₄×D₅₉	1	2	yes	—
186	C₄×C₈	C₂²×D₅₉	1	2	yes	—
187	#41	—	59	4	no	—
188	#41	—	59	4	no	—
189	#41	—	118	2	no	—
190	#41	—	118	2	no	—
191	C₂×Q₈	—	59	2	no	—
192	C₂×D₄	—	59	2	no	—
193	C₂×D₄	—	118	2	no	—
194	C₂²×C₄	—	118	2	no	—
195	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
196	Q₁₆	—	59	4	no	—
197	SD₁₆	—	59	8	no	—
198	D₈	—	59	4	no	—
199	M₁₆	—	59	4	no	—
200	M₁₆	—	118	2	no	—
201	M₁₆	—	118	4	no	—
202	M₁₆	—	118	4	no	—
203	M₁₆	C₈×D₅₉	1	4	yes	—
204	C₂×C₈	—	118	2	no	—
205	C₂×C₈	—	118	4	no	—
206	C₂×C₈	—	118	4	no	—
207	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
209	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
210	C₂×C₈	#11745685	1	2	yes	—
211	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂×C₄×D₅₉	1	2	yes	—
215	C₁₆	—	2	2	no	—
216	C₁₆	—	118	4	no	—
217	C₁₆	C₂×D₂₃₆	1	2	yes	—
218	C₁₆	C₂×D₂₃₆	1	1	yes	characteristic
219	C₁₆	C₂×D₂₃₆	1	1	yes	characteristic
220	C₂³	—	118	2	no	—
221	C₂³	#14273733	1	1	yes	characteristic
222	Q₈	—	59	4	no	—
223	Q₈	—	118	2	no	—
224	D₄	—	59	4	no	—
225	D₄	—	118	4	no	—
226	D₄	—	118	4	no	—
227	D₄	—	118	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	118	2	no	—
230	C₂×C₄	—	118	4	no	—
231	C₂×C₄	—	118	4	no	—
232	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
234	C₂×C₄	#14273733	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
236	C₈	—	118	4	no	—
237	C₈	C₄×D₂₃₆	1	2	yes	—
238	C₈	C₂×C₈×D₅₉	1	4	yes	—
239	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
240	C₈	C₂×#11745685	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	118	4	no	—
243	C₂²	—	236	4	no	—
244	C₂²	C₈×D₂₃₆	1	2	yes	—
245	C₂²	C₂×#14273733	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	118	4	no	—
248	C₄	C₈×D₂₃₆	1	2	yes	—
249	C₄	C₂×C₄×D₂₃₆	1	1	yes	characteristic
250	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	236	4	no	—
254	C₂	C₂×C₈×D₂₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂×C₅₉	C₂×C₈×D₄
prime radical	#16838202	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#33441
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. C₂×#16837818	C₂	semidirect
3. #21425939	C₂	semidirect
4. #21425934	C₂	semidirect
5. #21425862	C₂	semidirect
6. #21425723	C₂	semidirect
7. C₅₉×#2108	C₂	semidirect
12. #16837818	C₂²	semidirect
13. #16837818	C₂²	semidirect
21. #16837223	C₂²	semidirect
22. #16837185	C₂²	semidirect
26. C₅₉×#346	C₂²	semidirect
27. C₄×C₁₆×C₅₉	C₂²	semidirect
130. #2108	D₅₉	semidirect
153. #346	C₂×D₅₉	semidirect
154. C₄×C₁₆	C₂×D₅₉	semidirect
155. C₅₉	#33441	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
#21433000	2
#21425939	2
#21425934	2
#21425862	2
#21425723	2
C₅₉×#2108	2
#33441	59