Subgroups — #28488502

Order 15104.

694 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424903	C₂	1	2	yes	maximal, complemented
2	#21424903	C₂	1	2	yes	maximal, complemented
3	#21424224	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273648	—	2	4	no	—
7	C₂×#14273648	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837272	C₂²	1	2	yes	—
11	#16837165	—	2	2	no	—
12	#16837124	—	2	2	no	—
13	C₂²×D₈×C₅₉	—	2	2	no	—
14	C₂×#127×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#493	—	2	2	no	—
16	C₅₉×#493	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
19	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	#14273648	—	4	8	no	—
24	#14273648	—	4	8	no	—
25	C₄×C₅₉⋊C₈	—	2	1	no	—
26	C₄×C₅₉⋊C₈	—	2	1	no	—
27	C₄×C₅₉⋊C₈	—	2	1	no	—
28	C₄×C₅₉⋊C₈	—	2	1	no	—
29	C₂²×D₄×C₅₉	—	2	2	no	—
30	C₂²×D₄×C₅₉	—	2	2	no	—
31	C₂²×D₄×C₅₉	—	2	2	no	—
32	C₂×D₈×C₅₉	—	2	8	no	—
33	C₂×D₈×C₅₉	—	4	8	no	—
34	C₂²×C₈×C₅₉	D₄	1	2	yes	—
35	#127×C₅₉	—	2	2	no	—
36	#127×C₅₉	—	2	2	no	—
37	#127×C₅₉	—	2	2	no	—
38	#127×C₅₉	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₉	D₄	1	2	yes	—
42	C₂⁴×C₅₉	—	4	2	no	—
43	C₂×D₄×C₅₉	—	2	4	no	—
44	C₂×D₄×C₅₉	—	2	4	no	—
45	C₂×D₄×C₅₉	—	2	4	no	—
46	C₂×D₄×C₅₉	—	4	2	no	—
47	C₂×D₄×C₅₉	—	4	2	no	—
48	C₂×D₄×C₅₉	—	4	2	no	—
49	C₂×D₄×C₅₉	—	4	4	no	—
50	C₂×D₄×C₅₉	—	4	4	no	—
51	C₂×D₄×C₅₉	—	4	4	no	—
52	C₂×D₄×C₅₉	—	4	4	no	—
53	C₂×D₄×C₅₉	—	4	4	no	—
54	C₂×D₄×C₅₉	—	4	4	no	—
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₉	—	4	2	no	—
59	D₈×C₅₉	—	4	16	no	—
60	C₂×C₈×C₅₉	—	2	4	no	—
61	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
62	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₉	#41	1	1	yes	characteristic
65	C₄²×C₅₉	#41	1	1	yes	characteristic
66	C₄⋊Dic₅₉	—	4	4	no	—
67	C₄⋊Dic₅₉	—	4	4	no	—
68	C₄⋊Dic₅₉	—	4	4	no	—
69	C₂×C₅₉⋊C₈	—	2	2	no	—
70	C₂×C₅₉⋊C₈	—	2	2	no	—
71	C₂×C₅₉⋊C₈	—	2	2	no	—
72	C₂×C₅₉⋊C₈	—	2	4	no	—
73	C₂×C₅₉⋊C₈	D₈	1	4	yes	complemented
74	C₂³×C₅₉	—	4	4	no	—
75	C₂³×C₅₉	—	8	2	no	—
76	C₂³×C₅₉	—	8	2	no	—
77	C₂³×C₅₉	—	8	2	no	—
78	C₂³×C₅₉	—	8	2	no	—
79	C₂³×C₅₉	—	8	2	no	—
80	C₂³×C₅₉	—	8	2	no	—
81	C₂³×C₅₉	#127	1	1	yes	characteristic
82	D₄×C₅₉	—	4	8	no	—
83	D₄×C₅₉	—	4	8	no	—
84	D₄×C₅₉	—	4	8	no	—
85	D₄×C₅₉	—	8	4	no	—
86	D₄×C₅₉	—	8	4	no	—
87	D₄×C₅₉	—	8	4	no	—
88	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₅₉	#135	1	2	yes	—
96	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
97	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×Dic₅₉	—	8	2	no	—
99	C₂×Dic₅₉	—	8	2	no	—
100	C₂×Dic₅₉	—	8	2	no	—
101	C₈×C₅₉	—	2	4	no	—
102	C₅₉⋊C₈	—	2	4	no	—
103	C₅₉⋊C₈	—	4	2	no	—
104	#9832	—	59	1	no	maximal, Sylow
105	C₂²×C₅₉	—	8	4	no	—
106	C₂²×C₅₉	—	8	4	no	—
107	C₂²×C₅₉	—	8	4	no	—
108	C₂²×C₅₉	—	16	2	no	—
109	C₂²×C₅₉	—	16	2	no	—
110	C₂²×C₅₉	—	16	2	no	—
111	C₂²×C₅₉	—	16	2	no	—
112	C₂²×C₅₉	#390	1	1	yes	characteristic
113	C₂²×C₅₉	#493	1	1	yes	characteristic
114	C₂²×C₅₉	#495	1	1	yes	characteristic
115	C₂²×C₅₉	#496	1	2	yes	—
116	C₂²×C₅₉	#496	1	2	yes	—
117	C₄×C₅₉	#418	1	2	yes	—
118	C₄×C₅₉	C₄×D₈	1	2	yes	—
119	C₄×C₅₉	#442	1	2	yes	—
120	C₄×C₅₉	#466	1	2	yes	—
121	C₄×C₅₉	#469	1	2	yes	—
122	C₄×C₅₉	#490	1	2	yes	—
123	Dic₅₉	—	16	2	no	—
124	C₂×#493	D₅₉	1	1	yes	characteristic
125	#1957	—	59	2	no	—
126	#1957	—	59	2	no	—
127	#1834	—	59	1	no	—
128	C₂×#322	—	59	1	no	—
129	C₂×C₅₉	—	16	4	no	—
130	C₂×C₅₉	#1717	1	2	yes	—
131	C₂×C₅₉	#1721	1	2	yes	—
132	C₂×C₅₉	#1961	1	1	yes	characteristic
133	C₂×C₅₉	#1970	1	1	yes	characteristic
134	C₂×C₅₉	#1972	1	1	yes	characteristic
135	C₂²×D₈	—	2	2	no	—
136	C₂×#127	C₂×D₅₉	1	2	yes	—
137	#493	—	2	2	no	—
138	#493	Dic₅₉	1	4	yes	complemented
139	C₂×#102	—	118	4	no	—
140	C₂×#102	—	118	4	no	—
141	C₂×C₄×C₈	—	59	1	no	—
142	C₂×C₄×C₈	—	59	1	no	—
143	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
144	#384	—	59	2	no	—
145	#340	—	118	2	no	—
146	#322	—	118	2	no	—
147	C₅₉	#9832	1	1	yes	characteristic, complemented, Sylow
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	—	2	2	no	—
151	C₂×D₈	—	2	8	no	—
152	C₂×D₈	—	4	8	no	—
153	C₂²×C₈	—	59	2	no	—
154	C₂²×C₈	—	59	2	no	—
155	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	C₂×Dic₅₉	1	4	yes	—
160	C₂×C₄⋊C₄	—	118	2	no	—
161	C₂×C₄⋊C₄	—	118	2	no	—
162	C₂×C₄⋊C₄	—	118	2	no	—
163	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
164	#102	—	236	8	no	—
165	#102	—	236	8	no	—
166	C₄×C₈	—	118	1	no	—
167	C₄×C₈	—	118	1	no	—
168	C₄×C₈	—	118	1	no	—
169	C₄×C₈	—	118	1	no	—
170	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
171	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
172	C₂⁴	—	4	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂²×C₄	—	236	2	no	—
186	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
189	D₈	—	4	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	59	4	no	—
192	C₂×C₈	—	118	2	no	—
193	C₂×C₈	—	118	2	no	—
194	C₂×C₈	—	118	2	no	—
195	C₂×C₈	—	118	4	no	—
196	C₂×C₈	C₂²⋊Dic₅₉	1	4	yes	—
197	C₄⋊C₄	—	236	4	no	—
198	C₄⋊C₄	—	236	4	no	—
199	C₄⋊C₄	—	236	4	no	—
200	C₄²	#11745687	1	1	yes	characteristic
201	C₄²	#11745687	1	1	yes	characteristic
202	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
203	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
204	C₂³	—	4	4	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	2	no	—
207	C₂³	—	8	2	no	—
208	C₂³	—	8	2	no	—
209	C₂³	—	8	2	no	—
210	C₂³	—	8	2	no	—
211	C₂³	#14273723	1	1	yes	characteristic
212	D₄	—	4	8	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	4	no	—
217	D₄	—	8	4	no	—
218	C₂×C₄	—	472	2	no	—
219	C₂×C₄	—	472	2	no	—
220	C₂×C₄	—	472	2	no	—
221	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
222	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
223	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
224	C₂×C₄	#14273698	1	1	yes	characteristic
225	C₂×C₄	#14273698	1	1	yes	characteristic
226	C₂×C₄	#14273714	1	1	yes	characteristic
227	C₂×C₄	D₈×D₅₉	1	2	yes	—
228	C₂×C₄	#14273746	1	2	yes	—
229	C₂×C₄	#14273751	1	4	yes	—
230	C₂×C₄	#14273753	1	4	yes	—
231	C₈	—	2	4	no	—
232	C₈	—	118	4	no	—
233	C₈	—	236	2	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	—	16	2	no	—
238	C₂²	—	16	2	no	—
239	C₂²	—	16	2	no	—
240	C₂²	—	16	2	no	—
241	C₂²	#16837305	1	1	yes	characteristic
242	C₂²	#16837777	1	1	yes	characteristic
243	C₂²	#16837785	1	1	yes	characteristic
244	C₂²	#16837789	1	2	yes	—
245	C₂²	#16837790	1	2	yes	—
246	C₄	—	944	2	no	—
247	C₄	#16837407	1	2	yes	—
248	C₄	D₈×Dic₅₉	1	2	yes	—
249	C₄	#16837514	1	2	yes	—
250	C₄	#16837635	1	2	yes	—
251	C₄	#16837652	1	2	yes	—
252	C₄	#16837766	1	2	yes	—
253	C₂	—	16	4	no	—
254	C₂	#21423734	1	2	yes	—
255	C₂	#21423761	1	2	yes	—
256	C₂	#21424921	1	1	yes	characteristic
257	C₂	#21424963	1	1	yes	characteristic
258	C₂	#21424973	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting, prime radical	C₂×C₅₉×#493	C₂
socle	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#493	D₅₉
hypocenter	C₅₉	#9832
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #21424903	C₂	semidirect
2. #21424903	C₂	semidirect
3. #21424224	C₂	semidirect
4. C₂×#16837124	C₂	semidirect
16. C₅₉×#493	C₄	semidirect
73. C₂×C₅₉⋊C₈	D₈	semidirect
138. #493	Dic₅₉	semidirect
147. C₅₉	#9832	semidirect

Maximal subgroups

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

subgroup	index
#21424903	2
#21424224	2
C₂×#16837124	2
C₂×C₅₉×#493	2
#9832	59