Subgroups — #28488030

Order 15104.

406 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434610	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21422594	C₂	1	2	yes	maximal
4	#21422498	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	2	yes	—
7	#16837807	C₄	1	1	yes	characteristic, complemented
8	#16837807	C₄	1	1	yes	characteristic, complemented
9	#16837804	—	2	1	no	—
10	#16837804	—	2	1	no	—
11	#16837800	C₄	1	2	yes	complemented
12	#16837455	C₂²	1	2	yes	—
13	#16837440	—	2	1	no	—
14	#16837440	—	2	1	no	—
15	#16837155	—	2	1	no	—
16	#16837153	—	2	1	no	—
17	#16837138	—	2	4	no	—
18	#16837132	—	2	4	no	—
19	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₉	—	2	1	no	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#336	—	2	1	no	—
24	#14273727	C₂×C₄	1	2	yes	—
25	#14273727	C₂×C₄	1	2	yes	—
26	#14273719	—	2	2	no	—
27	#14273706	—	2	2	no	—
28	#14273706	—	2	2	no	—
29	C₄×Dic₁₁₈	—	2	2	no	—
30	C₄×Dic₁₁₈	—	2	2	no	—
31	#14273686	—	2	2	no	—
32	#14273661	—	2	1	no	—
33	#14273661	—	2	1	no	—
34	#14273661	—	2	1	no	—
35	#14273661	—	2	1	no	—
36	#14273659	—	2	2	no	—
37	#14273659	—	2	2	no	—
38	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
39	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
40	C₄×C₅₉⋊C₈	—	2	2	no	—
41	C₂²×C₈×C₅₉	—	2	1	no	—
42	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₂×C₄²×C₅₉	—	2	1	no	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₉	—	2	1	no	—
48	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
49	C₄×C₈×C₅₉	—	4	2	no	—
50	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
54	#95×C₅₉	D₄	1	2	yes	—
55	C₂²×C₄×C₅₉	—	2	2	no	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
59	C₂×Dic₁₁₈	—	2	2	no	—
60	C₂×Dic₁₁₈	—	2	2	no	—
61	C₂×C₈×C₅₉	—	2	1	no	—
62	C₂×C₈×C₅₉	—	2	1	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	—	4	1	no	—
65	C₂×C₈×C₅₉	—	4	1	no	—
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
71	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
72	C₄⋊C₄×C₅₉	—	2	4	no	—
73	C₄²×C₅₉	—	4	1	no	—
74	C₄²×C₅₉	—	4	1	no	—
75	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₉	#41	1	1	yes	characteristic
79	C₂²⋊Dic₅₉	—	4	2	no	—
80	C₄⋊Dic₅₉	—	2	2	no	—
81	C₄⋊Dic₅₉	—	2	2	no	—
82	C₄⋊Dic₅₉	—	2	2	no	—
83	C₄⋊Dic₅₉	—	2	2	no	—
84	Dic₅₉⋊C₄	—	4	2	no	—
85	Dic₅₉⋊C₄	—	4	2	no	—
86	C₄×Dic₅₉	—	4	2	no	—
87	C₂×C₅₉⋊C₈	—	4	2	no	—
88	C₂³×C₅₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	4	no	—
91	C₂×C₄×C₅₉	—	4	1	no	—
92	C₂×C₄×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	—	4	2	no	—
94	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
99	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
104	C₂×Dic₅₉	—	4	2	no	—
105	C₂×Dic₅₉	—	4	2	no	—
106	Dic₁₁₈	—	4	4	no	—
107	Dic₁₁₈	—	8	2	no	—
108	C₈×C₅₉	—	2	1	no	—
109	C₈×C₅₉	—	2	1	no	—
110	C₈×C₅₉	—	4	2	no	—
111	C₈×C₅₉	#103	1	2	yes	—
112	C₈×C₅₉	#103	1	2	yes	—
113	C₅₉⋊C₈	—	8	2	no	—
114	#9749	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	—	2	1	no	—
116	C₂²×C₅₉	—	2	1	no	—
117	C₂²×C₅₉	#390	1	1	yes	characteristic
118	C₂²×C₅₉	#461	1	1	yes	characteristic
119	C₂²×C₅₉	#462	1	1	yes	characteristic
120	C₄×C₅₉	—	2	1	no	—
121	C₄×C₅₉	—	2	1	no	—
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	4	4	no	—
124	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
125	C₄×C₅₉	#419	1	1	yes	characteristic
126	C₄×C₅₉	#443	1	2	yes	—
127	C₄×C₅₉	#461	1	1	yes	characteristic
128	C₄×C₅₉	#462	1	1	yes	characteristic
129	Dic₅₉	—	8	2	no	—
130	Dic₅₉	—	8	2	no	—
131	#3160	—	59	1	no	—
132	C₈×C₄⋊C₄	D₅₉	1	1	yes	characteristic
133	#1564	—	59	1	no	—
134	#1503	—	59	2	no	—
135	#1481	—	59	2	no	—
136	C₂×C₅₉	—	2	1	no	—
137	C₂×C₅₉	—	2	1	no	—
138	C₂×C₅₉	#1974	1	1	yes	characteristic
139	C₂×C₅₉	#1976	1	1	yes	characteristic
140	C₂×C₅₉	#1981	1	1	yes	characteristic
141	#533	—	59	2	no	—
142	#500	—	59	1	no	—
143	#500	—	59	1	no	—
144	#499	—	118	1	no	—
145	#499	—	118	1	no	—
146	#498	—	59	2	no	—
147	#431	—	59	2	no	—
148	#427	—	118	1	no	—
149	#427	—	118	1	no	—
150	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
151	C₂×C₄×C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#335	—	118	1	no	—
156	#334	—	118	1	no	—
157	#328	—	118	4	no	—
158	Q₈⋊C₈	—	118	4	no	—
159	C₅₉	#9749	1	1	yes	characteristic, complemented, Sylow
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
163	C₄⋊Q₈	—	59	2	no	—
164	C₄⋊Q₈	—	59	2	no	—
165	#125	—	118	2	no	—
166	C₂²⋊Q₈	—	118	2	no	—
167	C₂²⋊Q₈	—	118	2	no	—
168	C₄×Q₈	—	118	2	no	—
169	C₄×Q₈	—	118	2	no	—
170	#117	—	118	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
174	#107	—	118	1	no	—
175	#107	—	118	1	no	—
176	#107	—	118	1	no	—
177	#107	—	118	1	no	—
178	#106	—	118	2	no	—
179	#106	—	118	2	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	118	2	no	—
182	C₄⋊C₈	D₂₃₆	1	2	yes	—
183	C₂²⋊C₈	—	118	2	no	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	—	118	2	no	—
186	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
187	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
188	C₄×C₈	D₂₃₆	1	1	yes	characteristic
189	C₄×C₈	D₂₃₆	1	1	yes	characteristic
190	#95	D₂₃₆	1	2	yes	—
191	C₂×Q₈	—	118	2	no	—
192	C₂×Q₈	—	118	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
195	C₂²×C₄	#11745685	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	236	2	no	—
203	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	#11745670	1	2	yes	—
208	C₂×C₈	#11745672	1	2	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	118	2	no	—
211	C₄⋊C₄	—	118	2	no	—
212	C₄⋊C₄	—	118	2	no	—
213	C₄⋊C₄	—	118	2	no	—
214	C₄⋊C₄	—	236	2	no	—
215	C₄⋊C₄	—	236	2	no	—
216	C₂²⋊C₄	—	236	2	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	236	2	no	—
220	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
221	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
222	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
223	C₄²	#11745685	1	1	yes	characteristic
224	C₂³	#14273701	1	1	yes	characteristic
225	Q₈	—	472	2	no	—
226	Q₈	—	236	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	236	2	no	—
233	C₂×C₄	—	236	2	no	—
234	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
237	C₂×C₄	#14273701	1	1	yes	characteristic
238	C₂×C₄	#14273715	1	1	yes	characteristic
239	C₂×C₄	#14273724	1	1	yes	characteristic
240	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
241	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
242	C₂×C₄	#14273757	1	1	yes	characteristic
243	C₂×C₄	#14273757	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	472	2	no	—
248	C₈	#14273650	1	2	yes	—
249	C₈	#14273650	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#16837304	1	1	yes	characteristic
253	C₂²	#16837607	1	1	yes	characteristic
254	C₂²	#16837612	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	4	no	—
259	C₄	—	472	2	no	—
260	C₄	—	472	2	no	—
261	C₄	C₂×#14273650	1	1	yes	characteristic
262	C₄	#16837410	1	1	yes	characteristic
263	C₄	#16837520	1	2	yes	—
264	C₄	#16837607	1	1	yes	characteristic
265	C₄	#16837612	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#21424984	1	1	yes	characteristic
269	C₂	#21424994	1	1	yes	characteristic
270	C₂	#21425019	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₈×C₄⋊C₄×C₅₉	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#9749
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #16837807	C₄	semidirect
8. #16837807	C₄	semidirect
11. #16837800	C₄	semidirect
159. C₅₉	#9749	semidirect

Maximal subgroups

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

subgroup	index
#21434610	2
#21422844	2
#21422594	2
#21422498	2
C₈×C₄⋊C₄×C₅₉	2
#9749	59