Subgroups — #28481843

Order 15104.

434 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	#21424234	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422844	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422521	C₂	1	2	yes	maximal
4	#21421993	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#436	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837426	C₂²	1	2	yes	—
8	#16837167	—	2	1	no	—
9	#16837167	—	2	1	no	—
10	#16837155	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	D₄⋊(C₅₉⋊C₈)	—	2	4	no	—
13	C₂²×M₁₆×C₅₉	—	2	1	no	—
14	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#436	—	2	1	no	—
16	C₅₉×#436	—	2	1	no	—
17	C₅₉×#436	C₄	1	2	yes	—
18	C₅₉×#436	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₉	—	2	1	no	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#11745664	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄×C₅₉⋊C₈	—	2	2	no	—
28	C₂²×D₄×C₅₉	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₅₉	—	2	1	no	—
30	C₂×M₁₆×C₅₉	—	2	2	no	—
31	C₂×M₁₆×C₅₉	—	2	2	no	—
32	C₂×M₁₆×C₅₉	—	4	4	no	—
33	C₂²×C₈×C₅₉	—	2	1	no	—
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₄×D₄×C₅₉	—	2	1	no	—
37	C₄×D₄×C₅₉	—	2	1	no	—
38	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₅₉	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₉	Q₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₉	—	2	1	no	—
44	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₅₉	—	2	2	no	—
46	C₂²⋊C₈×C₅₉	—	2	2	no	—
47	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₂⁴×C₅₉	—	2	1	no	—
52	C₂×D₄×C₅₉	—	2	1	no	—
53	C₂×D₄×C₅₉	—	4	2	no	—
54	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₉	—	2	1	no	—
56	C₂²×C₄×C₅₉	—	2	2	no	—
57	C₂²×C₄×C₅₉	—	2	2	no	—
58	C₂²×C₄×C₅₉	—	4	2	no	—
59	C₂²×C₄×C₅₉	—	4	2	no	—
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
63	M₁₆×C₅₉	—	4	4	no	—
64	M₁₆×C₅₉	—	4	4	no	—
65	C₂×C₈×C₅₉	—	2	2	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	—	4	2	no	—
69	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₅₉	—	2	2	no	—
76	C₂²⋊C₄×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₉	#41	1	1	yes	characteristic
81	#11745664	—	4	2	no	—
82	#11745664	—	4	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂×C₅₉⋊C₈	—	4	2	no	—
85	C₂³×C₅₉	—	2	2	no	—
86	C₂³×C₅₉	—	4	2	no	—
87	C₂³×C₅₉	—	4	2	no	—
88	C₂³×C₅₉	—	4	2	no	—
89	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
90	D₄×C₅₉	—	2	4	no	—
91	D₄×C₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	4	1	no	—
95	C₂×C₄×C₅₉	—	4	2	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	4	4	no	—
99	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#135	1	2	yes	—
106	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
107	C₈×C₅₉	—	2	1	no	—
108	C₈×C₅₉	—	2	1	no	—
109	C₈×C₅₉	—	2	1	no	—
110	C₈×C₅₉	—	2	1	no	—
111	C₈×C₅₉	—	4	2	no	—
112	C₅₉⋊C₈	—	8	2	no	—
113	C₅₉⋊C₈	—	8	2	no	—
114	#8664	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	—	2	1	no	—
116	C₂²×C₅₉	—	2	1	no	—
117	C₂²×C₅₉	—	4	2	no	—
118	C₂²×C₅₉	—	4	2	no	—
119	C₂²×C₅₉	—	4	2	no	—
120	C₂²×C₅₉	—	8	4	no	—
121	C₂²×C₅₉	#380	1	1	yes	characteristic
122	C₂²×C₅₉	#478	1	2	yes	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	—	2	1	no	—
125	C₄×C₅₉	—	2	1	no	—
126	C₄×C₅₉	—	4	2	no	—
127	C₄×C₅₉	#416	1	1	yes	characteristic
128	C₄×C₅₉	#418	1	1	yes	characteristic
129	C₄×C₅₉	#430	1	2	yes	—
130	C₄×C₅₉	#463	1	2	yes	—
131	C₂×#436	D₅₉	1	1	yes	characteristic
132	#1837	—	59	1	no	—
133	#1564	—	59	1	no	—
134	#1486	—	59	2	no	—
135	#1301	—	59	2	no	—
136	C₂×C₅₉	—	2	2	no	—
137	C₂×C₅₉	—	8	2	no	—
138	C₂×C₅₉	#1804	1	1	yes	characteristic
139	C₂×C₅₉	#1811	1	1	yes	characteristic
140	C₂×C₅₉	#1814	1	1	yes	characteristic
141	C₂²×M₁₆	—	2	1	no	—
142	C₂×C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#436	Dic₅₉	1	2	yes	—
146	#436	Dic₅₉	1	2	yes	—
147	#431	—	59	2	no	—
148	#423	—	59	2	no	—
149	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
150	C₂×C₂²⋊C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
152	#341	—	118	1	no	—
153	#341	—	118	1	no	—
154	#335	—	118	1	no	—
155	#334	—	118	1	no	—
156	D₄⋊C₈	—	118	4	no	—
157	C₅₉	#8664	1	1	yes	characteristic, complemented, Sylow
158	C₂²×D₄	Dic₁₁₈	1	1	yes	characteristic
159	C₂³×C₄	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂×M₁₆	—	118	2	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	D₂₃₆	1	2	yes	—
170	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
171	C₂×C₄⋊C₄	Dic₁₁₈	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	118	2	no	—
176	C₄⋊C₈	—	118	2	no	—
177	C₄⋊C₈	—	118	2	no	—
178	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	118	2	no	—
182	C₄×C₈	—	118	2	no	—
183	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
184	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
187	C₂⁴	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	C₄⋊Dic₅₉	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
197	C₂²×C₄	#11745685	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	M₁₆	—	236	2	no	—
202	M₁₆	—	236	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	236	2	no	—
208	C₂×C₈	—	236	2	no	—
209	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
218	C₄²	#11745685	1	1	yes	characteristic
219	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	#14273706	1	1	yes	characteristic
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
237	C₂×C₄	#14273686	1	1	yes	characteristic
238	C₂×C₄	#14273701	1	1	yes	characteristic
239	C₂×C₄	#14273701	1	1	yes	characteristic
240	C₂×C₄	#14273706	1	1	yes	characteristic
241	C₂×C₄	#14273747	1	2	yes	—
242	C₂×C₄	#14273753	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	4	2	no	—
248	C₈	—	472	2	no	—
249	C₈	—	472	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	#16837262	1	1	yes	characteristic
257	C₂²	#16837702	1	2	yes	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	2	no	—
262	C₄	#16837398	1	1	yes	characteristic
263	C₄	#16837407	1	1	yes	characteristic
264	C₄	#16837451	1	2	yes	—
265	C₄	#16837617	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	#21424106	1	1	yes	characteristic
269	C₂	#21424138	1	1	yes	characteristic
270	C₂	#21424150	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#436	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂²×D₄×C₅₉	Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#436	D₅₉
hypocenter	C₅₉	#8664
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #21424234	C₂	semidirect
2. #21422844	C₂	semidirect
4. #21421993	C₂	semidirect
157. C₅₉	#8664	semidirect

Maximal subgroups

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

subgroup	index
#21424234	2
#21422844	2
#21422521	2
#21421993	2
C₂×C₅₉×#436	2
#8664	59