Subgroups — #28607007

Order 15104.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837509	C₂	1	2	yes	maximal
2	Q₈×C₄⋊Dic₅₉	C₂	1	2	yes	maximal
3	#21424835	C₂	1	2	yes	maximal
4	#21424366	C₂	1	4	yes	maximal
5	#21424253	C₂	1	1	yes	maximal, characteristic
6	#21424101	C₂	1	2	yes	maximal
7	#21423963	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₅₉	C₂²	1	4	yes	—
10	#16837509	—	2	8	no	—
11	C₂×#14273659	C₂²	1	2	yes	—
12	C₂×#14273659	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14273651	C₂²	1	4	yes	—
15	C₂×#14273637	C₂²	1	2	yes	—
16	#16837301	—	2	4	no	—
17	#16837301	C₂²	1	4	yes	—
18	#16837272	—	2	2	no	—
19	#16837272	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₅₉	—	2	2	no	—
21	C₄×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
22	#16837164	C₂²	1	4	yes	—
23	#16837156	C₂²	1	2	yes	—
24	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₅₉	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₅₉	C₄	1	8	yes	complemented
27	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₅₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
30	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₅₉	—	2	8	no	—
32	Q₈×Dic₅₉	—	4	4	no	—
33	C₂×C₄⋊Dic₅₉	—	2	2	no	—
34	C₂×C₄⋊Dic₅₉	—	2	4	no	—
35	C₂×C₄⋊Dic₅₉	—	2	2	no	—
36	C₂×C₄⋊Dic₅₉	—	2	4	no	—
37	C₂×C₄⋊Dic₅₉	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
39	C₂×C₄×Dic₅₉	—	2	4	no	—
40	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
41	#14273659	—	2	4	no	—
42	#14273659	—	4	2	no	—
43	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
44	#14273651	—	2	8	no	—
45	#14273637	—	2	4	no	—
46	C₅₉⋊#95	—	2	4	no	—
47	C₅₉⋊#95	D₄	1	2	yes	—
48	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
50	C₂×Q₁₆×C₅₉	D₄	1	4	yes	—
51	C₂×Q₁₆×C₅₉	Q₈	1	4	yes	—
52	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	—	2	4	no	—
55	C₄×Q₈×C₅₉	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
58	C₂×C₄²×C₅₉	—	2	2	no	—
59	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
60	#107×C₅₉	C₂×C₄	1	4	yes	—
61	#103×C₅₉	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
63	C₂×Q₈×C₅₉	—	2	2	no	—
64	C₂×Q₈×C₅₉	—	2	2	no	—
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
67	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
68	C₂×Q₈×C₅₉	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₅₉	—	2	2	no	—
70	C₂²×C₄×C₅₉	—	2	2	no	—
71	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₅₉	—	2	4	no	—
75	C₂²×Dic₅₉	C₂×D₄	1	4	yes	—
76	Q₁₆×C₅₉	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₅₉	—	2	1	no	—
78	C₂×C₈×C₅₉	—	2	1	no	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄×C₅₉	—	2	8	no	—
84	C₄⋊C₄×C₅₉	—	2	2	no	—
85	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
86	C₄²×C₅₉	—	2	8	no	—
87	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₉	#41	1	2	yes	—
89	C₄⋊Dic₅₉	—	2	4	no	—
90	C₄⋊Dic₅₉	—	2	4	no	—
91	C₄⋊Dic₅₉	—	2	4	no	—
92	C₄⋊Dic₅₉	—	4	8	no	—
93	C₄⋊Dic₅₉	—	4	4	no	—
94	C₄⋊Dic₅₉	—	8	4	no	—
95	C₄⋊Dic₅₉	—	8	2	no	—
96	C₄×Dic₅₉	—	2	2	no	—
97	C₄×Dic₅₉	—	2	2	no	—
98	C₄×Dic₅₉	—	4	8	no	—
99	C₂×C₅₉⋊C₈	—	2	2	no	—
100	C₂×C₅₉⋊C₈	—	2	2	no	—
101	C₂×C₅₉⋊C₈	—	2	2	no	—
102	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₅₉	—	2	4	no	—
104	Q₈×C₅₉	—	2	4	no	—
105	Q₈×C₅₉	—	2	4	no	—
106	Q₈×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	2	4	no	—
109	C₂×C₄×C₅₉	—	2	4	no	—
110	C₂×C₄×C₅₉	—	2	2	no	—
111	C₂×C₄×C₅₉	—	2	4	no	—
112	C₂×C₄×C₅₉	—	4	2	no	—
113	C₂×C₄×C₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₉	#137	1	4	yes	—
117	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₅₉	—	2	4	no	—
124	C₂×Dic₅₉	—	2	4	no	—
125	C₂×Dic₅₉	—	2	4	no	—
126	C₂×Dic₅₉	—	4	4	no	—
127	C₂×Dic₅₉	—	4	4	no	—
128	C₂×Dic₅₉	—	4	4	no	—
129	C₈×C₅₉	—	2	2	no	—
130	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
131	C₅₉⋊C₈	—	4	2	no	—
132	#22674	—	59	1	no	maximal, Sylow
133	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₅₉	D₄²	1	1	yes	characteristic
135	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
136	C₂²×C₅₉	C₂×#137	1	2	yes	—
137	C₂²×C₅₉	#577	1	2	yes	—
138	C₄×C₅₉	—	2	8	no	—
139	C₄×C₅₉	—	2	2	no	—
140	C₄×C₅₉	—	4	4	no	—
141	C₄×C₅₉	#441	1	4	yes	complemented
142	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₅₉	#520	1	1	yes	characteristic
144	C₄×C₅₉	#547	1	1	yes	characteristic
145	C₄×C₅₉	#554	1	1	yes	characteristic
146	Dic₅₉	—	4	4	no	—
147	Dic₅₉	—	8	4	no	—
148	C₂×#441	—	59	2	no	—
149	C₂×C₄×Q₁₆	D₅₉	1	1	yes	characteristic
150	Q₈×C₄⋊C₄	—	59	2	no	—
151	#1943	—	59	2	no	—
152	#1865	—	59	4	no	—
153	#1844	—	59	1	no	—
154	#1803	—	59	2	no	—
155	#1770	—	59	1	no	—
156	C₂×C₅₉	D₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₅₉	C₂×#441	1	1	yes	characteristic
158	C₂×C₅₉	#2952	1	1	yes	characteristic
159	C₂×C₅₉	#3278	1	2	yes	—
160	C₂×C₅₉	#3387	1	2	yes	—
161	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
162	C₂×C₄×Q₈	—	59	4	no	—
163	C₂×C₄×Q₈	C₂×D₅₉	1	2	yes	—
164	#441	—	118	8	no	—
165	C₄×Q₁₆	Dic₅₉	1	8	yes	complemented
166	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
167	C₂×#106	—	59	2	no	—
168	C₂×#106	—	59	2	no	—
169	C₂×C₄⋊C₈	—	59	1	no	—
170	C₂×#103	—	59	4	no	—
171	C₂×#103	C₂×D₅₉	1	2	yes	—
172	C₂×#97	—	59	2	no	—
173	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
174	#389	—	59	4	no	—
175	#389	—	118	4	no	—
176	#384	—	59	2	no	—
177	#384	—	118	2	no	—
178	C₄×C₄⋊C₄	—	59	1	no	—
179	C₄×C₄⋊C₄	—	118	2	no	—
180	#340	—	59	4	no	—
181	#336	—	59	2	no	—
182	C₅₉	#22674	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	C₂²×D₅₉	1	2	yes	—
184	C₂×Q₁₆	Dic₁₁₈	1	4	yes	—
185	C₂×Q₁₆	D₂₃₆	1	4	yes	—
186	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
187	C₂²×C₈	—	59	2	no	—
188	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	118	8	no	—
192	C₄×Q₈	—	236	4	no	—
193	C₄×Q₈	C₂×Dic₅₉	1	8	yes	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	59	2	no	—
196	C₂×C₄⋊C₄	—	59	4	no	—
197	C₂×C₄⋊C₄	—	118	4	no	—
198	C₂×C₄⋊C₄	—	118	2	no	—
199	C₂×C₄⋊C₄	—	118	4	no	—
200	C₂×C₄⋊C₄	—	118	2	no	—
201	C₂×C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	59	2	no	—
204	C₂×C₄²	—	118	4	no	—
205	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
206	#107	C₂×Dic₅₉	1	4	yes	—
207	#106	—	118	4	no	—
208	#106	—	236	2	no	—
209	C₄⋊C₈	—	118	2	no	—
210	#103	—	118	8	no	—
211	#103	C₂×Dic₅₉	1	8	yes	—
212	#97	—	118	4	no	—
213	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
214	#95	—	59	2	no	—
215	#95	—	118	4	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	C₂×Dic₁₁₈	1	4	yes	—
220	C₂×Q₈	C₂×D₂₃₆	1	4	yes	—
221	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	59	4	no	—
225	C₂²×C₄	—	118	4	no	—
226	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
229	Q₁₆	C₄⋊Dic₅₉	1	16	yes	complemented
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	118	2	no	—
233	C₂×C₈	—	118	2	no	—
234	C₂×C₈	—	118	2	no	—
235	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
236	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
237	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
238	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	118	4	no	—
242	C₄⋊C₄	—	118	4	no	—
243	C₄⋊C₄	—	118	4	no	—
244	C₄⋊C₄	—	472	2	no	—
245	C₄⋊C₄	—	472	4	no	—
246	C₄⋊C₄	—	236	8	no	—
247	C₄⋊C₄	—	236	4	no	—
248	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
249	C₄²	—	2	8	no	—
250	C₄²	—	118	2	no	—
251	C₄²	—	118	2	no	—
252	C₄²	—	236	8	no	—
253	C₄²	#11745687	1	2	yes	—
254	C₄²	C₂²×Dic₅₉	1	2	yes	—
255	C₂³	C₂×D₄×D₅₉	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	C₂×C₄⋊Dic₅₉	1	8	yes	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	118	4	no	—
268	C₂×C₄	—	118	4	no	—
269	C₂×C₄	—	118	4	no	—
270	C₂×C₄	—	236	4	no	—
271	C₂×C₄	—	236	4	no	—
272	C₂×C₄	—	236	4	no	—
273	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
274	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
275	C₂×C₄	#14273759	1	4	yes	—
276	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
278	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
279	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
280	C₂×C₄	#14273773	1	1	yes	characteristic
281	C₂×C₄	#14273774	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	236	2	no	—
284	C₈	C₂×C₄⋊Dic₅₉	1	4	yes	—
285	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
286	C₂²	D₄×D₂₃₆	1	1	yes	characteristic
287	C₂²	D₄×Dic₁₁₈	1	1	yes	characteristic
288	C₂²	C₂×#14273759	1	2	yes	—
289	C₂²	#16838209	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	472	4	no	—
294	C₄	—	236	4	no	—
295	C₄	#16837509	1	4	yes	complemented
296	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
297	C₄	#16837885	1	1	yes	characteristic
298	C₄	#16838056	1	1	yes	characteristic
299	C₄	#16838109	1	1	yes	characteristic
300	C₂	D₄×C₄⋊Dic₅₉	1	1	yes	characteristic
301	C₂	C₂×#16837509	1	1	yes	characteristic
302	C₂	#21432980	1	1	yes	characteristic
303	C₂	#21435763	1	2	yes	—
304	C₂	#21437026	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₄×Q₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₉
hypocenter	C₅₉	#22674
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×Q₁₆×C₅₉	C₄	semidirect
76. Q₁₆×C₅₉	C₄⋊C₄	semidirect
141. C₄×C₅₉	#441	semidirect
165. C₄×Q₁₆	Dic₅₉	semidirect
182. C₅₉	#22674	semidirect
229. Q₁₆	C₄⋊Dic₅₉	semidirect
295. C₄	#16837509	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837509	2
Q₈×C₄⋊Dic₅₉	2
#21424835	2
#21424366	2
#21424253	2
#21424101	2
#21423963	2
C₂×C₄×Q₁₆×C₅₉	2
#22674	59