Subgroups — #28741910

Order 15104.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837819	C₂	1	2	yes	maximal, complemented
2	#21425928	C₂	1	2	yes	maximal, complemented
3	#21425922	C₂	1	4	yes	maximal, complemented
4	D₄×C₅₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425710	C₂	1	2	yes	maximal, complemented
7	C₄×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2931	C₂	1	1	yes	maximal, characteristic
9	#16837819	—	2	8	no	—
10	C₂×#14273667	C₂²	1	4	yes	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
14	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
15	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
16	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
18	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
19	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
20	#16837178	C₂²	1	2	yes	—
21	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
22	C₂×#131×C₅₉	C₂²	1	2	yes	—
23	C₅₉×#532	C₄	1	1	yes	characteristic
24	C₅₉×#436	C₄	1	2	yes	—
25	C₅₉×#436	C₄	1	2	yes	—
26	C₈×D₄×C₅₉	C₂²	1	2	yes	—
27	C₈×D₄×C₅₉	C₂²	1	2	yes	—
28	C₅₉×#423	C₄	1	1	yes	characteristic
29	C₅₉×#408	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
31	#14273667	—	2	2	no	—
32	#14273667	—	2	8	no	—
33	#14273667	D₄	1	4	yes	complemented
34	C₂×C₅₉⋊C₁₆	—	2	4	no	—
35	C₂×C₅₉⋊C₁₆	—	2	8	no	—
36	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
37	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
38	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
39	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
40	C₂×#41×C₅₉	C₂×C₄	1	2	yes	—
41	#131×C₅₉	—	2	4	no	—
42	#131×C₅₉	—	2	4	no	—
43	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₅₉	C₈	1	1	yes	characteristic
49	#127×C₅₉	C₈	1	1	yes	characteristic
50	#124×C₅₉	C₈	1	2	yes	—
51	#121×C₅₉	C₈	1	2	yes	—
52	#121×C₅₉	C₈	1	2	yes	—
53	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
60	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	#41×C₅₉	—	2	4	no	—
65	#41×C₅₉	—	2	4	no	—
66	C₂×Q₈×C₅₉	C₂×C₈	1	2	yes	—
67	C₂×D₄×C₅₉	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₅₉	C₂×C₈	1	4	yes	—
69	C₂×D₄×C₅₉	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₅₉	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	M₁₆×C₅₉	—	2	4	no	—
76	M₁₆×C₅₉	—	2	8	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	—	2	4	no	—
79	C₂×C₈×C₅₉	—	2	4	no	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₅₉	#41	1	2	yes	—
86	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
90	C₂²⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
91	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₅₉⋊C₁₆	—	2	2	no	—
96	C₅₉⋊C₁₆	—	2	2	no	—
97	C₅₉⋊C₁₆	—	2	2	no	—
98	C₅₉⋊C₁₆	C₂×D₄	1	4	yes	—
99	C₂³×C₅₉	C₂²×C₈	1	4	yes	—
100	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
101	Q₈×C₅₉	—	2	4	no	—
102	D₄×C₅₉	—	2	8	no	—
103	D₄×C₅₉	—	2	4	no	—
104	D₄×C₅₉	—	2	4	no	—
105	D₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	2	4	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	2	2	no	—
109	C₂×C₄×C₅₉	—	4	1	no	—
110	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#131	1	2	yes	—
118	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₅₉	—	2	4	no	—
120	C₈×C₅₉	—	2	1	no	—
121	C₈×C₅₉	—	2	1	no	—
122	C₈×C₅₉	C₄×D₄	1	2	yes	—
123	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
125	#33438	—	59	1	no	maximal, Sylow
126	C₂²×C₅₉	—	2	1	no	—
127	C₂²×C₅₉	—	2	1	no	—
128	C₂²×C₅₉	—	2	4	no	—
129	C₂²×C₅₉	—	2	4	no	—
130	C₂²×C₅₉	—	2	4	no	—
131	C₂²×C₅₉	C₈×D₄	1	2	yes	—
132	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₅₉	—	2	1	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	2	4	no	—
137	C₄×C₅₉	C₈×D₄	1	2	yes	—
138	C₄×C₅₉	#504	1	2	yes	complemented
139	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
141	C₂×#504	—	59	2	no	—
142	#2931	D₅₉	1	1	yes	characteristic
143	#2169	—	59	2	no	—
144	#2168	—	59	4	no	—
145	D₄×C₁₆	—	59	2	no	—
146	C₂×C₄⋊C₁₆	—	59	1	no	—
147	#2113	—	59	2	no	—
148	C₄×M₃₂	—	59	1	no	—
149	C₂×C₅₉	—	2	2	no	—
150	C₂×C₅₉	—	4	4	no	—
151	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
152	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
153	C₂×C₅₉	#3407	1	1	yes	characteristic
154	C₂×#131	C₂×D₅₉	1	2	yes	—
155	#532	Dic₅₉	1	1	yes	characteristic
156	#504	—	118	8	no	—
157	C₂×M₃₂	—	59	1	no	—
158	C₂×M₃₂	—	59	1	no	—
159	C₂×M₃₂	—	59	4	no	—
160	C₂²×C₁₆	—	59	2	no	—
161	C₂²×C₁₆	—	59	4	no	—
162	#436	Dic₅₉	1	2	yes	—
163	#436	Dic₅₉	1	2	yes	—
164	C₈×D₄	C₂×D₅₉	1	2	yes	—
165	C₈×D₄	C₂×D₅₉	1	2	yes	—
166	#423	Dic₅₉	1	1	yes	characteristic
167	#408	Dic₅₉	1	2	yes	—
168	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
169	C₄⋊C₁₆	—	59	1	no	—
170	C₄⋊C₁₆	—	59	2	no	—
171	C₄⋊C₁₆	—	59	1	no	—
172	C₂²⋊C₁₆	—	59	4	no	—
173	C₂²⋊C₁₆	—	59	4	no	—
174	#346	—	59	2	no	—
175	C₄×C₁₆	—	59	2	no	—
176	C₅₉	#33438	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	C₂×Dic₅₉	1	2	yes	—
178	#131	—	2	4	no	—
179	#131	—	2	4	no	—
180	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
181	C₂×M₁₆	C₂×Dic₅₉	1	4	yes	—
182	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅₉	1	4	yes	—
184	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
185	C₄⋊Q₈	C₅₉⋊C₈	1	1	yes	characteristic
186	#127	C₅₉⋊C₈	1	1	yes	characteristic
187	#124	C₅₉⋊C₈	1	2	yes	—
188	#121	C₅₉⋊C₈	1	2	yes	—
189	#121	C₅₉⋊C₈	1	2	yes	—
190	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
191	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
192	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
193	M₃₂	—	59	4	no	—
194	M₃₂	—	118	8	no	—
195	M₃₂	—	118	2	no	—
196	C₂×C₁₆	—	59	2	no	—
197	C₂×C₁₆	—	59	2	no	—
198	C₂×C₁₆	—	59	2	no	—
199	C₂×C₁₆	—	59	2	no	—
200	C₂×C₁₆	—	118	8	no	—
201	C₂×C₁₆	—	118	4	no	—
202	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
203	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
204	C₂²⋊C₈	C₂×Dic₅₉	1	4	yes	—
205	C₂²⋊C₈	C₂×Dic₅₉	1	4	yes	—
206	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
207	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	C₂×C₅₉⋊C₈	1	2	yes	—
213	C₂×D₄	C₂×C₅₉⋊C₈	1	2	yes	—
214	C₂×D₄	C₂×C₅₉⋊C₈	1	4	yes	—
215	C₂×D₄	C₂×C₅₉⋊C₈	1	2	yes	—
216	C₂×D₄	C₂×C₅₉⋊C₈	1	2	yes	—
217	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₅₉	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	D₄×D₅₉	1	2	yes	—
227	C₂×C₈	#11745687	1	2	yes	—
228	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
231	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
233	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
234	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
235	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
236	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
237	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
238	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
239	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
240	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
241	C₁₆	—	59	4	no	—
242	C₁₆	—	118	2	no	—
243	C₁₆	—	118	2	no	—
244	C₁₆	—	118	2	no	—
245	C₂³	C₂²×C₅₉⋊C₈	1	4	yes	—
246	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	D₄	—	2	8	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
257	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₅₉⋊C₈	1	4	yes	—
261	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
263	C₂×C₄	#14273734	1	2	yes	—
264	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	4	no	—
268	C₈	D₄×Dic₅₉	1	2	yes	—
269	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
270	C₈	C₂×#11745687	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	D₄×C₅₉⋊C₈	1	2	yes	—
277	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	4	no	—
282	C₄	D₄×C₅₉⋊C₈	1	2	yes	—
283	C₄	#16837819	1	2	yes	complemented
284	C₄	C₂×D₄×Dic₅₉	1	1	yes	characteristic
285	C₄	C₂×#14273734	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×D₄×C₅₉⋊C₈	1	1	yes	characteristic
289	C₂	C₂×#16837819	1	1	yes	characteristic
290	C₂	#21437199	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₅₉×#2931	C₂
prime radical	C₅₉×#532	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2931	D₅₉
hypocenter	C₅₉	#33438
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×#16837819	C₂	semidirect
2. #21425928	C₂	semidirect
3. #21425922	C₂	semidirect
4. D₄×C₅₉⋊C₁₆	C₂	semidirect
5. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
6. #21425710	C₂	semidirect
7. C₄×#14273667	C₂	semidirect
33. #14273667	D₄	semidirect
138. C₄×C₅₉	#504	semidirect
176. C₅₉	#33438	semidirect
283. C₄	#16837819	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837819	2
#21425928	2
#21425922	2
D₄×C₅₉⋊C₁₆	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21425710	2
C₄×#14273667	2
C₅₉×#2931	2
#33438	59