Subgroups — C₄×#16837519 #28587245

Order 15104.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837519	C₂	1	2	yes	maximal
2	C₄×#14273734	C₂	1	2	yes	maximal, complemented
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic, complemented
4	#21423977	C₂	1	2	yes	maximal, complemented
5	#21423936	C₂	1	4	yes	maximal, complemented
6	C₄×#14273654	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14273734	C₂²	1	4	yes	—
10	#16837519	C₄	1	16	yes	direct product factor
11	C₂×#14273663	C₂²	1	2	yes	—
12	C₂×#14273654	C₂²	1	4	yes	—
13	#16837358	—	2	4	no	—
14	#16837358	C₂²	1	4	yes	—
15	C₄×#11745664	—	2	2	no	—
16	C₄×#11745664	C₂²	1	2	yes	—
17	C₂×C₄×C₅₉⋊C₈	—	2	2	no	—
18	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
20	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
21	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
22	#16837167	C₂²	1	2	yes	—
23	#16837161	C₂²	1	4	yes	—
24	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
25	C₄×#41×C₅₉	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₅₉	C₄	1	2	yes	complemented
27	C₄×SD₁₆×C₅₉	C₄	1	4	yes	complemented
28	C₄×D₈×C₅₉	C₄	1	2	yes	complemented
29	C₅₉×#427	C₂²	1	1	yes	characteristic
30	C₅₉×#416	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	#14273734	—	2	8	no	—
33	#14273734	C₂×C₄	1	16	yes	complemented
34	C₂×#11745664	—	2	4	no	—
35	C₂×#11745664	C₂³	1	4	yes	—
36	C₂²×C₅₉⋊C₈	—	2	4	no	—
37	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
38	C₂×C₄×Dic₅₉	—	2	2	no	—
39	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
40	#14273663	C₂×C₄	1	8	yes	complemented
41	#14273654	C₂×C₄	1	16	yes	complemented
42	#14273638	—	2	2	no	—
43	#14273638	—	2	4	no	—
44	C₄×C₅₉⋊C₈	—	2	2	no	—
45	C₄×C₅₉⋊C₈	—	2	4	no	—
46	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
47	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
48	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
49	C₈×Dic₅₉	—	2	4	no	—
50	C₈×Dic₅₉	C₂×C₄	1	8	yes	complemented
51	C₂×#41×C₅₉	C₂³	1	2	yes	—
52	#135×C₅₉	C₂×C₄	1	8	yes	—
53	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₅₉	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₅₉	—	2	4	no	—
60	C₄×D₄×C₅₉	C₂×C₄	1	4	yes	—
61	#117×C₅₉	—	2	2	no	—
62	#117×C₅₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₅₉	—	2	2	no	—
64	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
65	#107×C₅₉	C₂×C₄	1	2	yes	—
66	#106×C₅₉	C₂×C₄	1	2	yes	—
67	#103×C₅₉	C₂×C₄	1	4	yes	—
68	#102×C₅₉	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
71	#41×C₅₉	—	2	4	no	—
72	#41×C₅₉	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₅₉	—	2	2	no	—
75	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₉	—	2	2	no	—
77	C₂²×C₄×C₅₉	—	2	4	no	—
78	C₂²×C₄×C₅₉	#41	1	2	yes	—
79	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₅₉	—	2	2	no	—
81	Q₁₆×C₅₉	C₄²	1	4	yes	complemented
82	SD₁₆×C₅₉	C₄²	1	8	yes	complemented
83	D₈×C₅₉	C₄²	1	4	yes	complemented
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₅₉	—	2	4	no	—
90	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
91	C₂²⋊C₄×C₅₉	—	2	2	no	—
92	C₂²⋊C₄×C₅₉	—	2	4	no	—
93	C₄²×C₅₉	—	2	4	no	—
94	C₄²×C₅₉	—	2	2	no	—
95	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₄×Dic₅₉	—	2	8	no	—
99	C₄×Dic₅₉	—	2	2	no	—
100	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
101	#11745664	—	2	4	no	—
102	#11745664	—	2	16	no	—
103	#11745664	C₂²×C₄	1	8	yes	—
104	C₂×C₅₉⋊C₈	—	2	4	no	—
105	C₂×C₅₉⋊C₈	—	2	16	no	—
106	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
108	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
109	C₂×C₅₉⋊C₈	#41	1	4	yes	—
110	C₂³×C₅₉	—	2	2	no	—
111	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
112	Q₈×C₅₉	—	2	2	no	—
113	Q₈×C₅₉	C₂×C₄²	1	4	yes	—
114	D₄×C₅₉	—	2	8	no	—
115	D₄×C₅₉	—	2	2	no	—
116	D₄×C₅₉	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₅₉	—	2	2	no	—
118	C₂×C₄×C₅₉	—	2	8	no	—
119	C₂×C₄×C₅₉	—	2	4	no	—
120	C₂×C₄×C₅₉	—	2	4	no	—
121	C₂×C₄×C₅₉	—	4	8	no	—
122	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
128	C₂×Dic₅₉	—	2	8	no	—
129	C₂×Dic₅₉	—	2	4	no	—
130	C₈×C₅₉	—	2	2	no	—
131	C₈×C₅₉	C₂×C₄²	1	2	yes	—
132	C₈×C₅₉	C₂×C₄²	1	2	yes	—
133	C₅₉⋊C₈	—	2	8	no	—
134	C₅₉⋊C₈	C₄×D₄	1	4	yes	—
135	C₅₉⋊C₈	C₄×D₄	1	4	yes	—
136	C₄×#443	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	—	2	2	no	—
138	C₂²×C₅₉	—	2	4	no	—
139	C₂²×C₅₉	—	4	4	no	—
140	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₅₉	C₄×#41	1	2	yes	—
142	C₄×C₅₉	—	2	2	no	—
143	C₄×C₅₉	—	2	4	no	—
144	C₄×C₅₉	—	4	4	no	—
145	C₄×C₅₉	#443	1	4	yes	complemented
146	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₅₉	C₂×C₄×D₄	1	2	yes	—
148	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
149	Dic₅₉	—	2	8	no	—
150	C₂×#443	—	59	2	no	—
151	C₄×#135	D₅₉	1	1	yes	characteristic
152	C₄×#131	—	59	2	no	—
153	C₄×#108	—	59	1	no	—
154	#1772	—	59	2	no	—
155	#1764	—	59	4	no	—
156	C₄×#104	—	59	2	no	—
157	C₄²×C₈	—	59	1	no	—
158	C₂×C₅₉	—	2	2	no	—
159	C₂×C₅₉	—	4	4	no	—
160	C₂×C₅₉	C₄²×D₄	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#443	1	2	yes	—
162	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
163	C₂×#131	—	59	4	no	—
164	C₄×#41	C₂×D₅₉	1	2	yes	—
165	#443	—	59	16	no	—
166	C₄×Q₁₆	Dic₅₉	1	2	yes	complemented
167	C₄×SD₁₆	Dic₅₉	1	4	yes	complemented
168	C₄×D₈	Dic₅₉	1	2	yes	complemented
169	C₂×#108	—	59	2	no	—
170	#427	C₂×D₅₉	1	1	yes	characteristic
171	C₂×#104	—	59	4	no	—
172	#416	C₂×D₅₉	1	2	yes	—
173	#405	—	59	4	no	—
174	#405	—	118	4	no	—
175	C₄×M₁₆	—	59	2	no	—
176	C₄×M₁₆	—	118	2	no	—
177	C₂×C₄×C₈	—	59	2	no	—
178	C₂×C₄×C₈	—	59	1	no	—
179	C₂×C₄×C₈	—	59	2	no	—
180	C₂×C₄×C₈	—	118	2	no	—
181	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
182	C₄³	—	59	1	no	—
183	#341	—	59	2	no	—
184	#339	—	59	4	no	—
185	C₅₉	C₄×#443	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	C₂²×D₅₉	1	2	yes	—
187	#135	C₄×D₅₉	1	8	yes	—
188	C₂×Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
189	C₂×SD₁₆	C₂×Dic₅₉	1	2	yes	—
190	C₂×D₈	C₂×Dic₅₉	1	1	yes	characteristic
191	#131	—	59	16	no	—
192	#131	—	118	8	no	—
193	C₂×M₁₆	—	59	4	no	—
194	C₂×M₁₆	—	118	4	no	—
195	C₂²×C₈	—	59	2	no	—
196	C₂²×C₈	—	118	4	no	—
197	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
199	C₄×Q₈	C₂×Dic₅₉	1	4	yes	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	C₂×Dic₅₉	1	4	yes	—
202	#117	—	2	2	no	—
203	#117	C₂²×D₅₉	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	59	2	no	—
206	C₂×C₄²	—	118	2	no	—
207	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
208	#108	—	59	8	no	—
209	#107	C₂×Dic₅₉	1	2	yes	—
210	#106	C₂×Dic₅₉	1	2	yes	—
211	#104	—	59	16	no	—
212	#103	C₂×Dic₅₉	1	4	yes	—
213	#102	C₂×Dic₅₉	1	4	yes	—
214	#97	—	118	4	no	—
215	#97	—	118	2	no	—
216	C₄×C₈	—	59	8	no	—
217	C₄×C₈	—	59	2	no	—
218	C₄×C₈	—	59	1	no	—
219	C₄×C₈	—	59	1	no	—
220	C₄×C₈	—	118	4	no	—
221	C₄×C₈	—	118	4	no	—
222	C₄×C₈	—	118	2	no	—
223	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
224	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
225	#41	—	2	4	no	—
226	#41	C₂×C₄×D₅₉	1	8	yes	—
227	C₂×Q₈	C₂²×Dic₅₉	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₅₉	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	118	2	no	—
233	C₂²×C₄	#11745687	1	2	yes	—
234	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
235	Q₁₆	C₄×Dic₅₉	1	4	yes	complemented
236	SD₁₆	C₄×Dic₅₉	1	8	yes	complemented
237	D₈	C₄×Dic₅₉	1	4	yes	complemented
238	M₁₆	—	59	8	no	—
239	M₁₆	—	118	16	no	—
240	M₁₆	—	118	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	59	4	no	—
243	C₂×C₈	—	59	2	no	—
244	C₂×C₈	—	59	2	no	—
245	C₂×C₈	—	59	4	no	—
246	C₂×C₈	—	118	16	no	—
247	C₂×C₈	—	118	4	no	—
248	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
249	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
250	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	59	4	no	—
259	C₄²	—	118	8	no	—
260	C₄²	—	118	2	no	—
261	C₄²	D₄×D₅₉	1	1	yes	characteristic
262	C₄²	D₄×D₅₉	1	1	yes	characteristic
263	C₄²	C₂²×Dic₅₉	1	2	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	C₂×#11745687	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄×Dic₅₉	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄×Dic₅₉	1	4	yes	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	118	8	no	—
277	C₂×C₄	—	118	4	no	—
278	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
279	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₅₉	1	2	yes	—
281	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
283	C₂×C₄	#41×D₅₉	1	2	yes	—
284	C₈	—	2	2	no	—
285	C₈	—	59	4	no	—
286	C₈	—	59	4	no	—
287	C₈	—	118	8	no	—
288	C₈	C₂×C₄×Dic₅₉	1	2	yes	—
289	C₈	C₂×C₄×Dic₅₉	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
294	C₂²	C₄×#11745687	1	2	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	—	118	8	no	—
299	C₄	#16837519	1	4	yes	direct product factor
300	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
301	C₄	C₄×D₄×D₅₉	1	2	yes	—
302	C₄	#41×Dic₅₉	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄×D₄×Dic₅₉	1	1	yes	characteristic
306	C₂	C₂×#16837519	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₅₉
commutator	C₄×C₅₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₄×#135×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂×#135×C₅₉	C₂²
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₄×#135	D₅₉
hypocenter	C₅₉	C₄×#443
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₄×#14273734	C₂	semidirect
3. C₄×#14273663	C₂	semidirect
4. #21423977	C₂	semidirect
5. #21423936	C₂	semidirect
6. C₄×#14273654	C₂	semidirect
7. C₄×C₈×Dic₅₉	C₂	semidirect
10. #16837519	C₄	direct (N × H)
26. C₄×Q₁₆×C₅₉	C₄	semidirect
27. C₄×SD₁₆×C₅₉	C₄	semidirect
28. C₄×D₈×C₅₉	C₄	semidirect
33. #14273734	C₂×C₄	semidirect
40. #14273663	C₂×C₄	semidirect
41. #14273654	C₂×C₄	semidirect
50. C₈×Dic₅₉	C₂×C₄	semidirect
81. Q₁₆×C₅₉	C₄²	semidirect
82. SD₁₆×C₅₉	C₄²	semidirect
83. D₈×C₅₉	C₄²	semidirect
145. C₄×C₅₉	#443	semidirect
166. C₄×Q₁₆	Dic₅₉	semidirect
167. C₄×SD₁₆	Dic₅₉	semidirect
168. C₄×D₈	Dic₅₉	semidirect
185. C₅₉	C₄×#443	semidirect
235. Q₁₆	C₄×Dic₅₉	semidirect
236. SD₁₆	C₄×Dic₅₉	semidirect
237. D₈	C₄×Dic₅₉	semidirect
299. C₄	#16837519	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837519	2
C₄×#14273734	2
C₄×#14273663	2
#21423977	2
#21423936	2
C₄×#14273654	2
C₄×C₈×Dic₅₉	2
C₄×#135×C₅₉	2
C₄×#443	59

Subgroups — C4×#16837519 #28587245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837519 #28587245