Subgroups — C₄×#16837143 #28479608

Order 15104.

532 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273719	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273638	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837143	C₂	1	2	yes	maximal
4	#21421978	C₂	1	2	yes	maximal
5	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273719	C₂²	1	1	yes	characteristic
7	C₂×#14273638	—	2	1	no	—
8	C₂×#14273638	C₂²	1	2	yes	—
9	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
10	#16837300	C₄	1	1	yes	characteristic
11	#16837300	C₄	1	1	yes	characteristic
12	#16837286	—	2	1	no	—
13	#16837286	—	2	1	no	—
14	#16837268	C₂²	1	1	yes	characteristic
15	C₄×Dic₅₉⋊C₄	—	2	1	no	—
16	C₄×Dic₅₉⋊C₄	—	2	1	no	—
17	C₄×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
18	C₄×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
19	#16837143	C₄	1	8	yes	direct product factor
20	C₂×#97×C₅₉	—	2	1	no	—
21	C₂×#97×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	—	2	1	no	—
23	C₄³×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	—	2	2	no	—
25	#14273719	—	2	2	no	—
26	#14273719	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₅₉⋊C₄	—	2	1	no	—
32	C₂×Dic₅₉⋊C₄	—	2	1	no	—
33	C₂×Dic₅₉⋊C₄	—	2	1	no	—
34	C₂×Dic₅₉⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₅₉	—	2	1	no	—
36	C₂×C₄×Dic₅₉	—	2	1	no	—
37	#14273638	—	2	4	no	—
38	#14273638	—	2	2	no	—
39	#14273638	C₂×C₄	1	4	yes	—
40	C₄×C₅₉⋊C₈	—	2	4	no	—
41	C₅₉⋊#95	—	2	1	no	—
42	C₅₉⋊#95	—	2	1	no	—
43	C₅₉⋊#95	—	2	1	no	—
44	C₅₉⋊#95	—	2	1	no	—
45	C₅₉⋊#95	—	2	1	no	—
46	C₅₉⋊#95	—	2	1	no	—
47	C₂²×C₈×C₅₉	—	2	2	no	—
48	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	#97×C₅₉	—	2	4	no	—
56	#97×C₅₉	—	2	2	no	—
57	#97×C₅₉	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₅₉	—	2	4	no	—
59	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂²×Dic₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	4	no	—
71	C₂×C₈×C₅₉	—	2	4	no	—
72	C₂×C₈×C₅₉	—	2	4	no	—
73	C₄²×C₅₉	—	2	2	no	—
74	C₄²×C₅₉	—	2	2	no	—
75	C₄²×C₅₉	—	2	2	no	—
76	C₄²×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	—	2	2	no	—
78	C₄²×C₅₉	—	4	1	no	—
79	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₉	#41	1	2	yes	—
81	C₄⋊Dic₅₉	—	2	1	no	—
82	C₄⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	—	4	1	no	—
84	C₄⋊Dic₅₉	—	4	1	no	—
85	C₄⋊Dic₅₉	C₄²	1	2	yes	—
86	C₄⋊Dic₅₉	C₄²	1	2	yes	—
87	Dic₅₉⋊C₄	—	2	4	no	—
88	Dic₅₉⋊C₄	—	2	4	no	—
89	Dic₅₉⋊C₄	—	4	2	no	—
90	Dic₅₉⋊C₄	—	4	2	no	—
91	C₄×Dic₅₉	—	4	2	no	—
92	C₄×Dic₅₉	—	4	2	no	—
93	C₂×C₅₉⋊C₈	—	2	4	no	—
94	C₂×C₅₉⋊C₈	—	2	4	no	—
95	C₂×C₅₉⋊C₈	—	2	4	no	—
96	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	#101	1	2	yes	—
111	C₂×C₄×C₅₉	#104	1	2	yes	—
112	C₂×C₄×C₅₉	#104	1	2	yes	—
113	C₂×C₄×C₅₉	#104	1	2	yes	—
114	C₂×C₄×C₅₉	#104	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
119	C₂×Dic₅₉	—	2	2	no	—
120	C₂×Dic₅₉	—	2	2	no	—
121	C₂×Dic₅₉	—	4	1	no	—
122	C₂×Dic₅₉	—	4	1	no	—
123	C₂×Dic₅₉	—	4	1	no	—
124	C₂×Dic₅₉	—	4	1	no	—
125	C₂×Dic₅₉	—	4	1	no	—
126	C₂×Dic₅₉	—	4	1	no	—
127	C₂×Dic₅₉	—	4	1	no	—
128	C₂×Dic₅₉	—	4	1	no	—
129	C₂×Dic₅₉	—	4	1	no	—
130	C₂×Dic₅₉	—	4	1	no	—
131	C₈×C₅₉	—	2	8	no	—
132	C₅₉⋊C₈	—	2	8	no	—
133	C₄×#330	—	59	1	no	maximal, Sylow
134	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₉	C₂×#101	1	1	yes	characteristic
136	C₂²×C₅₉	#413	1	1	yes	characteristic
137	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
138	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
139	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
140	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	2	no	—
144	C₄×C₅₉	—	2	2	no	—
145	C₄×C₅₉	—	2	2	no	—
146	C₄×C₅₉	—	2	2	no	—
147	C₄×C₅₉	—	4	1	no	—
148	C₄×C₅₉	#330	1	4	yes	complemented
149	Dic₅₉	—	4	2	no	—
150	Dic₅₉	—	4	2	no	—
151	Dic₅₉	—	8	1	no	—
152	Dic₅₉	—	8	1	no	—
153	C₄×#125	—	59	1	no	—
154	C₄×#97	—	59	1	no	—
155	C₄×#97	D₅₉	1	1	yes	characteristic
156	C₂×#330	—	59	2	no	—
157	#1295	—	59	2	no	—
158	C₂×C₅₉	C₂×#330	1	1	yes	characteristic
159	C₂×C₅₉	#1524	1	1	yes	characteristic
160	C₂×C₅₉	#1524	1	1	yes	characteristic
161	C₂×C₅₉	#1526	1	1	yes	characteristic
162	C₂×C₅₉	C₄×#101	1	1	yes	characteristic
163	C₂×C₅₉	C₄×#104	1	1	yes	characteristic
164	C₂×C₅₉	C₄×#104	1	1	yes	characteristic
165	C₂×#125	—	59	1	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×#97	—	59	2	no	—
168	C₂×#97	—	118	1	no	—
169	C₂×#97	C₂×D₅₉	1	2	yes	—
170	C₂×C₄×C₈	—	2	1	no	—
171	C₂×C₄×C₈	—	118	1	no	—
172	#389	—	59	1	no	—
173	#389	—	59	1	no	—
174	#387	—	118	1	no	—
175	#387	—	118	1	no	—
176	#382	—	59	1	no	—
177	C₄×C₄⋊C₄	—	59	1	no	—
178	C₄×C₄⋊C₄	—	59	1	no	—
179	C₄×C₄⋊C₄	—	118	1	no	—
180	C₄×C₄⋊C₄	—	118	1	no	—
181	C₄³	C₂×D₅₉	1	1	yes	characteristic
182	#330	—	59	8	no	—
183	C₅₉	C₄×#330	1	1	yes	characteristic, complemented, Sylow
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	118	2	no	—
186	#125	—	59	4	no	—
187	#125	—	118	2	no	—
188	C₂×C₄⋊C₄	—	59	1	no	—
189	C₂×C₄⋊C₄	—	59	1	no	—
190	C₂×C₄⋊C₄	—	59	1	no	—
191	C₂×C₄⋊C₄	—	59	1	no	—
192	C₂×C₄⋊C₄	—	118	1	no	—
193	C₂×C₄⋊C₄	—	118	1	no	—
194	C₂×C₄⋊C₄	—	118	1	no	—
195	C₂×C₄⋊C₄	—	118	1	no	—
196	C₂×C₄²	—	118	1	no	—
197	C₂×C₄²	—	118	1	no	—
198	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
200	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
201	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
205	#97	—	2	4	no	—
206	#97	—	2	2	no	—
207	#97	—	59	4	no	—
208	#97	—	118	4	no	—
209	#97	—	118	2	no	—
210	#97	C₄×D₅₉	1	4	yes	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	—	118	4	no	—
213	#95	—	118	1	no	—
214	#95	—	118	1	no	—
215	#95	—	118	1	no	—
216	#95	—	118	1	no	—
217	#95	—	118	1	no	—
218	#95	—	118	1	no	—
219	C₂²×C₄	—	118	1	no	—
220	C₂²×C₄	—	118	1	no	—
221	C₂²×C₄	—	118	1	no	—
222	C₂²×C₄	—	118	1	no	—
223	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	118	4	no	—
234	C₂×C₈	—	118	4	no	—
235	C₂×C₈	—	118	4	no	—
236	C₄⋊C₄	—	59	2	no	—
237	C₄⋊C₄	—	59	2	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	4	no	—
241	C₄⋊C₄	—	118	4	no	—
242	C₄⋊C₄	—	236	1	no	—
243	C₄⋊C₄	—	236	1	no	—
244	C₄⋊C₄	—	236	2	no	—
245	C₄⋊C₄	—	236	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	236	2	no	—
253	C₄²	—	236	2	no	—
254	C₄²	C₂×C₄×D₅₉	1	2	yes	—
255	C₄²	#11745685	1	2	yes	—
256	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	118	2	no	—
271	C₂×C₄	—	118	2	no	—
272	C₂×C₄	—	236	1	no	—
273	C₂×C₄	—	236	1	no	—
274	C₂×C₄	—	236	1	no	—
275	C₂×C₄	—	236	1	no	—
276	C₂×C₄	—	236	1	no	—
277	C₂×C₄	—	236	1	no	—
278	C₂×C₄	—	236	1	no	—
279	C₂×C₄	—	236	1	no	—
280	C₂×C₄	—	236	1	no	—
281	C₂×C₄	—	236	1	no	—
282	C₂×C₄	#14273645	1	2	yes	—
283	C₂×C₄	#14273653	1	2	yes	—
284	C₂×C₄	#14273653	1	2	yes	—
285	C₂×C₄	#14273655	1	2	yes	—
286	C₂×C₄	#14273655	1	2	yes	—
287	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
288	C₂×C₄	#14273684	1	1	yes	characteristic
289	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
290	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
291	C₈	—	2	8	no	—
292	C₈	—	118	8	no	—
293	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
294	C₂²	C₂×#14273645	1	1	yes	characteristic
295	C₂²	#16837384	1	1	yes	characteristic
296	C₂²	C₂×#14273653	1	1	yes	characteristic
297	C₂²	C₂×#14273653	1	1	yes	characteristic
298	C₂²	C₂×#14273655	1	1	yes	characteristic
299	C₂²	C₂×#14273655	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	472	1	no	—
308	C₄	—	472	1	no	—
309	C₄	—	236	2	no	—
310	C₄	—	236	2	no	—
311	C₄	#16837143	1	4	yes	direct product factor
312	C₂	C₂×#16837143	1	1	yes	characteristic
313	C₂	#21422682	1	1	yes	characteristic
314	C₂	#21422685	1	1	yes	characteristic
315	C₂	#21422690	1	1	yes	characteristic
316	C₂	C₄×#14273645	1	1	yes	characteristic
317	C₂	C₄×#14273655	1	1	yes	characteristic
318	C₂	C₄×#14273653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×#97×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×#97	D₅₉
hypocenter	C₅₉	C₄×#330
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
19. #16837143	C₄	direct (N × H)
148. C₄×C₅₉	#330	semidirect
183. C₅₉	C₄×#330	semidirect
311. C₄	#16837143	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14273719	2
C₄×#14273638	2
C₂×#16837143	2
#21421978	2
C₄×#97×C₅₉	2
C₄×#330	59

Subgroups — C4×#16837143 #28479608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837143 #28479608