Subgroups — C₄×#16837835 #28741971

Order 15104.

690 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837835	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273750	C₂	1	2	yes	maximal, complemented
3	Q₃₂×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14273670	C₂	1	2	yes	maximal, complemented
5	#21425945	C₂	1	2	yes	maximal, complemented
6	C₄×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425942	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425877	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425802	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14273750	C₂²	1	2	yes	—
13	C₄×#11745689	—	2	2	no	—
14	#16837835	C₄	1	8	yes	direct product factor
15	C₂×#14273670	C₂²	1	2	yes	—
16	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
19	Q₁₆×Dic₅₉	C₂²	1	2	yes	—
20	C₄×#11745671	—	2	2	no	—
21	#16837487	—	2	2	no	—
22	C₄×D₄₇₂	C₂²	1	2	yes	—
23	#16837484	C₂²	1	2	yes	—
24	#16837439	C₂²	1	2	yes	—
25	#16837397	—	2	2	no	—
26	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
27	#16837229	C₂²	1	2	yes	—
28	#16837228	C₂²	1	1	yes	characteristic
29	#16837206	C₂²	1	2	yes	—
30	#16837204	C₂²	1	2	yes	—
31	#16837202	C₂²	1	2	yes	—
32	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
37	C₅₉×#366	C₂²	1	1	yes	characteristic
38	C₅₉×#358	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₂×#11745689	—	2	2	no	—
41	#14273750	—	2	4	no	—
42	#14273750	C₂×C₄	1	8	yes	complemented
43	C₂×#11745671	—	2	2	no	—
44	C₂×D₄₇₂	C₂³	1	2	yes	—
45	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅₉	—	2	2	no	—
48	C₄×D₂₃₆	—	2	2	no	—
49	C₄×D₂₃₆	—	4	2	no	—
50	#14273689	—	2	2	no	—
51	#14273689	—	4	2	no	—
52	#14273685	—	2	2	no	—
53	#14273685	—	4	2	no	—
54	C₄²×D₅₉	—	4	2	no	—
55	C₄²×D₅₉	D₄	1	1	yes	characteristic
56	#14273670	C₂×C₄	1	8	yes	complemented
57	D₉₄₄	C₂×C₄	1	4	yes	complemented
58	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₅₉	—	2	2	no	—
61	C₁₆×D₅₉	C₂×C₄	1	4	yes	complemented
62	#14273661	C₂³	1	2	yes	—
63	#14273658	—	2	2	no	—
64	#14273651	—	2	2	no	—
65	#14273649	—	2	2	no	—
66	#14273647	—	2	2	no	—
67	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
68	C₈×Dic₅₉	D₄	1	1	yes	characteristic
69	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₉	—	2	2	no	—
72	Q₃₂×C₅₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
75	#107×C₅₉	C₂³	1	2	yes	—
76	#103×C₅₉	—	2	2	no	—
77	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₅₉	—	2	2	no	—
79	#11745689	—	2	8	no	—
80	#11745689	—	4	4	no	—
81	C₂×D₂₃₆	—	2	2	no	—
82	C₂×D₂₃₆	—	4	2	no	—
83	C₂×C₄×D₅₉	—	4	2	no	—
84	C₂×C₄×D₅₉	—	4	2	no	—
85	C₂×C₄×D₅₉	—	4	2	no	—
86	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
87	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
88	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₅₉	—	2	2	no	—
90	Q₁₆×C₅₉	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₉	—	2	2	no	—
94	C₄⋊C₄×C₅₉	—	4	2	no	—
95	C₄²×C₅₉	—	4	2	no	—
96	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
97	#11745671	—	2	8	no	—
98	(C₂×D₅₉)⋊C₄	—	4	2	no	—
99	(C₂×D₅₉)⋊C₄	—	4	2	no	—
100	(C₂×D₅₉)⋊C₄	—	4	2	no	—
101	C₄⋊Dic₅₉	—	2	2	no	—
102	C₄⋊Dic₅₉	—	4	2	no	—
103	C₄×Dic₅₉	—	4	2	no	—
104	C₄×Dic₅₉	—	4	2	no	—
105	C₄×Dic₅₉	D₈	1	1	yes	characteristic
106	C₄×Dic₅₉	D₈	1	1	yes	characteristic
107	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
110	D₄₇₂	—	2	2	no	—
111	D₄₇₂	C₂²×C₄	1	4	yes	—
112	C₈×D₅₉	—	2	2	no	—
113	C₈×D₅₉	C₂²×C₄	1	2	yes	—
114	C₈×D₅₉	#41	1	2	yes	—
115	C₁₆×C₅₉	—	2	1	no	—
116	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
117	C₅₉⋊C₁₆	—	2	1	no	—
118	C₅₉⋊C₁₆	C₂²×C₄	1	2	yes	—
119	C₂²×D₅₉	—	4	2	no	—
120	C₂²×D₅₉	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₅₉	—	2	4	no	—
122	Q₈×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	4	2	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
132	D₂₃₆	—	2	4	no	—
133	D₂₃₆	—	4	8	no	—
134	D₂₃₆	—	4	2	no	—
135	C₄×D₅₉	—	2	2	no	—
136	C₄×D₅₉	—	2	2	no	—
137	C₄×D₅₉	—	4	8	no	—
138	C₄×D₅₉	—	8	4	no	—
139	C₄×D₅₉	—	8	4	no	—
140	C₄×D₅₉	C₄×D₄	1	2	yes	—
141	C₄×D₅₉	#135	1	2	yes	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
145	C₅₉⋊C₈	—	2	1	no	—
146	C₅₉⋊C₈	C₄×D₄	1	2	yes	—
147	C₄×#508	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₉	—	2	1	no	—
150	C₂×D₅₉	—	2	1	no	—
151	C₂×D₅₉	—	4	4	no	—
152	C₂×D₅₉	—	8	2	no	—
153	C₂×D₅₉	—	8	2	no	—
154	C₂×D₅₉	C₄×D₈	1	2	yes	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	4	4	no	—
157	C₄×C₅₉	—	8	2	no	—
158	C₄×C₅₉	#508	1	2	yes	complemented
159	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	8	2	no	—
163	Dic₅₉	C₄×D₈	1	2	yes	—
164	Dic₅₉	#508	1	2	yes	complemented
165	C₂×#508	—	59	1	no	—
166	C₄×#135	—	59	2	no	—
167	C₄×Q₃₂	—	59	1	no	—
168	C₄×Q₃₂	D₅₉	1	1	yes	characteristic, complemented
169	C₄×SD₃₂	—	59	2	no	—
170	C₄×SD₃₂	—	59	2	no	—
171	C₄×D₁₆	—	59	1	no	—
172	C₄×D₁₆	—	59	1	no	—
173	#2158	—	59	1	no	—
174	#2138	—	59	2	no	—
175	C₂×C₄×C₁₆	—	59	1	no	—
176	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
177	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
178	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
179	D₅₉	—	2	2	no	—
180	D₅₉	—	8	4	no	—
181	C₂×#135	—	59	2	no	—
182	C₄×#41	—	118	2	no	—
183	#508	—	59	8	no	—
184	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
185	C₂×SD₃₂	—	59	2	no	—
186	C₂×D₁₆	—	59	1	no	—
187	C₂²×C₁₆	—	59	1	no	—
188	C₂²×C₁₆	—	59	1	no	—
189	C₄×Q₁₆	—	59	2	no	—
190	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
191	C₄×SD₁₆	—	118	2	no	—
192	C₄×SD₁₆	—	118	2	no	—
193	C₄×D₈	—	59	2	no	—
194	C₄×D₈	—	59	2	no	—
195	#427	—	59	2	no	—
196	#416	—	118	2	no	—
197	C₂×C₄×C₈	—	59	1	no	—
198	#367	—	59	2	no	—
199	#366	—	59	1	no	—
200	#366	C₂×D₅₉	1	1	yes	characteristic
201	#358	—	59	2	no	—
202	#358	C₂×D₅₉	1	2	yes	—
203	#357	—	59	2	no	—
204	#357	—	59	2	no	—
205	C₄×C₁₆	—	59	1	no	—
206	C₄×C₁₆	—	59	1	no	—
207	C₄×C₁₆	—	59	1	no	—
208	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
209	C₅₉	C₄×#508	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	118	2	no	—
211	#135	—	59	8	no	—
212	#135	—	118	4	no	—
213	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
214	C₂×SD₁₆	—	118	2	no	—
215	C₂×D₈	—	59	2	no	—
216	C₂²×C₈	—	59	1	no	—
217	C₂²×C₈	—	59	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	118	2	no	—
220	C₄×D₄	—	118	2	no	—
221	C₄×D₄	—	118	2	no	—
222	C₄×D₄	—	236	2	no	—
223	C₄×D₄	—	236	2	no	—
224	#117	—	118	2	no	—
225	#117	—	236	2	no	—
226	C₂×C₄²	—	59	1	no	—
227	C₂×C₄²	—	236	2	no	—
228	Q₃₂	C₄×D₅₉	1	4	yes	complemented
229	SD₃₂	—	59	8	no	—
230	D₁₆	—	59	4	no	—
231	C₂×C₁₆	—	59	1	no	—
232	C₂×C₁₆	—	59	4	no	—
233	C₂×C₁₆	—	59	1	no	—
234	C₂×C₁₆	—	118	2	no	—
235	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
237	#107	—	59	2	no	—
238	#107	C₂²×D₅₉	1	2	yes	—
239	#106	—	118	2	no	—
240	#103	—	2	2	no	—
241	#103	—	118	2	no	—
242	#102	—	118	2	no	—
243	#102	—	118	2	no	—
244	C₄×C₈	—	59	1	no	—
245	C₄×C₈	—	59	1	no	—
246	C₄×C₈	—	59	1	no	—
247	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
248	#41	—	118	8	no	—
249	#41	—	236	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×D₄	—	118	2	no	—
252	C₂×D₄	—	236	2	no	—
253	C₂²×C₄	—	59	1	no	—
254	C₂²×C₄	—	59	1	no	—
255	C₂²×C₄	—	59	1	no	—
256	C₂²×C₄	—	236	2	no	—
257	C₂²×C₄	—	236	2	no	—
258	C₂²×C₄	—	236	2	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	C₂×C₄×D₅₉	1	4	yes	—
261	SD₁₆	—	118	8	no	—
262	D₈	—	59	4	no	—
263	D₈	—	118	2	no	—
264	C₂×C₈	—	59	1	no	—
265	C₂×C₈	—	59	2	no	—
266	C₂×C₈	—	59	2	no	—
267	C₂×C₈	—	59	1	no	—
268	C₂×C₈	—	118	2	no	—
269	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	118	2	no	—
274	C₄⋊C₄	—	236	2	no	—
275	C₂²⋊C₄	—	236	2	no	—
276	C₂²⋊C₄	—	236	2	no	—
277	C₂²⋊C₄	—	236	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	59	1	no	—
280	C₄²	—	59	1	no	—
281	C₄²	—	59	1	no	—
282	C₄²	—	236	2	no	—
283	C₄²	—	236	2	no	—
284	C₄²	D₄×D₅₉	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	59	2	no	—
287	C₁₆	—	118	1	no	—
288	C₁₆	C₂×C₄×D₅₉	1	2	yes	—
289	C₂³	—	59	1	no	—
290	C₂³	—	236	2	no	—
291	Q₈	—	2	4	no	—
292	Q₈	—	4	2	no	—
293	D₄	—	118	4	no	—
294	D₄	—	236	8	no	—
295	D₄	—	236	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	59	1	no	—
299	C₂×C₄	—	59	1	no	—
300	C₂×C₄	—	59	2	no	—
301	C₂×C₄	—	59	2	no	—
302	C₂×C₄	—	59	1	no	—
303	C₂×C₄	—	118	2	no	—
304	C₂×C₄	—	118	2	no	—
305	C₂×C₄	—	472	4	no	—
306	C₂×C₄	—	472	4	no	—
307	C₂×C₄	—	236	8	no	—
308	C₂×C₄	—	236	2	no	—
309	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
310	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	59	2	no	—
314	C₈	—	118	1	no	—
315	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
316	C₈	#41×D₅₉	1	1	yes	characteristic
317	C₂²	—	59	2	no	—
318	C₂²	—	118	1	no	—
319	C₂²	—	118	1	no	—
320	C₂²	—	472	2	no	—
321	C₂²	—	472	2	no	—
322	C₂²	—	236	4	no	—
323	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	4	4	no	—
326	C₄	—	8	2	no	—
327	C₄	—	59	2	no	—
328	C₄	—	59	2	no	—
329	C₄	—	118	1	no	—
330	C₄	—	472	2	no	—
331	C₄	#16837835	1	2	yes	direct product factor
332	C₄	C₄×D₄×D₅₉	1	1	yes	characteristic
333	C₄	#135×D₅₉	1	1	yes	characteristic
334	C₂	—	118	2	no	—
335	C₂	—	472	4	no	—
336	C₂	C₄×D₈×D₅₉	1	1	yes	characteristic
337	C₂	#508×D₅₉	1	1	yes	characteristic
338	C₂	C₂×#16837835	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₄×Q₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×#16837835	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₄×Q₃₂	D₅₉
hypocenter	C₅₉	C₄×#508
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₄×#14273750	C₂	semidirect
3. Q₃₂×Dic₅₉	C₂	semidirect
4. C₄×#14273670	C₂	semidirect
5. #21425945	C₂	semidirect
6. C₄×D₉₄₄	C₂	semidirect
7. #21425942	C₂	semidirect
8. #21425877	C₂	semidirect
9. #21425802	C₂	semidirect
10. C₄×C₁₆×D₅₉	C₂	semidirect
11. C₄×Q₃₂×C₅₉	C₂	semidirect
14. #16837835	C₄	direct (N × H)
42. #14273750	C₂×C₄	semidirect
56. #14273670	C₂×C₄	semidirect
57. D₉₄₄	C₂×C₄	semidirect
61. C₁₆×D₅₉	C₂×C₄	semidirect
72. Q₃₂×C₅₉	C₂×C₄	semidirect
158. C₄×C₅₉	#508	semidirect
164. Dic₅₉	#508	semidirect
168. C₄×Q₃₂	D₅₉	semidirect
209. C₅₉	C₄×#508	semidirect
228. Q₃₂	C₄×D₅₉	semidirect
331. C₄	#16837835	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×#16837835	2
C₄×#14273750	2
Q₃₂×Dic₅₉	2
C₄×#14273670	2
#21425945	2
C₄×D₉₄₄	2
#21425942	2
#21425877	2
#21425802	2
C₄×C₁₆×D₅₉	2
C₄×Q₃₂×C₅₉	2
C₄×#508	59

Subgroups — C4×#16837835 #28741971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837835 #28741971