Subgroups — C₅₉×#18404 #28430513

Order 15104.

910 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#3209	C₂	1	2	yes	maximal
3	C₅₉×#2567	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1605	C₂	1	8	yes	maximal
5	C₅₉×#1503	C₂	1	2	yes	maximal
6	C₅₉×#1461	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₅₉	—	2	2	no	—
8	C₅₉×#557	—	2	2	no	—
9	C₅₉×#557	—	2	4	no	—
10	C₅₉×#544	—	2	1	no	—
11	C₅₉×#544	C₂²	1	4	yes	—
12	C₅₉×#533	—	2	2	no	—
13	C₅₉×#533	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#519	—	2	1	no	—
16	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#494	—	2	2	no	—
18	C₅₉×#487	—	2	2	no	—
19	C₅₉×#462	—	2	4	no	—
20	C₅₉×#451	C₂²	1	8	yes	—
21	C₅₉×#431	C₂²	1	2	yes	—
22	C₅₉×#395	C₂²	1	4	yes	—
23	C₅₉×#391	—	2	1	no	—
24	C₅₉×#389	—	2	2	no	—
25	C₅₉×#384	—	2	1	no	—
26	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#328	C₂²	1	8	yes	—
28	C₅₉×#324	C₂²	1	4	yes	—
29	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₅₉	—	2	8	no	—
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₄⋊Q₈×C₅₉	—	2	2	no	—
34	C₄⋊Q₈×C₅₉	—	2	4	no	—
35	C₄⋊Q₈×C₅₉	—	2	8	no	—
36	C₄⋊Q₈×C₅₉	—	2	2	no	—
37	C₄⋊Q₈×C₅₉	—	2	2	no	—
38	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
39	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
40	#125×C₅₉	—	2	4	no	—
41	#125×C₅₉	—	2	4	no	—
42	#125×C₅₉	—	2	8	no	—
43	#125×C₅₉	—	2	2	no	—
44	C₂²⋊Q₈×C₅₉	—	2	2	no	—
45	C₂²⋊Q₈×C₅₉	—	2	4	no	—
46	C₂²⋊Q₈×C₅₉	C₂³	1	4	yes	—
47	C₄×Q₈×C₅₉	—	2	4	no	—
48	C₄×Q₈×C₅₉	—	2	4	no	—
49	C₄×Q₈×C₅₉	—	2	4	no	—
50	C₄×Q₈×C₅₉	—	2	4	no	—
51	C₄×Q₈×C₅₉	—	2	2	no	—
52	C₄×Q₈×C₅₉	—	2	2	no	—
53	C₄×Q₈×C₅₉	—	2	2	no	—
54	C₄×Q₈×C₅₉	D₄	1	4	yes	—
55	#117×C₅₉	—	2	4	no	—
56	#117×C₅₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
61	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₅₉	—	2	1	no	—
64	C₂×C₄²×C₅₉	D₄	1	2	yes	—
65	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₉	—	2	2	no	—
67	#103×C₅₉	—	2	8	no	—
68	C₂²⋊C₈×C₅₉	C₂³	1	4	yes	—
69	C₄×C₈×C₅₉	—	2	2	no	—
70	#95×C₅₉	—	2	4	no	—
71	#95×C₅₉	C₂³	1	2	yes	—
72	C₂×Q₈×C₅₉	—	2	4	no	—
73	C₂×Q₈×C₅₉	—	2	2	no	—
74	C₂×Q₈×C₅₉	—	2	2	no	—
75	C₂×Q₈×C₅₉	—	2	2	no	—
76	C₂×Q₈×C₅₉	—	2	4	no	—
77	C₂×Q₈×C₅₉	—	8	2	no	—
78	C₂×Q₈×C₅₉	Q₁₆	1	4	yes	complemented
79	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₅₉	—	2	2	no	—
81	C₂²×C₄×C₅₉	—	2	2	no	—
82	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
87	Q₁₆×C₅₉	—	8	8	no	—
88	C₂×C₈×C₅₉	—	2	4	no	—
89	C₄⋊C₄×C₅₉	—	2	4	no	—
90	C₄⋊C₄×C₅₉	—	2	4	no	—
91	C₄⋊C₄×C₅₉	—	2	4	no	—
92	C₄⋊C₄×C₅₉	—	2	4	no	—
93	C₄⋊C₄×C₅₉	—	2	8	no	—
94	C₄⋊C₄×C₅₉	—	2	2	no	—
95	C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₄⋊C₄×C₅₉	—	2	2	no	—
97	C₄⋊C₄×C₅₉	—	2	2	no	—
98	C₄⋊C₄×C₅₉	—	2	4	no	—
99	C₄⋊C₄×C₅₉	—	2	4	no	—
100	C₄⋊C₄×C₅₉	—	2	4	no	—
101	C₄⋊C₄×C₅₉	—	4	1	no	—
102	C₄⋊C₄×C₅₉	—	4	2	no	—
103	C₄⋊C₄×C₅₉	—	4	1	no	—
104	C₄⋊C₄×C₅₉	—	8	4	no	—
105	C₄⋊C₄×C₅₉	C₂×D₄	1	4	yes	—
106	C₂²⋊C₄×C₅₉	—	2	4	no	—
107	C₄²×C₅₉	—	2	8	no	—
108	C₄²×C₅₉	—	2	2	no	—
109	C₄²×C₅₉	—	2	2	no	—
110	C₄²×C₅₉	—	2	4	no	—
111	C₄²×C₅₉	—	4	2	no	—
112	C₄²×C₅₉	C₂×D₄	1	2	yes	—
113	C₄²×C₅₉	C₂×D₄	1	2	yes	—
114	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₅₉	—	2	8	no	—
116	Q₈×C₅₉	—	4	4	no	—
117	Q₈×C₅₉	—	4	8	no	—
118	Q₈×C₅₉	—	4	4	no	—
119	Q₈×C₅₉	—	8	4	no	—
120	C₂×C₄×C₅₉	—	2	2	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	2	2	no	—
123	C₂×C₄×C₅₉	—	2	4	no	—
124	C₂×C₄×C₅₉	—	2	4	no	—
125	C₂×C₄×C₅₉	—	2	4	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	4	1	no	—
128	C₂×C₄×C₅₉	—	4	2	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	4	2	no	—
131	C₂×C₄×C₅₉	—	4	4	no	—
132	C₂×C₄×C₅₉	—	4	2	no	—
133	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₅₉	#120	1	2	yes	—
135	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
137	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₅₉	#137	1	2	yes	—
139	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
140	C₈×C₅₉	—	4	4	no	—
141	#18404	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
145	C₂²×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
146	C₂²×C₅₉	C₂×#137	1	1	yes	characteristic
147	C₄×C₅₉	—	2	4	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	2	no	—
150	C₄×C₅₉	—	4	4	no	—
151	C₄×C₅₉	—	4	4	no	—
152	C₄×C₅₉	—	4	2	no	—
153	C₄×C₅₉	—	4	4	no	—
154	C₄×C₅₉	—	8	2	no	—
155	C₄×C₅₉	#451	1	4	yes	—
156	C₄×C₅₉	#456	1	2	yes	—
157	#3502	C₂×C₅₉	1	1	yes	characteristic
158	#3209	C₂×C₅₉	1	2	yes	—
159	#2567	C₂×C₅₉	1	1	yes	characteristic
160	#1605	C₂×C₅₉	1	8	yes	—
161	#1503	C₂×C₅₉	1	2	yes	—
162	#1461	C₂×C₅₉	1	1	yes	characteristic
163	C₂×C₅₉	—	2	1	no	—
164	C₂×C₅₉	—	2	1	no	—
165	C₂×C₅₉	C₂×#451	1	1	yes	characteristic
166	C₂×C₅₉	C₂×#456	1	1	yes	characteristic
167	C₂×C₅₉	#3027	1	1	yes	characteristic
168	Q₈²	—	2	2	no	—
169	#557	—	2	2	no	—
170	#557	—	2	4	no	—
171	#544	—	2	1	no	—
172	#544	C₂²×C₅₉	1	4	yes	—
173	#533	—	2	2	no	—
174	#533	C₂²×C₅₉	1	2	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
176	#519	—	2	1	no	—
177	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
178	#494	—	2	2	no	—
179	#487	—	2	2	no	—
180	#462	—	2	4	no	—
181	#451	C₂²×C₅₉	1	8	yes	—
182	#431	C₂²×C₅₉	1	2	yes	—
183	#395	C₂²×C₅₉	1	4	yes	—
184	#391	—	2	1	no	—
185	#389	—	2	2	no	—
186	#384	—	2	1	no	—
187	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
188	#328	C₂²×C₅₉	1	8	yes	—
189	#324	C₂²×C₅₉	1	4	yes	—
190	C₅₉	#18404	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×Q₈	D₄×C₅₉	1	1	yes	characteristic
192	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
193	C₂×Q₁₆	—	2	8	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	—	2	8	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
201	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
202	#125	—	2	4	no	—
203	#125	—	2	4	no	—
204	#125	—	2	8	no	—
205	#125	—	2	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	4	no	—
208	C₂²⋊Q₈	C₂³×C₅₉	1	4	yes	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	D₄×C₅₉	1	4	yes	—
217	#117	—	2	4	no	—
218	#117	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	D₄×C₅₉	1	2	yes	—
227	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	#103	—	2	8	no	—
230	C₂²⋊C₈	C₂³×C₅₉	1	4	yes	—
231	C₄×C₈	—	2	2	no	—
232	#95	—	2	4	no	—
233	#95	C₂³×C₅₉	1	2	yes	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	8	2	no	—
240	C₂×Q₈	Q₁₆×C₅₉	1	4	yes	complemented
241	C₂×Q₈	C₂×D₄×C₅₉	1	4	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
249	Q₁₆	—	8	8	no	—
250	C₂×C₈	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	8	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	8	4	no	—
267	C₄⋊C₄	C₂×D₄×C₅₉	1	4	yes	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₄²	—	2	8	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	4	no	—
273	C₄²	—	4	2	no	—
274	C₄²	C₂×D₄×C₅₉	1	2	yes	—
275	C₄²	C₂×D₄×C₅₉	1	2	yes	—
276	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	8	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	8	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
296	C₂×C₄	#120×C₅₉	1	2	yes	—
297	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
298	C₂×C₄	C₂×Q₁₆×C₅₉	1	4	yes	—
299	C₂×C₄	C₂×Q₁₆×C₅₉	1	2	yes	—
300	C₂×C₄	#137×C₅₉	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
306	C₂²	C₂²×Q₁₆×C₅₉	1	1	yes	characteristic
307	C₂²	C₂×#137×C₅₉	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	C₅₉×#451	1	4	yes	—
317	C₄	C₅₉×#456	1	2	yes	—
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	C₂×C₅₉×#451	1	1	yes	characteristic
321	C₂	C₂×C₅₉×#456	1	1	yes	characteristic
322	C₂	C₅₉×#3027	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂³×C₅₉	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
78. C₂×Q₈×C₅₉	Q₁₆	semidirect
141. #18404	C₅₉	direct (N × H)
190. C₅₉	#18404	direct (N × H)
240. C₂×Q₈	Q₁₆×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3502	2
C₅₉×#3209	2
C₅₉×#2567	2
C₅₉×#1605	2
C₅₉×#1503	2
C₅₉×#1461	2
#18404	59

Subgroups — C59×#18404 #28430513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#18404 #28430513