Subgroups — C₅₉×#33769 #28445878

Order 15104.

576 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3061	C₂	1	2	yes	maximal, complemented
2	C₅₉×#2236	C₂	1	2	yes	maximal, complemented
3	C₅₉×#2232	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2220	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2189	C₂	1	2	yes	maximal, complemented
8	C₅₉×#2186	C₂	1	2	yes	maximal, complemented
9	C₅₉×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₅₉	C₂²	1	2	yes	—
11	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#537	—	2	2	no	—
13	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×D₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#468	—	2	2	no	—
19	C₅₉×#468	—	2	2	no	—
20	C₅₉×#467	C₂²	1	2	yes	—
21	C₅₉×#467	C₂²	1	2	yes	—
22	C₅₉×#466	C₂²	1	2	yes	—
23	C₅₉×#466	C₂²	1	2	yes	—
24	C₂×#107×C₅₉	C₂²	1	2	yes	—
25	C₅₉×#417	—	2	2	no	—
26	C₅₉×#366	C₂²	1	1	yes	characteristic
27	C₅₉×#366	C₂²	1	2	yes	—
28	C₅₉×#366	C₂²	1	1	yes	characteristic
29	C₅₉×#358	C₂²	1	2	yes	—
30	C₅₉×#358	C₂²	1	2	yes	—
31	C₅₉×#357	C₂²	1	2	yes	—
32	C₅₉×#357	C₂²	1	2	yes	—
33	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₂×#41×C₅₉	—	2	2	no	—
36	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
37	#135×C₅₉	—	4	2	no	—
38	#135×C₅₉	—	4	2	no	—
39	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₅₉	—	2	2	no	—
41	C₂×D₈×C₅₉	C₂³	1	2	yes	—
42	#131×C₅₉	—	4	1	no	—
43	#131×C₅₉	—	4	1	no	—
44	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
45	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₂²⋊Q₈×C₅₉	—	2	2	no	—
49	C₂²⋊Q₈×C₅₉	—	2	2	no	—
50	C₂²⋊Q₈×C₅₉	—	4	2	no	—
51	#121×C₅₉	—	2	2	no	—
52	#121×C₅₉	—	2	2	no	—
53	#121×C₅₉	—	4	2	no	—
54	#121×C₅₉	—	4	2	no	—
55	#121×C₅₉	—	4	2	no	—
56	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
57	Q₃₂×C₅₉	—	4	1	no	—
58	SD₃₂×C₅₉	—	4	2	no	—
59	D₁₆×C₅₉	—	4	1	no	—
60	M₃₂×C₅₉	—	4	1	no	—
61	C₂×C₁₆×C₅₉	—	4	1	no	—
62	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
64	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
65	#107×C₅₉	—	2	2	no	—
66	#107×C₅₉	C₂³	1	2	yes	—
67	#107×C₅₉	C₂³	1	2	yes	—
68	#103×C₅₉	—	2	2	no	—
69	#103×C₅₉	—	2	2	no	—
70	#102×C₅₉	—	2	2	no	—
71	#102×C₅₉	—	2	2	no	—
72	#41×C₅₉	—	4	1	no	—
73	#41×C₅₉	—	4	1	no	—
74	#41×C₅₉	—	4	4	no	—
75	#41×C₅₉	—	4	4	no	—
76	C₂×Q₈×C₅₉	—	2	2	no	—
77	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
78	C₂×D₄×C₅₉	—	2	2	no	—
79	C₂×D₄×C₅₉	—	4	2	no	—
80	C₂×D₄×C₅₉	—	4	2	no	—
81	C₂×D₄×C₅₉	—	4	2	no	—
82	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
83	C₂×D₄×C₅₉	D₈	1	2	yes	—
84	C₂²×C₄×C₅₉	—	4	2	no	—
85	C₂²×C₄×C₅₉	—	4	2	no	—
86	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₅₉	—	2	2	no	—
89	Q₁₆×C₅₉	—	2	2	no	—
90	SD₁₆×C₅₉	—	4	4	no	—
91	D₈×C₅₉	—	2	2	no	—
92	D₈×C₅₉	—	2	2	no	—
93	M₁₆×C₅₉	—	4	2	no	—
94	M₁₆×C₅₉	—	4	1	no	—
95	C₂×C₈×C₅₉	—	2	1	no	—
96	C₂×C₈×C₅₉	—	2	1	no	—
97	C₂×C₈×C₅₉	—	2	2	no	—
98	C₂×C₈×C₅₉	—	2	2	no	—
99	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₅₉	—	2	2	no	—
103	C₄⋊C₄×C₅₉	—	2	2	no	—
104	C₄⋊C₄×C₅₉	—	4	2	no	—
105	C₄⋊C₄×C₅₉	—	4	2	no	—
106	C₄⋊C₄×C₅₉	—	4	2	no	—
107	C₂²⋊C₄×C₅₉	—	4	2	no	—
108	C₂²⋊C₄×C₅₉	—	4	2	no	—
109	C₂²⋊C₄×C₅₉	—	4	2	no	—
110	C₂²⋊C₄×C₅₉	—	4	2	no	—
111	C₁₆×C₅₉	—	2	1	no	—
112	C₁₆×C₅₉	—	2	1	no	—
113	C₁₆×C₅₉	—	2	2	no	—
114	C₂³×C₅₉	—	4	2	no	—
115	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
116	C₂³×C₅₉	C₂×D₈	1	2	yes	—
117	Q₈×C₅₉	—	4	1	no	—
118	Q₈×C₅₉	—	4	2	no	—
119	Q₈×C₅₉	—	4	2	no	—
120	D₄×C₅₉	—	4	2	no	—
121	D₄×C₅₉	—	4	1	no	—
122	D₄×C₅₉	—	4	2	no	—
123	D₄×C₅₉	—	4	2	no	—
124	D₄×C₅₉	—	8	4	no	—
125	D₄×C₅₉	—	16	2	no	—
126	D₄×C₅₉	—	16	2	no	—
127	C₂×C₄×C₅₉	—	2	1	no	—
128	C₂×C₄×C₅₉	—	2	2	no	—
129	C₂×C₄×C₅₉	—	2	1	no	—
130	C₂×C₄×C₅₉	—	2	2	no	—
131	C₂×C₄×C₅₉	—	4	2	no	—
132	C₂×C₄×C₅₉	—	4	2	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	—	8	4	no	—
135	C₂×C₄×C₅₉	—	16	2	no	—
136	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	2	no	—
141	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
143	#33769	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₅₉	—	2	2	no	—
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	2	no	—
148	C₂²×C₅₉	—	2	2	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	8	2	no	—
151	C₂²×C₅₉	—	8	2	no	—
152	C₂²×C₅₉	—	8	2	no	—
153	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₅₉	—	2	2	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	8	2	no	—
157	C₄×C₅₉	—	8	2	no	—
158	C₄×C₅₉	—	8	2	no	—
159	C₄×C₅₉	#534	1	1	yes	characteristic
160	C₄×C₅₉	#575	1	1	yes	characteristic
161	#3061	C₂×C₅₉	1	2	yes	complemented
162	#2236	C₂×C₅₉	1	2	yes	complemented
163	#2232	C₂×C₅₉	1	2	yes	complemented
164	#2220	C₂×C₅₉	1	2	yes	complemented
165	#2216	C₂×C₅₉	1	1	yes	characteristic, complemented
166	#2215	C₂×C₅₉	1	1	yes	characteristic, complemented
167	#2189	C₂×C₅₉	1	2	yes	complemented
168	#2186	C₂×C₅₉	1	2	yes	complemented
169	#2113	C₂×C₅₉	1	1	yes	characteristic, complemented
170	C₂×C₅₉	—	4	2	no	—
171	C₂×C₅₉	—	4	1	no	—
172	C₂×C₅₉	—	16	2	no	—
173	C₂×C₅₉	#3186	1	1	yes	characteristic
174	C₂×C₅₉	#3415	1	1	yes	characteristic
175	C₂×C₅₉	#3417	1	1	yes	characteristic
176	C₂×#135	C₂²×C₅₉	1	2	yes	—
177	C₂×#131	C₂²×C₅₉	1	1	yes	characteristic
178	#537	—	2	2	no	—
179	C₂×Q₃₂	C₂²×C₅₉	1	1	yes	characteristic
180	C₂×SD₃₂	C₂²×C₅₉	1	2	yes	—
181	C₂×D₁₆	C₂²×C₅₉	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₅₉	1	1	yes	characteristic
183	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
184	#468	—	2	2	no	—
185	#468	—	2	2	no	—
186	#467	C₂²×C₅₉	1	2	yes	—
187	#467	C₂²×C₅₉	1	2	yes	—
188	#466	C₂²×C₅₉	1	2	yes	—
189	#466	C₂²×C₅₉	1	2	yes	—
190	C₂×#107	C₂²×C₅₉	1	2	yes	—
191	#417	—	2	2	no	—
192	#366	C₂²×C₅₉	1	1	yes	characteristic
193	#366	C₂²×C₅₉	1	1	yes	characteristic
194	#366	C₂²×C₅₉	1	2	yes	—
195	#358	C₂²×C₅₉	1	2	yes	—
196	#358	C₂²×C₅₉	1	2	yes	—
197	#357	C₂²×C₅₉	1	2	yes	—
198	#357	C₂²×C₅₉	1	2	yes	—
199	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
200	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
201	C₅₉	#33769	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#41	—	2	2	no	—
203	C₂×#41	D₄×C₅₉	1	1	yes	characteristic
204	#135	—	4	2	no	—
205	#135	—	4	2	no	—
206	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
207	C₂×SD₁₆	—	2	2	no	—
208	C₂×D₈	C₂³×C₅₉	1	2	yes	—
209	#131	—	4	1	no	—
210	#131	—	4	1	no	—
211	C₂×M₁₆	D₄×C₅₉	1	2	yes	—
212	C₂×M₁₆	D₄×C₅₉	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
214	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	#121	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	4	2	no	—
221	#121	—	4	2	no	—
222	#121	—	4	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	Q₃₂	—	4	1	no	—
225	SD₃₂	—	4	2	no	—
226	D₁₆	—	4	1	no	—
227	M₃₂	—	4	1	no	—
228	C₂×C₁₆	—	4	1	no	—
229	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
231	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
232	#107	—	2	2	no	—
233	#107	C₂³×C₅₉	1	2	yes	—
234	#107	C₂³×C₅₉	1	2	yes	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#102	—	2	2	no	—
238	#102	—	2	2	no	—
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	#41	—	4	4	no	—
242	#41	—	4	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	D₈×C₅₉	1	1	yes	characteristic
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	D₈×C₅₉	1	1	yes	characteristic
250	C₂×D₄	D₈×C₅₉	1	2	yes	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	SD₁₆	—	4	4	no	—
258	D₈	—	2	2	no	—
259	D₈	—	2	2	no	—
260	M₁₆	—	4	2	no	—
261	M₁₆	—	4	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
268	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
283	C₂³	C₂×D₈×C₅₉	1	2	yes	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	16	2	no	—
293	D₄	—	16	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
309	C₈	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₅₉×#534	1	1	yes	characteristic
326	C₄	C₅₉×#575	1	1	yes	characteristic
327	C₂	—	4	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	16	2	no	—
330	C₂	C₅₉×#3186	1	1	yes	characteristic
331	C₂	C₅₉×#3415	1	1	yes	characteristic
332	C₂	C₅₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×#41×C₅₉ ◁ C₂×#131×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3061	C₂	semidirect
2. C₅₉×#2236	C₂	semidirect
3. C₅₉×#2232	C₂	semidirect
4. C₅₉×#2220	C₂	semidirect
5. C₅₉×#2216	C₂	semidirect
6. C₅₉×#2215	C₂	semidirect
7. C₅₉×#2189	C₂	semidirect
8. C₅₉×#2186	C₂	semidirect
9. C₅₉×#2113	C₂	semidirect
143. #33769	C₅₉	direct (N × H)
161. #3061	C₂×C₅₉	semidirect
162. #2236	C₂×C₅₉	semidirect
163. #2232	C₂×C₅₉	semidirect
164. #2220	C₂×C₅₉	semidirect
165. #2216	C₂×C₅₉	semidirect
166. #2215	C₂×C₅₉	semidirect
167. #2189	C₂×C₅₉	semidirect
168. #2186	C₂×C₅₉	semidirect
169. #2113	C₂×C₅₉	semidirect
201. C₅₉	#33769	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#3061	2
C₅₉×#2236	2
C₅₉×#2232	2
C₅₉×#2220	2
C₅₉×#2216	2
C₅₉×#2215	2
C₅₉×#2189	2
C₅₉×#2186	2
C₅₉×#2113	2
#33769	59

Subgroups — C59×#33769 #28445878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33769 #28445878