Subgroups — C₅₉×#33281 #28445390

Order 15104.

674 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	C₅₉×#2864	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2862	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1993	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1993	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1960	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1960	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1935	C₂	1	4	yes	maximal
8	C₅₉×#1933	C₂	1	4	yes	maximal
9	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×#103×C₅₉	C₂²	1	4	yes	—
13	C₂×#103×C₅₉	C₂²	1	4	yes	—
14	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
18	C₅₉×#391	—	2	2	no	—
19	C₅₉×#391	C₂²	1	2	yes	—
20	C₅₉×#391	C₂²	1	2	yes	—
21	C₅₉×#389	C₂²	1	2	yes	—
22	C₅₉×#389	C₂²	1	2	yes	—
23	C₅₉×#387	—	2	2	no	—
24	C₅₉×#387	—	2	2	no	—
25	C₅₉×#382	—	2	2	no	—
26	C₅₉×#382	C₂²	1	1	yes	characteristic
27	C₅₉×#382	C₂²	1	1	yes	characteristic
28	C₄³×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#340	C₂²	1	4	yes	—
30	C₅₉×#340	C₂²	1	4	yes	—
31	C₂²×Q₈×C₅₉	—	2	1	no	—
32	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
35	C₄⋊Q₈×C₅₉	—	2	2	no	—
36	C₄⋊Q₈×C₅₉	—	2	2	no	—
37	C₄⋊Q₈×C₅₉	—	2	2	no	—
38	C₄⋊Q₈×C₅₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₅₉	D₄	1	2	yes	—
48	C₂×C₄²×C₅₉	D₄	1	2	yes	—
49	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	#103×C₅₉	—	2	8	no	—
53	#103×C₅₉	—	2	8	no	—
54	#97×C₅₉	—	2	4	no	—
55	#97×C₅₉	—	2	2	no	—
56	#97×C₅₉	—	2	2	no	—
57	C₄×C₈×C₅₉	—	2	4	no	—
58	#95×C₅₉	—	2	4	no	—
59	#95×C₅₉	—	2	4	no	—
60	#95×C₅₉	—	2	1	no	—
61	#95×C₅₉	—	2	1	no	—
62	#95×C₅₉	—	2	1	no	—
63	#95×C₅₉	—	2	1	no	—
64	#95×C₅₉	—	2	2	no	—
65	#95×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	—	2	4	no	—
67	C₂×Q₈×C₅₉	—	2	4	no	—
68	C₂×Q₈×C₅₉	—	2	2	no	—
69	C₂×Q₈×C₅₉	—	2	2	no	—
70	C₂×Q₈×C₅₉	—	4	4	no	—
71	C₂×Q₈×C₅₉	—	4	2	no	—
72	C₂²×C₄×C₅₉	—	2	2	no	—
73	C₂²×C₄×C₅₉	—	2	2	no	—
74	C₂²×C₄×C₅₉	—	2	2	no	—
75	C₂²×C₄×C₅₉	—	2	2	no	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	2	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	#41	1	4	yes	—
86	C₂×C₈×C₅₉	#41	1	4	yes	—
87	C₄⋊C₄×C₅₉	—	2	4	no	—
88	C₄⋊C₄×C₅₉	—	2	4	no	—
89	C₄⋊C₄×C₅₉	—	2	2	no	—
90	C₄⋊C₄×C₅₉	—	2	2	no	—
91	C₄⋊C₄×C₅₉	—	4	4	no	—
92	C₄⋊C₄×C₅₉	—	4	2	no	—
93	C₄⋊C₄×C₅₉	—	4	2	no	—
94	C₄⋊C₄×C₅₉	—	4	2	no	—
95	C₄⋊C₄×C₅₉	—	4	4	no	—
96	C₄²×C₅₉	—	2	4	no	—
97	C₄²×C₅₉	—	2	2	no	—
98	C₄²×C₅₉	—	2	2	no	—
99	C₄²×C₅₉	—	4	2	no	—
100	C₄²×C₅₉	#41	1	2	yes	—
101	C₄²×C₅₉	#41	1	2	yes	—
102	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₅₉	—	4	4	no	—
104	Q₈×C₅₉	—	4	4	no	—
105	Q₈×C₅₉	—	8	4	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	2	no	—
108	C₂×C₄×C₅₉	—	2	2	no	—
109	C₂×C₄×C₅₉	—	2	2	no	—
110	C₂×C₄×C₅₉	—	2	2	no	—
111	C₂×C₄×C₅₉	—	2	2	no	—
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	—	4	2	no	—
115	C₂×C₄×C₅₉	—	4	2	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	4	4	no	—
124	C₂×C₄×C₅₉	#124	1	2	yes	—
125	C₂×C₄×C₅₉	#124	1	2	yes	—
126	C₂×C₄×C₅₉	#137	1	4	yes	—
127	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
130	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
131	C₈×C₅₉	—	2	4	no	—
132	C₈×C₅₉	—	2	4	no	—
133	#33281	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₉	C₂×#124	1	1	yes	characteristic
135	C₂²×C₅₉	#532	1	1	yes	characteristic
136	C₂²×C₅₉	#532	1	1	yes	characteristic
137	C₂²×C₅₉	C₂×#137	1	2	yes	—
138	C₂²×C₅₉	#575	1	2	yes	—
139	C₄×C₅₉	—	2	2	no	—
140	C₄×C₅₉	—	2	2	no	—
141	C₄×C₅₉	—	2	4	no	—
142	C₄×C₅₉	—	4	2	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	—	8	2	no	—
145	C₄×C₅₉	—	8	2	no	—
146	C₄×C₅₉	—	8	2	no	—
147	C₄×C₅₉	#529	1	2	yes	—
148	C₄×C₅₉	#529	1	2	yes	—
149	#2864	C₂×C₅₉	1	1	yes	characteristic
150	#2862	C₂×C₅₉	1	1	yes	characteristic
151	#1993	C₂×C₅₉	1	1	yes	characteristic
152	#1993	C₂×C₅₉	1	1	yes	characteristic
153	#1960	C₂×C₅₉	1	1	yes	characteristic
154	#1960	C₂×C₅₉	1	1	yes	characteristic
155	#1935	C₂×C₅₉	1	4	yes	—
156	#1933	C₂×C₅₉	1	4	yes	—
157	C₄×#97	C₂×C₅₉	1	1	yes	characteristic
158	C₂×C₅₉	#2861	1	1	yes	characteristic
159	C₂×C₅₉	#3139	1	1	yes	characteristic
160	C₂×C₅₉	#3140	1	1	yes	characteristic
161	C₂×C₅₉	#3172	1	2	yes	—
162	C₂×C₅₉	#3181	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
165	C₂×#103	C₂²×C₅₉	1	4	yes	—
166	C₂×#103	C₂²×C₅₉	1	4	yes	—
167	C₂×#97	C₂²×C₅₉	1	2	yes	—
168	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
169	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
171	#391	—	2	2	no	—
172	#391	C₂²×C₅₉	1	2	yes	—
173	#391	C₂²×C₅₉	1	2	yes	—
174	#389	C₂²×C₅₉	1	2	yes	—
175	#389	C₂²×C₅₉	1	2	yes	—
176	#387	—	2	2	no	—
177	#387	—	2	2	no	—
178	#382	—	2	2	no	—
179	#382	C₂²×C₅₉	1	1	yes	characteristic
180	#382	C₂²×C₅₉	1	1	yes	characteristic
181	C₄³	C₂²×C₅₉	1	1	yes	characteristic
182	#340	C₂²×C₅₉	1	4	yes	—
183	#340	C₂²×C₅₉	1	4	yes	—
184	C₅₉	#33281	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
187	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
188	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
201	C₂×C₄²	D₄×C₅₉	1	2	yes	—
202	C₂×C₄²	D₄×C₅₉	1	2	yes	—
203	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
206	#103	—	2	8	no	—
207	#103	—	2	8	no	—
208	#97	—	2	4	no	—
209	#97	—	2	2	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	4	no	—
212	#95	—	2	4	no	—
213	#95	—	2	4	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	#41×C₅₉	1	4	yes	—
240	C₂×C₈	#41×C₅₉	1	4	yes	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	#41×C₅₉	1	2	yes	—
255	C₄²	#41×C₅₉	1	2	yes	—
256	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	#124×C₅₉	1	2	yes	—
279	C₂×C₄	#124×C₅₉	1	2	yes	—
280	C₂×C₄	#137×C₅₉	1	4	yes	—
281	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
284	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
285	C₈	—	2	4	no	—
286	C₈	—	2	4	no	—
287	C₂²	C₂×#124×C₅₉	1	1	yes	characteristic
288	C₂²	C₅₉×#532	1	1	yes	characteristic
289	C₂²	C₅₉×#532	1	1	yes	characteristic
290	C₂²	C₂×#137×C₅₉	1	2	yes	—
291	C₂²	C₅₉×#575	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	C₅₉×#529	1	2	yes	—
301	C₄	C₅₉×#529	1	2	yes	—
302	C₂	C₅₉×#2861	1	1	yes	characteristic
303	C₂	C₅₉×#3139	1	1	yes	characteristic
304	C₂	C₅₉×#3140	1	1	yes	characteristic
305	C₂	C₅₉×#3172	1	2	yes	—
306	C₂	C₅₉×#3181	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
133. #33281	C₅₉	direct (N × H)
184. C₅₉	#33281	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×#2864	2
C₅₉×#2862	2
C₅₉×#1993	2
C₅₉×#1960	2
C₅₉×#1935	2
C₅₉×#1933	2
C₄×#97×C₅₉	2
#33281	59

Subgroups — C59×#33281 #28445390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33281 #28445390