Subgroups — C₅₉×#11240 #28423349

Order 15104.

638 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#344	C₂	1	2	yes	maximal
4	C₅₉×#1735	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1733	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₉	C₂²	1	2	yes	—
10	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₅₉	—	2	4	no	—
12	C₂×#108×C₅₉	—	2	2	no	—
13	C₅₉×#424	—	2	2	no	—
14	C₅₉×#424	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₅₉	—	2	2	no	—
16	C₅₉×#344	—	2	4	no	—
17	C₅₉×#340	—	2	2	no	—
18	C₅₉×#339	—	2	4	no	—
19	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	—	2	1	no	—
21	C₂×M₁₆×C₅₉	—	2	2	no	—
22	C₂×M₁₆×C₅₉	—	2	2	no	—
23	C₂×M₁₆×C₅₉	—	2	2	no	—
24	C₂×M₁₆×C₅₉	—	2	2	no	—
25	C₂×M₁₆×C₅₉	—	2	2	no	—
26	C₂×M₁₆×C₅₉	—	2	2	no	—
27	C₂×M₁₆×C₅₉	—	2	2	no	—
28	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₅₉	D₄	1	4	yes	—
30	C₂²×C₈×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	—	2	1	no	—
33	C₂²×C₈×C₅₉	—	2	2	no	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₉	D₄	1	2	yes	—
36	C₂²×C₈×C₅₉	D₄	1	2	yes	—
37	#117×C₅₉	—	2	1	no	—
38	#117×C₅₉	—	2	1	no	—
39	#117×C₅₉	—	2	2	no	—
40	#117×C₅₉	—	2	2	no	—
41	#117×C₅₉	—	2	2	no	—
42	#117×C₅₉	C₂×C₄	1	2	yes	—
43	#117×C₅₉	D₄	1	2	yes	—
44	#117×C₅₉	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₉	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
49	C₂×C₄²×C₅₉	—	2	1	no	—
50	C₂×C₄²×C₅₉	—	2	2	no	—
51	#108×C₅₉	—	2	8	no	—
52	C₄⋊C₈×C₅₉	—	2	8	no	—
53	C₂⁴×C₅₉	#41	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	—	2	2	no	—
55	C₂²×C₄×C₅₉	—	4	2	no	—
56	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅₉	#41	1	2	yes	—
64	C₂²×C₄×C₅₉	#41	1	2	yes	—
65	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
66	M₁₆×C₅₉	—	2	4	no	—
67	M₁₆×C₅₉	—	2	4	no	—
68	M₁₆×C₅₉	—	2	4	no	—
69	M₁₆×C₅₉	—	4	2	no	—
70	M₁₆×C₅₉	—	4	2	no	—
71	M₁₆×C₅₉	—	4	2	no	—
72	C₂×C₈×C₅₉	—	2	4	no	—
73	C₂×C₈×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	—	2	4	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	—	2	4	no	—
78	C₂×C₈×C₅₉	—	4	1	no	—
79	C₂×C₈×C₅₉	—	4	2	no	—
80	C₂×C₈×C₅₉	—	4	1	no	—
81	C₂×C₈×C₅₉	—	4	2	no	—
82	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₄⋊C₄×C₅₉	—	2	4	no	—
85	C₄⋊C₄×C₅₉	—	4	2	no	—
86	C₄⋊C₄×C₅₉	C₄⋊C₄	1	4	yes	—
87	C₂²⋊C₄×C₅₉	—	2	2	no	—
88	C₂²⋊C₄×C₅₉	—	2	2	no	—
89	C₂²⋊C₄×C₅₉	—	2	4	no	—
90	C₂²⋊C₄×C₅₉	—	4	2	no	—
91	C₄²×C₅₉	—	2	2	no	—
92	C₄²×C₅₉	—	2	2	no	—
93	C₄²×C₅₉	—	2	4	no	—
94	C₄²×C₅₉	—	4	2	no	—
95	C₂³×C₅₉	—	4	2	no	—
96	C₂³×C₅₉	#117	1	1	yes	characteristic
97	C₂³×C₅₉	#121	1	1	yes	characteristic
98	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₅₉	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₅₉	#126	1	2	yes	—
101	C₂×C₄×C₅₉	—	2	4	no	—
102	C₂×C₄×C₅₉	—	4	4	no	—
103	C₂×C₄×C₅₉	—	4	4	no	—
104	C₂×C₄×C₅₉	—	8	4	no	—
105	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#121	1	2	yes	—
111	C₂×C₄×C₅₉	#121	1	2	yes	—
112	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#123	1	2	yes	—
118	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₅₉	#125	1	2	yes	—
120	C₂×C₄×C₅₉	#125	1	2	yes	—
121	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₅₉	—	2	4	no	—
124	C₈×C₅₉	—	4	2	no	—
125	C₈×C₅₉	—	4	4	no	—
126	#11240	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₉	—	2	2	no	—
128	C₂²×C₅₉	—	2	2	no	—
129	C₂²×C₅₉	—	2	2	no	—
130	C₂²×C₅₉	—	4	2	no	—
131	C₂²×C₅₉	—	4	2	no	—
132	C₂²×C₅₉	#380	1	1	yes	characteristic
133	C₂²×C₅₉	#387	1	2	yes	—
134	C₂²×C₅₉	#395	1	1	yes	characteristic
135	C₂²×C₅₉	#396	1	2	yes	—
136	C₂²×C₅₉	#400	1	1	yes	characteristic
137	C₄×C₅₉	—	4	4	no	—
138	C₄×C₅₉	—	8	4	no	—
139	C₄×C₅₉	#386	1	1	yes	characteristic
140	C₄×C₅₉	#389	1	2	yes	—
141	C₄×C₅₉	#391	1	1	yes	characteristic
142	C₄×C₅₉	#394	1	1	yes	characteristic
143	C₄×C₅₉	#398	1	2	yes	—
144	C₄×C₅₉	#398	1	1	yes	characteristic
145	C₂²×#108	C₂×C₅₉	1	1	yes	characteristic
146	C₂×#424	C₂×C₅₉	1	1	yes	characteristic
147	C₂×#344	C₂×C₅₉	1	2	yes	—
148	#1735	C₂×C₅₉	1	1	yes	characteristic
149	#1733	C₂×C₅₉	1	2	yes	—
150	C₂×C₅₉	—	2	2	no	—
151	C₂×C₅₉	—	2	2	no	—
152	C₂×C₅₉	—	2	1	no	—
153	C₂×C₅₉	—	2	1	no	—
154	C₂×C₅₉	#1441	1	1	yes	characteristic
155	C₂×C₅₉	#1814	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
157	C₂²×M₁₆	C₂²×C₅₉	1	2	yes	—
158	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
159	C₂×#117	C₂²×C₅₉	1	2	yes	—
160	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
161	C₂×#108	—	2	4	no	—
162	C₂×#108	—	2	2	no	—
163	#424	—	2	2	no	—
164	#424	C₄×C₅₉	1	4	yes	complemented
165	C₂×C₄⋊C₈	—	2	2	no	—
166	#344	—	2	4	no	—
167	#340	—	2	2	no	—
168	#339	—	2	4	no	—
169	C₅₉	#11240	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	C₂×C₄×C₅₉	1	2	yes	—
180	C₂×M₁₆	D₄×C₅₉	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂×C₄×C₅₉	1	2	yes	—
186	C₂²×C₈	D₄×C₅₉	1	2	yes	—
187	C₂²×C₈	D₄×C₅₉	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	C₂×C₄×C₅₉	1	2	yes	—
194	#117	D₄×C₅₉	1	2	yes	—
195	#117	Q₈×C₅₉	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	Q₈×C₅₉	1	2	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	#108	—	2	8	no	—
203	C₄⋊C₈	—	2	8	no	—
204	C₂⁴	#41×C₅₉	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
212	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₅₉	1	2	yes	—
214	C₂²×C₄	#41×C₅₉	1	2	yes	—
215	C₂²×C₄	#41×C₅₉	1	2	yes	—
216	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₅₉	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	C₄⋊C₄×C₅₉	1	4	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	#117×C₅₉	1	1	yes	characteristic
248	C₂³	#121×C₅₉	1	1	yes	characteristic
249	C₂³	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
250	C₂³	C₂²⋊Q₈×C₅₉	1	2	yes	—
251	C₂³	#126×C₅₉	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₅₉	1	2	yes	—
259	C₂×C₄	C₄×Q₈×C₅₉	1	2	yes	—
260	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
261	C₂×C₄	#121×C₅₉	1	2	yes	—
262	C₂×C₄	#121×C₅₉	1	2	yes	—
263	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
265	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
266	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
267	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
268	C₂×C₄	#123×C₅₉	1	2	yes	—
269	C₂×C₄	#124×C₅₉	1	1	yes	characteristic
270	C₂×C₄	#125×C₅₉	1	2	yes	—
271	C₂×C₄	#125×C₅₉	1	2	yes	—
272	C₂×C₄	C₄⋊Q₈×C₅₉	1	2	yes	—
273	C₂×C₄	C₄⋊Q₈×C₅₉	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₅₉×#380	1	1	yes	characteristic
283	C₂²	C₅₉×#387	1	2	yes	—
284	C₂²	C₅₉×#395	1	1	yes	characteristic
285	C₂²	C₅₉×#396	1	2	yes	—
286	C₂²	C₅₉×#400	1	1	yes	characteristic
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	C₅₉×#386	1	1	yes	characteristic
290	C₄	C₅₉×#389	1	2	yes	—
291	C₄	C₅₉×#391	1	1	yes	characteristic
292	C₄	C₅₉×#394	1	1	yes	characteristic
293	C₄	C₅₉×#398	1	2	yes	—
294	C₄	C₅₉×#398	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	C₅₉×#1441	1	1	yes	characteristic
300	C₂	C₅₉×#1814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
socle	C₂²×C₅₉	#380
prime radical	C₂⁴×C₅₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₅₉×#424	C₄	semidirect
126. #11240	C₅₉	direct (N × H)
164. #424	C₄×C₅₉	semidirect
169. C₅₉	#11240	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#108×C₅₉	2
C₂×C₅₉×#424	2
C₂×C₅₉×#344	2
C₅₉×#1735	2
C₅₉×#1733	2
#11240	59

Subgroups — C59×#11240 #28423349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#11240 #28423349