Subgroups — C₅₉×#33896 #28446005

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3341	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2237	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2220	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2185	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2141	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#555	—	2	1	no	—
14	D₄×Q₈×C₅₉	—	2	1	no	—
15	C₂×SD₃₂×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×SD₃₂×C₅₉	C₂²	1	2	yes	—
17	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
18	C₅₉×#500	C₂²	1	1	yes	characteristic
19	C₅₉×#494	C₂²	1	1	yes	characteristic
20	C₅₉×#482	—	2	2	no	—
21	C₅₉×#476	—	2	1	no	—
22	C₅₉×#467	C₂²	1	2	yes	—
23	C₅₉×#466	C₂²	1	2	yes	—
24	C₅₉×#462	—	2	1	no	—
25	C₅₉×#451	—	2	2	no	—
26	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#427	C₂²	1	2	yes	—
30	C₅₉×#367	C₂²	1	1	yes	characteristic
31	C₅₉×#366	C₂²	1	2	yes	—
32	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₅₉×#358	C₂²	1	2	yes	—
34	C₅₉×#358	C₂²	1	2	yes	—
35	C₅₉×#358	C₂²	1	2	yes	—
36	C₅₉×#358	C₂²	1	2	yes	—
37	C₅₉×#358	C₂²	1	1	yes	characteristic
38	C₅₉×#357	C₂²	1	1	yes	characteristic
39	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₅₉	—	2	2	no	—
42	C₂×Q₁₆×C₅₉	—	2	4	no	—
43	C₂×Q₁₆×C₅₉	—	2	2	no	—
44	C₂×Q₁₆×C₅₉	—	2	1	no	—
45	C₂×Q₁₆×C₅₉	—	4	2	no	—
46	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₅₉	—	2	1	no	—
51	C₄⋊Q₈×C₅₉	—	2	1	no	—
52	C₄⋊Q₈×C₅₉	—	4	1	no	—
53	#124×C₅₉	—	4	2	no	—
54	C₂²⋊Q₈×C₅₉	—	2	2	no	—
55	C₂²⋊Q₈×C₅₉	—	4	2	no	—
56	#121×C₅₉	—	2	2	no	—
57	#121×C₅₉	—	4	1	no	—
58	C₄×Q₈×C₅₉	—	2	1	no	—
59	C₄×D₄×C₅₉	—	2	1	no	—
60	C₄×D₄×C₅₉	—	4	1	no	—
61	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
62	#117×C₅₉	—	2	2	no	—
63	#117×C₅₉	—	4	1	no	—
64	SD₃₂×C₅₉	—	2	2	no	—
65	SD₃₂×C₅₉	—	4	2	no	—
66	M₃₂×C₅₉	—	4	2	no	—
67	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
70	#107×C₅₉	—	2	1	no	—
71	#107×C₅₉	C₂³	1	1	yes	characteristic
72	#107×C₅₉	C₂³	1	2	yes	—
73	#107×C₅₉	C₂³	1	1	yes	characteristic
74	#106×C₅₉	—	2	2	no	—
75	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
76	#103×C₅₉	—	2	1	no	—
77	#103×C₅₉	—	2	2	no	—
78	#102×C₅₉	—	2	1	no	—
79	#102×C₅₉	—	2	2	no	—
80	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
81	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₉	—	2	1	no	—
83	C₂×Q₈×C₅₉	—	2	2	no	—
84	C₂×Q₈×C₅₉	—	4	4	no	—
85	C₂×Q₈×C₅₉	—	4	2	no	—
86	C₂×Q₈×C₅₉	—	4	1	no	—
87	C₂×Q₈×C₅₉	—	8	2	no	—
88	C₂×D₄×C₅₉	—	2	1	no	—
89	C₂×D₄×C₅₉	—	4	2	no	—
90	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₅₉	—	4	2	no	—
92	C₂²×C₄×C₅₉	—	4	1	no	—
93	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
94	Q₁₆×C₅₉	—	2	2	no	—
95	Q₁₆×C₅₉	—	2	1	no	—
96	Q₁₆×C₅₉	—	2	2	no	—
97	Q₁₆×C₅₉	—	4	4	no	—
98	Q₁₆×C₅₉	—	8	1	no	—
99	Q₁₆×C₅₉	#41	1	2	yes	—
100	D₈×C₅₉	—	2	1	no	—
101	D₈×C₅₉	—	2	1	no	—
102	C₂×C₈×C₅₉	—	2	2	no	—
103	C₂×C₈×C₅₉	—	2	2	no	—
104	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₅₉	—	2	1	no	—
108	C₄⋊C₄×C₅₉	—	2	2	no	—
109	C₄⋊C₄×C₅₉	—	2	1	no	—
110	C₄⋊C₄×C₅₉	—	4	1	no	—
111	C₄⋊C₄×C₅₉	—	4	1	no	—
112	C₄⋊C₄×C₅₉	—	4	1	no	—
113	C₄⋊C₄×C₅₉	—	4	2	no	—
114	C₄⋊C₄×C₅₉	—	4	1	no	—
115	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₅₉	—	4	2	no	—
117	C₂²⋊C₄×C₅₉	—	4	2	no	—
118	C₂²⋊C₄×C₅₉	—	4	2	no	—
119	C₂²⋊C₄×C₅₉	—	4	1	no	—
120	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
121	C₄²×C₅₉	—	4	2	no	—
122	C₄²×C₅₉	—	4	1	no	—
123	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₅₉	—	2	1	no	—
125	C₁₆×C₅₉	—	2	1	no	—
126	C₁₆×C₅₉	—	2	2	no	—
127	C₂³×C₅₉	—	4	1	no	—
128	C₂³×C₅₉	C₂×D₈	1	2	yes	—
129	Q₈×C₅₉	—	2	2	no	—
130	Q₈×C₅₉	—	4	1	no	—
131	Q₈×C₅₉	—	4	2	no	—
132	Q₈×C₅₉	—	4	2	no	—
133	Q₈×C₅₉	—	8	4	no	—
134	Q₈×C₅₉	—	16	1	no	—
135	D₄×C₅₉	—	2	2	no	—
136	D₄×C₅₉	—	4	1	no	—
137	D₄×C₅₉	—	4	1	no	—
138	D₄×C₅₉	—	16	2	no	—
139	C₂×C₄×C₅₉	—	2	2	no	—
140	C₂×C₄×C₅₉	—	2	2	no	—
141	C₂×C₄×C₅₉	—	4	1	no	—
142	C₂×C₄×C₅₉	—	4	1	no	—
143	C₂×C₄×C₅₉	—	4	1	no	—
144	C₂×C₄×C₅₉	—	4	2	no	—
145	C₂×C₄×C₅₉	—	4	2	no	—
146	C₂×C₄×C₅₉	—	8	4	no	—
147	C₂×C₄×C₅₉	—	8	2	no	—
148	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₅₉	—	2	2	no	—
153	C₈×C₅₉	—	2	1	no	—
154	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
156	#33896	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₅₉	—	2	2	no	—
158	C₂²×C₅₉	—	2	2	no	—
159	C₂²×C₅₉	—	2	2	no	—
160	C₂²×C₅₉	—	8	1	no	—
161	C₂²×C₅₉	—	8	1	no	—
162	C₂²×C₅₉	—	8	1	no	—
163	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₅₉	—	2	1	no	—
165	C₄×C₅₉	—	2	2	no	—
166	C₄×C₅₉	—	4	2	no	—
167	C₄×C₅₉	—	8	1	no	—
168	C₄×C₅₉	—	8	1	no	—
169	C₄×C₅₉	—	8	2	no	—
170	C₄×C₅₉	—	8	2	no	—
171	C₄×C₅₉	#510	1	2	yes	—
172	C₄×C₅₉	#546	1	1	yes	characteristic
173	C₄×C₅₉	#577	1	1	yes	characteristic
174	#3341	C₂×C₅₉	1	1	yes	characteristic
175	D₄×Q₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
176	#2241	C₂×C₅₉	1	1	yes	characteristic, complemented
177	#2237	C₂×C₅₉	1	2	yes	complemented
178	#2227	C₂×C₅₉	1	1	yes	characteristic, complemented
179	#2220	C₂×C₅₉	1	2	yes	complemented
180	#2208	C₂×C₅₉	1	1	yes	characteristic, complemented
181	#2185	C₂×C₅₉	1	2	yes	complemented
182	C₄×SD₃₂	C₂×C₅₉	1	1	yes	characteristic, complemented
183	#2169	C₂×C₅₉	1	1	yes	characteristic, complemented
184	#2141	C₂×C₅₉	1	2	yes	complemented
185	C₂×C₅₉	—	4	2	no	—
186	C₂×C₅₉	—	16	1	no	—
187	C₂×C₅₉	#3294	1	1	yes	characteristic
188	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
189	C₂×C₅₉	#3416	1	1	yes	characteristic
190	C₂²×Q₁₆	C₂²×C₅₉	1	2	yes	—
191	#555	—	2	1	no	—
192	D₄×Q₈	—	2	1	no	—
193	C₂×SD₃₂	C₂²×C₅₉	1	1	yes	characteristic
194	C₂×SD₃₂	C₂²×C₅₉	1	2	yes	—
195	C₂×M₃₂	C₂²×C₅₉	1	2	yes	—
196	#500	C₂²×C₅₉	1	1	yes	characteristic
197	#494	C₂²×C₅₉	1	1	yes	characteristic
198	#482	—	2	2	no	—
199	#476	—	2	1	no	—
200	#467	C₂²×C₅₉	1	2	yes	—
201	#466	C₂²×C₅₉	1	2	yes	—
202	#462	—	2	1	no	—
203	#451	—	2	2	no	—
204	C₄×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
205	C₄×D₈	C₂²×C₅₉	1	1	yes	characteristic
206	C₈×D₄	C₂²×C₅₉	1	1	yes	characteristic
207	#427	C₂²×C₅₉	1	2	yes	—
208	#367	C₂²×C₅₉	1	1	yes	characteristic
209	#366	C₂²×C₅₉	1	2	yes	—
210	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
211	#358	C₂²×C₅₉	1	1	yes	characteristic
212	#358	C₂²×C₅₉	1	2	yes	—
213	#358	C₂²×C₅₉	1	2	yes	—
214	#358	C₂²×C₅₉	1	2	yes	—
215	#358	C₂²×C₅₉	1	2	yes	—
216	#357	C₂²×C₅₉	1	1	yes	characteristic
217	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
218	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
219	C₅₉	#33896	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²×Q₈	—	2	2	no	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	4	2	no	—
225	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
226	C₂×Q₁₆	C₂³×C₅₉	1	1	yes	characteristic
227	C₂×D₈	C₂³×C₅₉	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	4	1	no	—
232	#124	—	4	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	4	2	no	—
235	#121	—	2	2	no	—
236	#121	—	4	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	4	1	no	—
240	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
241	#117	—	2	2	no	—
242	#117	—	4	1	no	—
243	SD₃₂	—	2	2	no	—
244	SD₃₂	—	4	2	no	—
245	M₃₂	—	4	2	no	—
246	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
247	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
249	#107	—	2	1	no	—
250	#107	C₂³×C₅₉	1	1	yes	characteristic
251	#107	C₂³×C₅₉	1	2	yes	—
252	#107	C₂³×C₅₉	1	1	yes	characteristic
253	#106	—	2	2	no	—
254	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
255	#103	—	2	1	no	—
256	#103	—	2	2	no	—
257	#102	—	2	1	no	—
258	#102	—	2	2	no	—
259	C₂²⋊C₈	D₄×C₅₉	1	2	yes	—
260	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	8	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	D₈×C₅₉	1	1	yes	characteristic
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
273	Q₁₆	—	2	2	no	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	4	4	no	—
277	Q₁₆	—	8	1	no	—
278	Q₁₆	#41×C₅₉	1	2	yes	—
279	D₈	—	2	1	no	—
280	D₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
285	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	D₈×C₅₉	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	C₂×D₈×C₅₉	1	2	yes	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	4	no	—
313	Q₈	—	16	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	16	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
328	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	1	no	—
333	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
334	C₈	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	1	no	—
349	C₄	C₅₉×#510	1	2	yes	—
350	C₄	C₅₉×#546	1	1	yes	characteristic
351	C₄	C₅₉×#577	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₅₉×#3294	1	1	yes	characteristic
355	C₂	C₂×C₅₉×#510	1	1	yes	characteristic
356	C₂	C₅₉×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₅₉×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×D₄×C₅₉ ◁ C₈×D₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×Q₁₆×C₅₉	C₂	semidirect
3. C₅₉×#2241	C₂	semidirect
4. C₅₉×#2237	C₂	semidirect
5. C₅₉×#2227	C₂	semidirect
6. C₅₉×#2220	C₂	semidirect
7. C₅₉×#2208	C₂	semidirect
8. C₅₉×#2185	C₂	semidirect
9. C₄×SD₃₂×C₅₉	C₂	semidirect
10. C₅₉×#2169	C₂	semidirect
11. C₅₉×#2141	C₂	semidirect
156. #33896	C₅₉	direct (N × H)
175. D₄×Q₁₆	C₂×C₅₉	semidirect
176. #2241	C₂×C₅₉	semidirect
177. #2237	C₂×C₅₉	semidirect
178. #2227	C₂×C₅₉	semidirect
179. #2220	C₂×C₅₉	semidirect
180. #2208	C₂×C₅₉	semidirect
181. #2185	C₂×C₅₉	semidirect
182. C₄×SD₃₂	C₂×C₅₉	semidirect
183. #2169	C₂×C₅₉	semidirect
184. #2141	C₂×C₅₉	semidirect
219. C₅₉	#33896	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×#3341	2
D₄×Q₁₆×C₅₉	2
C₅₉×#2241	2
C₅₉×#2237	2
C₅₉×#2227	2
C₅₉×#2220	2
C₅₉×#2208	2
C₅₉×#2185	2
C₄×SD₃₂×C₅₉	2
C₅₉×#2169	2
C₅₉×#2141	2
#33896	59

Subgroups — C59×#33896 #28446005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33896 #28446005