Subgroups — C₅₉×#35285 #28447394

Order 15104.

3206 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#127×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2858	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2641	C₂	1	4	yes	maximal, complemented
5	C₅₉×#2548	C₂	1	8	yes	maximal, complemented
6	C₅₉×#2435	C₂	1	8	yes	maximal, complemented
7	C₅₉×#2309	C₂	1	4	yes	maximal, complemented
8	C₄×#117×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₅₉	C₂²	1	2	yes	—
10	C₂³×D₄×C₅₉	C₂²	1	4	yes	—
11	C₅₉×#533	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	complemented
13	C₂×#127×C₅₉	—	2	8	no	—
14	C₂×#127×C₅₉	C₂²	1	4	yes	—
15	C₂×#127×C₅₉	C₂²	1	2	yes	—
16	C₂×#127×C₅₉	C₂²	1	2	yes	—
17	C₂×#125×C₅₉	C₂²	1	2	yes	complemented
18	C₂×#124×C₅₉	C₂²	1	16	yes	complemented
19	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₅₉	C₂²	1	4	yes	complemented
21	C₂×#117×C₅₉	C₂²	1	2	yes	—
22	C₂×#117×C₅₉	C₂²	1	2	yes	—
23	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#394	C₂²	1	32	yes	complemented
25	C₅₉×#390	C₂²	1	16	yes	complemented
26	C₅₉×#390	C₂²	1	16	yes	complemented
27	C₅₉×#390	C₂²	1	16	yes	complemented
28	C₅₉×#379	C₂²	1	16	yes	—
29	C₄×C₄⋊C₄×C₅₉	C₂²	1	4	yes	complemented
30	C₄×C₂²⋊C₄×C₅₉	C₂²	1	4	yes	—
31	C₅₉×#376	C₂²	1	2	yes	complemented
32	C₄³×C₅₉	C₂²	1	2	yes	complemented
33	C₂⁵×C₅₉	C₂³	1	4	yes	—
34	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
35	C₂²×D₄×C₅₉	—	2	16	no	—
36	C₂²×D₄×C₅₉	—	2	8	no	—
37	C₂²×D₄×C₅₉	—	2	16	no	—
38	C₂²×D₄×C₅₉	—	4	16	no	—
39	C₂²×D₄×C₅₉	C₂³	1	4	yes	—
40	C₂²×D₄×C₅₉	C₂³	1	4	yes	—
41	C₂²×D₄×C₅₉	C₂³	1	8	yes	—
42	C₂²×D₄×C₅₉	C₂³	1	8	yes	—
43	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
44	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₅₉	—	2	8	no	—
46	#127×C₅₉	—	2	16	no	—
47	#127×C₅₉	—	2	8	no	—
48	#127×C₅₉	—	2	8	no	—
49	#125×C₅₉	—	2	8	no	—
50	#124×C₅₉	—	4	32	no	—
51	C₂²⋊Q₈×C₅₉	—	2	16	no	—
52	C₄×Q₈×C₅₉	—	2	16	no	—
53	#117×C₅₉	—	2	8	no	—
54	#117×C₅₉	—	2	8	no	—
55	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
59	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₅₉	C₂³	1	64	yes	—
62	C₂×C₄²×C₅₉	D₄	1	8	yes	—
63	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
64	C₂×C₄²×C₅₉	C₂³	1	8	yes	—
65	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
66	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
69	#95×C₅₉	C₂³	1	8	yes	—
70	C₂⁴×C₅₉	—	2	8	no	—
71	C₂⁴×C₅₉	—	2	16	no	—
72	C₂⁴×C₅₉	—	4	16	no	—
73	C₂⁴×C₅₉	C₂⁴	1	8	yes	—
74	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
75	C₂×Q₈×C₅₉	—	2	8	no	—
76	C₂×Q₈×C₅₉	—	2	16	no	—
77	C₂×D₄×C₅₉	—	2	16	no	—
78	C₂×D₄×C₅₉	—	2	8	no	—
79	C₂×D₄×C₅₉	—	2	8	no	—
80	C₂×D₄×C₅₉	—	2	16	no	—
81	C₂×D₄×C₅₉	—	2	16	no	—
82	C₂×D₄×C₅₉	—	2	16	no	—
83	C₂×D₄×C₅₉	—	2	32	no	—
84	C₂×D₄×C₅₉	—	4	16	no	—
85	C₂×D₄×C₅₉	—	4	64	no	—
86	C₂×D₄×C₅₉	—	4	32	no	—
87	C₂²×C₄×C₅₉	—	2	8	no	—
88	C₂²×C₄×C₅₉	C₂×D₄	1	8	yes	—
89	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₅₉	#41	1	8	yes	—
91	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
92	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
95	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
97	C₄⋊C₄×C₅₉	—	2	8	no	—
98	C₄⋊C₄×C₅₉	—	2	16	no	—
99	C₄⋊C₄×C₅₉	—	2	8	no	—
100	C₄⋊C₄×C₅₉	—	2	8	no	—
101	C₂²⋊C₄×C₅₉	—	2	8	no	—
102	C₂²⋊C₄×C₅₉	—	2	8	no	—
103	C₂²⋊C₄×C₅₉	—	4	64	no	—
104	C₄²×C₅₉	—	2	8	no	—
105	C₄²×C₅₉	—	2	8	no	—
106	C₄²×C₅₉	—	2	16	no	—
107	C₄²×C₅₉	C₂×D₄	1	8	yes	—
108	C₄²×C₅₉	C₂×D₄	1	4	yes	—
109	C₄²×C₅₉	C₂×D₄	1	4	yes	—
110	C₂³×C₅₉	—	2	4	no	—
111	C₂³×C₅₉	—	2	8	no	—
112	C₂³×C₅₉	—	2	16	no	—
113	C₂³×C₅₉	—	2	32	no	—
114	C₂³×C₅₉	—	4	32	no	—
115	C₂³×C₅₉	—	4	16	no	—
116	C₂³×C₅₉	—	8	32	no	—
117	C₂³×C₅₉	C₂²×D₄	1	2	yes	—
118	C₂³×C₅₉	C₂×#41	1	4	yes	—
119	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
120	Q₈×C₅₉	—	4	16	no	—
121	D₄×C₅₉	—	4	32	no	—
122	D₄×C₅₉	—	4	16	no	—
123	D₄×C₅₉	—	4	16	no	—
124	D₄×C₅₉	—	4	32	no	—
125	C₂×C₄×C₅₉	—	2	4	no	—
126	C₂×C₄×C₅₉	—	2	16	no	—
127	C₂×C₄×C₅₉	—	2	8	no	—
128	C₂×C₄×C₅₉	—	2	8	no	—
129	C₂×C₄×C₅₉	—	2	16	no	—
130	C₂×C₄×C₅₉	—	2	16	no	—
131	C₂×C₄×C₅₉	#124	1	16	yes	complemented
132	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
136	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
137	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
138	C₂×C₄×C₅₉	H(5,𝔽₂)	1	8	yes	—
139	#35285	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₅₉	—	2	4	no	—
141	C₂²×C₅₉	—	2	8	no	—
142	C₂²×C₅₉	—	4	8	no	—
143	C₂²×C₅₉	—	4	16	no	—
144	C₂²×C₅₉	—	4	32	no	—
145	C₂²×C₅₉	—	8	32	no	—
146	C₂²×C₅₉	C₂×#124	1	4	yes	—
147	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
148	C₂²×C₅₉	C₂²×#41	1	2	yes	—
149	C₂²×C₅₉	C₂×H(5,𝔽₂)	1	4	yes	—
150	C₄×C₅₉	—	2	4	no	—
151	C₄×C₅₉	—	4	8	no	—
152	C₄×C₅₉	—	4	8	no	—
153	C₄×C₅₉	C₂×#124	1	4	yes	—
154	C₄×C₅₉	C₂×#124	1	4	yes	—
155	C₄×C₅₉	#535	1	8	yes	—
156	C₂×#533	C₂×C₅₉	1	1	yes	characteristic, complemented
157	C₂²×#127	C₂×C₅₉	1	1	yes	characteristic
158	#2858	C₂×C₅₉	1	4	yes	complemented
159	#2641	C₂×C₅₉	1	4	yes	complemented
160	#2548	C₂×C₅₉	1	8	yes	complemented
161	#2435	C₂×C₅₉	1	8	yes	complemented
162	#2309	C₂×C₅₉	1	4	yes	complemented
163	C₄×#117	C₂×C₅₉	1	1	yes	characteristic, complemented
164	C₂×C₅₉	—	2	4	no	—
165	C₂×C₅₉	—	8	8	no	—
166	C₂×C₅₉	C₂²×#124	1	1	yes	characteristic
167	C₂×C₅₉	C₂×#535	1	2	yes	—
168	C₂×C₅₉	#3508	1	4	yes	—
169	C₂³×D₄	C₂²×C₅₉	1	2	yes	—
170	C₂³×D₄	C₂²×C₅₉	1	4	yes	—
171	#533	—	2	8	no	—
172	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	complemented
173	C₂×#127	—	2	8	no	—
174	C₂×#127	C₂²×C₅₉	1	2	yes	—
175	C₂×#127	C₂²×C₅₉	1	2	yes	—
176	C₂×#127	C₂²×C₅₉	1	4	yes	—
177	C₂×#125	C₂²×C₅₉	1	2	yes	complemented
178	C₂×#124	C₂²×C₅₉	1	16	yes	complemented
179	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	4	yes	—
180	C₂×C₄×Q₈	C₂²×C₅₉	1	4	yes	complemented
181	C₂×#117	C₂²×C₅₉	1	2	yes	—
182	C₂×#117	C₂²×C₅₉	1	2	yes	—
183	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
184	#394	C₂²×C₅₉	1	32	yes	complemented
185	#390	C₂²×C₅₉	1	16	yes	complemented
186	#390	C₂²×C₅₉	1	16	yes	complemented
187	#390	C₂²×C₅₉	1	16	yes	complemented
188	#379	C₂²×C₅₉	1	16	yes	—
189	C₄×C₄⋊C₄	C₂²×C₅₉	1	4	yes	complemented
190	C₄×C₂²⋊C₄	C₂²×C₅₉	1	4	yes	—
191	#376	C₂²×C₅₉	1	2	yes	complemented
192	C₄³	C₂²×C₅₉	1	2	yes	complemented
193	C₅₉	#35285	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁵	C₂³×C₅₉	1	4	yes	—
195	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
196	C₂²×D₄	—	2	16	no	—
197	C₂²×D₄	—	2	8	no	—
198	C₂²×D₄	—	2	16	no	—
199	C₂²×D₄	—	4	16	no	—
200	C₂²×D₄	C₂³×C₅₉	1	8	yes	—
201	C₂²×D₄	C₂³×C₅₉	1	4	yes	—
202	C₂²×D₄	C₂³×C₅₉	1	4	yes	—
203	C₂²×D₄	C₂³×C₅₉	1	8	yes	—
204	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
205	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	8	no	—
207	#127	—	2	16	no	—
208	#127	—	2	8	no	—
209	#127	—	2	8	no	—
210	#125	—	2	8	no	—
211	#124	—	4	32	no	—
212	C₂²⋊Q₈	—	2	16	no	—
213	C₄×Q₈	—	2	16	no	—
214	#117	—	2	8	no	—
215	#117	—	2	8	no	—
216	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
217	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
218	C₂×C₄⋊C₄	C₂³×C₅₉	1	8	yes	—
219	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
222	C₂×C₂²⋊C₄	C₂³×C₅₉	1	64	yes	—
223	C₂×C₄²	D₄×C₅₉	1	8	yes	—
224	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
225	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
226	C₂×C₄²	C₂³×C₅₉	1	8	yes	—
227	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
228	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
229	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
230	#95	C₂³×C₅₉	1	8	yes	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	2	16	no	—
233	C₂⁴	—	4	16	no	—
234	C₂⁴	C₂⁴×C₅₉	1	8	yes	—
235	C₂⁴	C₂⁴×C₅₉	1	1	yes	characteristic
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	—	2	16	no	—
238	C₂×D₄	—	2	16	no	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	2	16	no	—
242	C₂×D₄	—	2	16	no	—
243	C₂×D₄	—	2	16	no	—
244	C₂×D₄	—	2	32	no	—
245	C₂×D₄	—	4	16	no	—
246	C₂×D₄	—	4	64	no	—
247	C₂×D₄	—	4	32	no	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	C₂×D₄×C₅₉	1	8	yes	—
250	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
251	C₂²×C₄	#41×C₅₉	1	8	yes	—
252	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
253	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
254	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
257	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₂²⋊C₄	—	2	8	no	—
264	C₂²⋊C₄	—	4	64	no	—
265	C₄²	—	2	8	no	—
266	C₄²	—	2	8	no	—
267	C₄²	—	2	16	no	—
268	C₄²	C₂×D₄×C₅₉	1	8	yes	—
269	C₄²	C₂×D₄×C₅₉	1	4	yes	—
270	C₄²	C₂×D₄×C₅₉	1	4	yes	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	8	no	—
273	C₂³	—	2	16	no	—
274	C₂³	—	2	32	no	—
275	C₂³	—	4	32	no	—
276	C₂³	—	4	16	no	—
277	C₂³	—	8	32	no	—
278	C₂³	C₂²×D₄×C₅₉	1	2	yes	—
279	C₂³	C₂×#41×C₅₉	1	4	yes	—
280	C₂³	C₂⁵×C₅₉	1	1	yes	characteristic
281	Q₈	—	4	16	no	—
282	D₄	—	4	32	no	—
283	D₄	—	4	16	no	—
284	D₄	—	4	16	no	—
285	D₄	—	4	32	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	16	no	—
291	C₂×C₄	—	2	16	no	—
292	C₂×C₄	#124×C₅₉	1	16	yes	complemented
293	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
296	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
297	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
298	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
299	C₂×C₄	H(5,𝔽₂)×C₅₉	1	8	yes	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	32	no	—
303	C₂²	—	4	8	no	—
304	C₂²	—	4	16	no	—
305	C₂²	—	8	32	no	—
306	C₂²	C₂×#124×C₅₉	1	4	yes	—
307	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
308	C₂²	C₂²×#41×C₅₉	1	2	yes	—
309	C₂²	C₂×H(5,𝔽₂)×C₅₉	1	4	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	8	no	—
312	C₄	—	4	8	no	—
313	C₄	C₂×#124×C₅₉	1	4	yes	—
314	C₄	C₂×#124×C₅₉	1	4	yes	—
315	C₄	C₅₉×#535	1	8	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	8	8	no	—
318	C₂	C₂²×#124×C₅₉	1	1	yes	characteristic
319	C₂	C₂×C₅₉×#535	1	2	yes	—
320	C₂	C₅₉×#3508	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₉
prime radical	C₂²×#127×C₅₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#533	C₂	semidirect
3. C₅₉×#2858	C₂	semidirect
4. C₅₉×#2641	C₂	semidirect
5. C₅₉×#2548	C₂	semidirect
6. C₅₉×#2435	C₂	semidirect
7. C₅₉×#2309	C₂	semidirect
8. C₄×#117×C₅₉	C₂	semidirect
12. C₂×C₄⋊Q₈×C₅₉	C₂²	semidirect
17. C₂×#125×C₅₉	C₂²	semidirect
18. C₂×#124×C₅₉	C₂²	semidirect
20. C₂×C₄×Q₈×C₅₉	C₂²	semidirect
24. C₅₉×#394	C₂²	semidirect
25. C₅₉×#390	C₂²	semidirect
26. C₅₉×#390	C₂²	semidirect
27. C₅₉×#390	C₂²	semidirect
29. C₄×C₄⋊C₄×C₅₉	C₂²	semidirect
31. C₅₉×#376	C₂²	semidirect
32. C₄³×C₅₉	C₂²	semidirect
131. C₂×C₄×C₅₉	#124	semidirect
139. #35285	C₅₉	direct (N × H)
156. C₂×#533	C₂×C₅₉	semidirect
158. #2858	C₂×C₅₉	semidirect
159. #2641	C₂×C₅₉	semidirect
160. #2548	C₂×C₅₉	semidirect
161. #2435	C₂×C₅₉	semidirect
162. #2309	C₂×C₅₉	semidirect
163. C₄×#117	C₂×C₅₉	semidirect
172. C₂×C₄⋊Q₈	C₂²×C₅₉	semidirect
177. C₂×#125	C₂²×C₅₉	semidirect
178. C₂×#124	C₂²×C₅₉	semidirect
180. C₂×C₄×Q₈	C₂²×C₅₉	semidirect
184. #394	C₂²×C₅₉	semidirect
185. #390	C₂²×C₅₉	semidirect
186. #390	C₂²×C₅₉	semidirect
187. #390	C₂²×C₅₉	semidirect
189. C₄×C₄⋊C₄	C₂²×C₅₉	semidirect
191. #376	C₂²×C₅₉	semidirect
192. C₄³	C₂²×C₅₉	semidirect
193. C₅₉	#35285	direct (N × H)
292. C₂×C₄	#124×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#533	2
C₂²×#127×C₅₉	2
C₅₉×#2858	2
C₅₉×#2641	2
C₅₉×#2548	2
C₅₉×#2435	2
C₅₉×#2309	2
C₄×#117×C₅₉	2
#35285	59

Subgroups — C59×#35285 #28447394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#35285 #28447394