Subgroups — C₅₉×#8235 #28420344

Order 15104.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#333	C₂	1	2	yes	maximal
5	C₅₉×#1296	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₅₉	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
10	C₂×#97×C₅₉	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	—	2	1	no	—
12	C₅₉×#391	C₄	1	2	yes	—
13	C₅₉×#389	—	2	1	no	—
14	C₅₉×#389	—	2	1	no	—
15	C₅₉×#384	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
18	C₄³×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#336	—	2	1	no	—
20	C₅₉×#336	—	2	1	no	—
21	C₅₉×#333	C₄	1	8	yes	complemented
22	C₂²×Q₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₉	—	2	2	no	—
24	C₂²×C₈×C₅₉	—	2	2	no	—
25	C₄⋊Q₈×C₅₉	—	2	2	no	—
26	C₄⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
27	C₄×Q₈×C₅₉	—	4	4	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₉	—	2	2	no	—
35	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₉	—	2	2	no	—
43	C₄⋊C₈×C₅₉	—	2	2	no	—
44	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
45	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
46	#97×C₅₉	—	4	2	no	—
47	C₄×C₈×C₅₉	—	4	2	no	—
48	#95×C₅₉	—	2	2	no	—
49	#95×C₅₉	—	2	2	no	—
50	C₂×Q₈×C₅₉	—	2	2	no	—
51	C₂×Q₈×C₅₉	—	4	2	no	—
52	C₂×Q₈×C₅₉	—	4	2	no	—
53	C₂×Q₈×C₅₉	C₄²	1	4	yes	—
54	C₂²×C₄×C₅₉	—	2	2	no	—
55	C₂²×C₄×C₅₉	—	2	2	no	—
56	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	—	2	4	no	—
64	C₂×C₈×C₅₉	—	2	4	no	—
65	C₂×C₈×C₅₉	—	4	2	no	—
66	C₂×C₈×C₅₉	—	4	2	no	—
67	C₂×C₈×C₅₉	—	4	2	no	—
68	C₂×C₈×C₅₉	—	4	2	no	—
69	C₄⋊C₄×C₅₉	—	2	4	no	—
70	C₄⋊C₄×C₅₉	—	2	4	no	—
71	C₄⋊C₄×C₅₉	—	4	2	no	—
72	C₄⋊C₄×C₅₉	—	4	4	no	—
73	C₄⋊C₄×C₅₉	—	4	2	no	—
74	C₄⋊C₄×C₅₉	—	4	1	no	—
75	C₄⋊C₄×C₅₉	—	4	1	no	—
76	C₄²×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	—	2	2	no	—
78	C₄²×C₅₉	—	2	1	no	—
79	C₄²×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	—	2	2	no	—
81	C₄²×C₅₉	—	4	1	no	—
82	C₄²×C₅₉	—	4	4	no	—
83	C₄²×C₅₉	SD₁₆	1	2	yes	—
84	C₄²×C₅₉	Q₁₆	1	2	yes	—
85	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₅₉	#41	1	2	yes	—
87	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	Q₈×C₅₉	—	4	4	no	—
89	Q₈×C₅₉	—	4	4	no	—
90	C₂×C₄×C₅₉	—	2	1	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	4	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	4	2	no	—
107	C₂×C₄×C₅₉	#100	1	2	yes	—
108	C₂×C₄×C₅₉	#103	1	2	yes	—
109	C₂×C₄×C₅₉	#103	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#135	1	2	yes	—
117	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#137	1	2	yes	—
119	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
120	C₈×C₅₉	—	4	4	no	—
121	C₈×C₅₉	—	4	4	no	—
122	#8235	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₅₉	C₂×#100	1	1	yes	characteristic
125	C₂²×C₅₉	#413	1	1	yes	characteristic
126	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
127	C₂²×C₅₉	#416	1	1	yes	characteristic
128	C₂²×C₅₉	#417	1	1	yes	characteristic
129	C₂²×C₅₉	#419	1	1	yes	characteristic
130	C₄×C₅₉	—	2	2	no	—
131	C₄×C₅₉	—	2	2	no	—
132	C₄×C₅₉	—	2	2	no	—
133	C₄×C₅₉	—	2	1	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	4	1	no	—
136	C₄×C₅₉	—	4	4	no	—
137	C₄×C₅₉	—	8	2	no	—
138	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
139	C₄×C₅₉	C₄×Q₁₆	1	2	yes	—
140	C₄×C₅₉	#440	1	2	yes	—
141	C₄×C₅₉	#441	1	2	yes	—
142	C₄×C₄⋊Q₈	C₂×C₅₉	1	1	yes	characteristic
143	#1767	C₂×C₅₉	1	1	yes	characteristic
144	C₄×C₄⋊C₈	C₂×C₅₉	1	1	yes	characteristic
145	C₂×#333	C₂×C₅₉	1	2	yes	—
146	#1296	C₂×C₅₉	1	2	yes	—
147	C₂×C₅₉	#1548	1	1	yes	characteristic
148	C₂×C₅₉	#1550	1	1	yes	characteristic
149	C₂×C₅₉	#1551	1	1	yes	characteristic
150	C₂×C₅₉	#1554	1	1	yes	characteristic
151	C₂×C₅₉	C₄×#100	1	1	yes	characteristic
152	C₂×C₅₉	C₄×#103	1	1	yes	characteristic
153	C₂×C₅₉	#1763	1	1	yes	characteristic
154	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₄×C₅₉	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
158	C₂×#97	—	2	1	no	—
159	C₂×C₄×C₈	—	2	1	no	—
160	#391	C₄×C₅₉	1	2	yes	—
161	#389	—	2	1	no	—
162	#389	—	2	1	no	—
163	#384	C₂²×C₅₉	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄×C₄⋊C₄	—	2	1	no	—
166	C₄³	C₂²×C₅₉	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#336	—	2	1	no	—
169	#333	C₄×C₅₉	1	8	yes	complemented
170	C₅₉	#8235	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×Q₈	C₂×C₄×C₅₉	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₂×C₄×C₅₉	1	4	yes	—
176	C₄×Q₈	—	4	4	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	2	yes	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂×C₄×C₅₉	1	4	yes	—
194	C₄⋊C₈	C₂×C₄×C₅₉	1	4	yes	—
195	#97	—	4	2	no	—
196	C₄×C₈	—	4	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	C₄²×C₅₉	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	4	no	—
232	C₄²	SD₁₆×C₅₉	1	2	yes	—
233	C₄²	Q₁₆×C₅₉	1	2	yes	—
234	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
235	C₄²	#41×C₅₉	1	2	yes	—
236	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	4	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	#100×C₅₉	1	2	yes	—
257	C₂×C₄	#103×C₅₉	1	2	yes	—
258	C₂×C₄	#103×C₅₉	1	2	yes	—
259	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
260	C₂×C₄	#117×C₅₉	1	1	yes	characteristic
261	C₂×C₄	C₄×D₄×C₅₉	1	2	yes	—
262	C₂×C₄	C₄×D₄×C₅₉	1	2	yes	—
263	C₂×C₄	C₂×SD₁₆×C₅₉	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₁₆×C₅₉	1	1	yes	characteristic
265	C₂×C₄	#135×C₅₉	1	2	yes	—
266	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
267	C₂×C₄	#137×C₅₉	1	2	yes	—
268	C₂×C₄	#137×C₅₉	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	—	4	4	no	—
271	C₂²	C₄×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
272	C₂²	C₂×#100×C₅₉	1	1	yes	characteristic
273	C₂²	C₅₉×#413	1	1	yes	characteristic
274	C₂²	C₂×#103×C₅₉	1	1	yes	characteristic
275	C₂²	C₅₉×#416	1	1	yes	characteristic
276	C₂²	C₅₉×#417	1	1	yes	characteristic
277	C₂²	C₅₉×#419	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	4	no	—
284	C₄	—	4	1	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄×SD₁₆×C₅₉	1	2	yes	—
287	C₄	C₄×Q₁₆×C₅₉	1	2	yes	—
288	C₄	C₅₉×#440	1	2	yes	—
289	C₄	C₅₉×#441	1	2	yes	—
290	C₂	C₅₉×#1548	1	1	yes	characteristic
291	C₂	C₅₉×#1550	1	1	yes	characteristic
292	C₂	C₅₉×#1551	1	1	yes	characteristic
293	C₂	C₅₉×#1554	1	1	yes	characteristic
294	C₂	C₄×#100×C₅₉	1	1	yes	characteristic
295	C₂	C₄×#103×C₅₉	1	1	yes	characteristic
296	C₂	C₅₉×#1763	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₅₉×#333	C₄	semidirect
122. #8235	C₅₉	direct (N × H)
169. #333	C₄×C₅₉	semidirect
170. C₅₉	#8235	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₉	2
C₅₉×#1767	2
C₄×C₄⋊C₈×C₅₉	2
C₂×C₅₉×#333	2
C₅₉×#1296	2
#8235	59

Subgroups — C59×#8235 #28420344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8235 #28420344