Subgroups — C₅₉×#29209 #28441318

Order 15104.

686 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2058	C₂	1	2	yes	maximal
4	C₅₉×#2057	C₂	1	2	yes	maximal
5	C₅₉×#1991	C₂	1	2	yes	maximal
6	C₅₉×#1949	C₂	1	4	yes	maximal
7	C₅₉×#1846	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
10	C₂×#125×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₅₉	C₂²	1	4	yes	—
12	C₂×#106×C₅₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#391	—	2	2	no	—
16	C₅₉×#391	—	2	2	no	—
17	C₅₉×#389	C₂²	1	2	yes	—
18	C₅₉×#387	—	2	2	no	—
19	C₅₉×#382	C₂²	1	2	yes	—
20	C₅₉×#382	C₂²	1	1	yes	characteristic
21	C₅₉×#382	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
23	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
24	C₄³×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#340	C₂²	1	4	yes	—
26	C₅₉×#340	C₂²	1	4	yes	—
27	C₅₉×#336	C₂²	1	1	yes	characteristic
28	C₅₉×#336	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₅₉	—	4	4	no	—
33	#125×C₅₉	—	2	2	no	—
34	#125×C₅₉	Q₈	1	4	yes	complemented
35	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
41	C₂×C₄²×C₅₉	—	2	2	no	—
42	C₂×C₄²×C₅₉	D₄	1	2	yes	—
43	C₂×C₄²×C₅₉	D₄	1	2	yes	—
44	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	#107×C₅₉	—	4	4	no	—
48	#106×C₅₉	—	4	4	no	—
49	C₄⋊C₈×C₅₉	Q₈	1	8	yes	complemented
50	C₄×C₈×C₅₉	—	2	4	no	—
51	#95×C₅₉	—	2	2	no	—
52	#95×C₅₉	—	2	2	no	—
53	#95×C₅₉	—	2	2	no	—
54	#95×C₅₉	—	2	4	no	—
55	#95×C₅₉	—	2	2	no	—
56	#95×C₅₉	—	2	2	no	—
57	C₂×Q₈×C₅₉	—	4	4	no	—
58	C₂×Q₈×C₅₉	—	4	4	no	—
59	C₂×Q₈×C₅₉	—	4	4	no	—
60	C₂²×C₄×C₅₉	—	2	2	no	—
61	C₂²×C₄×C₅₉	—	2	2	no	—
62	C₂²×C₄×C₅₉	—	2	4	no	—
63	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₅₉	—	2	2	no	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
74	C₄⋊C₄×C₅₉	—	2	8	no	—
75	C₄⋊C₄×C₅₉	—	2	2	no	—
76	C₄⋊C₄×C₅₉	—	2	4	no	—
77	C₄⋊C₄×C₅₉	—	2	4	no	—
78	C₄⋊C₄×C₅₉	—	2	4	no	—
79	C₄⋊C₄×C₅₉	—	4	4	no	—
80	C₄⋊C₄×C₅₉	—	4	4	no	—
81	C₄⋊C₄×C₅₉	C₂×Q₈	1	4	yes	—
82	C₄²×C₅₉	—	2	4	no	—
83	C₄²×C₅₉	—	2	2	no	—
84	C₄²×C₅₉	—	4	2	no	—
85	C₄²×C₅₉	—	4	4	no	—
86	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
87	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
88	C₄²×C₅₉	#41	1	2	yes	—
89	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₅₉	—	8	8	no	—
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	—	4	2	no	—
99	C₂×C₄×C₅₉	—	4	2	no	—
100	C₂×C₄×C₅₉	—	4	4	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	—	4	4	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	—	4	4	no	—
106	C₂×C₄×C₅₉	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₉	#135	1	4	yes	—
110	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
114	C₈×C₅₉	—	2	4	no	—
115	C₈×C₅₉	—	2	4	no	—
116	#29209	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
118	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₂²×C₅₉	#535	1	1	yes	characteristic
120	C₂²×C₅₉	C₂×#135	1	2	yes	—
121	C₂²×C₅₉	#575	1	2	yes	—
122	C₄×C₅₉	—	2	4	no	—
123	C₄×C₅₉	—	2	2	no	—
124	C₄×C₅₉	—	2	2	no	—
125	C₄×C₅₉	—	2	4	no	—
126	C₄×C₅₉	—	4	4	no	—
127	C₄×C₅₉	—	8	2	no	—
128	C₄×C₅₉	—	8	4	no	—
129	C₄×C₅₉	#554	1	2	yes	—
130	C₄×C₅₉	Q₈²	1	2	yes	—
131	#2862	C₂×C₅₉	1	1	yes	characteristic
132	C₄×#125	C₂×C₅₉	1	1	yes	characteristic
133	#2058	C₂×C₅₉	1	2	yes	—
134	#2057	C₂×C₅₉	1	2	yes	—
135	#1991	C₂×C₅₉	1	2	yes	—
136	#1949	C₂×C₅₉	1	4	yes	—
137	#1846	C₂×C₅₉	1	2	yes	—
138	C₄×C₄⋊C₈	C₂×C₅₉	1	1	yes	characteristic
139	C₂×C₅₉	#2567	1	1	yes	characteristic
140	C₂×C₅₉	#3076	1	2	yes	—
141	C₂×C₅₉	#3160	1	1	yes	characteristic
142	C₂×C₅₉	#3166	1	1	yes	characteristic
143	C₂×C₅₉	#3211	1	2	yes	—
144	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
145	C₂×#125	C₂²×C₅₉	1	1	yes	characteristic
146	C₂×#107	C₂²×C₅₉	1	4	yes	—
147	C₂×#106	C₂²×C₅₉	1	4	yes	—
148	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
150	#391	—	2	2	no	—
151	#391	—	2	2	no	—
152	#389	C₂²×C₅₉	1	2	yes	—
153	#387	—	2	2	no	—
154	#382	C₂²×C₅₉	1	1	yes	characteristic
155	#382	C₂²×C₅₉	1	2	yes	—
156	#382	C₂²×C₅₉	1	2	yes	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
159	C₄³	C₂²×C₅₉	1	1	yes	characteristic
160	#340	C₂²×C₅₉	1	4	yes	—
161	#340	C₂²×C₅₉	1	4	yes	—
162	#336	C₂²×C₅₉	1	1	yes	characteristic
163	#336	C₂²×C₅₉	1	1	yes	characteristic
164	C₅₉	#29209	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
167	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
168	C₄⋊Q₈	—	4	4	no	—
169	#125	—	2	2	no	—
170	#125	Q₈×C₅₉	1	4	yes	complemented
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
175	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	D₄×C₅₉	1	2	yes	—
179	C₂×C₄²	D₄×C₅₉	1	2	yes	—
180	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
183	#107	—	4	4	no	—
184	#106	—	4	4	no	—
185	C₄⋊C₈	Q₈×C₅₉	1	8	yes	complemented
186	C₄×C₈	—	2	4	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	4	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
203	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
209	C₂×C₈	C₂×Q₈×C₅₉	1	4	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂×Q₈×C₅₉	1	4	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	C₂×Q₈×C₅₉	1	2	yes	—
223	C₄²	C₂×Q₈×C₅₉	1	4	yes	—
224	C₄²	#41×C₅₉	1	2	yes	—
225	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
226	Q₈	—	8	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₂²⋊Q₈×C₅₉	1	4	yes	—
243	C₂×C₄	C₄⋊Q₈×C₅₉	1	2	yes	—
244	C₂×C₄	C₄⋊Q₈×C₅₉	1	2	yes	—
245	C₂×C₄	#135×C₅₉	1	4	yes	—
246	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
247	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
249	C₂×C₄	H(5,𝔽₂)×C₅₉	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
253	C₂²	C₂×C₄⋊Q₈×C₅₉	1	1	yes	characteristic
254	C₂²	C₅₉×#535	1	1	yes	characteristic
255	C₂²	C₂×#135×C₅₉	1	2	yes	—
256	C₂²	C₅₉×#575	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	C₅₉×#554	1	2	yes	—
265	C₄	Q₈²×C₅₉	1	2	yes	—
266	C₂	C₅₉×#2567	1	1	yes	characteristic
267	C₂	C₅₉×#3076	1	2	yes	—
268	C₂	C₅₉×#3160	1	1	yes	characteristic
269	C₂	C₅₉×#3166	1	1	yes	characteristic
270	C₂	C₅₉×#3211	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
34. #125×C₅₉	Q₈	semidirect
49. C₄⋊C₈×C₅₉	Q₈	semidirect
116. #29209	C₅₉	direct (N × H)
164. C₅₉	#29209	direct (N × H)
170. #125	Q₈×C₅₉	semidirect
185. C₄⋊C₈	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2862	2
C₄×#125×C₅₉	2
C₅₉×#2058	2
C₅₉×#2057	2
C₅₉×#1991	2
C₅₉×#1949	2
C₅₉×#1846	2
C₄×C₄⋊C₈×C₅₉	2
#29209	59

Subgroups — C59×#29209 #28441318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#29209 #28441318