Subgroups — C₅₉×#32679 #28444788

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2568	C₂	1	2	yes	maximal
2	C₅₉×#2056	C₂	1	2	yes	maximal
3	C₅₉×#2016	C₂	1	4	yes	maximal
4	C₅₉×#1985	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1971	C₂	1	2	yes	maximal
6	C₅₉×#1927	C₂	1	4	yes	maximal
7	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#106×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#103×C₅₉	C₂²	1	4	yes	—
14	C₂×#103×C₅₉	C₂²	1	4	yes	—
15	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#400	—	2	2	no	—
17	C₅₉×#400	—	2	2	no	—
18	C₅₉×#391	—	2	2	no	—
19	C₅₉×#391	C₂²	1	4	yes	—
20	C₅₉×#391	C₂²	1	4	yes	—
21	C₅₉×#382	—	2	2	no	—
22	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
23	C₅₉×#340	C₂²	1	2	yes	—
24	C₅₉×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
27	C₂×Q₁₆×C₅₉	—	2	4	no	—
28	C₂×Q₁₆×C₅₉	—	4	2	no	—
29	C₂×Q₁₆×C₅₉	—	4	2	no	—
30	C₂×Q₁₆×C₅₉	—	4	4	no	—
31	C₂×Q₁₆×C₅₉	—	4	2	no	—
32	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₅₉	—	4	2	no	—
35	C₄⋊Q₈×C₅₉	—	4	2	no	—
36	C₄×Q₈×C₅₉	—	4	1	no	—
37	C₄×Q₈×C₅₉	—	4	1	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₅₉	—	2	2	no	—
44	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	#106×C₅₉	—	4	2	no	—
47	C₄⋊C₈×C₅₉	—	4	1	no	—
48	#103×C₅₉	—	2	8	no	—
49	#103×C₅₉	—	4	4	no	—
50	#97×C₅₉	—	4	1	no	—
51	#95×C₅₉	—	2	2	no	—
52	#95×C₅₉	—	2	4	no	—
53	#95×C₅₉	—	2	4	no	—
54	#95×C₅₉	—	2	4	no	—
55	#95×C₅₉	—	2	4	no	—
56	C₂×Q₈×C₅₉	—	2	4	no	—
57	C₂×Q₈×C₅₉	—	2	4	no	—
58	C₂×Q₈×C₅₉	—	2	4	no	—
59	C₂×Q₈×C₅₉	—	2	4	no	—
60	C₂×Q₈×C₅₉	—	2	4	no	—
61	C₂×Q₈×C₅₉	—	2	4	no	—
62	C₂×Q₈×C₅₉	—	2	4	no	—
63	C₂×Q₈×C₅₉	—	4	1	no	—
64	C₂²×C₄×C₅₉	—	2	4	no	—
65	C₂²×C₄×C₅₉	—	2	4	no	—
66	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
71	Q₁₆×C₅₉	—	4	8	no	—
72	Q₁₆×C₅₉	—	4	8	no	—
73	C₂×C₈×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	—	2	4	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	#41	1	4	yes	—
78	C₄⋊C₄×C₅₉	—	2	4	no	—
79	C₄⋊C₄×C₅₉	—	2	4	no	—
80	C₄⋊C₄×C₅₉	—	4	2	no	—
81	C₄⋊C₄×C₅₉	—	4	1	no	—
82	C₄⋊C₄×C₅₉	—	4	4	no	—
83	C₄⋊C₄×C₅₉	—	4	4	no	—
84	C₄²×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	2	2	no	—
87	C₄²×C₅₉	—	2	2	no	—
88	C₄²×C₅₉	—	4	4	no	—
89	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₅₉	—	4	4	no	—
91	Q₈×C₅₉	—	4	4	no	—
92	Q₈×C₅₉	—	4	4	no	—
93	Q₈×C₅₉	—	4	4	no	—
94	Q₈×C₅₉	—	4	4	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	2	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	2	4	no	—
103	C₂×C₄×C₅₉	—	4	4	no	—
104	C₂×C₄×C₅₉	—	4	4	no	—
105	C₂×C₄×C₅₉	—	4	4	no	—
106	C₂×C₄×C₅₉	—	4	4	no	—
107	C₂×C₄×C₅₉	—	4	4	no	—
108	C₂×C₄×C₅₉	—	4	4	no	—
109	C₂×C₄×C₅₉	#137	1	4	yes	—
110	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
111	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
113	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₈×C₅₉	—	2	4	no	—
115	C₈×C₅₉	—	4	2	no	—
116	#32679	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₅₉	#534	1	1	yes	characteristic
118	C₂²×C₅₉	#550	1	2	yes	—
119	C₂²×C₅₉	C₂×#137	1	2	yes	—
120	C₂²×C₅₉	#578	1	2	yes	—
121	C₄×C₅₉	—	2	4	no	—
122	C₄×C₅₉	—	4	1	no	—
123	C₄×C₅₉	—	4	2	no	—
124	C₄×C₅₉	—	4	1	no	—
125	C₄×C₅₉	—	8	4	no	—
126	C₄×C₅₉	—	8	4	no	—
127	C₄×C₅₉	#538	1	2	yes	—
128	C₄×C₅₉	#555	1	1	yes	characteristic
129	C₄×C₅₉	H(3,𝔽₄)	1	1	yes	characteristic
130	#2568	C₂×C₅₉	1	2	yes	—
131	#2056	C₂×C₅₉	1	2	yes	—
132	#2016	C₂×C₅₉	1	4	yes	—
133	#1985	C₂×C₅₉	1	1	yes	characteristic
134	#1971	C₂×C₅₉	1	2	yes	—
135	#1927	C₂×C₅₉	1	4	yes	—
136	C₂×C₅₉	#2733	1	1	yes	characteristic
137	C₂×C₅₉	#3195	1	1	yes	characteristic
138	C₂×C₅₉	#3202	1	1	yes	characteristic
139	C₂×C₅₉	#3398	1	4	yes	—
140	C₂²×Q₁₆	C₂²×C₅₉	1	2	yes	—
141	C₂²×Q₁₆	C₂²×C₅₉	1	2	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
143	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
144	C₂×#106	C₂²×C₅₉	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
146	C₂×#103	C₂²×C₅₉	1	4	yes	—
147	C₂×#103	C₂²×C₅₉	1	4	yes	—
148	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
149	#400	—	2	2	no	—
150	#400	—	2	2	no	—
151	#391	—	2	2	no	—
152	#391	C₂²×C₅₉	1	4	yes	—
153	#391	C₂²×C₅₉	1	4	yes	—
154	#382	—	2	2	no	—
155	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
156	#340	C₂²×C₅₉	1	2	yes	—
157	#336	C₂²×C₅₉	1	4	yes	—
158	C₅₉	#32679	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×Q₈	D₄×C₅₉	1	1	yes	characteristic
160	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	4	2	no	—
163	C₂×Q₁₆	—	4	2	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×Q₁₆	—	4	2	no	—
166	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
167	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
168	C₄⋊Q₈	—	4	2	no	—
169	C₄⋊Q₈	—	4	2	no	—
170	C₄×Q₈	—	4	1	no	—
171	C₄×Q₈	—	4	1	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
175	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
179	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
180	#106	—	4	2	no	—
181	C₄⋊C₈	—	4	1	no	—
182	#103	—	2	8	no	—
183	#103	—	4	4	no	—
184	#97	—	4	1	no	—
185	#95	—	2	2	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	#95	—	2	4	no	—
189	#95	—	2	4	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
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	C₂×D₄×C₅₉	1	2	yes	—
204	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
205	Q₁₆	—	4	8	no	—
206	Q₁₆	—	4	8	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	#41×C₅₉	1	4	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	4	no	—
223	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	Q₈	—	4	4	no	—
226	Q₈	—	4	4	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	#137×C₅₉	1	4	yes	—
244	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
245	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
246	C₂×C₄	H(5,𝔽₂)×C₅₉	1	2	yes	—
247	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	—	4	2	no	—
250	C₂²	C₅₉×#534	1	1	yes	characteristic
251	C₂²	C₅₉×#550	1	2	yes	—
252	C₂²	C₂×#137×C₅₉	1	2	yes	—
253	C₂²	C₅₉×#578	1	2	yes	—
254	C₄	—	2	4	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	4	no	—
259	C₄	—	8	4	no	—
260	C₄	C₅₉×#538	1	2	yes	—
261	C₄	C₅₉×#555	1	1	yes	characteristic
262	C₄	C₅₉×H(3,𝔽₄)	1	1	yes	characteristic
263	C₂	C₅₉×#2733	1	1	yes	characteristic
264	C₂	C₅₉×#3195	1	1	yes	characteristic
265	C₂	C₅₉×#3202	1	1	yes	characteristic
266	C₂	C₅₉×#3398	1	4	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 ◁ C₂×C₄×Q₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
116. #32679	C₅₉	direct (N × H)
158. C₅₉	#32679	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×#2568	2
C₅₉×#2056	2
C₅₉×#2016	2
C₅₉×#1985	2
C₅₉×#1971	2
C₅₉×#1927	2
#32679	59

Subgroups — C59×#32679 #28444788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#32679 #28444788