Subgroups — C₂×C₅₉×#1561 #28429845

Order 15104.

1042 conjugacy classes of subgroups.

The subgroup lattice has 262 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
2	C₂×C₅₉×#423	C₂	1	2	yes	maximal
3	C₅₉×#1561	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₉×#332	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₉	—	2	2	no	—
6	C₅₉×#533	—	2	4	no	—
7	C₅₉×#533	C₄	1	4	yes	—
8	C₅₉×#533	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
10	C₂×#125×C₅₉	C₄	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₅₉	—	2	2	no	—
12	C₂×C₄×Q₈×C₅₉	—	2	2	no	—
13	C₂×#117×C₅₉	C₄	1	2	yes	—
14	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#423	—	2	4	no	—
16	C₅₉×#423	C₂²	1	8	yes	—
17	C₂×C₄⋊C₈×C₅₉	—	2	2	no	—
18	C₂×C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
19	C₅₉×#332	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₅₉	—	2	2	no	—
21	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₉	D₄	1	2	yes	—
23	C₂×M₁₆×C₅₉	—	2	8	no	—
24	C₂×M₁₆×C₅₉	—	4	8	no	—
25	C₂²×C₈×C₅₉	—	2	4	no	—
26	C₄⋊Q₈×C₅₉	—	2	2	no	—
27	C₄⋊Q₈×C₅₉	—	2	2	no	—
28	C₄⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
29	C₄⋊Q₈×C₅₉	C₂³	1	4	yes	—
30	#125×C₅₉	—	2	4	no	—
31	#125×C₅₉	C₂×C₄	1	8	yes	—
32	C₂²⋊Q₈×C₅₉	—	2	8	no	—
33	C₂²⋊Q₈×C₅₉	—	2	8	no	—
34	C₄×Q₈×C₅₉	—	2	8	no	—
35	C₄×Q₈×C₅₉	—	2	8	no	—
36	#117×C₅₉	—	2	4	no	—
37	#117×C₅₉	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
42	C₂×C₄²×C₅₉	—	2	2	no	—
43	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₉	D₄	1	2	yes	—
46	C₂×C₄²×C₅₉	D₄	1	4	yes	—
47	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
49	C₄⋊C₈×C₅₉	—	2	8	no	—
50	C₄⋊C₈×C₅₉	—	2	4	no	—
51	C₄⋊C₈×C₅₉	C₂³	1	8	yes	—
52	C₂⁴×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈×C₅₉	—	2	4	no	—
54	C₂×Q₈×C₅₉	—	2	4	no	—
55	C₂×Q₈×C₅₉	—	4	4	no	—
56	C₂×Q₈×C₅₉	—	4	4	no	—
57	C₂²×C₄×C₅₉	—	2	4	no	—
58	C₂²×C₄×C₅₉	—	2	4	no	—
59	C₂²×C₄×C₅₉	—	2	4	no	—
60	C₂²×C₄×C₅₉	—	2	4	no	—
61	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
69	M₁₆×C₅₉	—	4	16	no	—
70	C₂×C₈×C₅₉	—	2	8	no	—
71	C₂×C₈×C₅₉	—	4	8	no	—
72	C₄⋊C₄×C₅₉	—	2	16	no	—
73	C₄⋊C₄×C₅₉	—	2	4	no	—
74	C₄⋊C₄×C₅₉	—	2	4	no	—
75	C₄⋊C₄×C₅₉	—	2	4	no	—
76	C₄⋊C₄×C₅₉	C₂²×C₄	1	8	yes	—
77	C₂²⋊C₄×C₅₉	—	2	8	no	—
78	C₄²×C₅₉	—	2	8	no	—
79	C₄²×C₅₉	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×D₄	1	4	yes	—
84	C₄²×C₅₉	C₂×D₄	1	2	yes	—
85	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₂³×C₅₉	—	2	2	no	—
87	C₂³×C₅₉	—	2	4	no	—
88	C₂³×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₅₉	—	4	4	no	—
91	Q₈×C₅₉	—	4	4	no	—
92	Q₈×C₅₉	—	4	8	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	4	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	—	2	4	no	—
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	—	2	4	no	—
99	C₂×C₄×C₅₉	—	2	8	no	—
100	C₂×C₄×C₅₉	—	2	8	no	—
101	C₂×C₄×C₅₉	—	4	8	no	—
102	C₂×C₄×C₅₉	#101	1	4	yes	—
103	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
108	C₂×C₄×C₅₉	#137	1	4	yes	—
109	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
111	C₈×C₅₉	—	4	8	no	—
112	C₂×#1561	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₉	—	2	2	no	—
114	C₂²×C₅₉	—	2	4	no	—
115	C₂²×C₅₉	—	4	4	no	—
116	C₂²×C₅₉	C₂×#101	1	2	yes	—
117	C₂²×C₅₉	#417	1	4	yes	—
118	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	2	1	no	—
121	C₄×C₅₉	—	2	4	no	—
122	C₄×C₅₉	—	4	8	no	—
123	C₄×C₅₉	C₂×#101	1	4	yes	—
124	C₄×C₅₉	#417	1	4	yes	—
125	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₅₉	C₂×#137	1	2	yes	—
127	C₂×#533	C₂×C₅₉	1	1	yes	characteristic
128	C₂×#423	C₂×C₅₉	1	2	yes	—
129	#1561	C₂×C₅₉	1	8	yes	direct product factor
130	C₂×#332	C₂×C₅₉	1	4	yes	complemented
131	C₂×C₅₉	—	4	2	no	—
132	C₂×C₅₉	#1561	1	4	yes	direct product factor
133	C₂×C₅₉	C₂²×#101	1	1	yes	characteristic
134	C₂×C₅₉	C₂×#417	1	2	yes	—
135	C₂²×M₁₆	—	2	2	no	—
136	#533	—	2	4	no	—
137	#533	C₄×C₅₉	1	4	yes	—
138	#533	C₂²×C₅₉	1	4	yes	—
139	C₂×C₄⋊Q₈	C₂²×C₅₉	1	2	yes	—
140	C₂×#125	C₄×C₅₉	1	2	yes	—
141	C₂×C₂²⋊Q₈	—	2	2	no	—
142	C₂×C₄×Q₈	—	2	2	no	—
143	C₂×#117	C₄×C₅₉	1	2	yes	—
144	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
145	#423	—	2	4	no	—
146	#423	C₂²×C₅₉	1	8	yes	—
147	C₂×C₄⋊C₈	—	2	2	no	—
148	C₂×C₄⋊C₈	C₂²×C₅₉	1	4	yes	—
149	#332	C₂²×C₅₉	1	16	yes	complemented
150	C₅₉	C₂×#1561	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×Q₈	—	2	2	no	—
152	C₂³×C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₅₉	1	2	yes	—
154	C₂×M₁₆	—	2	8	no	—
155	C₂×M₁₆	—	4	8	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	C₂×C₄×C₅₉	1	4	yes	—
160	C₄⋊Q₈	C₂³×C₅₉	1	4	yes	—
161	#125	—	2	4	no	—
162	#125	C₂×C₄×C₅₉	1	8	yes	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₂²⋊Q₈	—	2	8	no	—
165	C₄×Q₈	—	2	8	no	—
166	C₄×Q₈	—	2	8	no	—
167	#117	—	2	4	no	—
168	#117	C₂×C₄×C₅₉	1	8	yes	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
176	C₂×C₄²	D₄×C₅₉	1	2	yes	—
177	C₂×C₄²	D₄×C₅₉	1	4	yes	—
178	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
180	C₄⋊C₈	—	2	8	no	—
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	C₂³×C₅₉	1	8	yes	—
183	C₂⁴	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₅₉	1	4	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
195	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₅₉	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
200	M₁₆	—	4	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	4	8	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×C₄×C₅₉	1	8	yes	—
208	C₂²⋊C₄	—	2	8	no	—
209	C₄²	—	2	8	no	—
210	C₄²	C₂²⋊C₄×C₅₉	1	4	yes	—
211	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
212	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
213	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₅₉	1	4	yes	—
215	C₄²	C₂×D₄×C₅₉	1	2	yes	—
216	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
220	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	#101×C₅₉	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
238	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	4	yes	—
239	C₂×C₄	#137×C₅₉	1	4	yes	—
240	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
242	C₈	—	4	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×#101×C₅₉	1	2	yes	—
247	C₂²	C₅₉×#417	1	4	yes	—
248	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×#101×C₅₉	1	4	yes	—
254	C₄	C₅₉×#417	1	4	yes	—
255	C₄	C₂×AGL(1,ℤ/8)×C₅₉	1	2	yes	—
256	C₄	C₂×#137×C₅₉	1	2	yes	—
257	C₂	—	4	2	no	—
258	C₂	C₅₉×#1561	1	4	yes	direct product factor
259	C₂	C₂²×#101×C₅₉	1	1	yes	characteristic
260	C₂	C₂×C₅₉×#417	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₄²	C₂⁴×C₅₉
prime radical	C₂⁴×C₅₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₉×#1561	C₂	direct (N × H)
4. C₂×C₅₉×#332	C₂	semidirect
19. C₅₉×#332	C₂²	semidirect
112. C₂×#1561	C₅₉	direct (N × H)
129. #1561	C₂×C₅₉	direct (N × H)
130. C₂×#332	C₂×C₅₉	semidirect
132. C₂×C₅₉	#1561	direct (N × H)
149. #332	C₂²×C₅₉	semidirect
150. C₅₉	C₂×#1561	direct (N × H)
258. C₂	C₅₉×#1561	direct (N × H)

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
C₂×C₅₉×#423	2
C₅₉×#1561	2
C₂×C₅₉×#332	2
C₂×#1561	59

Subgroups — C2×C59×#1561 #28429845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#1561 #28429845