Subgroups — C₂²×C₅₉×#419 #28472190

Order 15104.

5070 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#117×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#419	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#103×C₅₉	C₂	1	4	yes	maximal, complemented
6	C₂³×Q₈×C₅₉	—	2	8	no	—
7	C₂³×Q₈×C₅₉	—	2	6	no	—
8	C₂³×Q₈×C₅₉	—	2	1	no	—
9	C₂³×Q₈×C₅₉	C₄	1	16	yes	complemented
10	C₂³×Q₈×C₅₉	C₂²	1	2	yes	—
11	C₂³×Q₈×C₅₉	C₂²	1	12	yes	—
12	C₂⁴×C₄×C₅₉	—	2	1	no	—
13	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₅₉	—	2	8	no	—
15	C₂²×M₁₆×C₅₉	C₂²	1	12	yes	—
16	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
17	C₂×#117×C₅₉	—	2	6	no	—
18	C₂×#117×C₅₉	C₂²	1	12	yes	—
19	C₂²×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₅₉	—	2	1	no	—
21	C₂²×C₄²×C₅₉	—	2	1	no	—
22	C₅₉×#419	C₂²	1	64	yes	direct product factor
23	C₂×#103×C₅₉	C₂²	1	48	yes	complemented
24	C₂⁵×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₅₉	—	2	16	no	—
26	C₂²×Q₈×C₅₉	—	2	96	no	—
27	C₂²×Q₈×C₅₉	—	2	8	no	—
28	C₂²×Q₈×C₅₉	—	2	48	no	—
29	C₂²×Q₈×C₅₉	—	2	12	no	—
30	C₂²×Q₈×C₅₉	—	2	6	no	—
31	C₂²×Q₈×C₅₉	—	2	24	no	—
32	C₂²×Q₈×C₅₉	C₂×C₄	1	16	yes	—
33	C₂²×Q₈×C₅₉	C₂×C₄	1	96	yes	complemented
34	C₂²×Q₈×C₅₉	C₂³	1	12	yes	—
35	C₂²×Q₈×C₅₉	C₂³	1	16	yes	—
36	C₂³×C₄×C₅₉	—	2	16	no	—
37	C₂³×C₄×C₅₉	—	2	12	no	—
38	C₂³×C₄×C₅₉	—	2	2	no	—
39	C₂³×C₄×C₅₉	—	2	2	no	—
40	C₂³×C₄×C₅₉	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₅₉	D₄	1	6	yes	—
42	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₉	D₄	1	8	yes	—
44	C₂³×C₄×C₅₉	C₂³	1	6	yes	—
45	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₅₉	—	2	48	no	—
47	C₂×M₁₆×C₅₉	C₂³	1	16	yes	—
48	C₂²×C₈×C₅₉	—	2	8	no	—
49	C₂²×C₈×C₅₉	C₂³	1	12	yes	—
50	#117×C₅₉	—	2	24	no	—
51	#117×C₅₉	C₂³	1	16	yes	—
52	C₂×C₄⋊C₄×C₅₉	—	2	6	no	—
53	C₂×C₄⋊C₄×C₅₉	C₂³	1	12	yes	—
54	C₂×C₂²⋊C₄×C₅₉	—	2	12	no	—
55	C₂×C₄²×C₅₉	—	2	12	no	—
56	#103×C₅₉	C₂³	1	64	yes	complemented
57	C₂⁴×C₅₉	—	2	8	no	—
58	C₂⁴×C₅₉	C₂²⋊C₄	1	8	yes	—
59	C₂⁴×C₅₉	C₂×D₄	1	6	yes	—
60	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₅₉	—	2	48	no	—
62	C₂×Q₈×C₅₉	—	2	96	no	—
63	C₂×Q₈×C₅₉	—	2	192	no	—
64	C₂×Q₈×C₅₉	—	2	24	no	—
65	C₂×Q₈×C₅₉	—	2	32	no	—
66	C₂×Q₈×C₅₉	—	2	16	no	—
67	C₂×Q₈×C₅₉	—	2	12	no	—
68	C₂×Q₈×C₅₉	C₂²×C₄	1	48	yes	—
69	C₂×Q₈×C₅₉	C₂²×C₄	1	64	yes	complemented
70	C₂×Q₈×C₅₉	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₅₉	—	2	16	no	—
72	C₂²×C₄×C₅₉	—	2	96	no	—
73	C₂²×C₄×C₅₉	—	2	12	no	—
74	C₂²×C₄×C₅₉	—	2	16	no	—
75	C₂²×C₄×C₅₉	—	2	12	no	—
76	C₂²×C₄×C₅₉	—	4	8	no	—
77	C₂²×C₄×C₅₉	C₂²⋊C₄	1	48	yes	—
78	C₂²×C₄×C₅₉	C₂²⋊C₄	1	8	yes	—
79	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₅₉	C₂²×C₄	1	24	yes	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
82	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₅₉	C₂×D₄	1	12	yes	—
84	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₅₉	C₂×D₄	1	24	yes	—
86	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
87	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
88	M₁₆×C₅₉	—	2	32	no	—
89	C₂×C₈×C₅₉	—	2	24	no	—
90	C₂×C₈×C₅₉	C₂⁴	1	8	yes	—
91	C₄⋊C₄×C₅₉	—	2	12	no	—
92	C₄⋊C₄×C₅₉	C₂⁴	1	8	yes	—
93	C₂²⋊C₄×C₅₉	—	2	16	no	—
94	C₄²×C₅₉	—	2	16	no	—
95	C₂³×C₅₉	—	2	24	no	—
96	C₂³×C₅₉	—	2	32	no	—
97	C₂³×C₅₉	C₂×C₂²⋊C₄	1	24	yes	—
98	C₂³×C₅₉	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂³×C₅₉	#137	1	8	yes	—
100	C₂³×C₅₉	C₂²×D₄	1	4	yes	—
101	C₂³×C₅₉	C₂²×D₄	1	3	yes	—
102	Q₈×C₅₉	—	2	48	no	—
103	Q₈×C₅₉	—	2	64	no	—
104	Q₈×C₅₉	—	2	8	no	—
105	Q₈×C₅₉	C₂³×C₄	1	16	yes	—
106	C₂×C₄×C₅₉	—	2	48	no	—
107	C₂×C₄×C₅₉	—	2	64	no	—
108	C₂×C₄×C₅₉	—	2	8	no	—
109	C₂×C₄×C₅₉	—	2	8	no	—
110	C₂×C₄×C₅₉	—	4	24	no	—
111	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	48	yes	—
112	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	12	yes	—
113	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	24	yes	—
114	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
115	C₂×C₄×C₅₉	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
117	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
118	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
119	C₂×C₄×C₅₉	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
121	C₈×C₅₉	—	2	8	no	—
122	C₂²×#419	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₉	—	2	48	no	—
124	C₂²×C₅₉	—	2	8	no	—
125	C₂²×C₅₉	#419	1	16	yes	direct product factor
126	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	8	yes	—
127	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	6	yes	—
128	C₂²×C₅₉	C₂×#137	1	12	yes	—
129	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₅₉	—	2	16	no	—
131	C₄×C₅₉	—	4	8	no	—
132	C₄×C₅₉	C₂²×C₂²⋊C₄	1	6	yes	—
133	C₄×C₅₉	C₂²×C₂²⋊C₄	1	8	yes	—
134	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
135	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
136	C₂⁴×Q₈	C₂×C₅₉	1	1	yes	characteristic
137	C₂³×M₁₆	C₂×C₅₉	1	1	yes	characteristic
138	C₂²×#117	C₂×C₅₉	1	1	yes	characteristic
139	C₂×#419	C₂×C₅₉	1	24	yes	direct product factor
140	C₂²×#103	C₂×C₅₉	1	4	yes	complemented
141	C₂×C₅₉	—	2	8	no	—
142	C₂×C₅₉	C₂×#419	1	12	yes	direct product factor
143	C₂×C₅₉	C₂³×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₅₉	C₂²×#137	1	2	yes	—
145	C₂³×Q₈	—	2	8	no	—
146	C₂³×Q₈	—	2	6	no	—
147	C₂³×Q₈	—	2	1	no	—
148	C₂³×Q₈	C₄×C₅₉	1	16	yes	complemented
149	C₂³×Q₈	C₂²×C₅₉	1	2	yes	—
150	C₂³×Q₈	C₂²×C₅₉	1	12	yes	—
151	C₂⁴×C₄	—	2	1	no	—
152	C₂⁴×C₄	C₂²×C₅₉	1	1	yes	characteristic
153	C₂²×M₁₆	—	2	8	no	—
154	C₂²×M₁₆	C₂²×C₅₉	1	12	yes	—
155	C₂³×C₈	C₂²×C₅₉	1	2	yes	—
156	C₂×#117	—	2	6	no	—
157	C₂×#117	C₂²×C₅₉	1	12	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
159	C₂²×C₂²⋊C₄	—	2	1	no	—
160	C₂²×C₄²	—	2	1	no	—
161	#419	C₂²×C₅₉	1	64	yes	direct product factor
162	C₂×#103	C₂²×C₅₉	1	48	yes	complemented
163	C₅₉	C₂²×#419	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁵	D₄×C₅₉	1	1	yes	characteristic
165	C₂²×Q₈	—	2	16	no	—
166	C₂²×Q₈	—	2	96	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	48	no	—
169	C₂²×Q₈	—	2	12	no	—
170	C₂²×Q₈	—	2	6	no	—
171	C₂²×Q₈	—	2	24	no	—
172	C₂²×Q₈	C₂×C₄×C₅₉	1	16	yes	—
173	C₂²×Q₈	C₂×C₄×C₅₉	1	96	yes	complemented
174	C₂²×Q₈	C₂³×C₅₉	1	12	yes	—
175	C₂²×Q₈	C₂³×C₅₉	1	16	yes	—
176	C₂³×C₄	—	2	16	no	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	—	2	12	no	—
179	C₂³×C₄	—	2	2	no	—
180	C₂³×C₄	C₂×C₄×C₅₉	1	8	yes	—
181	C₂³×C₄	D₄×C₅₉	1	6	yes	—
182	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
183	C₂³×C₄	D₄×C₅₉	1	8	yes	—
184	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₅₉	1	6	yes	—
186	C₂×M₁₆	—	2	48	no	—
187	C₂×M₁₆	C₂³×C₅₉	1	16	yes	—
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	C₂³×C₅₉	1	12	yes	—
190	#117	—	2	24	no	—
191	#117	C₂³×C₅₉	1	16	yes	—
192	C₂×C₄⋊C₄	—	2	6	no	—
193	C₂×C₄⋊C₄	C₂³×C₅₉	1	12	yes	—
194	C₂×C₂²⋊C₄	—	2	12	no	—
195	C₂×C₄²	—	2	12	no	—
196	#103	C₂³×C₅₉	1	64	yes	complemented
197	C₂⁴	—	2	8	no	—
198	C₂⁴	C₂²⋊C₄×C₅₉	1	8	yes	—
199	C₂⁴	C₂×D₄×C₅₉	1	6	yes	—
200	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
201	C₂×Q₈	—	2	48	no	—
202	C₂×Q₈	—	2	96	no	—
203	C₂×Q₈	—	2	192	no	—
204	C₂×Q₈	—	2	24	no	—
205	C₂×Q₈	—	2	32	no	—
206	C₂×Q₈	—	2	16	no	—
207	C₂×Q₈	—	2	12	no	—
208	C₂×Q₈	C₂²×C₄×C₅₉	1	48	yes	—
209	C₂×Q₈	C₂²×C₄×C₅₉	1	64	yes	complemented
210	C₂×Q₈	C₂⁴×C₅₉	1	8	yes	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	96	no	—
213	C₂²×C₄	—	2	12	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	2	12	no	—
216	C₂²×C₄	—	4	8	no	—
217	C₂²×C₄	C₂²⋊C₄×C₅₉	1	48	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₅₉	1	8	yes	—
219	C₂²×C₄	C₂²×C₄×C₅₉	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₅₉	1	24	yes	—
221	C₂²×C₄	C₂×D₄×C₅₉	1	6	yes	—
222	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
223	C₂²×C₄	C₂×D₄×C₅₉	1	12	yes	—
224	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₅₉	1	24	yes	—
226	C₂²×C₄	C₂⁴×C₅₉	1	3	yes	—
227	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
228	M₁₆	—	2	32	no	—
229	C₂×C₈	—	2	24	no	—
230	C₂×C₈	C₂⁴×C₅₉	1	8	yes	—
231	C₄⋊C₄	—	2	12	no	—
232	C₄⋊C₄	C₂⁴×C₅₉	1	8	yes	—
233	C₂²⋊C₄	—	2	16	no	—
234	C₄²	—	2	16	no	—
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	C₂×C₂²⋊C₄×C₅₉	1	24	yes	—
238	C₂³	C₂×C₂²⋊C₄×C₅₉	1	4	yes	—
239	C₂³	#137×C₅₉	1	8	yes	—
240	C₂³	C₂²×D₄×C₅₉	1	4	yes	—
241	C₂³	C₂²×D₄×C₅₉	1	3	yes	—
242	Q₈	—	2	48	no	—
243	Q₈	—	2	64	no	—
244	Q₈	—	2	8	no	—
245	Q₈	C₂³×C₄×C₅₉	1	16	yes	—
246	C₂×C₄	—	2	48	no	—
247	C₂×C₄	—	2	64	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	24	no	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	48	yes	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	12	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	24	yes	—
254	C₂×C₄	C₂³×C₄×C₅₉	1	6	yes	—
255	C₂×C₄	C₂³×C₄×C₅₉	1	8	yes	—
256	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
257	C₂×C₄	C₂²×D₄×C₅₉	1	3	yes	—
258	C₂×C₄	C₂²×D₄×C₅₉	1	6	yes	—
259	C₂×C₄	C₂²×D₄×C₅₉	1	8	yes	—
260	C₂×C₄	C₂⁵×C₅₉	1	1	yes	characteristic
261	C₈	—	2	8	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	48	no	—
264	C₂²	C₅₉×#419	1	16	yes	direct product factor
265	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	8	yes	—
266	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	6	yes	—
267	C₂²	C₂×#137×C₅₉	1	12	yes	—
268	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
269	C₄	—	2	16	no	—
270	C₄	—	4	8	no	—
271	C₄	C₂²×C₂²⋊C₄×C₅₉	1	6	yes	—
272	C₄	C₂²×C₂²⋊C₄×C₅₉	1	8	yes	—
273	C₄	C₂⁴×C₄×C₅₉	1	1	yes	characteristic
274	C₄	C₂³×D₄×C₅₉	1	1	yes	characteristic
275	C₂	—	2	8	no	—
276	C₂	C₂×C₅₉×#419	1	12	yes	direct product factor
277	C₂	C₂³×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
278	C₂	C₂²×#137×C₅₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₉	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₅₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₅₉
prime radical	C₂⁵×C₅₉	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₉ ◁ C₂⁴×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅₉×#419	C₂	direct (N × H)
5. C₂²×#103×C₅₉	C₂	semidirect
9. C₂³×Q₈×C₅₉	C₄	semidirect
22. C₅₉×#419	C₂²	direct (N × H)
23. C₂×#103×C₅₉	C₂²	semidirect
33. C₂²×Q₈×C₅₉	C₂×C₄	semidirect
56. #103×C₅₉	C₂³	semidirect
69. C₂×Q₈×C₅₉	C₂²×C₄	semidirect
122. C₂²×#419	C₅₉	direct (N × H)
125. C₂²×C₅₉	#419	direct (N × H)
139. C₂×#419	C₂×C₅₉	direct (N × H)
140. C₂²×#103	C₂×C₅₉	semidirect
142. C₂×C₅₉	C₂×#419	direct (N × H)
148. C₂³×Q₈	C₄×C₅₉	semidirect
161. #419	C₂²×C₅₉	direct (N × H)
162. C₂×#103	C₂²×C₅₉	semidirect
163. C₅₉	C₂²×#419	direct (N × H)
173. C₂²×Q₈	C₂×C₄×C₅₉	semidirect
196. #103	C₂³×C₅₉	semidirect
209. C₂×Q₈	C₂²×C₄×C₅₉	semidirect
264. C₂²	C₅₉×#419	direct (N × H)
276. C₂	C₂×C₅₉×#419	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₅₉	2
C₂³×M₁₆×C₅₉	2
C₂²×#117×C₅₉	2
C₂×C₅₉×#419	2
C₂²×#103×C₅₉	2
C₂²×#419	59

Subgroups — C22×C59×#419 #28472190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₉×#419 #28472190