Subgroups — C₂×#21425860 #28741687

Order 15104.

698 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₉	C₂	1	2	yes	maximal
2	C₂×#16837356	C₂	1	1	yes	maximal, characteristic
3	#21425860	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837222	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11745659	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
9	C₂×#14273684	C₄	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	2	yes	—
11	C₂×#14273666	—	2	2	no	—
12	M₃₂×D₅₉	—	2	4	no	—
13	M₃₂×D₅₉	C₂²	1	8	yes	—
14	C₂×C₁₆×D₅₉	—	2	2	no	—
15	#16837356	C₄	1	4	yes	—
16	#16837356	C₂²	1	4	yes	—
17	C₂×#14273638	C₄	1	1	yes	characteristic
18	C₂×#14273637	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
20	#16837223	C₂²	1	4	yes	complemented
21	#16837222	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#364	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×#11745659	C₈	1	4	yes	—
27	C₂×#11745659	C₈	1	4	yes	—
28	C₂×#11745659	C₂×C₄	1	4	yes	—
29	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₉	D₄	1	2	yes	—
33	C₂×C₈×D₅₉	D₄	1	2	yes	—
34	C₂×C₈×D₅₉	Q₈	1	2	yes	—
35	C₂×C₈×D₅₉	Q₈	1	2	yes	—
36	C₂×C₈×D₅₉	C₂³	1	2	yes	—
37	#14273684	—	2	2	no	—
38	#14273684	C₂×C₄	1	4	yes	—
39	C₂×Dic₅₉⋊C₄	—	2	1	no	—
40	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
41	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
42	#14273667	—	2	2	no	—
43	#14273667	C₂³	1	4	yes	—
44	#14273666	—	2	8	no	—
45	C₂×C₅₉⋊C₁₆	—	2	2	no	—
46	C₁₆×D₅₉	—	2	8	no	—
47	#14273638	C₂×C₄	1	4	yes	—
48	#14273637	C₂×C₄	1	4	yes	—
49	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
50	C₈×Dic₅₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₅₉	—	2	2	no	—
55	M₃₂×C₅₉	C₂³	1	4	yes	—
56	C₂×C₁₆×C₅₉	—	2	2	no	—
57	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₉	C₂³	1	2	yes	—
59	C₂³×D₅₉	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	M₁₆	1	1	yes	characteristic
64	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₅₉	M₁₆	1	2	yes	—
69	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₄²×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
81	(C₂×D₅₉)⋊C₄	—	2	4	no	—
82	Dic₅₉⋊C₄	—	2	4	no	—
83	C₄×Dic₅₉	—	2	1	no	—
84	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
85	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
88	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
89	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
91	#11745659	C₂×C₈	1	16	yes	—
92	C₈×D₅₉	C₄⋊C₄	1	4	yes	—
93	C₈×D₅₉	C₄⋊C₄	1	4	yes	—
94	C₈×D₅₉	C₂×D₄	1	4	yes	—
95	C₈×D₅₉	C₂×Q₈	1	4	yes	—
96	C₁₆×C₅₉	—	2	4	no	—
97	C₅₉⋊C₁₆	—	2	4	no	—
98	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
99	C₂²×D₅₉	—	4	2	no	—
100	C₂²×D₅₉	C₄⋊C₈	1	2	yes	—
101	C₂²×D₅₉	C₄⋊C₈	1	2	yes	—
102	C₂²×D₅₉	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₅₉	—	2	2	no	—
112	C₂×Dic₅₉	—	4	2	no	—
113	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
114	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
115	C₂×Dic₅₉	C₂×M₁₆	1	2	yes	—
116	C₄×D₅₉	C₄⋊C₈	1	4	yes	—
117	C₄×D₅₉	C₄⋊C₈	1	4	yes	—
118	C₄×D₅₉	C₂×C₄⋊C₄	1	4	yes	—
119	C₄×D₅₉	C₂²×C₈	1	4	yes	—
120	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₅₉	C₂²×C₈	1	4	yes	—
123	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
125	C₅₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
126	C₅₉⋊C₈	C₂²×C₈	1	4	yes	—
127	C₂×#2153	—	59	1	no	maximal, Sylow
128	C₂²×C₅₉	—	2	2	no	—
129	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
132	C₂×D₅₉	—	2	4	no	—
133	C₂×D₅₉	—	4	2	no	—
134	C₂×D₅₉	—	4	2	no	—
135	C₂×D₅₉	C₂×C₄⋊C₈	1	4	yes	—
136	C₄×C₅₉	—	4	2	no	—
137	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
141	Dic₅₉	—	4	2	no	—
142	Dic₅₉	C₂×C₄⋊C₈	1	4	yes	—
143	C₂²×M₃₂	—	59	2	no	—
144	C₂×#405	—	59	1	no	—
145	#2153	—	59	8	no	—
146	C₂×#364	—	59	2	no	—
147	C₂×#364	—	59	1	no	—
148	C₂×#364	D₅₉	1	1	yes	characteristic, complemented
149	C₂×C₅₉	—	2	1	no	—
150	C₂×C₅₉	—	2	1	no	—
151	C₂×C₅₉	#2153	1	2	yes	complemented
152	C₂×C₅₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
153	D₅₉	—	2	4	no	—
154	C₂²×M₁₆	—	59	1	no	—
155	C₂³×C₈	—	59	1	no	—
156	C₂×#117	—	59	1	no	—
157	C₂×M₃₂	—	59	2	no	—
158	C₂×M₃₂	—	59	8	no	—
159	C₂×M₃₂	—	118	2	no	—
160	C₂×M₃₂	—	118	4	no	—
161	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
162	C₂²×C₁₆	—	118	2	no	—
163	#405	—	59	4	no	—
164	#405	—	59	4	no	—
165	C₂×#97	—	59	1	no	—
166	C₂×#97	—	59	1	no	—
167	C₂×C₄×C₈	—	59	1	no	—
168	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
169	#364	—	59	8	no	—
170	#364	—	59	4	no	—
171	#364	C₂×D₅₉	1	4	yes	complemented
172	C₅₉	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	59	1	no	—
174	C₂×M₁₆	—	59	4	no	—
175	C₂×M₁₆	—	59	4	no	—
176	C₂×M₁₆	—	59	4	no	—
177	C₂²×C₈	—	59	1	no	—
178	C₂²×C₈	—	59	1	no	—
179	C₂²×C₈	—	59	2	no	—
180	C₂²×C₈	—	59	2	no	—
181	C₂²×C₈	—	59	2	no	—
182	C₂²×C₈	—	59	2	no	—
183	C₂²×C₈	—	59	2	no	—
184	C₂²×C₈	—	59	2	no	—
185	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
187	#117	—	59	4	no	—
188	#117	—	118	2	no	—
189	C₂×C₄⋊C₄	—	118	1	no	—
190	C₂×C₂²⋊C₄	—	118	1	no	—
191	C₂×C₄²	—	59	1	no	—
192	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
193	M₃₂	—	2	2	no	—
194	M₃₂	—	59	4	no	—
195	M₃₂	—	118	8	no	—
196	M₃₂	—	118	2	no	—
197	M₃₂	C₂²×D₅₉	1	4	yes	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	118	8	no	—
200	C₂×C₁₆	—	118	2	no	—
201	#97	—	59	4	no	—
202	#97	—	59	4	no	—
203	C₄×C₈	—	59	2	no	—
204	C₄×C₈	—	59	2	no	—
205	C₄×C₈	C₄×D₅₉	1	2	yes	—
206	C₄×C₈	C₂²×D₅₉	1	2	yes	—
207	C₂⁴	—	59	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	59	1	no	—
210	C₂²×C₄	—	59	2	no	—
211	C₂²×C₄	—	59	2	no	—
212	C₂²×C₄	—	59	2	no	—
213	C₂²×C₄	—	59	2	no	—
214	C₂²×C₄	—	59	2	no	—
215	C₂²×C₄	—	59	2	no	—
216	C₂²×C₄	—	118	1	no	—
217	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
218	M₁₆	—	59	16	no	—
219	C₂×C₈	—	59	2	no	—
220	C₂×C₈	—	59	2	no	—
221	C₂×C₈	—	59	2	no	—
222	C₂×C₈	—	59	2	no	—
223	C₂×C₈	—	59	2	no	—
224	C₂×C₈	—	59	2	no	—
225	C₂×C₈	—	59	4	no	—
226	C₂×C₈	—	59	4	no	—
227	C₂×C₈	—	59	4	no	—
228	C₂×C₈	—	59	4	no	—
229	C₂×C₈	C₈×D₅₉	1	2	yes	—
230	C₂×C₈	C₈×D₅₉	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
233	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
234	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
236	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
238	C₄⋊C₄	—	118	4	no	—
239	C₂²⋊C₄	—	118	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	59	2	no	—
242	C₄²	—	118	1	no	—
243	C₄²	C₂×C₄×D₅₉	1	2	yes	—
244	C₁₆	—	2	4	no	—
245	C₁₆	—	118	4	no	—
246	C₂³	—	59	2	no	—
247	C₂³	—	59	2	no	—
248	C₂³	—	59	2	no	—
249	C₂³	—	236	2	no	—
250	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	59	2	no	—
254	C₂×C₄	—	59	2	no	—
255	C₂×C₄	—	59	2	no	—
256	C₂×C₄	—	59	4	no	—
257	C₂×C₄	—	59	4	no	—
258	C₂×C₄	—	59	4	no	—
259	C₂×C₄	—	59	4	no	—
260	C₂×C₄	—	118	2	no	—
261	C₂×C₄	—	236	2	no	—
262	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
263	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
268	C₈	—	59	4	no	—
269	C₈	—	59	4	no	—
270	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
271	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
272	C₈	C₂×C₈×D₅₉	1	4	yes	—
273	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	59	4	no	—
277	C₂²	—	118	4	no	—
278	C₂²	—	236	2	no	—
279	C₂²	—	236	2	no	—
280	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
281	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
282	C₂²	C₂×M₁₆×D₅₉	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	59	4	no	—
285	C₄	—	236	2	no	—
286	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
287	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
288	C₄	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
289	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	118	4	no	—
293	C₂	#21425860	1	2	yes	direct product factor
294	C₂	C₂×C₄⋊C₈×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×D₅₉	M₁₆
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	C₂×#2153
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #21425860	C₂	direct (N × H)
4. C₂×#16837223	C₂	semidirect
5. C₂×#16837222	C₂	semidirect
6. C₂×C₅₉×#364	C₂	semidirect
20. #16837223	C₂²	semidirect
21. #16837222	C₂²	semidirect
24. C₅₉×#364	C₂²	semidirect
148. C₂×#364	D₅₉	semidirect
151. C₂×C₅₉	#2153	semidirect
171. #364	C₂×D₅₉	semidirect
172. C₅₉	C₂×#2153	semidirect
293. C₂	#21425860	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×M₃₂×D₅₉	2
C₂×#16837356	2
#21425860	2
C₂×#16837223	2
C₂×#16837222	2
C₂×C₅₉×#364	2
C₂×#2153	59

Subgroups — C2×#21425860 #28741687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425860 #28741687