Subgroups — C₂×#21444732 #29649782

Order 15616.

806 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	C₂×C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839839	C₂	1	2	yes	maximal, complemented
3	#21444732	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839489	C₂	1	2	yes	maximal, complemented
5	C₂×#16839483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₈₈	C₄	1	1	yes	characteristic
8	C₂²×#11758189	—	2	2	no	—
9	C₂×C₄×D₂₄₄	C₂²	1	2	yes	—
10	C₄×D₄₈₈	C₄	1	4	yes	—
11	C₄×D₄₈₈	C₂²	1	4	yes	—
12	#16839839	—	2	4	no	—
13	#16839839	C₂²	1	8	yes	complemented
14	C₂×#14519711	C₄	1	1	yes	characteristic
15	C₂×#14519699	C₄	1	2	yes	—
16	C₂×#14519689	—	2	2	no	—
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
19	#16839489	C₂²	1	8	yes	complemented
20	#16839483	C₂²	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#334	C₂²	1	4	yes	complemented
24	C₂²×D₂₄₄	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₆₁	—	2	2	no	—
26	C₂×D₄₈₈	—	2	4	no	—
27	C₂×D₄₈₈	C₂×C₄	1	4	yes	—
28	C₂×#11758189	—	2	8	no	—
29	C₂×#11758189	—	4	8	no	—
30	C₂²×#92418	—	2	1	no	—
31	C₂²×#92418	D₄	1	2	yes	—
32	C₄×D₂₄₄	—	2	2	no	—
33	C₄×D₂₄₄	—	2	2	no	—
34	C₄×D₂₄₄	C₂×C₄	1	4	yes	—
35	C₄×D₂₄₄	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
37	C₂×#11758198	—	2	2	no	—
38	#14519711	C₂×C₄	1	4	yes	—
39	#14519699	C₂×C₄	1	8	yes	—
40	#14519689	—	2	8	no	—
41	C₄×#92418	—	2	1	no	—
42	C₄×#92418	D₄	1	2	yes	—
43	C₄×#92418	C₂³	1	2	yes	—
44	C₄×#92418	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₁	—	2	2	no	—
46	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₁	—	2	2	no	—
50	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
51	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₆₁	C₂³	1	2	yes	—
53	C₂³×D₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
57	C₂²×Dic₆₁	—	2	2	no	—
58	C₂×D₂₄₄	—	2	4	no	—
59	C₂×D₂₄₄	—	2	4	no	—
60	C₂×D₂₄₄	—	2	2	no	—
61	C₂×D₂₄₄	C₂²×C₄	1	4	yes	—
62	C₂×C₄×D₆₁	—	2	4	no	—
63	C₂×C₄×D₆₁	—	2	4	no	—
64	C₂×C₄×D₆₁	—	4	4	no	—
65	C₂×C₄×D₆₁	—	4	4	no	—
66	C₂×C₈×C₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	—	2	4	no	—
68	C₂×C₈×C₆₁	—	4	4	no	—
69	C₂×C₈×C₆₁	M₁₆	1	4	yes	complemented
70	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
72	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₁	#41	1	1	yes	characteristic
75	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
76	#11758198	—	2	4	no	—
77	#11758198	—	2	4	no	—
78	C₄⋊Dic₆₁	—	2	2	no	—
79	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
80	C₂×#92418	—	2	4	no	—
81	C₂×#92418	—	2	2	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂×#92418	C₂×D₄	1	4	yes	—
84	D₄₈₈	—	2	8	no	—
85	#11758189	—	4	16	no	—
86	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
87	C₂²×D₆₁	—	2	4	no	—
88	C₂²×D₆₁	—	4	4	no	—
89	C₂²×D₆₁	—	4	4	no	—
90	C₂²×D₆₁	—	4	4	no	—
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₆₁	#131	1	2	yes	—
97	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
101	C₂×Dic₆₁	—	2	4	no	—
102	C₂×Dic₆₁	—	4	4	no	—
103	D₂₄₄	—	2	4	no	—
104	D₂₄₄	—	2	4	no	—
105	D₂₄₄	—	4	4	no	—
106	C₄×D₆₁	—	4	4	no	—
107	C₄×D₆₁	—	4	4	no	—
108	C₄×D₆₁	—	4	8	no	—
109	C₈×C₆₁	—	2	2	no	—
110	C₈×C₆₁	—	4	4	no	—
111	C₈×C₆₁	C₂×M₁₆	1	4	yes	complemented
112	#92418	—	2	4	no	—
113	#92418	—	4	2	no	—
114	C₂×#1594	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	#436	1	2	yes	—
116	C₂²×C₆₁	#442	1	2	yes	—
117	C₂²×C₆₁	#444	1	2	yes	—
118	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×D₆₁	—	4	4	no	—
120	C₂×D₆₁	—	4	4	no	—
121	C₂×D₆₁	—	4	4	no	—
122	C₂×D₆₁	—	8	8	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	#436	1	2	yes	—
126	C₄×C₆₁	#442	1	2	yes	—
127	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
128	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
129	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	Dic₆₁	—	4	4	no	—
131	C₂×C₄×D₈	—	61	1	no	—
132	C₂×#436	—	61	2	no	—
133	#1594	—	61	8	no	—
134	C₂×#334	D₆₁	1	1	yes	characteristic, complemented
135	C₂×D₄⋊C₈	—	61	2	no	—
136	C₂×#322	—	61	1	no	—
137	C₂×C₆₁	#1594	1	4	yes	complemented
138	C₂×C₆₁	C₂×#436	1	1	yes	characteristic
139	C₂×C₆₁	C₂×#442	1	1	yes	characteristic
140	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
141	D₆₁	—	8	4	no	—
142	C₂²×D₈	—	61	1	no	—
143	C₂²×M₁₆	—	122	2	no	—
144	C₂×C₄×D₄	—	61	2	no	—
145	C₄×D₈	—	61	4	no	—
146	C₄×D₈	—	61	4	no	—
147	#436	—	61	8	no	—
148	#436	—	122	4	no	—
149	C₂×#107	—	61	1	no	—
150	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
151	C₂×#102	—	61	2	no	—
152	C₂×C₂²⋊C₈	—	122	2	no	—
153	C₂×C₄×C₈	—	61	1	no	—
154	C₂×C₄×C₈	—	61	1	no	—
155	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
156	#334	C₂×D₆₁	1	4	yes	complemented
157	D₄⋊C₈	—	61	8	no	—
158	#322	—	61	4	no	—
159	C₆₁	C₂×#1594	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	—	61	2	no	—
161	C₂³×C₄	—	122	2	no	—
162	C₂×D₈	—	61	4	no	—
163	C₂×D₈	—	122	4	no	—
164	C₂×M₁₆	—	122	8	no	—
165	C₂×M₁₆	—	244	8	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	61	2	no	—
168	C₂²×C₈	—	122	1	no	—
169	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
171	C₄×D₄	—	61	4	no	—
172	C₄×D₄	—	61	4	no	—
173	C₄×D₄	—	122	2	no	—
174	C₄×D₄	—	122	2	no	—
175	C₂×C₄⋊C₄	—	61	2	no	—
176	C₂×C₂²⋊C₄	—	122	2	no	—
177	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
178	#107	—	61	4	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂²×D₆₁	1	4	yes	—
181	#102	—	61	8	no	—
182	C₂²⋊C₈	—	122	8	no	—
183	C₄×C₈	—	61	2	no	—
184	C₄×C₈	—	61	2	no	—
185	C₄×C₈	—	61	2	no	—
186	C₄×C₈	—	122	1	no	—
187	C₄×C₈	C₄×D₆₁	1	2	yes	—
188	C₄×C₈	C₂²×D₆₁	1	2	yes	—
189	C₂⁴	—	122	2	no	—
190	C₂×D₄	—	61	4	no	—
191	C₂×D₄	—	122	2	no	—
192	C₂×D₄	—	122	4	no	—
193	C₂×D₄	—	122	4	no	—
194	C₂²×C₄	—	122	2	no	—
195	C₂²×C₄	—	122	4	no	—
196	C₂²×C₄	—	122	4	no	—
197	C₂²×C₄	—	244	4	no	—
198	C₂²×C₄	—	244	4	no	—
199	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
201	C₂²×C₄	#11758226	1	1	yes	characteristic
202	D₈	—	122	8	no	—
203	M₁₆	—	244	16	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	61	4	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	—	122	2	no	—
210	C₂×C₈	—	244	2	no	—
211	C₂×C₈	#11758189	1	4	yes	complemented
212	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
214	C₄⋊C₄	—	61	4	no	—
215	C₄⋊C₄	—	122	2	no	—
216	C₂²⋊C₄	—	122	4	no	—
217	C₂²⋊C₄	—	122	4	no	—
218	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
219	C₄²	D₄×D₆₁	1	1	yes	characteristic
220	C₄²	#11758226	1	1	yes	characteristic
221	C₄²	C₂³×D₆₁	1	1	yes	characteristic
222	C₂³	—	122	4	no	—
223	C₂³	—	244	4	no	—
224	C₂³	—	244	4	no	—
225	C₂³	—	244	4	no	—
226	C₂³	#14519739	1	1	yes	characteristic
227	D₄	—	122	4	no	—
228	D₄	—	122	4	no	—
229	D₄	—	244	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	122	4	no	—
232	C₂×C₄	—	244	4	no	—
233	C₂×C₄	—	244	4	no	—
234	C₂×C₄	—	244	4	no	—
235	C₂×C₄	—	244	8	no	—
236	C₂×C₄	#14519739	1	1	yes	characteristic
237	C₂×C₄	#14519739	1	1	yes	characteristic
238	C₂×C₄	#14519739	1	1	yes	characteristic
239	C₂×C₄	C₂×#11758189	1	2	yes	—
240	C₂×C₄	#14519785	1	2	yes	—
241	C₂×C₄	#14519804	1	4	yes	—
242	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
244	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	4	4	no	—
247	C₈	—	122	4	no	—
248	C₈	—	244	2	no	—
249	C₈	C₂×#11758189	1	4	yes	complemented
250	C₂²	—	488	8	no	—
251	C₂²	—	244	4	no	—
252	C₂²	—	244	4	no	—
253	C₂²	—	244	4	no	—
254	C₂²	#16839839	1	2	yes	—
255	C₂²	#16839873	1	2	yes	—
256	C₂²	#16839883	1	2	yes	—
257	C₂²	C₂×#14519739	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	244	4	no	—
261	C₄	#16839839	1	2	yes	—
262	C₄	#16839873	1	2	yes	—
263	C₄	C₂²×#11758189	1	1	yes	characteristic
264	C₄	C₂×#14519785	1	1	yes	characteristic
265	C₄	C₂×#14519804	1	2	yes	—
266	C₂	—	488	4	no	—
267	C₂	#21444732	1	4	yes	direct product factor
268	C₂	C₂×#16839839	1	1	yes	characteristic
269	C₂	C₂×#16839873	1	1	yes	characteristic
270	C₂	C₂×#16839883	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle	C₂³×C₆₁	C₄×D₄
prime radical	C₂²×D₄₈₈	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	C₂×#1594
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. C₂×#16839839	C₂	semidirect
3. #21444732	C₂	direct (N × H)
4. C₂×#16839489	C₂	semidirect
5. C₂×#16839483	C₂	semidirect
6. C₂×C₆₁×#334	C₂	semidirect
13. #16839839	C₂²	semidirect
19. #16839489	C₂²	semidirect
20. #16839483	C₂²	semidirect
23. C₆₁×#334	C₂²	semidirect
69. C₂×C₈×C₆₁	M₁₆	semidirect
111. C₈×C₆₁	C₂×M₁₆	semidirect
134. C₂×#334	D₆₁	semidirect
137. C₂×C₆₁	#1594	semidirect
156. #334	C₂×D₆₁	semidirect
159. C₆₁	C₂×#1594	semidirect
211. C₂×C₈	#11758189	semidirect
249. C₈	C₂×#11758189	semidirect
267. C₂	#21444732	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×C₄×D₄₈₈	2
C₂×#16839839	2
#21444732	2
C₂×#16839489	2
C₂×#16839483	2
C₂×C₆₁×#334	2
C₂×#1594	61

Subgroups — C2×#21444732 #29649782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21444732 #29649782