Subgroups — C₂×C₄×#14519716 #29832641

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519716	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#14519716	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839537	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₆₁	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₆₁	C₄	1	1	yes	characteristic
10	C₂×#14519716	C₄	1	16	yes	direct product factor
11	C₂×#14519716	C₂²	1	8	yes	—
12	C₂²×#11758185	C₂²	1	2	yes	—
13	C₂×C₄×#92418	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₆₁	C₄	1	4	yes	—
17	C₄×C₈×D₆₁	C₂²	1	4	yes	—
18	#16839537	C₂²	1	8	yes	complemented
19	C₄×#11758185	C₂²	1	4	yes	complemented
20	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₆₁	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₆₁	C₈	1	2	yes	—
24	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₆₁	C₈	1	8	yes	—
28	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₆₁	C₂×C₄	1	8	yes	—
30	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₆₁	C₂³	1	2	yes	—
32	C₂×C₄×Dic₆₁	C₈	1	2	yes	—
33	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₆₁	C₈	1	4	yes	—
35	C₄²×D₆₁	C₂×C₄	1	4	yes	—
36	#14519716	C₂×C₄	1	32	yes	direct product factor
37	C₂×#11758185	C₂×C₄	1	8	yes	complemented
38	C₂×#11758185	C₂³	1	4	yes	—
39	C₄×#92418	C₂×C₄	1	4	yes	—
40	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
41	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
42	C₈×Dic₆₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
47	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
48	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₁	C₂³	1	2	yes	—
50	C₂³×D₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
54	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×D₆₁	—	2	8	no	—
56	C₂×C₄×D₆₁	C₄²	1	4	yes	—
57	C₂×C₄×D₆₁	C₂×C₈	1	8	yes	—
58	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₆₁	C₂×C₈	1	4	yes	—
60	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
66	C₄²×C₆₁	C₂×C₈	1	2	yes	—
67	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
68	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
69	C₄×Dic₆₁	C₂×C₈	1	8	yes	—
70	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
71	C₂×#92418	C₄²	1	4	yes	—
72	C₂×#92418	C₂×C₈	1	4	yes	—
73	C₂×#92418	C₂²×C₄	1	2	yes	—
74	C₂×#92418	C₂²×C₄	1	2	yes	—
75	C₈×D₆₁	C₄²	1	8	yes	—
76	C₈×D₆₁	C₂×C₈	1	16	yes	—
77	C₈×D₆₁	C₂²×C₄	1	8	yes	—
78	C₁₆×C₆₁	C₂²×C₄	1	8	yes	complemented
79	#11758185	C₂²×C₄	1	8	yes	complemented
80	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
81	C₂²×D₆₁	—	2	4	no	—
82	C₂²×D₆₁	C₄×C₈	1	4	yes	—
83	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₆₁	M₃₂	1	8	yes	complemented
85	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₆₁	—	2	4	no	—
91	C₂×Dic₆₁	C₄×C₈	1	4	yes	—
92	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
93	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
94	C₄×D₆₁	—	2	16	no	—
95	C₄×D₆₁	C₄×C₈	1	8	yes	—
96	C₄×D₆₁	C₂×C₄²	1	4	yes	—
97	C₄×D₆₁	C₂²×C₈	1	4	yes	—
98	C₈×C₆₁	C₂×C₄²	1	2	yes	—
99	C₈×C₆₁	C₂²×C₈	1	4	yes	—
100	C₈×C₆₁	C₂³×C₄	1	2	yes	—
101	#92418	C₂×C₄²	1	4	yes	—
102	#92418	C₂²×C₈	1	4	yes	—
103	C₂×C₄×M₃₂	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
105	C₂²×C₆₁	C₂×M₃₂	1	4	yes	—
106	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
107	C₂×D₆₁	—	2	8	no	—
108	C₂×D₆₁	—	2	4	no	—
109	C₂×D₆₁	C₂×C₄×C₈	1	4	yes	—
110	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
111	C₄×C₆₁	C₂×M₃₂	1	8	yes	complemented
112	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
114	Dic₆₁	—	2	4	no	—
115	Dic₆₁	C₂×C₄×C₈	1	4	yes	—
116	C₂²×M₃₂	—	61	2	no	—
117	C₂²×C₄×C₈	—	61	1	no	—
118	C₄×M₃₂	—	61	8	no	—
119	C₂×#346	—	61	2	no	—
120	C₂×C₄×C₁₆	—	61	1	no	—
121	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
122	C₂×C₆₁	C₄×M₃₂	1	4	yes	complemented
123	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
124	C₂×C₆₁	C₂²×M₃₂	1	2	yes	—
125	D₆₁	—	2	4	no	—
126	C₂³×C₈	—	61	1	no	—
127	C₂³×C₈	—	61	1	no	—
128	C₂²×C₄²	—	61	1	no	—
129	C₂×M₃₂	—	61	16	no	—
130	C₂×M₃₂	—	61	8	no	—
131	C₂²×C₁₆	—	61	2	no	—
132	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
133	C₂×C₄×C₈	—	61	1	no	—
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	—	61	1	no	—
136	C₂×C₄×C₈	—	61	4	no	—
137	C₂×C₄×C₈	—	61	4	no	—
138	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
139	#346	—	61	8	no	—
140	C₄×C₁₆	—	61	4	no	—
141	C₄×C₁₆	C₂×D₆₁	1	4	yes	complemented
142	C₆₁	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₄	—	61	1	no	—
144	C₂³×C₄	—	61	2	no	—
145	C₂²×C₈	—	61	1	no	—
146	C₂²×C₈	—	61	1	no	—
147	C₂²×C₈	—	61	8	no	—
148	C₂²×C₈	—	61	2	no	—
149	C₂²×C₈	—	61	4	no	—
150	C₂²×C₈	—	61	8	no	—
151	C₂²×C₈	—	61	2	no	—
152	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
154	C₂×C₄²	—	61	2	no	—
155	C₂×C₄²	—	61	1	no	—
156	C₂×C₄²	—	61	4	no	—
157	C₂×C₄²	—	61	4	no	—
158	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
159	M₃₂	—	61	32	no	—
160	C₂×C₁₆	—	61	8	no	—
161	C₂×C₁₆	—	61	4	no	—
162	C₂×C₁₆	C₄×D₆₁	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
164	C₄×C₈	—	61	2	no	—
165	C₄×C₈	—	61	4	no	—
166	C₄×C₈	—	61	2	no	—
167	C₄×C₈	—	61	4	no	—
168	C₄×C₈	C₄×D₆₁	1	2	yes	—
169	C₄×C₈	C₂²×D₆₁	1	2	yes	—
170	C₂⁴	—	61	1	no	—
171	C₂²×C₄	—	61	1	no	—
172	C₂²×C₄	—	61	2	no	—
173	C₂²×C₄	—	61	4	no	—
174	C₂²×C₄	—	61	2	no	—
175	C₂²×C₄	—	61	4	no	—
176	C₂²×C₄	—	61	2	no	—
177	C₂²×C₄	—	61	8	no	—
178	C₂²×C₄	—	122	8	no	—
179	C₂²×C₄	C₈×D₆₁	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
181	C₂×C₈	—	61	2	no	—
182	C₂×C₈	—	61	2	no	—
183	C₂×C₈	—	61	4	no	—
184	C₂×C₈	—	61	4	no	—
185	C₂×C₈	—	61	8	no	—
186	C₂×C₈	—	61	8	no	—
187	C₂×C₈	—	61	16	no	—
188	C₂×C₈	C₈×D₆₁	1	4	yes	—
189	C₂×C₈	C₂×C₄×D₆₁	1	4	yes	—
190	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
192	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
193	C₄²	—	61	2	no	—
194	C₄²	—	61	8	no	—
195	C₄²	—	61	2	no	—
196	C₄²	C₈×D₆₁	1	2	yes	—
197	C₄²	C₂×C₄×D₆₁	1	2	yes	—
198	C₁₆	—	61	8	no	—
199	C₁₆	C₂×C₄×D₆₁	1	8	yes	complemented
200	C₂³	—	61	4	no	—
201	C₂³	—	61	2	no	—
202	C₂³	—	122	4	no	—
203	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
204	C₂×C₄	—	61	4	no	—
205	C₂×C₄	—	61	2	no	—
206	C₂×C₄	—	61	4	no	—
207	C₂×C₄	—	61	8	no	—
208	C₂×C₄	—	61	4	no	—
209	C₂×C₄	—	61	4	no	—
210	C₂×C₄	—	122	4	no	—
211	C₂×C₄	—	122	16	no	—
212	C₂×C₄	#14519716	1	8	yes	direct product factor
213	C₂×C₄	C₄²×D₆₁	1	2	yes	—
214	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
215	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
216	C₂×C₄	C₂×C₈×D₆₁	1	4	yes	—
217	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
218	C₈	—	61	4	no	—
219	C₈	—	61	4	no	—
220	C₈	C₄²×D₆₁	1	2	yes	—
221	C₈	C₂×C₈×D₆₁	1	4	yes	—
222	C₈	C₂²×C₄×D₆₁	1	2	yes	—
223	C₂²	—	61	4	no	—
224	C₂²	—	122	4	no	—
225	C₂²	—	122	8	no	—
226	C₂²	C₄×C₈×D₆₁	1	2	yes	—
227	C₂²	C₂×#14519716	1	4	yes	—
228	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
229	C₄	—	61	4	no	—
230	C₄	—	122	4	no	—
231	C₄	C₄×C₈×D₆₁	1	2	yes	—
232	C₄	C₂×#14519716	1	8	yes	direct product factor
233	C₄	C₂×C₄²×D₆₁	1	1	yes	characteristic
234	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
235	C₂	—	122	4	no	—
236	C₂	C₄×#14519716	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₈×D₆₁	1	1	yes	characteristic
238	C₂	C₂²×#14519716	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₆₁
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂³×D₆₁	C₂×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#14519716	C₂	direct (N × H)
4. C₂×#16839537	C₂	semidirect
5. C₂×C₄×#11758185	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
10. C₂×#14519716	C₄	direct (N × H)
18. #16839537	C₂²	semidirect
19. C₄×#11758185	C₂²	semidirect
22. C₄×C₁₆×C₆₁	C₂²	semidirect
36. #14519716	C₂×C₄	direct (N × H)
37. C₂×#11758185	C₂×C₄	semidirect
46. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
78. C₁₆×C₆₁	C₂²×C₄	semidirect
79. #11758185	C₂²×C₄	semidirect
84. C₂×C₄×C₆₁	M₃₂	semidirect
111. C₄×C₆₁	C₂×M₃₂	semidirect
121. C₂×C₄×C₁₆	D₆₁	semidirect
122. C₂×C₆₁	C₄×M₃₂	semidirect
141. C₄×C₁₆	C₂×D₆₁	semidirect
142. C₆₁	C₂×C₄×M₃₂	semidirect
162. C₂×C₁₆	C₄×D₆₁	semidirect
199. C₁₆	C₂×C₄×D₆₁	semidirect
212. C₂×C₄	#14519716	direct (N × H)
232. C₄	C₂×#14519716	direct (N × H)
236. C₂	C₄×#14519716	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519716	2
C₂×C₄×C₈×D₆₁	2
C₄×#14519716	2
C₂×#16839537	2
C₂×C₄×#11758185	2
C₂×C₄×C₁₆×C₆₁	2
C₂×C₄×M₃₂	61

Subgroups — C2×C4×#14519716 #29832641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14519716 #29832641