Subgroups — C₄×M₁₆×Dic₆₁ #29677343

Order 15616.

1734 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11758194	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₆₁	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₄×#14519687	C₂	1	2	yes	maximal, complemented
5	#97×Dic₆₁	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×Dic₆₁	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11758194	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₆₁	C₄	1	4	yes	—
10	C₂²×C₄×Dic₆₁	C₂²	1	2	yes	—
11	C₄×#11758194	C₄	1	8	yes	complemented
12	M₁₆×Dic₆₁	C₄	1	32	yes	direct product factor
13	C₂×#14519688	C₂²	1	2	yes	—
14	C₂×#14519687	C₂²	1	8	yes	—
15	C₂×C₄×#92418	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₁	C₂²	1	8	yes	—
17	C₄²×Dic₆₁	C₄	1	4	yes	—
18	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₄²×Dic₆₁	C₂²	1	2	yes	—
20	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₆₁	C₄	1	8	yes	complemented
24	C₂×#97×C₆₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
26	C₂³×Dic₆₁	C₂×C₄	1	4	yes	—
27	C₂×#11758194	C₂×C₄	1	16	yes	—
28	C₂×#11758194	C₂×C₄	1	8	yes	—
29	C₂²×#92418	C₂³	1	4	yes	—
30	C₂×C₄×Dic₆₁	—	2	16	no	—
31	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₆₁	C₂×C₄	1	8	yes	—
33	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₆₁	C₂×C₄	1	16	yes	—
35	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
37	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
38	#14519688	C₂×C₄	1	8	yes	complemented
39	#14519687	C₂×C₄	1	32	yes	complemented
40	C₄×#92418	C₂×C₄	1	8	yes	complemented
41	C₈×Dic₆₁	C₂×C₄	1	32	yes	complemented
42	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₁	C₂×C₄	1	16	yes	—
45	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
52	#97×C₆₁	C₂×C₄	1	8	yes	complemented
53	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	complemented
54	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	—	2	8	no	—
56	C₂²×C₄×C₆₁	C₄²	1	8	yes	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₆₁	—	2	16	no	—
65	C₂²×Dic₆₁	C₄²	1	16	yes	—
66	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
67	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
68	M₁₆×C₆₁	C₄²	1	32	yes	complemented
69	C₂×C₈×C₆₁	C₂²×C₄	1	16	yes	—
70	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
71	C₄²×C₆₁	C₄²	1	8	yes	—
72	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
76	C₄×Dic₆₁	—	2	8	no	—
77	C₄×Dic₆₁	—	2	16	no	—
78	C₄×Dic₆₁	C₄²	1	16	yes	—
79	C₄×Dic₆₁	M₁₆	1	16	yes	direct product factor
80	C₄×Dic₆₁	C₂²×C₄	1	8	yes	—
81	C₄×Dic₆₁	C₂²×C₄	1	8	yes	—
82	#11758194	C₄²	1	32	yes	complemented
83	C₂×#92418	C₂²×C₄	1	16	yes	—
84	C₂×#92418	C₂²×C₄	1	8	yes	—
85	C₂³×C₆₁	—	2	4	no	—
86	C₂³×C₆₁	C₂×C₄²	1	4	yes	—
87	C₂³×C₆₁	C₂×C₄²	1	2	yes	—
88	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	2	16	no	—
91	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
93	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₆₁	C₂×C₄²	1	8	yes	—
95	C₂×C₄×C₆₁	C₂×C₄²	1	8	yes	—
96	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
97	C₂×C₄×C₆₁	C₂×M₁₆	1	8	yes	—
98	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₆₁	—	2	8	no	—
102	C₂×Dic₆₁	—	2	32	no	—
103	C₂×Dic₆₁	C₂×C₄²	1	8	yes	—
104	C₂×Dic₆₁	C₂×C₄²	1	8	yes	—
105	C₂×Dic₆₁	C₂×M₁₆	1	16	yes	—
106	C₈×C₆₁	C₂×C₄²	1	16	yes	complemented
107	#92418	C₂×C₄²	1	16	yes	complemented
108	C₄²×M₁₆	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	8	no	—
110	C₂²×C₆₁	—	2	4	no	—
111	C₂²×C₆₁	C₄³	1	4	yes	—
112	C₂²×C₆₁	C₂²×C₄²	1	2	yes	—
113	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₆₁	C₂²×M₁₆	1	4	yes	—
115	C₄×C₆₁	—	2	4	no	—
116	C₄×C₆₁	C₄³	1	4	yes	—
117	C₄×C₆₁	C₄×M₁₆	1	8	yes	complemented
118	C₄×C₆₁	C₂²×C₄²	1	2	yes	—
119	C₄×C₆₁	C₂²×C₄²	1	2	yes	—
120	Dic₆₁	—	2	8	no	—
121	Dic₆₁	C₄×M₁₆	1	16	yes	direct product factor
122	C₂×C₄×M₁₆	—	61	1	no	—
123	C₂×C₄×M₁₆	—	61	4	no	—
124	C₂×C₄×M₁₆	D₆₁	1	1	yes	characteristic
125	C₂×C₄³	—	61	1	no	—
126	C₄×#97	—	61	4	no	—
127	C₄×#97	—	61	2	no	—
128	C₄²×C₈	—	61	2	no	—
129	C₂×C₆₁	—	2	4	no	—
130	C₂×C₆₁	C₂×C₄³	1	1	yes	characteristic
131	C₂×C₆₁	C₂×C₄×M₁₆	1	4	yes	—
132	C₂×C₆₁	C₂×C₄×M₁₆	1	2	yes	—
133	C₂²×M₁₆	—	61	2	no	—
134	C₂²×M₁₆	C₂×D₆₁	1	2	yes	—
135	C₂²×C₄²	—	61	2	no	—
136	C₂²×C₄²	—	61	4	no	—
137	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
138	C₄×M₁₆	—	61	32	no	—
139	C₄×M₁₆	—	61	8	no	—
140	C₄×M₁₆	Dic₆₁	1	8	yes	direct product factor
141	C₂×#97	—	61	8	no	—
142	C₂×#97	—	61	2	no	—
143	C₂×#97	C₂×D₆₁	1	2	yes	—
144	C₂×C₄×C₈	—	61	8	no	—
145	C₂×C₄×C₈	—	61	2	no	—
146	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
147	C₄³	—	61	4	no	—
148	C₄³	—	61	2	no	—
149	C₄³	—	61	1	no	—
150	C₄³	—	61	1	no	—
151	C₆₁	C₄²×M₁₆	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	—	61	4	no	—
153	C₂³×C₄	C₄×D₆₁	1	2	yes	—
154	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
155	C₂×M₁₆	—	61	8	no	—
156	C₂×M₁₆	—	61	16	no	—
157	C₂×M₁₆	C₄×D₆₁	1	16	yes	—
158	C₂×M₁₆	C₂×Dic₆₁	1	8	yes	—
159	C₂²×C₈	—	61	4	no	—
160	C₂²×C₈	C₂²×D₆₁	1	4	yes	—
161	C₂×C₄²	—	61	4	no	—
162	C₂×C₄²	—	61	4	no	—
163	C₂×C₄²	—	61	16	no	—
164	C₂×C₄²	—	61	2	no	—
165	C₂×C₄²	—	61	2	no	—
166	C₂×C₄²	—	61	8	no	—
167	C₂×C₄²	—	61	4	no	—
168	C₂×C₄²	—	122	16	no	—
169	C₂×C₄²	C₄×D₆₁	1	2	yes	—
170	C₂×C₄²	C₄×D₆₁	1	4	yes	—
171	C₂×C₄²	C₂×Dic₆₁	1	4	yes	—
172	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
174	#97	—	61	32	no	—
175	#97	—	61	8	no	—
176	#97	C₂×Dic₆₁	1	8	yes	complemented
177	C₄×C₈	—	61	32	no	—
178	C₄×C₈	—	61	8	no	—
179	C₄×C₈	C₂×Dic₆₁	1	8	yes	complemented
180	C₂⁴	C₂×C₄×D₆₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	61	4	no	—
183	C₂²×C₄	—	61	4	no	—
184	C₂²×C₄	—	61	16	no	—
185	C₂²×C₄	—	122	16	no	—
186	C₂²×C₄	C₄×Dic₆₁	1	8	yes	—
187	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
188	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₆₁	1	8	yes	—
190	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
193	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
194	M₁₆	—	61	32	no	—
195	M₁₆	C₄×Dic₆₁	1	32	yes	direct product factor
196	C₂×C₈	—	61	8	no	—
197	C₂×C₈	—	61	16	no	—
198	C₂×C₈	C₂×C₄×D₆₁	1	16	yes	—
199	C₂×C₈	C₂²×Dic₆₁	1	8	yes	—
200	C₄²	—	61	8	no	—
201	C₄²	—	61	8	no	—
202	C₄²	—	61	16	no	—
203	C₄²	—	61	16	no	—
204	C₄²	—	122	16	no	—
205	C₄²	—	122	8	no	—
206	C₄²	C₄×Dic₆₁	1	8	yes	—
207	C₄²	C₂×C₄×D₆₁	1	2	yes	—
208	C₄²	C₂×C₄×D₆₁	1	2	yes	—
209	C₄²	C₂²×Dic₆₁	1	2	yes	—
210	C₄²	C₂²×Dic₆₁	1	2	yes	—
211	C₂³	—	2	4	no	—
212	C₂³	C₄²×D₆₁	1	4	yes	—
213	C₂³	C₂×C₄×Dic₆₁	1	2	yes	—
214	C₂³	C₂²×C₄×D₆₁	1	1	yes	characteristic
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	61	8	no	—
218	C₂×C₄	—	61	16	no	—
219	C₂×C₄	—	61	8	no	—
220	C₂×C₄	—	122	32	no	—
221	C₂×C₄	—	122	8	no	—
222	C₂×C₄	C₄²×D₆₁	1	8	yes	—
223	C₂×C₄	C₄²×D₆₁	1	4	yes	—
224	C₂×C₄	C₂×C₄×Dic₆₁	1	4	yes	—
225	C₂×C₄	C₂×C₄×Dic₆₁	1	8	yes	—
226	C₂×C₄	C₂×C₄×Dic₆₁	1	4	yes	—
227	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
228	C₂×C₄	M₁₆×D₆₁	1	8	yes	—
229	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₄×D₆₁	1	4	yes	—
231	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
232	C₈	—	61	16	no	—
233	C₈	C₂×C₄×Dic₆₁	1	16	yes	complemented
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	C₄²×Dic₆₁	1	4	yes	—
237	C₂²	C₂×C₄²×D₆₁	1	2	yes	—
238	C₂²	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
239	C₂²	C₂×M₁₆×D₆₁	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	61	16	no	—
242	C₄	—	122	8	no	—
243	C₄	C₄²×Dic₆₁	1	4	yes	—
244	C₄	M₁₆×Dic₆₁	1	8	yes	direct product factor
245	C₄	C₂×C₄²×D₆₁	1	2	yes	—
246	C₄	C₂²×C₄×Dic₆₁	1	2	yes	—
247	C₂	—	2	4	no	—
248	C₂	C₂×C₄²×Dic₆₁	1	1	yes	characteristic
249	C₂	C₄×M₁₆×D₆₁	1	4	yes	—
250	C₂	C₂×M₁₆×Dic₆₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₁
commutator	C₂×C₆₁	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₄×M₁₆×C₆₁	C₂
socle	C₂³×C₆₁	C₂³×C₄
prime radical	C₂⁴×C₆₁	C₂²×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₁
hypocenter	C₆₁	C₄²×M₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₄×#14519687	C₂	semidirect
5. #97×Dic₆₁	C₂	semidirect
6. C₄×C₈×Dic₆₁	C₂	semidirect
11. C₄×#11758194	C₄	semidirect
12. M₁₆×Dic₆₁	C₄	direct (N × H)
23. C₄×M₁₆×C₆₁	C₄	semidirect
38. #14519688	C₂×C₄	semidirect
39. #14519687	C₂×C₄	semidirect
40. C₄×#92418	C₂×C₄	semidirect
41. C₈×Dic₆₁	C₂×C₄	semidirect
52. #97×C₆₁	C₂×C₄	semidirect
53. C₄×C₈×C₆₁	C₂×C₄	semidirect
68. M₁₆×C₆₁	C₄²	semidirect
79. C₄×Dic₆₁	M₁₆	direct (N × H)
82. #11758194	C₄²	semidirect
106. C₈×C₆₁	C₂×C₄²	semidirect
107. #92418	C₂×C₄²	semidirect
117. C₄×C₆₁	C₄×M₁₆	semidirect
121. Dic₆₁	C₄×M₁₆	direct (N × H)
140. C₄×M₁₆	Dic₆₁	direct (N × H)
151. C₆₁	C₄²×M₁₆	semidirect
176. #97	C₂×Dic₆₁	semidirect
179. C₄×C₈	C₂×Dic₆₁	semidirect
195. M₁₆	C₄×Dic₆₁	direct (N × H)
233. C₈	C₂×C₄×Dic₆₁	semidirect
244. C₄	M₁₆×Dic₆₁	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×#11758194	2
C₂×M₁₆×Dic₆₁	2
C₂×C₄²×Dic₆₁	2
C₄×#14519687	2
#97×Dic₆₁	2
C₄×C₈×Dic₆₁	2
C₂×C₄×M₁₆×C₆₁	2
C₄²×M₁₆	61

Subgroups — C4×M16×Dic61 #29677343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₁₆×Dic₆₁ #29677343