Subgroups — C₂³×D₆₁⋊M₁₆ #30626738

Order 15616.

13310 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758213	C₂	1	2	yes	maximal, complemented
2	C₂²×D₆₁⋊M₁₆	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11758217	C₂	1	2	yes	maximal, complemented
4	C₂³×D₆₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758213	C₂²	1	56	yes	complemented
7	C₂×D₆₁⋊M₁₆	C₂²	1	448	yes	direct product factor
8	C₂²×#11758217	C₂²	1	56	yes	complemented
9	C₂²×D₆₁⋊C₈	C₂²	1	56	yes	complemented
10	C₂³×#92420	C₂²	1	4	yes	complemented
11	C₂⁵×D₆₁	C₄	1	1	yes	characteristic
12	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×D₆₁	C₄	1	28	yes	—
14	C₂³×C₄×D₆₁	C₄	1	1	yes	characteristic
15	C₂³×C₄×D₆₁	C₄	1	1	yes	characteristic
16	C₂³×C₄×D₆₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×D₆₁	C₂²	1	28	yes	—
18	C₂³×C₄×D₆₁	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
20	C₂×#11758213	C₂³	1	224	yes	complemented
21	D₆₁⋊M₁₆	C₂³	1	512	yes	direct product factor
22	C₂×#11758217	C₂³	1	224	yes	complemented
23	C₂×D₆₁⋊C₈	C₂³	1	224	yes	complemented
24	C₂²×#92420	C₂³	1	56	yes	complemented
25	C₂⁴×D₆₁	—	2	16	no	—
26	C₂⁴×D₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂⁴×D₆₁	C₂×C₄	1	28	yes	—
28	C₂⁴×D₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₆₁	C₂×C₄	1	14	yes	—
30	C₂³×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂³×Dic₆₁	C₂³	1	1	yes	characteristic
32	C₂³×Dic₆₁	C₂³	1	14	yes	—
33	C₂²×C₄×D₆₁	C₂×C₄	1	336	yes	—
34	C₂²×C₄×D₆₁	C₂×C₄	1	28	yes	—
35	C₂²×C₄×D₆₁	C₂×C₄	1	14	yes	—
36	C₂²×C₄×D₆₁	C₂×C₄	1	14	yes	—
37	C₂²×C₄×D₆₁	C₂×C₄	1	28	yes	—
38	C₂²×C₄×D₆₁	C₂³	1	14	yes	—
39	C₂²×C₄×D₆₁	C₂³	1	112	yes	—
40	C₂²×C₄×D₆₁	C₂³	1	14	yes	—
41	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₁	C₂×C₄	1	28	yes	—
43	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂⁴×C₆₁	—	2	8	no	—
46	C₂⁴×C₆₁	C₂²×C₄	1	14	yes	—
47	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂³×D₆₁	—	2	8	no	—
49	C₂³×D₆₁	—	2	224	no	—
50	C₂³×D₆₁	M₁₆	1	16	yes	complemented
51	C₂³×D₆₁	C₂²×C₄	1	14	yes	—
52	C₂³×D₆₁	C₂²×C₄	1	112	yes	—
53	C₂³×D₆₁	C₂²×C₄	1	14	yes	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	112	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	14	yes	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	14	yes	—
57	C₂²×Dic₆₁	C₂²×C₄	1	84	yes	—
58	C₂²×Dic₆₁	C₂²×C₄	1	7	yes	—
59	C₂²×Dic₆₁	C₂²×C₄	1	14	yes	—
60	C₂²×Dic₆₁	C₂⁴	1	7	yes	—
61	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
62	C₂×C₄×D₆₁	C₂²×C₄	1	112	yes	—
63	C₂×C₄×D₆₁	C₂²×C₄	1	448	yes	—
64	C₂×C₄×D₆₁	C₂²×C₄	1	84	yes	—
65	C₂×C₄×D₆₁	C₂²×C₄	1	84	yes	—
66	C₂×C₄×D₆₁	C₂²×C₄	1	112	yes	—
67	C₂×C₄×D₆₁	C₂⁴	1	28	yes	—
68	C₂×C₄×D₆₁	C₂⁴	1	64	yes	—
69	C₂×C₄×D₆₁	C₂⁴	1	28	yes	—
70	#11758217	C₂⁴	1	128	yes	complemented
71	C₂×#92420	C₂⁴	1	112	yes	complemented
72	#11758213	C₂⁴	1	128	yes	complemented
73	D₆₁⋊C₈	C₂⁴	1	128	yes	complemented
74	C₂³×C₆₁	—	2	56	no	—
75	C₂³×C₆₁	C₂×M₁₆	1	8	yes	complemented
76	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
77	C₂³×C₆₁	C₂³×C₄	1	7	yes	—
78	C₂²×D₆₁	—	2	56	no	—
79	C₂²×D₆₁	—	2	448	no	—
80	C₂²×D₆₁	C₂×M₁₆	1	112	yes	complemented
81	C₂²×D₆₁	C₂³×C₄	1	28	yes	—
82	C₂²×D₆₁	C₂³×C₄	1	64	yes	—
83	C₂²×D₆₁	C₂³×C₄	1	28	yes	—
84	C₂×C₄×C₆₁	C₂³×C₄	1	64	yes	—
85	C₂×C₄×C₆₁	C₂³×C₄	1	28	yes	—
86	C₂×C₄×C₆₁	C₂³×C₄	1	28	yes	—
87	C₂×Dic₆₁	C₂³×C₄	1	56	yes	—
88	C₂×Dic₆₁	C₂³×C₄	1	21	yes	—
89	C₂×Dic₆₁	C₂³×C₄	1	28	yes	—
90	C₂×Dic₆₁	C₂⁵	1	7	yes	—
91	C₂×Dic₆₁	C₂⁵	1	8	yes	—
92	C₄×D₆₁	C₂³×C₄	1	64	yes	—
93	C₄×D₆₁	C₂³×C₄	1	56	yes	—
94	C₄×D₆₁	C₂³×C₄	1	56	yes	—
95	C₄×D₆₁	C₂³×C₄	1	64	yes	—
96	C₄×D₆₁	C₂⁵	1	8	yes	—
97	C₄×D₆₁	C₂⁵	1	8	yes	—
98	#92420	C₂⁵	1	32	yes	complemented
99	C₂⁴×M₁₆	—	61	1	no	maximal, Sylow
100	C₂²×C₆₁	—	2	56	no	—
101	C₂²×C₆₁	C₂²×M₁₆	1	28	yes	complemented
102	C₂²×C₆₁	C₂⁴×C₄	1	8	yes	—
103	C₂²×C₆₁	C₂⁴×C₄	1	7	yes	—
104	C₂×D₆₁	—	2	56	no	—
105	C₂×D₆₁	—	2	128	no	—
106	C₂×D₆₁	C₂²×M₁₆	1	112	yes	complemented
107	C₂×D₆₁	C₂⁴×C₄	1	8	yes	—
108	C₂×D₆₁	C₂⁴×C₄	1	8	yes	—
109	C₄×C₆₁	C₂⁴×C₄	1	8	yes	—
110	C₄×C₆₁	C₂⁴×C₄	1	8	yes	—
111	Dic₆₁	C₂⁴×C₄	1	7	yes	—
112	Dic₆₁	C₂⁴×C₄	1	8	yes	—
113	Dic₆₁	C₂⁶	1	1	yes	characteristic
114	C₂⁵×C₄	—	61	1	no	—
115	C₂³×M₁₆	—	61	2	no	—
116	C₂³×M₁₆	—	61	56	no	—
117	C₂³×M₁₆	—	61	2	no	—
118	C₂⁴×C₈	—	61	2	no	—
119	C₂×C₆₁	—	2	8	no	—
120	C₂×C₆₁	C₂³×M₁₆	1	14	yes	complemented
121	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
122	D₆₁	—	2	8	no	—
123	D₆₁	C₂³×M₁₆	1	16	yes	complemented
124	C₂⁶	—	61	1	no	—
125	C₂⁴×C₄	—	61	1	no	—
126	C₂⁴×C₄	—	61	1	no	—
127	C₂⁴×C₄	—	61	28	no	—
128	C₂⁴×C₄	—	61	1	no	—
129	C₂⁴×C₄	—	61	28	no	—
130	C₂⁴×C₄	—	61	1	no	—
131	C₂⁴×C₄	—	61	1	no	—
132	C₂⁴×C₄	#6923	1	1	yes	characteristic
133	C₂²×M₁₆	—	61	56	no	—
134	C₂²×M₁₆	—	61	448	no	—
135	C₂²×M₁₆	—	61	56	no	—
136	C₂³×C₈	—	61	4	no	—
137	C₂³×C₈	—	61	56	no	—
138	C₆₁	C₂⁴×M₁₆	1	1	yes	characteristic, complemented, Sylow
139	C₂⁵	—	61	1	no	—
140	C₂⁵	—	61	28	no	—
141	C₂⁵	—	61	1	no	—
142	C₂⁵	—	122	16	no	—
143	C₂⁵	C₂×#6923	1	1	yes	characteristic
144	C₂³×C₄	—	61	14	no	—
145	C₂³×C₄	—	61	1	no	—
146	C₂³×C₄	—	61	1	no	—
147	C₂³×C₄	—	61	14	no	—
148	C₂³×C₄	—	61	28	no	—
149	C₂³×C₄	—	61	336	no	—
150	C₂³×C₄	—	61	14	no	—
151	C₂³×C₄	—	61	14	no	—
152	C₂³×C₄	—	61	28	no	—
153	C₂³×C₄	—	61	14	no	—
154	C₂³×C₄	—	61	14	no	—
155	C₂³×C₄	—	61	112	no	—
156	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
157	C₂³×C₄	C₂×#6923	1	28	yes	—
158	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
159	C₂×M₁₆	—	61	224	no	—
160	C₂×M₁₆	—	61	512	no	—
161	C₂×M₁₆	—	61	224	no	—
162	C₂²×C₈	—	61	56	no	—
163	C₂²×C₈	—	61	224	no	—
164	C₂⁴	—	2	8	no	—
165	C₂⁴	—	61	16	no	—
166	C₂⁴	—	61	14	no	—
167	C₂⁴	—	61	112	no	—
168	C₂⁴	—	61	14	no	—
169	C₂⁴	—	122	224	no	—
170	C₂⁴	—	122	8	no	—
171	C₂⁴	C₂²×#6923	1	14	yes	—
172	C₂⁴	C₂²×#6923	1	1	yes	characteristic
173	C₂²×C₄	—	61	7	no	—
174	C₂²×C₄	—	61	28	no	—
175	C₂²×C₄	—	61	84	no	—
176	C₂²×C₄	—	61	7	no	—
177	C₂²×C₄	—	61	14	no	—
178	C₂²×C₄	—	61	28	no	—
179	C₂²×C₄	—	61	28	no	—
180	C₂²×C₄	—	61	64	no	—
181	C₂²×C₄	—	61	84	no	—
182	C₂²×C₄	—	61	112	no	—
183	C₂²×C₄	—	61	448	no	—
184	C₂²×C₄	—	61	84	no	—
185	C₂²×C₄	—	61	112	no	—
186	C₂²×C₄	C₂²×#6923	1	112	yes	—
187	C₂²×C₄	C₂²×#6923	1	14	yes	—
188	C₂²×C₄	C₂²×#6923	1	14	yes	—
189	M₁₆	—	61	128	no	—
190	M₁₆	—	61	128	no	—
191	C₂×C₈	—	61	128	no	—
192	C₂×C₈	—	61	112	no	—
193	C₂³	—	2	56	no	—
194	C₂³	—	61	112	no	—
195	C₂³	—	61	28	no	—
196	C₂³	—	61	64	no	—
197	C₂³	—	61	28	no	—
198	C₂³	—	122	448	no	—
199	C₂³	—	122	56	no	—
200	C₂³	D₆₁⋊M₁₆	1	8	yes	direct product factor
201	C₂³	C₂³×#6923	1	28	yes	—
202	C₂³	C₂³×#6923	1	7	yes	—
203	C₂×C₄	—	61	56	no	—
204	C₂×C₄	—	61	21	no	—
205	C₂×C₄	—	61	28	no	—
206	C₂×C₄	—	61	7	no	—
207	C₂×C₄	—	61	8	no	—
208	C₂×C₄	—	61	56	no	—
209	C₂×C₄	—	61	64	no	—
210	C₂×C₄	—	61	56	no	—
211	C₂×C₄	—	61	64	no	—
212	C₂×C₄	—	61	8	no	—
213	C₂×C₄	—	61	8	no	—
214	C₂×C₄	C₂³×#6923	1	64	yes	—
215	C₂×C₄	C₂³×#6923	1	28	yes	—
216	C₂×C₄	C₂³×#6923	1	28	yes	—
217	C₈	—	61	32	no	—
218	C₂²	—	2	56	no	—
219	C₂²	—	61	112	no	—
220	C₂²	—	61	8	no	—
221	C₂²	—	61	8	no	—
222	C₂²	—	122	128	no	—
223	C₂²	—	122	56	no	—
224	C₂²	C₂×D₆₁⋊M₁₆	1	28	yes	direct product factor
225	C₂²	C₂⁴×#6923	1	8	yes	—
226	C₂²	C₂⁴×#6923	1	7	yes	—
227	C₄	—	61	7	no	—
228	C₄	—	61	8	no	—
229	C₄	—	61	1	no	—
230	C₄	C₂⁴×#6923	1	8	yes	—
231	C₄	C₂⁴×#6923	1	8	yes	—
232	C₂	—	2	8	no	—
233	C₂	—	61	16	no	—
234	C₂	—	122	8	no	—
235	C₂	C₂²×D₆₁⋊M₁₆	1	14	yes	direct product factor
236	C₂	C₂⁵×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#6923
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂	C₂⁵×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle	C₂⁴×C₆₁	C₂²×C₄
prime radical	C₂⁵×D₆₁	C₄
cosocle	Dic₆₁	C₂⁶
hypercenter	C₂⁴×C₄	#6923
hypocenter	C₆₁	C₂⁴×M₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11758213	C₂	semidirect
2. C₂²×D₆₁⋊M₁₆	C₂	direct (N × H)
3. C₂³×#11758217	C₂	semidirect
4. C₂³×D₆₁⋊C₈	C₂	semidirect
6. C₂²×#11758213	C₂²	semidirect
7. C₂×D₆₁⋊M₁₆	C₂²	direct (N × H)
8. C₂²×#11758217	C₂²	semidirect
9. C₂²×D₆₁⋊C₈	C₂²	semidirect
10. C₂³×#92420	C₂²	semidirect
20. C₂×#11758213	C₂³	semidirect
21. D₆₁⋊M₁₆	C₂³	direct (N × H)
22. C₂×#11758217	C₂³	semidirect
23. C₂×D₆₁⋊C₈	C₂³	semidirect
24. C₂²×#92420	C₂³	semidirect
50. C₂³×D₆₁	M₁₆	semidirect
70. #11758217	C₂⁴	semidirect
71. C₂×#92420	C₂⁴	semidirect
72. #11758213	C₂⁴	semidirect
73. D₆₁⋊C₈	C₂⁴	semidirect
75. C₂³×C₆₁	C₂×M₁₆	semidirect
80. C₂²×D₆₁	C₂×M₁₆	semidirect
98. #92420	C₂⁵	semidirect
101. C₂²×C₆₁	C₂²×M₁₆	semidirect
106. C₂×D₆₁	C₂²×M₁₆	semidirect
120. C₂×C₆₁	C₂³×M₁₆	semidirect
123. D₆₁	C₂³×M₁₆	semidirect
138. C₆₁	C₂⁴×M₁₆	semidirect
200. C₂³	D₆₁⋊M₁₆	direct (N × H)
224. C₂²	C₂×D₆₁⋊M₁₆	direct (N × H)
235. C₂	C₂²×D₆₁⋊M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11758213	2
C₂²×D₆₁⋊M₁₆	2
C₂³×#11758217	2
C₂³×D₆₁⋊C₈	2
C₂⁴×C₄×D₆₁	2
C₂⁴×M₁₆	61

Subgroups — C23×D61⋊M16 #30626738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₆₁⋊M₁₆ #30626738