Subgroups — C₂×#127×Dic₆₁ #29838528

Order 15616.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×Dic₆₁	C₂	1	6	yes	maximal, complemented
2	#127×Dic₆₁	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16839665	C₂	1	6	yes	maximal, complemented
4	C₂×#16839632	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₆₁	C₂²	1	4	yes	—
8	C₂×D₄×Dic₆₁	—	2	48	no	—
9	C₂×D₄×Dic₆₁	C₂²	1	48	yes	complemented
10	C₂²×C₄⋊Dic₆₁	C₂²	1	6	yes	—
11	C₂²×C₂²⋊Dic₆₁	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₆₁	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₆₁	C₂²	1	3	yes	—
14	#16839665	C₂²	1	48	yes	complemented
15	#16839632	C₂²	1	8	yes	complemented
16	C₄²×Dic₆₁	C₂²	1	8	yes	complemented
17	C₂³×D₄×C₆₁	C₂²	1	6	yes	—
18	C₂×#127×C₆₁	C₄	1	16	yes	complemented
19	C₂×#127×C₆₁	C₂²	1	8	yes	—
20	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	C₂³×Dic₆₁	—	2	48	no	—
22	C₂³×Dic₆₁	C₂³	1	4	yes	—
23	C₂³×Dic₆₁	C₂³	1	16	yes	—
24	C₂³×Dic₆₁	C₂³	1	3	yes	—
25	C₂³×Dic₆₁	C₂³	1	1	yes	characteristic
26	D₄×Dic₆₁	—	2	192	no	—
27	C₂×C₄⋊Dic₆₁	—	2	24	no	—
28	C₂×C₄⋊Dic₆₁	C₂³	1	24	yes	—
29	C₂×C₂²⋊Dic₆₁	—	2	48	no	—
30	C₂×C₂²⋊Dic₆₁	C₂³	1	48	yes	—
31	C₂×C₄×Dic₆₁	—	2	12	no	—
32	C₂×C₄×Dic₆₁	D₄	1	24	yes	complemented
33	C₂×C₄×Dic₆₁	C₂³	1	12	yes	—
34	C₂×C₄×Dic₆₁	C₂³	1	12	yes	—
35	C₂⁵×C₆₁	C₂³	1	4	yes	—
36	C₂²×D₄×C₆₁	—	2	24	no	—
37	C₂²×D₄×C₆₁	—	2	24	no	—
38	C₂²×D₄×C₆₁	C₂×C₄	1	48	yes	complemented
39	C₂²×D₄×C₆₁	C₂³	1	24	yes	—
40	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
41	#127×C₆₁	C₂×C₄	1	64	yes	complemented
42	C₂×C₄²×C₆₁	C₂×C₄	1	8	yes	complemented
43	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
44	C₂⁴×C₆₁	—	2	24	no	—
45	C₂⁴×C₆₁	—	2	24	no	—
46	C₂⁴×C₆₁	C₂²×C₄	1	16	yes	—
47	C₂⁴×C₆₁	C₂⁴	1	8	yes	—
48	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₂×D₄×C₆₁	—	2	192	no	—
50	C₂×D₄×C₆₁	—	2	48	no	—
51	C₂×D₄×C₆₁	—	2	48	no	—
52	C₂×D₄×C₆₁	C₂²×C₄	1	96	yes	complemented
53	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₆₁	#41	1	12	yes	—
56	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
57	C₂²×Dic₆₁	—	2	12	no	—
58	C₂²×Dic₆₁	—	2	24	no	—
59	C₂²×Dic₆₁	—	2	96	no	—
60	C₂²×Dic₆₁	—	4	64	no	—
61	C₂²×Dic₆₁	C₂×D₄	1	12	yes	—
62	C₂²×Dic₆₁	C₂×D₄	1	12	yes	—
63	C₂²×Dic₆₁	C₂⁴	1	8	yes	—
64	C₂²×Dic₆₁	C₂⁴	1	6	yes	—
65	C₂²×Dic₆₁	C₂⁴	1	2	yes	—
66	C₄²×C₆₁	C₂²×C₄	1	16	yes	complemented
67	C₂²⋊Dic₆₁	—	2	96	no	—
68	C₄⋊Dic₆₁	—	2	48	no	—
69	C₄×Dic₆₁	—	2	24	no	—
70	C₄×Dic₆₁	C₂×D₄	1	48	yes	complemented
71	C₂³×C₆₁	—	2	96	no	—
72	C₂³×C₆₁	—	2	24	no	—
73	C₂³×C₆₁	—	2	24	no	—
74	C₂³×C₆₁	—	4	16	no	—
75	C₂³×C₆₁	—	4	48	no	—
76	C₂³×C₆₁	C₂³×C₄	1	16	yes	—
77	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
78	C₂³×C₆₁	C₂²×D₄	1	6	yes	—
79	C₂³×C₆₁	C₂×#41	1	6	yes	—
80	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
81	D₄×C₆₁	—	2	192	no	—
82	C₂×C₄×C₆₁	C₄×D₄	1	48	yes	complemented
83	C₂×C₄×C₆₁	C₂³×C₄	1	12	yes	—
84	C₂×C₄×C₆₁	C₂²×D₄	1	12	yes	—
85	C₂×C₄×C₆₁	C₂×#41	1	12	yes	—
86	C₂×Dic₆₁	—	2	24	no	—
87	C₂×Dic₆₁	—	2	12	no	—
88	C₂×Dic₆₁	—	2	24	no	—
89	C₂×Dic₆₁	—	4	16	no	—
90	C₂×Dic₆₁	—	4	64	no	—
91	C₂×Dic₆₁	#127	1	16	yes	direct product factor
92	C₂×Dic₆₁	C₂²×D₄	1	12	yes	—
93	C₂×Dic₆₁	C₂²×D₄	1	12	yes	—
94	C₂×C₄×#127	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	48	no	—
96	C₂²×C₆₁	—	4	64	no	—
97	C₂²×C₆₁	—	4	8	no	—
98	C₂²×C₆₁	—	4	24	no	—
99	C₂²×C₆₁	C₂×C₄×D₄	1	12	yes	—
100	C₂²×C₆₁	C₂×#127	1	4	yes	—
101	C₂²×C₆₁	#532	1	12	yes	—
102	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
103	C₂²×C₆₁	C₂³×D₄	1	3	yes	—
104	C₂²×C₆₁	C₂²×#41	1	3	yes	—
105	C₄×C₆₁	C₂×C₄×D₄	1	24	yes	complemented
106	Dic₆₁	—	2	12	no	—
107	Dic₆₁	—	4	8	no	—
108	Dic₆₁	C₂×#127	1	8	yes	direct product factor
109	C₂²×#127	D₆₁	1	1	yes	characteristic
110	C₂²×C₄×D₄	—	61	6	no	—
111	C₄×#127	—	61	16	no	—
112	C₂×#390	—	61	6	no	—
113	C₂×#384	—	61	1	no	—
114	C₂×C₄³	—	61	1	no	—
115	C₂×C₆₁	—	4	16	no	—
116	C₂×C₆₁	C₄×#127	1	8	yes	complemented
117	C₂×C₆₁	C₂²×C₄×D₄	1	3	yes	—
118	C₂×C₆₁	C₂²×#127	1	1	yes	characteristic
119	C₂×C₆₁	C₂×#532	1	3	yes	—
120	C₂³×D₄	C₂×D₆₁	1	6	yes	—
121	C₂⁴×C₄	—	61	4	no	—
122	C₂×#127	Dic₆₁	1	16	yes	direct product factor
123	C₂×#127	C₂×D₆₁	1	8	yes	—
124	C₂×C₄×D₄	—	61	48	no	—
125	C₂×C₄×D₄	—	122	48	no	—
126	C₂²×C₄⋊C₄	—	61	6	no	—
127	C₂²×C₂²⋊C₄	—	61	12	no	—
128	C₂²×C₄²	—	61	3	no	—
129	C₂²×C₄²	—	61	3	no	—
130	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
131	#390	—	61	48	no	—
132	#384	—	61	8	no	—
133	C₄³	—	61	8	no	—
134	C₆₁	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
135	C₂⁵	C₂²×D₆₁	1	4	yes	—
136	C₂²×D₄	—	2	24	no	—
137	C₂²×D₄	—	2	24	no	—
138	C₂²×D₄	C₂×Dic₆₁	1	48	yes	complemented
139	C₂²×D₄	C₂²×D₆₁	1	24	yes	—
140	C₂³×C₄	—	61	1	no	—
141	C₂³×C₄	—	61	3	no	—
142	C₂³×C₄	—	61	16	no	—
143	C₂³×C₄	—	61	4	no	—
144	C₂³×C₄	—	122	48	no	—
145	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
146	#127	C₂×Dic₆₁	1	64	yes	direct product factor
147	C₄×D₄	—	122	192	no	—
148	C₂×C₄⋊C₄	—	61	24	no	—
149	C₂×C₄⋊C₄	—	122	24	no	—
150	C₂×C₂²⋊C₄	—	61	48	no	—
151	C₂×C₂²⋊C₄	—	122	48	no	—
152	C₂×C₄²	—	61	24	no	—
153	C₂×C₄²	—	61	12	no	—
154	C₂×C₄²	—	61	12	no	—
155	C₂×C₄²	—	122	12	no	—
156	C₂×C₄²	C₂×Dic₆₁	1	8	yes	complemented
157	C₂×C₄²	C₂²×D₆₁	1	4	yes	—
158	C₂⁴	—	2	24	no	—
159	C₂⁴	—	2	24	no	—
160	C₂⁴	C₂²×Dic₆₁	1	16	yes	—
161	C₂⁴	C₂³×D₆₁	1	8	yes	—
162	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
163	C₂×D₄	—	2	48	no	—
164	C₂×D₄	—	2	48	no	—
165	C₂×D₄	—	2	192	no	—
166	C₂×D₄	C₂²×Dic₆₁	1	96	yes	complemented
167	C₂²×C₄	—	61	12	no	—
168	C₂²×C₄	—	61	2	no	—
169	C₂²×C₄	—	61	12	no	—
170	C₂²×C₄	—	61	6	no	—
171	C₂²×C₄	—	61	8	no	—
172	C₂²×C₄	—	122	96	no	—
173	C₂²×C₄	—	122	12	no	—
174	C₂²×C₄	—	122	24	no	—
175	C₂²×C₄	—	244	64	no	—
176	C₂²×C₄	D₄×D₆₁	1	12	yes	—
177	C₂²×C₄	#11758224	1	12	yes	—
178	C₂²×C₄	C₂²×Dic₆₁	1	12	yes	—
179	C₂²×C₄	C₂³×D₆₁	1	6	yes	—
180	C₄⋊C₄	—	122	48	no	—
181	C₂²⋊C₄	—	122	96	no	—
182	C₄²	—	61	48	no	—
183	C₄²	—	122	24	no	—
184	C₄²	C₂²×Dic₆₁	1	16	yes	complemented
185	C₂³	—	2	96	no	—
186	C₂³	—	2	24	no	—
187	C₂³	—	2	24	no	—
188	C₂³	—	4	16	no	—
189	C₂³	—	4	48	no	—
190	C₂³	C₂³×Dic₆₁	1	16	yes	—
191	C₂³	C₂³×Dic₆₁	1	2	yes	—
192	C₂³	C₂×D₄×D₆₁	1	6	yes	—
193	C₂³	C₂×#11758224	1	6	yes	—
194	C₂³	C₂⁴×D₆₁	1	1	yes	characteristic
195	D₄	—	2	192	no	—
196	C₂×C₄	—	61	16	no	—
197	C₂×C₄	—	61	12	no	—
198	C₂×C₄	—	61	12	no	—
199	C₂×C₄	—	122	24	no	—
200	C₂×C₄	—	122	12	no	—
201	C₂×C₄	—	122	24	no	—
202	C₂×C₄	—	244	64	no	—
203	C₂×C₄	—	244	16	no	—
204	C₂×C₄	D₄×Dic₆₁	1	48	yes	complemented
205	C₂×C₄	C₂³×Dic₆₁	1	12	yes	—
206	C₂×C₄	C₂×D₄×D₆₁	1	12	yes	—
207	C₂×C₄	C₂×#11758224	1	12	yes	—
208	C₂²	—	2	48	no	—
209	C₂²	—	4	8	no	—
210	C₂²	—	4	24	no	—
211	C₂²	—	4	64	no	—
212	C₂²	C₂×D₄×Dic₆₁	1	12	yes	—
213	C₂²	#127×D₆₁	1	4	yes	—
214	C₂²	#16840310	1	12	yes	—
215	C₂²	C₂⁴×Dic₆₁	1	1	yes	characteristic
216	C₂²	C₂²×D₄×D₆₁	1	3	yes	—
217	C₂²	C₂²×#11758224	1	3	yes	—
218	C₄	—	61	8	no	—
219	C₄	—	122	12	no	—
220	C₄	—	244	8	no	—
221	C₄	C₂×D₄×Dic₆₁	1	24	yes	complemented
222	C₂	—	4	16	no	—
223	C₂	#127×Dic₆₁	1	8	yes	direct product factor
224	C₂	C₂²×D₄×Dic₆₁	1	3	yes	—
225	C₂	C₂×#127×D₆₁	1	1	yes	characteristic
226	C₂	C₂×#16840310	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₆₁
Fitting, prime radical	C₂²×#127×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
cosocle	C₂³×C₆₁	C₂⁵
hypercenter	C₂²×#127	D₆₁
hypocenter	C₆₁	C₂×C₄×#127
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×Dic₆₁	C₂	semidirect
2. #127×Dic₆₁	C₂	direct (N × H)
3. C₂×#16839665	C₂	semidirect
4. C₂×#16839632	C₂	semidirect
5. C₂×C₄²×Dic₆₁	C₂	semidirect
9. C₂×D₄×Dic₆₁	C₂²	semidirect
14. #16839665	C₂²	semidirect
15. #16839632	C₂²	semidirect
16. C₄²×Dic₆₁	C₂²	semidirect
18. C₂×#127×C₆₁	C₄	semidirect
32. C₂×C₄×Dic₆₁	D₄	semidirect
38. C₂²×D₄×C₆₁	C₂×C₄	semidirect
41. #127×C₆₁	C₂×C₄	semidirect
42. C₂×C₄²×C₆₁	C₂×C₄	semidirect
52. C₂×D₄×C₆₁	C₂²×C₄	semidirect
66. C₄²×C₆₁	C₂²×C₄	semidirect
70. C₄×Dic₆₁	C₂×D₄	semidirect
82. C₂×C₄×C₆₁	C₄×D₄	semidirect
91. C₂×Dic₆₁	#127	direct (N × H)
105. C₄×C₆₁	C₂×C₄×D₄	semidirect
108. Dic₆₁	C₂×#127	direct (N × H)
116. C₂×C₆₁	C₄×#127	semidirect
122. C₂×#127	Dic₆₁	direct (N × H)
134. C₆₁	C₂×C₄×#127	semidirect
138. C₂²×D₄	C₂×Dic₆₁	semidirect
146. #127	C₂×Dic₆₁	direct (N × H)
156. C₂×C₄²	C₂×Dic₆₁	semidirect
166. C₂×D₄	C₂²×Dic₆₁	semidirect
184. C₄²	C₂²×Dic₆₁	semidirect
204. C₂×C₄	D₄×Dic₆₁	semidirect
221. C₄	C₂×D₄×Dic₆₁	semidirect
223. C₂	#127×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
C₂²×D₄×Dic₆₁	2
#127×Dic₆₁	2
C₂×#16839665	2
C₂×#16839632	2
C₂×C₄²×Dic₆₁	2
C₂²×#127×C₆₁	2
C₂×C₄×#127	61

Subgroups — C2×#127×Dic61 #29838528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#127×Dic₆₁ #29838528