Subgroups — C₂×C₄×#127×C₆₁ #29537469

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

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₆₁	C₂	semidirect
3. C₄×#127×C₆₁	C₂	direct (N × H)
4. C₂×C₆₁×#390	C₂	semidirect
5. C₂×C₆₁×#384	C₂	semidirect
6. C₂×C₄³×C₆₁	C₂	semidirect
9. C₂×#127×C₆₁	C₄	direct (N × H)
12. C₂×C₄×D₄×C₆₁	C₂²	semidirect
18. C₆₁×#390	C₂²	semidirect
19. C₆₁×#384	C₂²	semidirect
20. C₄³×C₆₁	C₂²	semidirect
24. C₂²×D₄×C₆₁	C₂×C₄	semidirect
32. #127×C₆₁	C₂×C₄	direct (N × H)
39. C₂×C₄²×C₆₁	C₂×C₄	semidirect
40. C₂×C₄²×C₆₁	D₄	semidirect
52. C₂×D₄×C₆₁	C₂²×C₄	semidirect
69. C₄²×C₆₁	C₂²×C₄	semidirect
70. C₄²×C₆₁	C₂×D₄	semidirect
87. C₂×C₄×C₆₁	C₄×D₄	semidirect
88. C₂×C₄×C₆₁	#127	direct (N × H)
94. C₂×C₄×#127	C₆₁	direct (N × H)
107. C₄×C₆₁	C₂×C₄×D₄	semidirect
108. C₄×C₆₁	C₂×#127	direct (N × H)
110. C₂²×C₄×D₄	C₂×C₆₁	semidirect
111. C₄×#127	C₂×C₆₁	direct (N × H)
112. C₂×#390	C₂×C₆₁	semidirect
113. C₂×#384	C₂×C₆₁	semidirect
114. C₂×C₄³	C₂×C₆₁	semidirect
116. C₂×C₆₁	C₄×#127	direct (N × H)
122. C₂×#127	C₄×C₆₁	direct (N × H)
125. C₂×C₄×D₄	C₂²×C₆₁	semidirect
131. #390	C₂²×C₆₁	semidirect
132. #384	C₂²×C₆₁	semidirect
133. C₄³	C₂²×C₆₁	semidirect
134. C₆₁	C₂×C₄×#127	direct (N × H)
138. C₂²×D₄	C₂×C₄×C₆₁	semidirect
146. #127	C₂×C₄×C₆₁	direct (N × H)
153. C₂×C₄²	C₂×C₄×C₆₁	semidirect
154. C₂×C₄²	D₄×C₆₁	semidirect
166. C₂×D₄	C₂²×C₄×C₆₁	semidirect
183. C₄²	C₂²×C₄×C₆₁	semidirect
184. C₄²	C₂×D₄×C₆₁	semidirect
201. C₂×C₄	C₄×D₄×C₆₁	semidirect
202. C₂×C₄	#127×C₆₁	direct (N × H)
220. C₄	C₂×C₄×D₄×C₆₁	semidirect
221. C₄	C₂×#127×C₆₁	direct (N × H)
223. C₂	C₄×#127×C₆₁	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×#127×C₆₁	2
C₂²×C₄×D₄×C₆₁	2
C₄×#127×C₆₁	2
C₂×C₆₁×#390	2
C₂×C₆₁×#384	2
C₂×C₄³×C₆₁	2
C₂×C₄×#127	61

Subgroups — C2×C4×#127×C61 #29537469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#127×C₆₁ #29537469