Subgroups — C₄×C₆₁×#427 #29523882

Order 15616.

1182 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#427	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₆₁	C₂	1	2	yes	maximal
4	C₄×#107×C₆₁	C₂	1	4	yes	maximal, complemented
5	C₄×#106×C₆₁	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1735	C₂	1	2	yes	maximal
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₁	C₂²	1	4	yes	—
10	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#427	C₄	1	16	yes	direct product factor
12	C₂×#107×C₆₁	C₂²	1	4	yes	—
13	C₂×#106×C₆₁	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₆₁	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₁	C₂²	1	8	yes	—
18	C₄×C₂²⋊C₄×C₆₁	—	2	4	no	—
19	C₆₁×#376	—	2	2	no	—
20	C₄³×C₆₁	—	2	2	no	—
21	C₆₁×#340	C₂²	1	8	yes	—
22	C₂³×C₄×C₆₁	D₄	1	2	yes	—
23	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	—	2	4	no	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	#117×C₆₁	—	2	8	no	—
32	#117×C₆₁	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
34	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
35	C₂×C₄²×C₆₁	—	2	4	no	—
36	C₂×C₄²×C₆₁	—	2	8	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
42	#107×C₆₁	C₂×C₄	1	16	yes	complemented
43	#106×C₆₁	C₂×C₄	1	16	yes	complemented
44	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	—
46	#95×C₆₁	—	2	8	no	—
47	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	8	no	—
49	C₂²×C₄×C₆₁	—	2	8	no	—
50	C₂²×C₄×C₆₁	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₆₁	#41	1	4	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₆₁	—	2	8	no	—
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	C₄²	1	16	yes	complemented
62	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
67	C₄⋊C₄×C₆₁	—	2	8	no	—
68	C₄⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
69	C₂²⋊C₄×C₆₁	—	2	16	no	—
70	C₄²×C₆₁	—	2	16	no	—
71	C₄²×C₆₁	—	4	16	no	—
72	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
73	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
74	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
75	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
76	C₂³×C₆₁	—	2	4	no	—
77	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₆₁	C₄×D₄	1	4	yes	—
79	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	16	no	—
82	C₂×C₄×C₆₁	—	2	16	no	—
83	C₂×C₄×C₆₁	—	4	16	no	—
84	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₆₁	#117	1	4	yes	—
89	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
90	C₂×C₄×C₆₁	C₄×Q₈	1	8	yes	—
91	C₂×C₄×C₆₁	#135	1	8	yes	—
92	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
96	C₈×C₆₁	—	2	4	no	—
97	C₈×C₆₁	C₂×C₄²	1	4	yes	—
98	C₈×C₆₁	C₂×C₄²	1	4	yes	—
99	C₄×#427	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₆₁	—	2	4	no	—
101	C₂²×C₆₁	—	2	8	no	—
102	C₂²×C₆₁	C₄×C₄⋊C₄	1	4	yes	—
103	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₆₁	C₂×C₄×D₄	1	2	yes	—
105	C₂²×C₆₁	C₂×#135	1	4	yes	—
106	C₄×C₆₁	—	2	4	no	—
107	C₄×C₆₁	—	4	16	no	—
108	C₄×C₆₁	C₄×C₄⋊C₄	1	4	yes	—
109	C₄×C₆₁	#427	1	8	yes	direct product factor
110	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
111	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
112	C₄×C₆₁	C₂×C₄×Q₈	1	2	yes	—
113	C₂×#427	C₂×C₆₁	1	2	yes	—
114	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
115	C₄×#117	C₂×C₆₁	1	2	yes	—
116	C₄×#107	C₂×C₆₁	1	4	yes	complemented
117	C₄×#106	C₂×C₆₁	1	4	yes	complemented
118	#1735	C₂×C₆₁	1	2	yes	—
119	C₂×C₆₁	—	2	4	no	—
120	C₂×C₆₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₆₁	C₂×#427	1	2	yes	—
122	C₂×C₆₁	C₄×#135	1	4	yes	—
123	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
125	C₂×#117	C₂²×C₆₁	1	4	yes	—
126	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
127	#427	C₄×C₆₁	1	16	yes	direct product factor
128	C₂×#107	C₂²×C₆₁	1	4	yes	—
129	C₂×#106	C₂²×C₆₁	1	4	yes	—
130	C₂×C₄×C₈	C₄×C₆₁	1	8	yes	complemented
131	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
133	C₄×C₄⋊C₄	C₂²×C₆₁	1	8	yes	—
134	C₄×C₂²⋊C₄	—	2	4	no	—
135	#376	—	2	2	no	—
136	C₄³	—	2	2	no	—
137	#340	C₂²×C₆₁	1	8	yes	—
138	C₆₁	C₄×#427	1	1	yes	characteristic, direct product factor, Sylow
139	C₂³×C₄	D₄×C₆₁	1	2	yes	—
140	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
144	C₂²×C₈	C₂×C₄×C₆₁	1	8	yes	—
145	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
146	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
148	#117	—	2	8	no	—
149	#117	C₂×C₄×C₆₁	1	16	yes	—
150	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
151	C₂×C₂²⋊C₄	—	2	4	no	—
152	C₂×C₄²	—	2	4	no	—
153	C₂×C₄²	—	2	8	no	—
154	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
155	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
156	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
158	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
159	#107	C₂×C₄×C₆₁	1	16	yes	complemented
160	#106	C₂×C₄×C₆₁	1	16	yes	complemented
161	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	—
162	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	—
163	#95	—	2	8	no	—
164	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	C₄⋊C₄×C₆₁	1	8	yes	—
168	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
170	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
174	C₂²×C₄	#41×C₆₁	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄²×C₆₁	1	16	yes	complemented
179	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
180	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
183	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
186	C₂²⋊C₄	—	2	16	no	—
187	C₄²	—	2	16	no	—
188	C₄²	—	4	16	no	—
189	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
190	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
191	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
192	C₄²	C₂×Q₈×C₆₁	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
195	C₂³	C₄×D₄×C₆₁	1	4	yes	—
196	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	4	16	no	—
201	C₂×C₄	C₂×C₄²×C₆₁	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
204	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
205	C₂×C₄	#117×C₆₁	1	4	yes	—
206	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
207	C₂×C₄	C₄×Q₈×C₆₁	1	8	yes	—
208	C₂×C₄	#135×C₆₁	1	8	yes	—
209	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
210	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
211	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄²×C₆₁	1	4	yes	—
215	C₈	C₂×C₄²×C₆₁	1	4	yes	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	8	no	—
218	C₂²	C₄×C₄⋊C₄×C₆₁	1	4	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
220	C₂²	C₂×C₄×D₄×C₆₁	1	2	yes	—
221	C₂²	C₂×#135×C₆₁	1	4	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	4	16	no	—
224	C₄	C₄×C₄⋊C₄×C₆₁	1	4	yes	—
225	C₄	C₆₁×#427	1	8	yes	direct product factor
226	C₄	C₂²×C₄²×C₆₁	1	1	yes	characteristic
227	C₄	C₂×#117×C₆₁	1	1	yes	characteristic
228	C₄	C₂×C₄×Q₈×C₆₁	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₂×C₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
231	C₂	C₂×C₆₁×#427	1	2	yes	—
232	C₂	C₄×#135×C₆₁	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₁	D₄
commutator	C₄	C₂²×C₄²×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂²×D₄
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×#107×C₆₁	C₂	semidirect
5. C₄×#106×C₆₁	C₂	semidirect
11. C₆₁×#427	C₄	direct (N × H)
14. C₂×C₄×C₈×C₆₁	C₄	semidirect
42. #107×C₆₁	C₂×C₄	semidirect
43. #106×C₆₁	C₂×C₄	semidirect
61. C₂×C₈×C₆₁	C₄²	semidirect
99. C₄×#427	C₆₁	direct (N × H)
109. C₄×C₆₁	#427	direct (N × H)
116. C₄×#107	C₂×C₆₁	semidirect
117. C₄×#106	C₂×C₆₁	semidirect
127. #427	C₄×C₆₁	direct (N × H)
130. C₂×C₄×C₈	C₄×C₆₁	semidirect
138. C₆₁	C₄×#427	direct (N × H)
159. #107	C₂×C₄×C₆₁	semidirect
160. #106	C₂×C₄×C₆₁	semidirect
178. C₂×C₈	C₄²×C₆₁	semidirect
225. C₄	C₆₁×#427	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×C₆₁×#427	2
C₂²×C₄×C₈×C₆₁	2
C₄×#117×C₆₁	2
C₄×#107×C₆₁	2
C₄×#106×C₆₁	2
C₆₁×#1735	2
C₄×#427	61

Subgroups — C4×C61×#427 #29523882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#427 #29523882