Subgroups — C₄×C₆₁×#331 #29511661

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#331	C₂	1	2	yes	maximal
4	C₆₁×#1296	C₂	1	2	yes	maximal, complemented
5	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×D₄×C₆₁	—	2	2	no	—
7	C₂×C₄×D₄×C₆₁	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₆₁	—	2	2	no	—
10	C₆₁×#390	—	2	2	no	—
11	C₆₁×#390	C₄	1	2	yes	—
12	C₆₁×#384	C₂²	1	1	yes	characteristic
13	C₄³×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#336	—	2	2	no	—
15	C₆₁×#331	C₄	1	8	yes	direct product factor
16	C₂²×D₄×C₆₁	—	2	2	no	—
17	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₆₁	—	2	2	no	—
19	C₂²×C₈×C₆₁	—	2	4	no	—
20	#127×C₆₁	—	2	2	no	—
21	#127×C₆₁	C₂×C₄	1	4	yes	—
22	C₄×D₄×C₆₁	—	4	4	no	—
23	C₄×D₄×C₆₁	—	4	4	no	—
24	C₄×D₄×C₆₁	—	4	4	no	—
25	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	D₄	1	2	yes	—
32	C₂×C₄²×C₆₁	D₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	4	no	—
35	C₄⋊C₈×C₆₁	C₂×C₄	1	8	yes	complemented
36	C₄×C₈×C₆₁	—	4	4	no	—
37	C₂⁴×C₆₁	—	2	2	no	—
38	C₂×D₄×C₆₁	—	2	2	no	—
39	C₂×D₄×C₆₁	—	2	8	no	—
40	C₂×D₄×C₆₁	—	4	2	no	—
41	C₂×D₄×C₆₁	—	4	2	no	—
42	C₂×D₄×C₆₁	—	4	2	no	—
43	C₂×D₄×C₆₁	—	4	2	no	—
44	C₂×D₄×C₆₁	—	4	4	no	—
45	C₂×D₄×C₆₁	C₄²	1	4	yes	—
46	C₂²×C₄×C₆₁	—	2	2	no	—
47	C₂²×C₄×C₆₁	—	4	4	no	—
48	C₂²×C₄×C₆₁	—	4	8	no	—
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₆₁	—	2	8	no	—
55	C₂×C₈×C₆₁	—	4	4	no	—
56	C₂×C₈×C₆₁	—	4	4	no	—
57	C₄⋊C₄×C₆₁	—	4	2	no	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₂²⋊C₄×C₆₁	—	4	4	no	—
61	C₂²⋊C₄×C₆₁	—	4	8	no	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄²×C₆₁	—	2	4	no	—
64	C₄²×C₆₁	—	4	1	no	—
65	C₄²×C₆₁	D₈	1	4	yes	—
66	C₄²×C₆₁	SD₁₆	1	4	yes	—
67	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
68	C₄²×C₆₁	#41	1	2	yes	—
69	C₂³×C₆₁	—	2	4	no	—
70	C₂³×C₆₁	—	4	2	no	—
71	C₂³×C₆₁	—	4	2	no	—
72	C₂³×C₆₁	—	4	4	no	—
73	C₂³×C₆₁	—	4	4	no	—
74	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	D₄×C₆₁	—	4	4	no	—
76	D₄×C₆₁	—	4	4	no	—
77	D₄×C₆₁	—	4	8	no	—
78	D₄×C₆₁	—	4	8	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	2	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	4	2	no	—
86	C₂×C₄×C₆₁	—	4	4	no	—
87	C₂×C₄×C₆₁	—	8	8	no	—
88	C₂×C₄×C₆₁	#100	1	2	yes	—
89	C₂×C₄×C₆₁	#102	1	4	yes	—
90	C₂×C₄×C₆₁	#102	1	4	yes	—
91	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₆₁	#135	1	4	yes	—
97	C₈×C₆₁	—	4	8	no	—
98	C₄×#331	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₆₁	—	4	4	no	—
100	C₂²×C₆₁	—	4	8	no	—
101	C₂²×C₆₁	—	8	2	no	—
102	C₂²×C₆₁	—	8	4	no	—
103	C₂²×C₆₁	—	8	2	no	—
104	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
106	C₂²×C₆₁	#413	1	1	yes	characteristic
107	C₂²×C₆₁	C₂×#102	1	2	yes	—
108	C₂²×C₆₁	#416	1	2	yes	—
109	C₄×C₆₁	—	2	4	no	—
110	C₄×C₆₁	—	2	1	no	—
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	4	1	no	—
113	C₄×C₆₁	—	8	2	no	—
114	C₄×C₆₁	#331	1	4	yes	direct product factor
115	C₄×C₆₁	C₄×D₈	1	4	yes	—
116	C₄×C₆₁	C₄×SD₁₆	1	4	yes	—
117	C₄×#127	C₂×C₆₁	1	1	yes	characteristic
118	C₄×C₄⋊C₈	C₂×C₆₁	1	2	yes	complemented
119	C₂×#331	C₂×C₆₁	1	2	yes	—
120	#1296	C₂×C₆₁	1	2	yes	complemented
121	C₂×C₆₁	—	8	4	no	—
122	C₂×C₆₁	C₂×#331	1	1	yes	characteristic
123	C₂×C₆₁	#1540	1	2	yes	—
124	C₂×C₆₁	#1543	1	1	yes	characteristic
125	C₂×C₆₁	C₄×#100	1	1	yes	characteristic
126	C₂×C₆₁	C₄×#102	1	2	yes	—
127	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
128	C₂×C₄×D₄	—	2	2	no	—
129	C₂×C₄×D₄	C₄×C₆₁	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
131	C₂×C₄×C₈	—	2	2	no	—
132	#390	—	2	2	no	—
133	#390	C₄×C₆₁	1	2	yes	—
134	#384	C₂²×C₆₁	1	1	yes	characteristic
135	C₄³	C₂²×C₆₁	1	1	yes	characteristic
136	#336	—	2	2	no	—
137	#331	C₄×C₆₁	1	8	yes	direct product factor
138	C₆₁	C₄×#331	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂²×C₈	—	2	4	no	—
143	#127	—	2	2	no	—
144	#127	C₂×C₄×C₆₁	1	4	yes	—
145	C₄×D₄	—	4	4	no	—
146	C₄×D₄	—	4	4	no	—
147	C₄×D₄	—	4	4	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₂²⋊C₄	—	2	2	no	—
152	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
153	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₆₁	1	2	yes	—
155	C₂×C₄²	D₄×C₆₁	1	2	yes	—
156	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	C₂×C₄×C₆₁	1	8	yes	complemented
159	C₄×C₈	—	4	4	no	—
160	C₂⁴	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	8	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	C₄²×C₆₁	1	4	yes	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	4	8	no	—
172	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₂²⋊C₄	—	4	8	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	1	no	—
188	C₄²	D₈×C₆₁	1	4	yes	—
189	C₄²	SD₁₆×C₆₁	1	4	yes	—
190	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
191	C₄²	#41×C₆₁	1	2	yes	—
192	C₂³	—	2	4	no	—
193	C₂³	—	4	2	no	—
194	C₂³	—	4	2	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	4	4	no	—
197	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	8	no	—
201	D₄	—	4	8	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	8	no	—
211	C₂×C₄	#100×C₆₁	1	2	yes	—
212	C₂×C₄	#102×C₆₁	1	4	yes	—
213	C₂×C₄	#102×C₆₁	1	4	yes	—
214	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
215	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
216	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
217	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
218	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
219	C₂×C₄	#135×C₆₁	1	4	yes	—
220	C₈	—	4	8	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	8	2	no	—
226	C₂²	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
227	C₂²	C₂×#100×C₆₁	1	1	yes	characteristic
228	C₂²	C₆₁×#413	1	1	yes	characteristic
229	C₂²	C₂×#102×C₆₁	1	2	yes	—
230	C₂²	C₆₁×#416	1	2	yes	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	1	no	—
235	C₄	—	8	2	no	—
236	C₄	C₆₁×#331	1	4	yes	direct product factor
237	C₄	C₄×D₈×C₆₁	1	4	yes	—
238	C₄	C₄×SD₁₆×C₆₁	1	4	yes	—
239	C₂	—	8	4	no	—
240	C₂	C₂×C₆₁×#331	1	1	yes	characteristic
241	C₂	C₆₁×#1540	1	2	yes	—
242	C₂	C₆₁×#1543	1	1	yes	characteristic
243	C₂	C₄×#100×C₆₁	1	1	yes	characteristic
244	C₂	C₄×#102×C₆₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂³×C₆₁	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊C₈×C₆₁	C₂	semidirect
4. C₆₁×#1296	C₂	semidirect
15. C₆₁×#331	C₄	direct (N × H)
35. C₄⋊C₈×C₆₁	C₂×C₄	semidirect
98. C₄×#331	C₆₁	direct (N × H)
114. C₄×C₆₁	#331	direct (N × H)
118. C₄×C₄⋊C₈	C₂×C₆₁	semidirect
120. #1296	C₂×C₆₁	semidirect
137. #331	C₄×C₆₁	direct (N × H)
138. C₆₁	C₄×#331	direct (N × H)
158. C₄⋊C₈	C₂×C₄×C₆₁	semidirect
236. C₄	C₆₁×#331	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#127×C₆₁	2
C₄×C₄⋊C₈×C₆₁	2
C₂×C₆₁×#331	2
C₆₁×#1296	2
C₄×#331	61

Subgroups — C4×C61×#331 #29511661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#331 #29511661