Subgroups — C₄×C₆₁×#340 #29514135

Order 15616.

1182 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	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal
3	C₂×C₆₁×#340	C₂	1	4	yes	maximal
4	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
5	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₆₁	—	2	2	no	—
7	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₁	—	2	4	no	—
9	C₂×C₄×C₈×C₆₁	C₄	1	4	yes	complemented
10	C₄×C₄⋊C₄×C₆₁	—	2	4	no	—
11	C₄×C₄⋊C₄×C₆₁	C₄	1	8	yes	complemented
12	C₂×#95×C₆₁	—	2	2	no	—
13	C₆₁×#340	C₄	1	16	yes	direct product factor
14	C₂³×C₄×C₆₁	—	2	4	no	—
15	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₆₁	—	2	4	no	—
19	C₂²×C₈×C₆₁	—	2	4	no	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
23	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
24	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	16	yes	—
25	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₆₁	—	2	8	no	—
27	C₂×C₄²×C₆₁	—	4	4	no	—
28	C₂×C₄²×C₆₁	—	4	4	no	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₁	D₄	1	4	yes	—
32	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
33	C₄×C₈×C₆₁	—	2	8	no	—
34	#95×C₆₁	—	2	8	no	—
35	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	—	2	4	no	—
37	C₂²×C₄×C₆₁	—	2	8	no	—
38	C₂²×C₄×C₆₁	—	2	16	no	—
39	C₂²×C₄×C₆₁	—	4	4	no	—
40	C₂²×C₄×C₆₁	—	4	4	no	—
41	C₂²×C₄×C₆₁	—	4	8	no	—
42	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₁	D₈	1	2	yes	—
47	C₂²×C₄×C₆₁	SD₁₆	1	4	yes	—
48	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
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	4	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	#41	1	4	yes	—
55	C₂×C₈×C₆₁	—	2	4	no	—
56	C₂×C₈×C₆₁	—	2	4	no	—
57	C₂×C₈×C₆₁	—	2	16	no	—
58	C₂×C₈×C₆₁	C₄²	1	8	yes	complemented
59	C₄⋊C₄×C₆₁	—	2	8	no	—
60	C₄⋊C₄×C₆₁	—	2	16	no	—
61	C₄⋊C₄×C₆₁	C₄²	1	16	yes	complemented
62	C₄²×C₆₁	—	4	16	no	—
63	C₄²×C₆₁	C₄²	1	4	yes	—
64	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₆₁	C₄⋊C₄	1	8	yes	—
66	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₆₁	C₄×D₄	1	4	yes	—
69	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
70	C₂³×C₆₁	C₂×SD₁₆	1	2	yes	—
71	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂³×C₆₁	#135	1	4	yes	—
73	C₂×C₄×C₆₁	—	2	8	no	—
74	C₂×C₄×C₆₁	—	2	4	no	—
75	C₂×C₄×C₆₁	—	2	16	no	—
76	C₂×C₄×C₆₁	—	4	8	no	—
77	C₂×C₄×C₆₁	—	4	8	no	—
78	C₂×C₄×C₆₁	—	4	16	no	—
79	C₂×C₄×C₆₁	—	4	16	no	—
80	C₂×C₄×C₆₁	#95	1	4	yes	—
81	C₂×C₄×C₆₁	#95	1	4	yes	—
82	C₂×C₄×C₆₁	#102	1	8	yes	—
83	C₂×C₄×C₆₁	#103	1	8	yes	—
84	C₂×C₄×C₆₁	#106	1	4	yes	—
85	C₂×C₄×C₆₁	#107	1	4	yes	—
86	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₆₁	#117	1	2	yes	—
92	C₂×C₄×C₆₁	#117	1	4	yes	—
93	C₂×C₄×C₆₁	C₄×D₄	1	8	yes	—
94	C₂×C₄×C₆₁	C₄×Q₈	1	4	yes	—
95	C₈×C₆₁	—	2	8	no	—
96	C₄×#340	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
98	C₂²×C₆₁	C₄×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
100	C₂²×C₆₁	C₂×#102	1	2	yes	—
101	C₂²×C₆₁	C₂×#103	1	2	yes	—
102	C₂²×C₆₁	#416	1	4	yes	—
103	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
104	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
105	C₂²×C₆₁	#427	1	2	yes	—
106	C₂²×C₆₁	C₄×D₈	1	4	yes	—
107	C₂²×C₆₁	C₄×SD₁₆	1	8	yes	—
108	C₂²×C₆₁	C₄×Q₁₆	1	4	yes	—
109	C₄×C₆₁	—	2	4	no	—
110	C₄×C₆₁	—	4	16	no	—
111	C₄×C₆₁	#340	1	8	yes	direct product factor
112	C₄×C₆₁	C₄³	1	1	yes	characteristic
113	C₄×C₆₁	#376	1	1	yes	characteristic
114	C₄×C₆₁	C₄×C₂²⋊C₄	1	2	yes	—
115	C₄×C₆₁	C₄×C₄⋊C₄	1	4	yes	—
116	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
117	C₂×C₄×C₄⋊C₄	C₂×C₆₁	1	2	yes	—
118	C₂×#340	C₂×C₆₁	1	4	yes	—
119	C₂×C₆₁	C₄×#95	1	1	yes	characteristic
120	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
121	C₂×C₆₁	#1735	1	1	yes	characteristic
122	C₂×C₆₁	C₄×#102	1	4	yes	—
123	C₂×C₆₁	C₄×#103	1	4	yes	—
124	C₂×C₆₁	C₄×#106	1	2	yes	—
125	C₂×C₆₁	C₄×#107	1	2	yes	—
126	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
128	C₂²×C₄²	—	2	2	no	—
129	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	4	no	—
131	C₂×C₄×C₈	C₄×C₆₁	1	4	yes	complemented
132	C₄×C₄⋊C₄	—	2	4	no	—
133	C₄×C₄⋊C₄	C₄×C₆₁	1	8	yes	complemented
134	C₂×#95	—	2	2	no	—
135	#340	C₄×C₆₁	1	16	yes	direct product factor
136	C₆₁	C₄×#340	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×C₄	—	2	4	no	—
138	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
139	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
140	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₂×C₄×C₆₁	1	8	yes	—
144	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	8	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	16	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	—
149	C₂×C₄²	—	2	8	no	—
150	C₂×C₄²	—	4	4	no	—
151	C₂×C₄²	—	4	4	no	—
152	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
153	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
154	C₂×C₄²	D₄×C₆₁	1	4	yes	—
155	C₂×C₄²	Q₈×C₆₁	1	2	yes	—
156	C₄×C₈	—	2	8	no	—
157	#95	—	2	8	no	—
158	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	16	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
167	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
168	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
169	C₂²×C₄	D₈×C₆₁	1	2	yes	—
170	C₂²×C₄	SD₁₆×C₆₁	1	4	yes	—
171	C₂²×C₄	Q₁₆×C₆₁	1	2	yes	—
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	4	yes	—
175	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
176	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
177	C₂²×C₄	#41×C₆₁	1	4	yes	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	16	no	—
181	C₂×C₈	C₄²×C₆₁	1	8	yes	complemented
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	16	no	—
184	C₄⋊C₄	C₄²×C₆₁	1	16	yes	complemented
185	C₄²	—	4	16	no	—
186	C₄²	C₄²×C₆₁	1	4	yes	—
187	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
188	C₄²	C₄⋊C₄×C₆₁	1	8	yes	—
189	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
190	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
191	C₂³	C₄×D₄×C₆₁	1	4	yes	—
192	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
193	C₂³	C₂×SD₁₆×C₆₁	1	2	yes	—
194	C₂³	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
195	C₂³	#135×C₆₁	1	4	yes	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	—	4	16	no	—
203	C₂×C₄	#95×C₆₁	1	4	yes	—
204	C₂×C₄	#95×C₆₁	1	4	yes	—
205	C₂×C₄	#102×C₆₁	1	8	yes	—
206	C₂×C₄	#103×C₆₁	1	8	yes	—
207	C₂×C₄	#106×C₆₁	1	4	yes	—
208	C₂×C₄	#107×C₆₁	1	4	yes	—
209	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
211	C₂×C₄	C₂×C₄²×C₆₁	1	4	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
214	C₂×C₄	#117×C₆₁	1	2	yes	—
215	C₂×C₄	#117×C₆₁	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₆₁	1	8	yes	—
217	C₂×C₄	C₄×Q₈×C₆₁	1	4	yes	—
218	C₈	—	2	8	no	—
219	C₂²	C₂×#95×C₆₁	1	1	yes	characteristic
220	C₂²	C₄×C₂²⋊C₄×C₆₁	1	4	yes	—
221	C₂²	C₄×C₄⋊C₄×C₆₁	1	2	yes	—
222	C₂²	C₂×#102×C₆₁	1	2	yes	—
223	C₂²	C₂×#103×C₆₁	1	2	yes	—
224	C₂²	C₆₁×#416	1	4	yes	—
225	C₂²	C₂×#106×C₆₁	1	1	yes	characteristic
226	C₂²	C₂×#107×C₆₁	1	1	yes	characteristic
227	C₂²	C₆₁×#427	1	2	yes	—
228	C₂²	C₄×D₈×C₆₁	1	4	yes	—
229	C₂²	C₄×SD₁₆×C₆₁	1	8	yes	—
230	C₂²	C₄×Q₁₆×C₆₁	1	4	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	16	no	—
233	C₄	C₆₁×#340	1	8	yes	direct product factor
234	C₄	C₄³×C₆₁	1	1	yes	characteristic
235	C₄	C₆₁×#376	1	1	yes	characteristic
236	C₄	C₄×C₂²⋊C₄×C₆₁	1	2	yes	—
237	C₄	C₄×C₄⋊C₄×C₆₁	1	4	yes	—
238	C₂	C₄×#95×C₆₁	1	1	yes	characteristic
239	C₂	C₂×C₆₁×#340	1	1	yes	characteristic
240	C₂	C₆₁×#1735	1	1	yes	characteristic
241	C₂	C₄×#102×C₆₁	1	4	yes	—
242	C₂	C₄×#103×C₆₁	1	4	yes	—
243	C₂	C₄×#106×C₆₁	1	2	yes	—
244	C₂	C₄×#107×C₆₁	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₁	D₄
commutator	C₄	C₄³×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
socle, prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₈×C₆₁	C₄	semidirect
11. C₄×C₄⋊C₄×C₆₁	C₄	semidirect
13. C₆₁×#340	C₄	direct (N × H)
58. C₂×C₈×C₆₁	C₄²	semidirect
61. C₄⋊C₄×C₆₁	C₄²	semidirect
96. C₄×#340	C₆₁	direct (N × H)
111. C₄×C₆₁	#340	direct (N × H)
131. C₂×C₄×C₈	C₄×C₆₁	semidirect
133. C₄×C₄⋊C₄	C₄×C₆₁	semidirect
135. #340	C₄×C₆₁	direct (N × H)
136. C₆₁	C₄×#340	direct (N × H)
181. C₂×C₈	C₄²×C₆₁	semidirect
184. C₄⋊C₄	C₄²×C₆₁	semidirect
233. C₄	C₆₁×#340	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×C₄×C₈×C₆₁	2
C₂×C₄×C₄⋊C₄×C₆₁	2
C₂×C₆₁×#340	2
C₄×#340	61

Subgroups — C4×C61×#340 #29514135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#340 #29514135