Subgroups — C₂×C₆₁×#1290 #29511418

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#432	C₂	1	2	yes	maximal
2	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1290	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₆₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
6	C₆₁×#432	—	2	2	no	—
7	C₆₁×#432	—	2	2	no	—
8	C₆₁×#432	C₄	1	4	yes	—
9	C₆₁×#432	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
11	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₁	—	2	2	no	—
16	C₂×#97×C₆₁	C₄	1	4	yes	—
17	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	—	2	2	no	—
21	C₂²×C₈×C₆₁	—	2	2	no	—
22	C₂²×C₈×C₆₁	—	2	4	no	—
23	C₂²×C₈×C₆₁	—	4	4	no	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	D₄	1	2	yes	—
30	C₂×C₄²×C₆₁	D₄	1	2	yes	—
31	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	—	2	4	no	—
34	C₄⋊C₈×C₆₁	—	2	4	no	—
35	C₄⋊C₈×C₆₁	—	2	4	no	—
36	C₄⋊C₈×C₆₁	—	2	4	no	—
37	C₂²⋊C₈×C₆₁	—	2	4	no	—
38	C₂²⋊C₈×C₆₁	—	2	4	no	—
39	#97×C₆₁	—	2	4	no	—
40	#97×C₆₁	C₂×C₄	1	8	yes	—
41	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	—	2	2	no	—
44	C₂²×C₄×C₆₁	—	2	2	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₁	—	2	8	no	—
61	C₂×C₈×C₆₁	—	2	4	no	—
62	C₂×C₈×C₆₁	—	4	8	no	—
63	C₂×C₈×C₆₁	—	4	4	no	—
64	C₂×C₈×C₆₁	—	4	4	no	—
65	C₂×C₈×C₆₁	—	4	4	no	—
66	C₄²×C₆₁	C₄²	1	2	yes	—
67	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₁	C₂×D₄	1	2	yes	—
74	C₂³×C₆₁	—	2	2	no	—
75	C₂³×C₆₁	—	2	2	no	—
76	C₂³×C₆₁	#95	1	1	yes	characteristic
77	C₂³×C₆₁	#95	1	1	yes	characteristic
78	C₂³×C₆₁	#108	1	4	yes	—
79	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	4	no	—
82	C₂×C₄×C₆₁	—	2	4	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	4	1	no	—
94	C₂×C₄×C₆₁	—	4	1	no	—
95	C₂×C₄×C₆₁	#95	1	2	yes	—
96	C₂×C₄×C₆₁	#95	1	2	yes	—
97	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#100	1	2	yes	—
100	C₂×C₄×C₆₁	#101	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₆₁	—	4	8	no	—
106	C₈×C₆₁	—	4	4	no	—
107	C₂×#1290	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₆₁	—	2	8	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	1	no	—
111	C₂²×C₆₁	—	2	1	no	—
112	C₂²×C₆₁	—	2	1	no	—
113	C₂²×C₆₁	H(3,ℤ/4)	1	2	yes	—
114	C₂²×C₆₁	#341	1	2	yes	—
115	C₂²×C₆₁	#341	1	2	yes	—
116	C₂²×C₆₁	#343	1	2	yes	—
117	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×#108	1	2	yes	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	—	2	2	no	—
122	C₄×C₆₁	—	2	2	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	#343	1	2	yes	—
126	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
127	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
128	C₂×#432	C₂×C₆₁	1	2	yes	—
129	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
130	#1290	C₂×C₆₁	1	4	yes	direct product factor
131	C₂×C₆₁	—	2	2	no	—
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	#1290	1	4	yes	direct product factor
134	C₂×C₆₁	C₂×H(3,ℤ/4)	1	1	yes	characteristic
135	C₂×C₆₁	C₂×#341	1	1	yes	characteristic
136	C₂×C₆₁	C₂×#343	1	1	yes	characteristic
137	C₂³×C₈	C₄×C₆₁	1	2	yes	—
138	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
139	#432	—	2	2	no	—
140	#432	—	2	2	no	—
141	#432	C₄×C₆₁	1	4	yes	—
142	#432	C₂²×C₆₁	1	4	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	—	2	2	no	—
145	C₂×C₄⋊C₈	—	2	2	no	—
146	C₂×C₄⋊C₈	C₄×C₆₁	1	2	yes	—
147	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
148	C₂×C₂²⋊C₈	—	2	2	no	—
149	C₂×#97	C₄×C₆₁	1	4	yes	—
150	C₆₁	C₂×#1290	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
153	C₂³×C₄	Q₈×C₆₁	1	1	yes	characteristic
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	4	4	no	—
158	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
159	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
162	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₆₁	1	2	yes	—
164	C₂×C₄²	D₄×C₆₁	1	2	yes	—
165	C₂×C₄²	Q₈×C₆₁	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	#97	—	2	4	no	—
174	#97	C₂×C₄×C₆₁	1	8	yes	—
175	C₂⁴	C₄⋊C₄×C₆₁	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
181	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
189	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
190	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
194	C₂×C₈	—	2	8	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₄²	C₄²×C₆₁	1	2	yes	—
201	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
202	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
203	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
204	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
205	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
206	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
207	C₄²	C₂×D₄×C₆₁	1	2	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	#95×C₆₁	1	1	yes	characteristic
211	C₂³	#95×C₆₁	1	1	yes	characteristic
212	C₂³	#108×C₆₁	1	4	yes	—
213	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#95×C₆₁	1	2	yes	—
230	C₂×C₄	#95×C₆₁	1	2	yes	—
231	C₂×C₄	#95×C₆₁	1	1	yes	characteristic
232	C₂×C₄	#95×C₆₁	1	1	yes	characteristic
233	C₂×C₄	#100×C₆₁	1	2	yes	—
234	C₂×C₄	#101×C₆₁	1	2	yes	—
235	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
239	C₈	—	4	8	no	—
240	C₈	—	4	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	C₆₁×H(3,ℤ/4)	1	2	yes	—
247	C₂²	C₆₁×#341	1	2	yes	—
248	C₂²	C₆₁×#341	1	2	yes	—
249	C₂²	C₆₁×#343	1	2	yes	—
250	C₂²	C₂×#95×C₆₁	1	1	yes	characteristic
251	C₂²	C₂×#108×C₆₁	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	C₆₁×#343	1	2	yes	—
259	C₄	C₂×#100×C₆₁	1	1	yes	characteristic
260	C₄	C₂×#101×C₆₁	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	C₆₁×#1290	1	4	yes	direct product factor
264	C₂	C₂×C₆₁×H(3,ℤ/4)	1	1	yes	characteristic
265	C₂	C₂×C₆₁×#341	1	1	yes	characteristic
266	C₂	C₂×C₆₁×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	#95
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1290	C₂	direct (N × H)
107. C₂×#1290	C₆₁	direct (N × H)
130. #1290	C₂×C₆₁	direct (N × H)
133. C₂×C₆₁	#1290	direct (N × H)
150. C₆₁	C₂×#1290	direct (N × H)
263. C₂	C₆₁×#1290	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×#432	2
C₂²×C₄⋊C₈×C₆₁	2
C₆₁×#1290	2
C₂×#1290	61

Subgroups — C2×C61×#1290 #29511418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1290 #29511418