Subgroups — C₂×#21448172 #29838218

Order 15616.

5790 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊Dic₆₁	C₂	1	2	yes	maximal
2	#21448172	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂²⋊C₄×Dic₆₁	C₂	1	8	yes	maximal, complemented
4	C₂×C₄×C₂²⋊Dic₆₁	C₂	1	4	yes	maximal, complemented
5	C₂³×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₆₁	C₂²	1	4	yes	—
7	C₂²×C₂²⋊Dic₆₁	C₄	1	32	yes	complemented
8	C₂²×C₂²⋊Dic₆₁	C₂²	1	4	yes	—
9	C₂²×C₂²⋊Dic₆₁	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₆₁	C₂²	1	16	yes	—
11	C₂²×C₄×Dic₆₁	C₂²	1	8	yes	complemented
12	C₂²⋊C₄×Dic₆₁	C₂²	1	64	yes	complemented
13	C₄×C₂²⋊Dic₆₁	C₂²	1	32	yes	complemented
14	C₂×#14519684	C₂²	1	8	yes	complemented
15	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₆₁	C₄	1	16	yes	complemented
18	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	—
20	C₂³×Dic₆₁	—	4	16	no	—
21	C₂³×Dic₆₁	C₂×C₄	1	32	yes	complemented
22	C₂³×Dic₆₁	C₂³	1	8	yes	—
23	C₂³×Dic₆₁	C₂³	1	16	yes	—
24	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	32	yes	complemented
25	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	32	yes	complemented
26	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	128	yes	complemented
27	C₂×C₂²⋊Dic₆₁	C₂³	1	16	yes	—
28	C₂×C₂²⋊Dic₆₁	C₂³	1	16	yes	—
29	C₂×C₄×Dic₆₁	—	4	16	no	—
30	C₂×C₄×Dic₆₁	C₂³	1	32	yes	complemented
31	#14519684	C₂³	1	32	yes	complemented
32	C₂⁵×C₆₁	—	4	8	no	—
33	C₂⁵×C₆₁	C₂×C₄	1	16	yes	—
34	C₂⁵×C₆₁	C₂×C₄	1	8	yes	—
35	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
36	C₂⁵×C₆₁	C₂³	1	2	yes	—
37	C₂⁵×C₆₁	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₁	—	4	8	no	—
39	C₂³×C₄×C₆₁	C₂×C₄	1	16	yes	complemented
40	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
41	C₂³×C₄×C₆₁	C₂³	1	8	yes	—
42	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	32	yes	complemented
43	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	64	yes	complemented
44	C₂×C₂²⋊C₄×C₆₁	C₂³	1	16	yes	—
45	C₂⁴×C₆₁	—	2	16	no	—
46	C₂⁴×C₆₁	—	2	8	no	—
47	C₂⁴×C₆₁	—	4	64	no	—
48	C₂⁴×C₆₁	—	4	32	no	—
49	C₂⁴×C₆₁	—	4	16	no	—
50	C₂⁴×C₆₁	C₄²	1	64	yes	complemented
51	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
52	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
53	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
55	C₂⁴×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂⁴×C₆₁	C₂²×C₄	1	32	yes	—
57	C₂⁴×C₆₁	C₂²×C₄	1	16	yes	—
58	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
59	C₂⁴×C₆₁	C₂⁴	1	2	yes	—
60	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	—	2	16	no	—
62	C₂²×C₄×C₆₁	—	4	32	no	—
63	C₂²×C₄×C₆₁	—	4	16	no	—
64	C₂²×C₄×C₆₁	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	32	yes	complemented
66	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
67	C₂²×Dic₆₁	—	2	32	no	—
68	C₂²×Dic₆₁	—	4	64	no	—
69	C₂²×Dic₆₁	—	4	32	no	—
70	C₂²×Dic₆₁	C₂²×C₄	1	32	yes	—
71	C₂²×Dic₆₁	C₂²×C₄	1	64	yes	complemented
72	C₂²×Dic₆₁	C₂⁴	1	16	yes	—
73	C₂²⋊C₄×C₆₁	C₂²×C₄	1	64	yes	complemented
74	C₂²⋊Dic₆₁	C₂²×C₄	1	64	yes	complemented
75	C₂²⋊Dic₆₁	C₂²×C₄	1	64	yes	complemented
76	C₄×Dic₆₁	—	4	32	no	—
77	C₂³×C₆₁	—	2	32	no	—
78	C₂³×C₆₁	—	2	16	no	—
79	C₂³×C₆₁	—	2	32	no	—
80	C₂³×C₆₁	—	2	32	no	—
81	C₂³×C₆₁	—	2	32	no	—
82	C₂³×C₆₁	—	2	16	no	—
83	C₂³×C₆₁	—	2	8	no	—
84	C₂³×C₆₁	—	4	128	no	—
85	C₂³×C₆₁	—	4	64	no	—
86	C₂³×C₆₁	—	4	32	no	—
87	C₂³×C₆₁	C₂×C₄²	1	32	yes	—
88	C₂³×C₆₁	C₂×C₄²	1	16	yes	—
89	C₂³×C₆₁	C₂×C₄²	1	64	yes	complemented
90	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
91	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
92	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
93	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
94	C₂³×C₆₁	C₂³×C₄	1	4	yes	—
95	C₂³×C₆₁	C₂³×C₄	1	4	yes	—
96	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
97	C₂³×C₆₁	H(5,𝔽₂)	1	8	yes	—
98	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
99	C₂×C₄×C₆₁	—	2	32	no	—
100	C₂×C₄×C₆₁	—	2	16	no	—
101	C₂×C₄×C₆₁	—	4	32	no	—
102	C₂×C₄×C₆₁	C₂³×C₄	1	16	yes	—
103	C₂×Dic₆₁	—	2	64	no	—
104	C₂×Dic₆₁	—	2	32	no	—
105	C₂×Dic₆₁	—	4	64	no	—
106	C₂×Dic₆₁	C₂³×C₄	1	32	yes	—
107	C₂×#2273	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	—	2	16	no	—
109	C₂²×C₆₁	—	2	8	no	—
110	C₂²×C₆₁	—	2	16	no	—
111	C₂²×C₆₁	—	2	16	no	—
112	C₂²×C₆₁	—	2	16	no	—
113	C₂²×C₆₁	—	2	64	no	—
114	C₂²×C₆₁	—	2	32	no	—
115	C₂²×C₆₁	—	4	64	no	—
116	C₂²×C₆₁	C₂²×C₄²	1	8	yes	—
117	C₂²×C₆₁	C₂²×C₄²	1	16	yes	—
118	C₂²×C₆₁	C₂²×C₄²	1	4	yes	—
119	C₂²×C₆₁	#518	1	16	yes	—
120	C₂²×C₆₁	#518	1	8	yes	—
121	C₂²×C₆₁	C₂⁴×C₄	1	2	yes	—
122	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
123	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	4	yes	—
124	C₄×C₆₁	—	2	16	no	—
125	Dic₆₁	—	2	32	no	—
126	C₂³×C₂²⋊C₄	—	61	2	no	—
127	C₂³×C₂²⋊C₄	D₆₁	1	1	yes	characteristic
128	#2273	—	61	16	no	—
129	C₂×C₄×C₂²⋊C₄	—	61	4	no	—
130	C₂×C₄×C₂²⋊C₄	—	61	8	no	—
131	C₂×C₆₁	—	2	16	no	—
132	C₂×C₆₁	—	2	8	no	—
133	C₂×C₆₁	#2273	1	8	yes	complemented
134	C₂×C₆₁	C₂³×C₄²	1	1	yes	characteristic
135	C₂×C₆₁	C₂×#518	1	4	yes	—
136	C₂×C₆₁	C₂×#518	1	2	yes	—
137	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
138	C₂⁴×C₄	—	61	4	no	—
139	C₂⁴×C₄	C₂×D₆₁	1	2	yes	—
140	C₂²×C₂²⋊C₄	—	61	32	no	—
141	C₂²×C₂²⋊C₄	—	61	4	no	—
142	C₂²×C₂²⋊C₄	—	61	4	no	—
143	C₂²×C₂²⋊C₄	—	61	16	no	—
144	C₂²×C₂²⋊C₄	Dic₆₁	1	16	yes	complemented
145	C₂²×C₂²⋊C₄	C₂×D₆₁	1	4	yes	—
146	C₂²×C₂²⋊C₄	C₂×D₆₁	1	8	yes	—
147	C₂²×C₄²	—	61	8	no	—
148	C₄×C₂²⋊C₄	—	61	32	no	—
149	C₄×C₂²⋊C₄	—	61	64	no	—
150	C₂×#95	—	61	8	no	—
151	C₆₁	C₂×#2273	1	1	yes	characteristic, complemented, Sylow
152	C₂⁵	—	4	8	no	—
153	C₂⁵	C₄×D₆₁	1	16	yes	—
154	C₂⁵	C₂×Dic₆₁	1	8	yes	—
155	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
156	C₂⁵	C₂²×D₆₁	1	2	yes	—
157	C₂⁵	C₂²×D₆₁	1	4	yes	—
158	C₂³×C₄	—	4	8	no	—
159	C₂³×C₄	—	61	8	no	—
160	C₂³×C₄	—	61	32	no	—
161	C₂³×C₄	—	61	16	no	—
162	C₂³×C₄	—	244	16	no	—
163	C₂³×C₄	C₂×Dic₆₁	1	16	yes	complemented
164	C₂³×C₄	C₂²×D₆₁	1	4	yes	—
165	C₂³×C₄	C₂²×D₆₁	1	8	yes	—
166	C₂×C₂²⋊C₄	—	61	32	no	—
167	C₂×C₂²⋊C₄	—	61	32	no	—
168	C₂×C₂²⋊C₄	—	61	128	no	—
169	C₂×C₂²⋊C₄	—	61	16	no	—
170	C₂×C₂²⋊C₄	—	61	16	no	—
171	C₂×C₂²⋊C₄	C₂×Dic₆₁	1	32	yes	complemented
172	C₂×C₂²⋊C₄	C₂×Dic₆₁	1	64	yes	complemented
173	C₂×C₂²⋊C₄	C₂²×D₆₁	1	16	yes	—
174	C₂×C₄²	—	61	32	no	—
175	C₂×C₄²	—	244	16	no	—
176	#95	—	61	32	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	2	16	no	—
179	C₂⁴	—	4	64	no	—
180	C₂⁴	—	4	32	no	—
181	C₂⁴	—	4	16	no	—
182	C₂⁴	C₄×Dic₆₁	1	64	yes	complemented
183	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
184	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
185	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
186	C₂⁴	C₂×C₄×D₆₁	1	32	yes	—
187	C₂⁴	C₂²×Dic₆₁	1	8	yes	—
188	C₂⁴	C₂²×Dic₆₁	1	4	yes	—
189	C₂⁴	C₂²×Dic₆₁	1	16	yes	—
190	C₂⁴	C₂³×D₆₁	1	2	yes	—
191	C₂⁴	C₂³×D₆₁	1	4	yes	—
192	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
193	C₂²×C₄	—	2	16	no	—
194	C₂²×C₄	—	4	32	no	—
195	C₂²×C₄	—	4	16	no	—
196	C₂²×C₄	—	61	32	no	—
197	C₂²×C₄	—	61	16	no	—
198	C₂²×C₄	—	61	64	no	—
199	C₂²×C₄	—	122	32	no	—
200	C₂²×C₄	—	244	64	no	—
201	C₂²×C₄	—	244	32	no	—
202	C₂²×C₄	C₂²×Dic₆₁	1	16	yes	—
203	C₂²×C₄	C₂²×Dic₆₁	1	32	yes	complemented
204	C₂²×C₄	C₂³×D₆₁	1	8	yes	—
205	C₂²⋊C₄	—	61	64	no	—
206	C₂²⋊C₄	—	61	64	no	—
207	C₂²⋊C₄	C₂²×Dic₆₁	1	64	yes	complemented
208	C₄²	—	244	32	no	—
209	C₂³	—	2	32	no	—
210	C₂³	—	2	32	no	—
211	C₂³	—	2	32	no	—
212	C₂³	—	2	16	no	—
213	C₂³	—	2	32	no	—
214	C₂³	—	2	8	no	—
215	C₂³	—	2	16	no	—
216	C₂³	—	4	128	no	—
217	C₂³	—	4	64	no	—
218	C₂³	—	4	32	no	—
219	C₂³	C₂×C₄×Dic₆₁	1	16	yes	—
220	C₂³	C₂×C₄×Dic₆₁	1	32	yes	—
221	C₂³	C₂×C₄×Dic₆₁	1	64	yes	complemented
222	C₂³	C₂²×C₄×D₆₁	1	8	yes	—
223	C₂³	C₂²×C₄×D₆₁	1	8	yes	—
224	C₂³	C₂²×C₄×D₆₁	1	8	yes	—
225	C₂³	C₂²×C₄×D₆₁	1	4	yes	—
226	C₂³	C₂³×Dic₆₁	1	8	yes	—
227	C₂³	C₂³×Dic₆₁	1	4	yes	—
228	C₂³	C₂³×Dic₆₁	1	2	yes	—
229	C₂³	#14519822	1	8	yes	—
230	C₂³	C₂⁴×D₆₁	1	1	yes	characteristic
231	C₂×C₄	—	2	32	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	—	4	32	no	—
234	C₂×C₄	—	61	32	no	—
235	C₂×C₄	—	122	64	no	—
236	C₂×C₄	—	122	32	no	—
237	C₂×C₄	—	244	64	no	—
238	C₂×C₄	C₂³×Dic₆₁	1	16	yes	—
239	C₂²	—	2	16	no	—
240	C₂²	—	2	16	no	—
241	C₂²	—	2	16	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	16	no	—
244	C₂²	—	2	64	no	—
245	C₂²	—	2	32	no	—
246	C₂²	—	4	64	no	—
247	C₂²	C₂²×C₄×Dic₆₁	1	8	yes	—
248	C₂²	C₂²×C₄×Dic₆₁	1	16	yes	—
249	C₂²	C₂²×C₄×Dic₆₁	1	4	yes	—
250	C₂²	#16840235	1	16	yes	—
251	C₂²	#16840237	1	8	yes	—
252	C₂²	C₂³×C₄×D₆₁	1	2	yes	—
253	C₂²	C₂⁴×Dic₆₁	1	1	yes	characteristic
254	C₂²	C₂×#14519822	1	4	yes	—
255	C₄	—	2	16	no	—
256	C₄	—	122	32	no	—
257	C₂	—	2	16	no	—
258	C₂	—	2	8	no	—
259	C₂	#21448172	1	8	yes	direct product factor
260	C₂	C₂³×C₄×Dic₆₁	1	1	yes	characteristic
261	C₂	C₂×#16840235	1	4	yes	—
262	C₂	C₂×#16840237	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator	C₂×C₆₁	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₆₁
Fitting	C₂³×C₂²⋊C₄×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
prime radical	C₂⁶×C₆₁	C₂²
cosocle	C₂³×C₆₁	C₂⁵
hypercenter	C₂³×C₂²⋊C₄	D₆₁
hypocenter	C₆₁	C₂×#2273
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #21448172	C₂	direct (N × H)
3. C₂×C₂²⋊C₄×Dic₆₁	C₂	semidirect
4. C₂×C₄×C₂²⋊Dic₆₁	C₂	semidirect
7. C₂²×C₂²⋊Dic₆₁	C₄	semidirect
11. C₂²×C₄×Dic₆₁	C₂²	semidirect
12. C₂²⋊C₄×Dic₆₁	C₂²	semidirect
13. C₄×C₂²⋊Dic₆₁	C₂²	semidirect
14. C₂×#14519684	C₂²	semidirect
17. C₂²×C₂²⋊C₄×C₆₁	C₄	semidirect
21. C₂³×Dic₆₁	C₂×C₄	semidirect
24. C₂×C₂²⋊Dic₆₁	C₂×C₄	semidirect
25. C₂×C₂²⋊Dic₆₁	C₂×C₄	semidirect
26. C₂×C₂²⋊Dic₆₁	C₂×C₄	semidirect
30. C₂×C₄×Dic₆₁	C₂³	semidirect
31. #14519684	C₂³	semidirect
39. C₂³×C₄×C₆₁	C₂×C₄	semidirect
42. C₂×C₂²⋊C₄×C₆₁	C₂×C₄	semidirect
43. C₂×C₂²⋊C₄×C₆₁	C₂×C₄	semidirect
50. C₂⁴×C₆₁	C₄²	semidirect
65. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
71. C₂²×Dic₆₁	C₂²×C₄	semidirect
73. C₂²⋊C₄×C₆₁	C₂²×C₄	semidirect
74. C₂²⋊Dic₆₁	C₂²×C₄	semidirect
75. C₂²⋊Dic₆₁	C₂²×C₄	semidirect
89. C₂³×C₆₁	C₂×C₄²	semidirect
133. C₂×C₆₁	#2273	semidirect
144. C₂²×C₂²⋊C₄	Dic₆₁	semidirect
151. C₆₁	C₂×#2273	semidirect
163. C₂³×C₄	C₂×Dic₆₁	semidirect
171. C₂×C₂²⋊C₄	C₂×Dic₆₁	semidirect
172. C₂×C₂²⋊C₄	C₂×Dic₆₁	semidirect
182. C₂⁴	C₄×Dic₆₁	semidirect
203. C₂²×C₄	C₂²×Dic₆₁	semidirect
207. C₂²⋊C₄	C₂²×Dic₆₁	semidirect
221. C₂³	C₂×C₄×Dic₆₁	semidirect
259. C₂	#21448172	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊Dic₆₁	2
#21448172	2
C₂×C₂²⋊C₄×Dic₆₁	2
C₂×C₄×C₂²⋊Dic₆₁	2
C₂³×C₂²⋊C₄×C₆₁	2
C₂×#2273	61

Subgroups — C2×#21448172 #29838218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21448172 #29838218