Subgroups — C₂²×#16840237 #30586594

Order 15616.

14590 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840237	C₂	1	48	yes	maximal, direct product factor
2	C₂²×D₄×Dic₆₁	C₂	1	8	yes	maximal, complemented
3	C₂²×#14519736	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊Dic₆₁	C₂	1	4	yes	maximal, complemented
5	C₂⁴×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₆₁	C₂²	1	8	yes	complemented
7	#16840237	C₂²	1	256	yes	direct product factor
8	C₂×D₄×Dic₆₁	C₂²	1	192	yes	complemented
9	C₂×#14519736	C₂²	1	48	yes	complemented
10	C₂²×C₄⋊Dic₆₁	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊Dic₆₁	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₆₁	C₂²	1	8	yes	complemented
13	C₂²×C₂²⋊Dic₆₁	C₂²	1	96	yes	complemented
14	C₂²×C₄×Dic₆₁	C₂²	1	4	yes	complemented
15	C₂⁶×C₆₁	C₂²	1	2	yes	—
16	C₂³×D₄×C₆₁	C₄	1	32	yes	complemented
17	C₂³×D₄×C₆₁	C₂²	1	4	yes	—
18	C₂³×D₄×C₆₁	C₂²	1	24	yes	—
19	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂³×Dic₆₁	—	4	32	no	—
21	C₂³×Dic₆₁	C₂³	1	8	yes	—
22	C₂³×Dic₆₁	C₂³	1	96	yes	complemented
23	D₄×Dic₆₁	C₂³	1	512	yes	complemented
24	#14519736	C₂³	1	128	yes	complemented
25	C₂×C₄⋊Dic₆₁	C₂³	1	48	yes	complemented
26	C₂×C₂²⋊Dic₆₁	C₂³	1	48	yes	—
27	C₂×C₂²⋊Dic₆₁	C₂³	1	256	yes	complemented
28	C₂×C₂²⋊Dic₆₁	C₂³	1	96	yes	complemented
29	C₂×C₄×Dic₆₁	C₂³	1	48	yes	complemented
30	C₂⁵×C₆₁	—	4	16	no	—
31	C₂⁵×C₆₁	C₂×C₄	1	32	yes	complemented
32	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
33	C₂⁵×C₆₁	C₂³	1	4	yes	—
34	C₂⁵×C₆₁	C₂³	1	24	yes	—
35	C₂²×D₄×C₆₁	C₂×C₄	1	64	yes	complemented
36	C₂²×D₄×C₆₁	C₂×C₄	1	384	yes	complemented
37	C₂²×D₄×C₆₁	C₂³	1	48	yes	—
38	C₂²×D₄×C₆₁	C₂³	1	64	yes	—
39	C₂³×C₄×C₆₁	C₂×C₄	1	16	yes	complemented
40	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
41	C₂³×C₄×C₆₁	C₂³	1	12	yes	—
42	C₂⁴×C₆₁	—	2	32	no	—
43	C₂⁴×C₆₁	—	2	8	no	—
44	C₂⁴×C₆₁	—	4	128	no	—
45	C₂⁴×C₆₁	—	4	96	no	—
46	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
47	C₂⁴×C₆₁	C₂²×C₄	1	32	yes	—
48	C₂⁴×C₆₁	C₂²×C₄	1	192	yes	complemented
49	C₂⁴×C₆₁	C₂⁴	1	24	yes	—
50	C₂⁴×C₆₁	C₂⁴	1	6	yes	—
51	C₂⁴×C₆₁	C₂⁴	1	32	yes	—
52	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
53	C₂×D₄×C₆₁	C₂²×C₄	1	384	yes	complemented
54	C₂×D₄×C₆₁	C₂²×C₄	1	512	yes	complemented
55	C₂×D₄×C₆₁	C₂⁴	1	64	yes	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	96	yes	complemented
58	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
59	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
60	C₂²×Dic₆₁	—	2	32	no	—
61	C₂²×Dic₆₁	—	4	192	no	—
62	C₂²×Dic₆₁	C₂⁴	1	48	yes	—
63	C₂²×Dic₆₁	C₂⁴	1	128	yes	complemented
64	C₂²⋊Dic₆₁	C₂⁴	1	64	yes	—
65	C₂²⋊Dic₆₁	C₂⁴	1	128	yes	complemented
66	C₄⋊Dic₆₁	C₂⁴	1	64	yes	complemented
67	C₄×Dic₆₁	C₂⁴	1	64	yes	complemented
68	C₂³×C₆₁	—	2	32	no	—
69	C₂³×C₆₁	—	2	128	no	—
70	C₂³×C₆₁	—	2	96	no	—
71	C₂³×C₆₁	—	2	24	no	—
72	C₂³×C₆₁	—	4	384	no	—
73	C₂³×C₆₁	—	4	64	no	—
74	C₂³×C₆₁	C₂³×C₄	1	24	yes	—
75	C₂³×C₆₁	C₂³×C₄	1	96	yes	—
76	C₂³×C₆₁	C₂³×C₄	1	4	yes	—
77	C₂³×C₆₁	C₂³×C₄	1	128	yes	complemented
78	C₂³×C₆₁	H(5,𝔽₂)	1	8	yes	—
79	C₂³×C₆₁	C₂⁵	1	4	yes	—
80	C₂³×C₆₁	C₂⁵	1	16	yes	—
81	C₂³×C₆₁	C₂⁵	1	3	yes	—
82	D₄×C₆₁	C₂³×C₄	1	256	yes	complemented
83	C₂×C₄×C₆₁	C₂³×C₄	1	48	yes	—
84	C₂×C₄×C₆₁	C₂³×C₄	1	64	yes	complemented
85	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
86	C₂×Dic₆₁	—	2	96	no	—
87	C₂×Dic₆₁	—	4	128	no	—
88	C₂×Dic₆₁	C₂⁵	1	32	yes	—
89	C₂²×#518	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	32	no	—
91	C₂²×C₆₁	—	2	8	no	—
92	C₂²×C₆₁	—	2	48	no	—
93	C₂²×C₆₁	—	2	192	no	—
94	C₂²×C₆₁	—	4	128	no	—
95	C₂²×C₆₁	#518	1	16	yes	complemented
96	C₂²×C₆₁	C₂⁴×C₄	1	8	yes	—
97	C₂²×C₆₁	C₂⁴×C₄	1	32	yes	—
98	C₂²×C₆₁	C₂⁴×C₄	1	6	yes	—
99	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	12	yes	—
100	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
101	C₄×C₆₁	C₂⁴×C₄	1	16	yes	—
102	Dic₆₁	—	2	32	no	—
103	C₂⁴×D₄	D₆₁	1	1	yes	characteristic
104	C₂×#518	—	61	48	no	—
105	C₂²×C₄×D₄	—	61	8	no	—
106	C₂²×#117	—	61	2	no	—
107	C₂³×C₂²⋊C₄	—	61	4	no	—
108	C₂×C₆₁	—	2	8	no	—
109	C₂×C₆₁	—	2	32	no	—
110	C₂×C₆₁	C₂×#518	1	12	yes	complemented
111	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
112	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	2	yes	—
113	C₂⁶	C₂×D₆₁	1	2	yes	—
114	C₂³×D₄	Dic₆₁	1	32	yes	complemented
115	C₂³×D₄	C₂×D₆₁	1	4	yes	—
116	C₂³×D₄	C₂×D₆₁	1	24	yes	—
117	C₂⁴×C₄	—	61	8	no	—
118	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
119	#518	—	61	256	no	—
120	C₂×C₄×D₄	—	61	192	no	—
121	C₂×#117	—	61	48	no	—
122	C₂²×C₄⋊C₄	—	61	4	no	—
123	C₂²×C₂²⋊C₄	—	61	4	no	—
124	C₂²×C₂²⋊C₄	—	61	8	no	—
125	C₂²×C₂²⋊C₄	—	61	96	no	—
126	C₂²×C₄²	—	61	4	no	—
127	C₆₁	C₂²×#518	1	1	yes	characteristic, complemented, Sylow
128	C₂⁵	—	4	16	no	—
129	C₂⁵	C₂×Dic₆₁	1	32	yes	complemented
130	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
131	C₂⁵	C₂²×D₆₁	1	4	yes	—
132	C₂⁵	C₂²×D₆₁	1	24	yes	—
133	C₂²×D₄	C₂×Dic₆₁	1	64	yes	complemented
134	C₂²×D₄	C₂×Dic₆₁	1	384	yes	complemented
135	C₂²×D₄	C₂²×D₆₁	1	48	yes	—
136	C₂²×D₄	C₂²×D₆₁	1	64	yes	—
137	C₂³×C₄	—	61	8	no	—
138	C₂³×C₄	—	61	96	no	—
139	C₂³×C₄	—	244	32	no	—
140	C₂³×C₄	C₂×Dic₆₁	1	16	yes	complemented
141	C₂³×C₄	C₂²×D₆₁	1	2	yes	—
142	C₂³×C₄	C₂²×D₆₁	1	12	yes	—
143	C₄×D₄	—	61	512	no	—
144	#117	—	61	128	no	—
145	C₂×C₄⋊C₄	—	61	48	no	—
146	C₂×C₂²⋊C₄	—	61	48	no	—
147	C₂×C₂²⋊C₄	—	61	96	no	—
148	C₂×C₂²⋊C₄	—	61	256	no	—
149	C₂×C₄²	—	61	48	no	—
150	C₂⁴	—	2	32	no	—
151	C₂⁴	—	2	8	no	—
152	C₂⁴	—	4	128	no	—
153	C₂⁴	—	4	96	no	—
154	C₂⁴	C₂²×Dic₆₁	1	8	yes	—
155	C₂⁴	C₂²×Dic₆₁	1	32	yes	—
156	C₂⁴	C₂²×Dic₆₁	1	192	yes	complemented
157	C₂⁴	C₂³×D₆₁	1	6	yes	—
158	C₂⁴	C₂³×D₆₁	1	24	yes	—
159	C₂⁴	C₂³×D₆₁	1	32	yes	—
160	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
161	C₂×D₄	C₂²×Dic₆₁	1	384	yes	complemented
162	C₂×D₄	C₂²×Dic₆₁	1	512	yes	complemented
163	C₂×D₄	C₂³×D₆₁	1	64	yes	—
164	C₂²×C₄	—	61	48	no	—
165	C₂²×C₄	—	61	128	no	—
166	C₂²×C₄	—	122	32	no	—
167	C₂²×C₄	—	244	192	no	—
168	C₂²×C₄	C₂²×Dic₆₁	1	16	yes	—
169	C₂²×C₄	C₂²×Dic₆₁	1	96	yes	complemented
170	C₂²×C₄	C₂³×D₆₁	1	12	yes	—
171	C₂²×C₄	C₂³×D₆₁	1	16	yes	—
172	C₄⋊C₄	—	61	64	no	—
173	C₂²⋊C₄	—	61	64	no	—
174	C₂²⋊C₄	—	61	128	no	—
175	C₄²	—	61	64	no	—
176	C₂³	—	2	32	no	—
177	C₂³	—	2	128	no	—
178	C₂³	—	2	96	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	4	384	no	—
181	C₂³	—	4	64	no	—
182	C₂³	C₂³×Dic₆₁	1	24	yes	—
183	C₂³	C₂³×Dic₆₁	1	96	yes	—
184	C₂³	C₂³×Dic₆₁	1	4	yes	—
185	C₂³	C₂³×Dic₆₁	1	128	yes	complemented
186	C₂³	#14519822	1	8	yes	—
187	C₂³	C₂⁴×D₆₁	1	4	yes	—
188	C₂³	C₂⁴×D₆₁	1	16	yes	—
189	C₂³	C₂⁴×D₆₁	1	3	yes	—
190	D₄	C₂³×Dic₆₁	1	256	yes	complemented
191	C₂×C₄	—	61	32	no	—
192	C₂×C₄	—	122	96	no	—
193	C₂×C₄	—	244	128	no	—
194	C₂×C₄	C₂³×Dic₆₁	1	48	yes	—
195	C₂×C₄	C₂³×Dic₆₁	1	64	yes	complemented
196	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
197	C₂²	—	2	48	no	—
198	C₂²	—	2	192	no	—
199	C₂²	—	2	32	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	128	no	—
202	C₂²	#16840237	1	16	yes	direct product factor
203	C₂²	C₂⁴×Dic₆₁	1	8	yes	—
204	C₂²	C₂⁴×Dic₆₁	1	32	yes	—
205	C₂²	C₂⁴×Dic₆₁	1	6	yes	—
206	C₂²	C₂×#14519822	1	12	yes	—
207	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
208	C₄	—	122	32	no	—
209	C₄	C₂⁴×Dic₆₁	1	16	yes	—
210	C₂	—	2	8	no	—
211	C₂	—	2	32	no	—
212	C₂	C₂×#16840237	1	12	yes	direct product factor
213	C₂	C₂⁵×Dic₆₁	1	1	yes	characteristic
214	C₂	C₂²×#14519822	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting, prime radical	C₂⁴×D₄×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂⁴×D₄	D₆₁
hypocenter	C₆₁	C₂²×#518
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16840237	C₂	direct (N × H)
2. C₂²×D₄×Dic₆₁	C₂	semidirect
3. C₂²×#14519736	C₂	semidirect
4. C₂³×C₂²⋊Dic₆₁	C₂	semidirect
6. C₂⁴×Dic₆₁	C₂²	semidirect
7. #16840237	C₂²	direct (N × H)
8. C₂×D₄×Dic₆₁	C₂²	semidirect
9. C₂×#14519736	C₂²	semidirect
10. C₂²×C₄⋊Dic₆₁	C₂²	semidirect
12. C₂²×C₂²⋊Dic₆₁	C₂²	semidirect
13. C₂²×C₂²⋊Dic₆₁	C₂²	semidirect
14. C₂²×C₄×Dic₆₁	C₂²	semidirect
16. C₂³×D₄×C₆₁	C₄	semidirect
22. C₂³×Dic₆₁	C₂³	semidirect
23. D₄×Dic₆₁	C₂³	semidirect
24. #14519736	C₂³	semidirect
25. C₂×C₄⋊Dic₆₁	C₂³	semidirect
27. C₂×C₂²⋊Dic₆₁	C₂³	semidirect
28. C₂×C₂²⋊Dic₆₁	C₂³	semidirect
29. C₂×C₄×Dic₆₁	C₂³	semidirect
31. C₂⁵×C₆₁	C₂×C₄	semidirect
35. C₂²×D₄×C₆₁	C₂×C₄	semidirect
36. C₂²×D₄×C₆₁	C₂×C₄	semidirect
39. C₂³×C₄×C₆₁	C₂×C₄	semidirect
48. C₂⁴×C₆₁	C₂²×C₄	semidirect
53. C₂×D₄×C₆₁	C₂²×C₄	semidirect
54. C₂×D₄×C₆₁	C₂²×C₄	semidirect
57. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
63. C₂²×Dic₆₁	C₂⁴	semidirect
65. C₂²⋊Dic₆₁	C₂⁴	semidirect
66. C₄⋊Dic₆₁	C₂⁴	semidirect
67. C₄×Dic₆₁	C₂⁴	semidirect
77. C₂³×C₆₁	C₂³×C₄	semidirect
82. D₄×C₆₁	C₂³×C₄	semidirect
84. C₂×C₄×C₆₁	C₂³×C₄	semidirect
95. C₂²×C₆₁	#518	semidirect
110. C₂×C₆₁	C₂×#518	semidirect
114. C₂³×D₄	Dic₆₁	semidirect
127. C₆₁	C₂²×#518	semidirect
129. C₂⁵	C₂×Dic₆₁	semidirect
133. C₂²×D₄	C₂×Dic₆₁	semidirect
134. C₂²×D₄	C₂×Dic₆₁	semidirect
140. C₂³×C₄	C₂×Dic₆₁	semidirect
156. C₂⁴	C₂²×Dic₆₁	semidirect
161. C₂×D₄	C₂²×Dic₆₁	semidirect
162. C₂×D₄	C₂²×Dic₆₁	semidirect
169. C₂²×C₄	C₂²×Dic₆₁	semidirect
185. C₂³	C₂³×Dic₆₁	semidirect
190. D₄	C₂³×Dic₆₁	semidirect
195. C₂×C₄	C₂³×Dic₆₁	semidirect
202. C₂²	#16840237	direct (N × H)
212. C₂	C₂×#16840237	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×#16840237	2
C₂²×D₄×Dic₆₁	2
C₂²×#14519736	2
C₂³×C₂²⋊Dic₆₁	2
C₂⁴×D₄×C₆₁	2
C₂²×#518	61

Subgroups — C22×#16840237 #30586594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840237 #30586594