Subgroups — C₂³×C₄×D₂₄₄ #30586579

Order 15616.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₆₁	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₂₄₄	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11758198	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₁	C₂²	1	2	yes	—
8	C₂³×D₂₄₄	C₄	1	32	yes	direct product factor
9	C₂³×D₂₄₄	C₂²	1	28	yes	—
10	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
11	C₂³×C₄×D₆₁	—	2	32	no	—
12	C₂³×C₄×D₆₁	C₂²	1	56	yes	complemented
13	C₂×C₄×D₂₄₄	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₆₁	C₂²	1	28	yes	complemented
15	C₂²×#11758198	C₂²	1	56	yes	complemented
16	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₆₁	C₂²	1	28	yes	complemented
20	C₂⁴×D₆₁	—	2	16	no	—
21	C₂⁴×D₆₁	—	2	16	no	—
22	C₂⁴×D₆₁	C₂×C₄	1	32	yes	complemented
23	C₂⁴×D₆₁	C₂³	1	28	yes	—
24	C₂²×D₂₄₄	C₂×C₄	1	448	yes	direct product factor
25	C₂²×D₂₄₄	C₂³	1	112	yes	—
26	C₂³×Dic₆₁	—	2	16	no	—
27	C₂³×Dic₆₁	C₂³	1	28	yes	—
28	C₂²×C₄×D₆₁	—	2	448	no	—
29	C₂²×C₄×D₆₁	C₂³	1	224	yes	complemented
30	C₄×D₂₄₄	C₂³	1	512	yes	direct product factor
31	C₂×C₄⋊Dic₆₁	C₂³	1	112	yes	complemented
32	C₂×#11758198	C₂³	1	224	yes	complemented
33	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	—	2	8	no	—
35	C₂³×C₄×C₆₁	C₂×C₄	1	16	yes	complemented
36	C₂³×C₄×C₆₁	D₄	1	16	yes	complemented
37	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
38	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
39	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
40	C₂×C₄²×C₆₁	C₂³	1	112	yes	complemented
41	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
43	C₂⁴×C₆₁	#41	1	8	yes	—
44	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
45	C₂³×D₆₁	—	2	256	no	—
46	C₂³×D₆₁	—	2	112	no	—
47	C₂³×D₆₁	—	2	112	no	—
48	C₂³×D₆₁	C₂²×C₄	1	224	yes	complemented
49	C₂³×D₆₁	C₂⁴	1	56	yes	—
50	C₂²×C₄×C₆₁	—	2	56	no	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	112	yes	complemented
52	C₂²×C₄×C₆₁	C₂×D₄	1	112	yes	complemented
53	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
54	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
55	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
56	C₂²×Dic₆₁	—	2	112	no	—
57	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
58	C₂×D₂₄₄	C₂²×C₄	1	896	yes	direct product factor
59	C₂×D₂₄₄	C₂⁴	1	64	yes	—
60	C₂×C₄×D₆₁	—	2	896	no	—
61	C₂×C₄×D₆₁	C₂⁴	1	128	yes	complemented
62	C₄²×C₆₁	C₂⁴	1	64	yes	complemented
63	#11758198	C₂⁴	1	128	yes	complemented
64	C₄⋊Dic₆₁	C₂⁴	1	64	yes	complemented
65	C₂³×C₆₁	C₄×D₄	1	64	yes	complemented
66	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
67	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
68	C₂³×C₆₁	C₂×#41	1	28	yes	—
69	C₂³×C₆₁	C₂⁵	1	7	yes	—
70	C₂²×D₆₁	—	2	896	no	—
71	C₂²×D₆₁	—	2	112	no	—
72	C₂²×D₆₁	—	2	112	no	—
73	C₂²×D₆₁	C₂³×C₄	1	224	yes	complemented
74	C₂²×D₆₁	C₂⁵	1	16	yes	—
75	C₂×C₄×C₆₁	—	2	56	no	—
76	C₂×C₄×C₆₁	C₂³×C₄	1	112	yes	complemented
77	C₂×C₄×C₆₁	C₂²×D₄	1	112	yes	complemented
78	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
79	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
80	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
81	C₂×Dic₆₁	—	2	112	no	—
82	C₂×Dic₆₁	C₂⁵	1	16	yes	—
83	D₂₄₄	C₂³×C₄	1	256	yes	direct product factor
84	C₄×D₆₁	—	2	256	no	—
85	C₂³×C₄×D₄	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	C₂×C₄×D₄	1	112	yes	complemented
87	C₂²×C₆₁	C₂⁴×C₄	1	14	yes	—
88	C₂²×C₆₁	C₂³×D₄	1	14	yes	—
89	C₂²×C₆₁	C₂²×#41	1	14	yes	—
90	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
91	C₂×D₆₁	—	2	16	no	—
92	C₂×D₆₁	—	2	16	no	—
93	C₂×D₆₁	—	2	448	no	—
94	C₂×D₆₁	C₂⁴×C₄	1	32	yes	complemented
95	C₄×C₆₁	—	2	8	no	—
96	C₄×C₆₁	C₂⁴×C₄	1	16	yes	complemented
97	C₄×C₆₁	C₂³×D₄	1	16	yes	complemented
98	Dic₆₁	—	2	16	no	—
99	C₂⁴×D₄	—	61	1	no	—
100	C₂⁵×C₄	—	61	2	no	—
101	C₂²×C₄×D₄	—	61	56	no	—
102	C₂³×C₄⋊C₄	—	61	1	no	—
103	C₂³×C₂²⋊C₄	—	61	2	no	—
104	C₂³×C₄²	D₆₁	1	1	yes	characteristic, complemented
105	C₂×C₆₁	C₂²×C₄×D₄	1	28	yes	complemented
106	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
107	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
108	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
109	D₆₁	—	2	32	no	—
110	C₂⁶	—	61	2	no	—
111	C₂³×D₄	—	61	32	no	—
112	C₂³×D₄	—	61	28	no	—
113	C₂⁴×C₄	—	61	2	no	—
114	C₂⁴×C₄	—	61	56	no	—
115	C₂⁴×C₄	—	122	32	no	—
116	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
117	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
118	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
119	C₂×C₄×D₄	—	61	448	no	—
120	C₂²×C₄⋊C₄	—	61	28	no	—
121	C₂²×C₂²⋊C₄	—	61	56	no	—
122	C₂²×C₄²	C₂×D₆₁	1	28	yes	complemented
123	C₆₁	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	61	32	no	—
125	C₂⁵	—	61	28	no	—
126	C₂⁵	—	122	16	no	—
127	C₂⁵	—	122	16	no	—
128	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
129	C₂²×D₄	—	61	448	no	—
130	C₂²×D₄	—	61	112	no	—
131	C₂³×C₄	—	2	8	no	—
132	C₂³×C₄	—	61	28	no	—
133	C₂³×C₄	—	61	224	no	—
134	C₂³×C₄	—	122	16	no	—
135	C₂³×C₄	—	122	448	no	—
136	C₂³×C₄	C₄×D₆₁	1	16	yes	complemented
137	C₂³×C₄	D₂₄₄	1	16	yes	direct product factor
138	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
139	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
140	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
141	C₄×D₄	—	61	512	no	—
142	C₂×C₄⋊C₄	—	61	112	no	—
143	C₂×C₂²⋊C₄	—	61	224	no	—
144	C₂×C₄²	C₂²×D₆₁	1	112	yes	complemented
145	C₂⁴	—	61	224	no	—
146	C₂⁴	—	61	56	no	—
147	C₂⁴	—	122	112	no	—
148	C₂⁴	—	122	112	no	—
149	C₂⁴	—	122	256	no	—
150	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
151	C₂⁴	C₂×D₂₄₄	1	8	yes	—
152	C₂⁴	#11758222	1	8	yes	—
153	C₂⁴	C₂³×D₆₁	1	7	yes	—
154	C₂×D₄	—	61	896	no	—
155	C₂×D₄	—	61	64	no	—
156	C₂²×C₄	—	2	56	no	—
157	C₂²×C₄	—	61	56	no	—
158	C₂²×C₄	—	61	128	no	—
159	C₂²×C₄	—	122	112	no	—
160	C₂²×C₄	—	122	896	no	—
161	C₂²×C₄	C₂×C₄×D₆₁	1	112	yes	complemented
162	C₂²×C₄	C₂×D₂₄₄	1	112	yes	direct product factor
163	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
164	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
165	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
166	C₄⋊C₄	—	61	64	no	—
167	C₂²⋊C₄	—	61	128	no	—
168	C₄²	C₂³×D₆₁	1	64	yes	complemented
169	C₂³	—	61	224	no	—
170	C₂³	—	61	16	no	—
171	C₂³	—	122	896	no	—
172	C₂³	—	122	112	no	—
173	C₂³	—	122	112	no	—
174	C₂³	C₄×D₂₄₄	1	64	yes	direct product factor
175	C₂³	C₂²×C₄×D₆₁	1	28	yes	—
176	C₂³	C₂²×D₂₄₄	1	28	yes	—
177	C₂³	C₂×#11758222	1	28	yes	—
178	C₂³	C₂⁴×D₆₁	1	7	yes	—
179	D₄	—	61	256	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	61	16	no	—
182	C₂×C₄	—	122	112	no	—
183	C₂×C₄	—	122	256	no	—
184	C₂×C₄	C₂²×C₄×D₆₁	1	112	yes	complemented
185	C₂×C₄	C₂²×D₂₄₄	1	112	yes	direct product factor
186	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
187	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
188	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
189	C₂²	—	61	32	no	—
190	C₂²	—	122	448	no	—
191	C₂²	—	122	16	no	—
192	C₂²	—	122	16	no	—
193	C₂²	C₂×C₄×D₂₄₄	1	112	yes	direct product factor
194	C₂²	C₂³×C₄×D₆₁	1	14	yes	—
195	C₂²	C₂³×D₂₄₄	1	14	yes	—
196	C₂²	C₂²×#11758222	1	14	yes	—
197	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	122	16	no	—
200	C₄	C₂³×C₄×D₆₁	1	16	yes	complemented
201	C₄	C₂³×D₂₄₄	1	16	yes	direct product factor
202	C₂	—	122	32	no	—
203	C₂	C₂²×C₄×D₂₄₄	1	28	yes	direct product factor
204	C₂	C₂⁴×C₄×D₆₁	1	1	yes	characteristic
205	C₂	C₂⁴×D₂₄₄	1	1	yes	characteristic
206	C₂	C₂³×#11758222	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₄²×C₆₁	C₂
socle	C₂⁵×C₆₁	C₂³
prime radical	C₂⁴×D₂₄₄	C₂
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂³×C₄²	D₆₁
hypocenter	C₆₁	C₂³×C₄×D₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₆₁	C₂	semidirect
3. C₂²×C₄×D₂₄₄	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₆₁	C₂	semidirect
5. C₂³×#11758198	C₂	semidirect
6. C₂³×C₄²×C₆₁	C₂	semidirect
8. C₂³×D₂₄₄	C₄	direct (N × H)
12. C₂³×C₄×D₆₁	C₂²	semidirect
13. C₂×C₄×D₂₄₄	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₆₁	C₂²	semidirect
15. C₂²×#11758198	C₂²	semidirect
19. C₂²×C₄²×C₆₁	C₂²	semidirect
22. C₂⁴×D₆₁	C₂×C₄	semidirect
24. C₂²×D₂₄₄	C₂×C₄	direct (N × H)
29. C₂²×C₄×D₆₁	C₂³	semidirect
30. C₄×D₂₄₄	C₂³	direct (N × H)
31. C₂×C₄⋊Dic₆₁	C₂³	semidirect
32. C₂×#11758198	C₂³	semidirect
35. C₂³×C₄×C₆₁	C₂×C₄	semidirect
36. C₂³×C₄×C₆₁	D₄	semidirect
40. C₂×C₄²×C₆₁	C₂³	semidirect
48. C₂³×D₆₁	C₂²×C₄	semidirect
51. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
52. C₂²×C₄×C₆₁	C₂×D₄	semidirect
58. C₂×D₂₄₄	C₂²×C₄	direct (N × H)
61. C₂×C₄×D₆₁	C₂⁴	semidirect
62. C₄²×C₆₁	C₂⁴	semidirect
63. #11758198	C₂⁴	semidirect
64. C₄⋊Dic₆₁	C₂⁴	semidirect
65. C₂³×C₆₁	C₄×D₄	semidirect
73. C₂²×D₆₁	C₂³×C₄	semidirect
76. C₂×C₄×C₆₁	C₂³×C₄	semidirect
77. C₂×C₄×C₆₁	C₂²×D₄	semidirect
83. D₂₄₄	C₂³×C₄	direct (N × H)
86. C₂²×C₆₁	C₂×C₄×D₄	semidirect
94. C₂×D₆₁	C₂⁴×C₄	semidirect
96. C₄×C₆₁	C₂⁴×C₄	semidirect
97. C₄×C₆₁	C₂³×D₄	semidirect
104. C₂³×C₄²	D₆₁	semidirect
105. C₂×C₆₁	C₂²×C₄×D₄	semidirect
122. C₂²×C₄²	C₂×D₆₁	semidirect
123. C₆₁	C₂³×C₄×D₄	semidirect
136. C₂³×C₄	C₄×D₆₁	semidirect
137. C₂³×C₄	D₂₄₄	direct (N × H)
144. C₂×C₄²	C₂²×D₆₁	semidirect
161. C₂²×C₄	C₂×C₄×D₆₁	semidirect
162. C₂²×C₄	C₂×D₂₄₄	direct (N × H)
168. C₄²	C₂³×D₆₁	semidirect
174. C₂³	C₄×D₂₄₄	direct (N × H)
184. C₂×C₄	C₂²×C₄×D₆₁	semidirect
185. C₂×C₄	C₂²×D₂₄₄	direct (N × H)
193. C₂²	C₂×C₄×D₂₄₄	direct (N × H)
200. C₄	C₂³×C₄×D₆₁	semidirect
201. C₄	C₂³×D₂₄₄	direct (N × H)
203. C₂	C₂²×C₄×D₂₄₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₄₄	2
C₂⁴×C₄×D₆₁	2
C₂²×C₄×D₂₄₄	2
C₂³×C₄⋊Dic₆₁	2
C₂³×#11758198	2
C₂³×C₄²×C₆₁	2
C₂³×C₄×D₄	61

Subgroups — C23×C4×D244 #30586579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₂₄₄ #30586579