Subgroups — C₂³×#14519733 #30586570

Order 15616.

14334 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14519733	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11758196	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×Dic₆₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₆₁	C₄	1	32	yes	complemented
8	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
9	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₆₁	C₂²	1	28	yes	—
12	C₂×#14519733	C₂²	1	448	yes	direct product factor
13	C₂²×#11758196	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊Dic₆₁	C₂²	1	28	yes	—
15	C₂²×C₄×Dic₆₁	C₂²	1	56	yes	complemented
16	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	28	yes	—
19	C₂³×Dic₆₁	—	2	16	no	—
20	C₂³×Dic₆₁	C₂×C₄	1	16	yes	—
21	C₂³×Dic₆₁	C₂×C₄	1	16	yes	—
22	C₂³×Dic₆₁	C₂×C₄	1	448	yes	complemented
23	C₂³×Dic₆₁	C₂³	1	28	yes	—
24	C₂³×Dic₆₁	C₂³	1	14	yes	—
25	C₂³×Dic₆₁	C₂³	1	14	yes	—
26	C₂³×Dic₆₁	C₂³	1	112	yes	—
27	#14519733	C₂³	1	512	yes	direct product factor
28	C₂×#11758196	C₂³	1	224	yes	complemented
29	C₂×C₂²⋊Dic₆₁	C₂³	1	112	yes	—
30	C₂×C₄×Dic₆₁	C₂³	1	224	yes	complemented
31	C₂⁵×C₆₁	—	2	16	no	—
32	C₂⁵×C₆₁	C₂×C₄	1	16	yes	—
33	C₂⁵×C₆₁	C₂³	1	14	yes	—
34	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₆₁	—	2	16	no	—
36	C₂³×C₄×C₆₁	C₂³	1	28	yes	—
37	C₂×C₂²⋊C₄×C₆₁	C₂³	1	112	yes	—
38	C₂⁴×C₆₁	—	2	128	no	—
39	C₂⁴×C₆₁	—	2	112	no	—
40	C₂⁴×C₆₁	C₂²×C₄	1	112	yes	—
41	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₆₁	#41	1	16	yes	—
43	C₂⁴×C₆₁	C₂⁴	1	28	yes	—
44	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
45	C₂²×C₄×C₆₁	—	2	112	no	—
46	C₂²×C₄×C₆₁	C₂⁴	1	56	yes	—
47	C₂²×Dic₆₁	—	2	112	no	—
48	C₂²×Dic₆₁	C₂²×C₄	1	112	yes	—
49	C₂²×Dic₆₁	C₂²×C₄	1	112	yes	—
50	C₂²×Dic₆₁	C₂²×C₄	1	896	yes	complemented
51	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
52	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
53	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
54	C₂²×Dic₆₁	C₂⁴	1	64	yes	—
55	C₂²⋊C₄×C₆₁	C₂⁴	1	64	yes	—
56	C₂²⋊Dic₆₁	C₂⁴	1	64	yes	—
57	#11758196	C₂⁴	1	128	yes	complemented
58	C₄×Dic₆₁	C₂⁴	1	128	yes	complemented
59	C₂³×C₆₁	—	2	448	no	—
60	C₂³×C₆₁	—	2	112	no	—
61	C₂³×C₆₁	#117	1	64	yes	complemented
62	C₂³×C₆₁	C₂³×C₄	1	112	yes	—
63	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
64	C₂³×C₆₁	C₂×#41	1	56	yes	—
65	C₂³×C₆₁	C₂⁵	1	8	yes	—
66	C₂³×C₆₁	C₂⁵	1	7	yes	—
67	C₂×C₄×C₆₁	—	2	112	no	—
68	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
69	C₂×Dic₆₁	—	2	112	no	—
70	C₂×Dic₆₁	C₂³×C₄	1	112	yes	—
71	C₂×Dic₆₁	C₂³×C₄	1	112	yes	—
72	C₂×Dic₆₁	C₂³×C₄	1	256	yes	complemented
73	C₂×Dic₆₁	C₂⁵	1	16	yes	—
74	C₂×Dic₆₁	C₂⁵	1	8	yes	—
75	C₂×Dic₆₁	C₂⁵	1	8	yes	—
76	C₂³×#117	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	—	2	16	no	—
78	C₂²×C₆₁	—	2	224	no	—
79	C₂²×C₆₁	C₂×#117	1	112	yes	complemented
80	C₂²×C₆₁	C₂⁴×C₄	1	16	yes	—
81	C₂²×C₆₁	C₂⁴×C₄	1	14	yes	—
82	C₂²×C₆₁	C₂²×#41	1	28	yes	—
83	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
84	C₄×C₆₁	—	2	16	no	—
85	Dic₆₁	—	2	16	no	—
86	Dic₆₁	C₂⁴×C₄	1	16	yes	—
87	Dic₆₁	C₂⁴×C₄	1	16	yes	—
88	C₂⁵×C₄	—	61	1	no	—
89	C₂²×#117	—	61	56	no	—
90	C₂³×C₄⋊C₄	—	61	2	no	—
91	C₂³×C₂²⋊C₄	—	61	1	no	—
92	C₂³×C₂²⋊C₄	D₆₁	1	1	yes	characteristic
93	C₂³×C₄²	—	61	2	no	—
94	C₂×C₆₁	—	2	16	no	—
95	C₂×C₆₁	C₂²×#117	1	28	yes	complemented
96	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
97	C₂×C₆₁	C₂³×#41	1	2	yes	—
98	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
99	C₂⁴×C₄	—	61	32	no	—
100	C₂⁴×C₄	—	61	28	no	—
101	C₂⁴×C₄	—	61	1	no	—
102	C₂⁴×C₄	—	61	1	no	—
103	C₂⁴×C₄	—	61	2	no	—
104	C₂⁴×C₄	C₂×D₆₁	1	2	yes	—
105	C₂×#117	—	61	448	no	—
106	C₂²×C₄⋊C₄	—	61	56	no	—
107	C₂²×C₂²⋊C₄	—	61	28	no	—
108	C₂²×C₂²⋊C₄	C₂×D₆₁	1	28	yes	—
109	C₂²×C₄²	—	61	56	no	—
110	C₆₁	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
111	C₂⁵	—	2	16	no	—
112	C₂⁵	C₄×D₆₁	1	16	yes	—
113	C₂⁵	C₂²×D₆₁	1	14	yes	—
114	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
115	C₂³×C₄	—	2	16	no	—
116	C₂³×C₄	—	61	16	no	—
117	C₂³×C₄	—	61	16	no	—
118	C₂³×C₄	—	61	448	no	—
119	C₂³×C₄	—	61	14	no	—
120	C₂³×C₄	—	61	112	no	—
121	C₂³×C₄	—	61	14	no	—
122	C₂³×C₄	—	61	28	no	—
123	C₂³×C₄	—	122	16	no	—
124	C₂³×C₄	C₂²×D₆₁	1	28	yes	—
125	#117	—	61	512	no	—
126	C₂×C₄⋊C₄	—	61	224	no	—
127	C₂×C₂²⋊C₄	—	61	112	no	—
128	C₂×C₂²⋊C₄	C₂²×D₆₁	1	112	yes	—
129	C₂×C₄²	—	61	224	no	—
130	C₂⁴	—	2	112	no	—
131	C₂⁴	—	2	128	no	—
132	C₂⁴	C₂×C₄×D₆₁	1	112	yes	—
133	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
134	C₂⁴	#11758224	1	16	yes	—
135	C₂⁴	C₂³×D₆₁	1	28	yes	—
136	C₂⁴	C₂³×D₆₁	1	7	yes	—
137	C₂²×C₄	—	2	112	no	—
138	C₂²×C₄	—	61	112	no	—
139	C₂²×C₄	—	61	112	no	—
140	C₂²×C₄	—	61	896	no	—
141	C₂²×C₄	—	61	28	no	—
142	C₂²×C₄	—	61	28	no	—
143	C₂²×C₄	—	61	64	no	—
144	C₂²×C₄	—	61	56	no	—
145	C₂²×C₄	—	122	112	no	—
146	C₂²×C₄	C₂³×D₆₁	1	56	yes	—
147	C₄⋊C₄	—	61	128	no	—
148	C₂²⋊C₄	—	61	64	no	—
149	C₂²⋊C₄	C₂³×D₆₁	1	64	yes	—
150	C₄²	—	61	128	no	—
151	C₂³	—	2	448	no	—
152	C₂³	—	2	112	no	—
153	C₂³	#14519733	1	64	yes	direct product factor
154	C₂³	C₂²×C₄×D₆₁	1	112	yes	—
155	C₂³	C₂²×C₄×D₆₁	1	28	yes	—
156	C₂³	C₂×#11758224	1	56	yes	—
157	C₂³	C₂⁴×D₆₁	1	8	yes	—
158	C₂³	C₂⁴×D₆₁	1	7	yes	—
159	C₂×C₄	—	2	112	no	—
160	C₂×C₄	—	61	112	no	—
161	C₂×C₄	—	61	112	no	—
162	C₂×C₄	—	61	256	no	—
163	C₂×C₄	—	61	8	no	—
164	C₂×C₄	—	61	8	no	—
165	C₂×C₄	—	61	16	no	—
166	C₂×C₄	—	122	112	no	—
167	C₂×C₄	C₂⁴×D₆₁	1	16	yes	—
168	C₂²	—	2	224	no	—
169	C₂²	—	2	16	no	—
170	C₂²	C₂×#14519733	1	112	yes	direct product factor
171	C₂²	C₂³×C₄×D₆₁	1	16	yes	—
172	C₂²	C₂³×C₄×D₆₁	1	14	yes	—
173	C₂²	C₂²×#11758224	1	28	yes	—
174	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
175	C₄	—	2	16	no	—
176	C₄	—	61	16	no	—
177	C₄	—	61	16	no	—
178	C₄	—	122	16	no	—
179	C₂	—	2	16	no	—
180	C₂	C₂²×#14519733	1	28	yes	direct product factor
181	C₂	C₂⁴×C₄×D₆₁	1	1	yes	characteristic
182	C₂	C₂³×#11758224	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₂²⋊C₄×C₆₁	C₂
socle	C₂⁵×C₆₁	C₂³
prime radical	C₂⁶×C₆₁	C₂²
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₆₁
hypocenter	C₆₁	C₂³×#117
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#14519733	C₂	direct (N × H)
3. C₂³×#11758196	C₂	semidirect
5. C₂³×C₄×Dic₆₁	C₂	semidirect
7. C₂⁴×Dic₆₁	C₄	semidirect
12. C₂×#14519733	C₂²	direct (N × H)
13. C₂²×#11758196	C₂²	semidirect
15. C₂²×C₄×Dic₆₁	C₂²	semidirect
22. C₂³×Dic₆₁	C₂×C₄	semidirect
27. #14519733	C₂³	direct (N × H)
28. C₂×#11758196	C₂³	semidirect
30. C₂×C₄×Dic₆₁	C₂³	semidirect
50. C₂²×Dic₆₁	C₂²×C₄	semidirect
57. #11758196	C₂⁴	semidirect
58. C₄×Dic₆₁	C₂⁴	semidirect
61. C₂³×C₆₁	#117	semidirect
72. C₂×Dic₆₁	C₂³×C₄	semidirect
79. C₂²×C₆₁	C₂×#117	semidirect
95. C₂×C₆₁	C₂²×#117	semidirect
110. C₆₁	C₂³×#117	semidirect
153. C₂³	#14519733	direct (N × H)
170. C₂²	C₂×#14519733	direct (N × H)
180. C₂	C₂²×#14519733	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₆₁	2
C₂²×#14519733	2
C₂³×#11758196	2
C₂³×C₂²⋊Dic₆₁	2
C₂³×C₄×Dic₆₁	2
C₂³×C₂²⋊C₄×C₆₁	2
C₂³×#117	61

Subgroups — C23×#14519733 #30586570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519733 #30586570