Subgroups — C₄×#16837272 #28509915

Order 15104.

2094 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837272	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊Dic₅₉	C₂	1	6	yes	maximal
3	#21422326	C₂	1	6	yes	maximal
4	C₂×C₄³×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₉	C₂²	1	12	yes	—
6	C₂²×C₄×Dic₅₉	C₂²	1	4	yes	—
7	#16837272	C₄	1	16	yes	direct product factor
8	C₄×C₄⋊Dic₅₉	—	2	24	no	—
9	C₂×C₅₉⋊#95	C₂²	1	12	yes	—
10	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
11	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
13	C₄³×C₅₉	C₄	1	8	yes	complemented
14	C₂³×Dic₅₉	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₅₉	—	2	24	no	—
16	C₂×C₄⋊Dic₅₉	—	2	24	no	—
17	C₂×C₄⋊Dic₅₉	C₂×C₄	1	48	yes	—
18	C₂×C₄×Dic₅₉	—	2	24	no	—
19	C₅₉⋊#95	—	2	24	no	—
20	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
21	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
22	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	—	2	12	no	—
24	C₂×C₄²×C₅₉	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₅₉	C₂×C₄	1	12	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₉	C₂×C₄	1	12	yes	—
28	C₂×C₄²×C₅₉	D₄	1	12	yes	—
29	C₂×C₄²×C₅₉	Q₈	1	12	yes	—
30	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	—	2	12	no	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₅₉	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
39	C₂²×C₄×C₅₉	C₂×Q₈	1	6	yes	—
40	C₂²×C₄×C₅₉	C₂×Q₈	1	6	yes	—
41	C₂²×C₄×C₅₉	C₂×Q₈	1	6	yes	—
42	C₂²×C₄×C₅₉	#41	1	12	yes	—
43	C₂²×Dic₅₉	—	2	24	no	—
44	C₂²×Dic₅₉	—	2	24	no	—
45	C₂²×Dic₅₉	C₂²×C₄	1	16	yes	—
46	C₄²×C₅₉	—	2	24	no	—
47	C₄²×C₅₉	C₄²	1	16	yes	complemented
48	C₄²×C₅₉	C₄⋊C₄	1	48	yes	complemented
49	C₄⋊Dic₅₉	—	2	96	no	—
50	C₄×Dic₅₉	—	4	16	no	—
51	C₂³×C₅₉	C₂³×C₄	1	2	yes	—
52	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₅₉	C₂²×D₄	1	3	yes	—
54	C₂³×C₅₉	C₂²×Q₈	1	3	yes	—
55	C₂³×C₅₉	C₂×#41	1	6	yes	—
56	C₂×C₄×C₅₉	—	2	12	no	—
57	C₂×C₄×C₅₉	—	2	12	no	—
58	C₂×C₄×C₅₉	—	2	12	no	—
59	C₂×C₄×C₅₉	C₂×C₄²	1	12	yes	—
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	12	yes	—
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	12	yes	—
62	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	12	yes	—
63	C₂×C₄×C₅₉	#117	1	12	yes	—
64	C₂×C₄×C₅₉	C₄×D₄	1	24	yes	—
65	C₂×C₄×C₅₉	C₄×Q₈	1	24	yes	—
66	C₂×C₄×C₅₉	#127	1	4	yes	—
67	C₂×C₄×C₅₉	C₄⋊Q₈	1	12	yes	—
68	C₂×Dic₅₉	—	2	48	no	—
69	C₂×Dic₅₉	—	4	8	no	—
70	C₂×Dic₅₉	—	4	24	no	—
71	C₄×#384	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
73	C₂²×C₅₉	C₂²×C₄⋊C₄	1	3	yes	—
74	C₂²×C₅₉	C₂×#117	1	3	yes	—
75	C₂²×C₅₉	C₂×C₄×D₄	1	6	yes	—
76	C₂²×C₅₉	C₂×C₄×Q₈	1	6	yes	—
77	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
78	C₂²×C₅₉	C₂×C₄⋊Q₈	1	3	yes	—
79	C₂²×C₅₉	#532	1	6	yes	—
80	C₂²×C₅₉	#533	1	6	yes	—
81	C₄×C₅₉	—	2	12	no	—
82	C₄×C₅₉	C₄×C₄⋊C₄	1	24	yes	complemented
83	C₄×C₅₉	#384	1	8	yes	complemented
84	Dic₅₉	—	4	16	no	—
85	C₂×#384	—	59	2	no	—
86	C₂×C₄×C₄⋊C₄	—	59	6	no	—
87	C₂×C₄³	D₅₉	1	1	yes	characteristic
88	C₄⋊#95	—	59	6	no	—
89	C₂×C₅₉	C₂×C₄×C₄⋊C₄	1	3	yes	—
90	C₂×C₅₉	C₂×#384	1	1	yes	characteristic
91	C₂×C₅₉	#2285	1	3	yes	—
92	C₂×C₅₉	C₄×#127	1	2	yes	—
93	C₂×C₅₉	C₄×C₄⋊Q₈	1	6	yes	—
94	C₂²×C₄⋊C₄	—	59	12	no	—
95	C₂²×C₄²	—	59	4	no	—
96	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
97	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
98	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
99	#384	—	59	16	no	—
100	C₄×C₄⋊C₄	—	118	24	no	—
101	C₂×#95	—	59	12	no	—
102	C₄³	Dic₅₉	1	8	yes	complemented
103	C₅₉	C₄×#384	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₄	—	59	8	no	—
105	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
106	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
107	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	59	48	no	—
109	C₂×C₄⋊C₄	—	118	24	no	—
110	C₂×C₄⋊C₄	—	118	24	no	—
111	C₂×C₄²	—	2	12	no	—
112	C₂×C₄²	—	118	24	no	—
113	C₂×C₄²	Dic₁₁₈	1	12	yes	—
114	C₂×C₄²	C₄×D₅₉	1	8	yes	—
115	C₂×C₄²	D₂₃₆	1	12	yes	—
116	C₂×C₄²	C₂×Dic₅₉	1	12	yes	—
117	C₂×C₄²	C₂×Dic₅₉	1	4	yes	—
118	C₂×C₄²	C₂×Dic₅₉	1	12	yes	—
119	#95	—	118	24	no	—
120	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
121	C₂²×C₄	—	2	12	no	—
122	C₂²×C₄	—	59	16	no	—
123	C₂²×C₄	—	118	24	no	—
124	C₂²×C₄	—	118	24	no	—
125	C₂²×C₄	C₂×Dic₁₁₈	1	6	yes	—
126	C₂²×C₄	C₂×Dic₁₁₈	1	6	yes	—
127	C₂²×C₄	C₂×Dic₁₁₈	1	6	yes	—
128	C₂²×C₄	C₂×C₄×D₅₉	1	12	yes	—
129	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
130	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
131	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
132	C₂²×C₄	#11745685	1	12	yes	—
133	C₂²×C₄	C₂²×Dic₅₉	1	6	yes	—
134	C₂²×C₄	C₂²×Dic₅₉	1	6	yes	—
135	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
136	C₄⋊C₄	—	118	96	no	—
137	C₄²	—	2	24	no	—
138	C₄²	—	236	16	no	—
139	C₄²	C₄×Dic₅₉	1	16	yes	complemented
140	C₄²	C₄⋊Dic₅₉	1	48	yes	complemented
141	C₂³	C₂²×C₄×D₅₉	1	2	yes	—
142	C₂³	C₂³×Dic₅₉	1	1	yes	characteristic
143	C₂³	C₂²×D₂₃₆	1	3	yes	—
144	C₂³	C₂²×Dic₁₁₈	1	3	yes	—
145	C₂³	C₂×#11745685	1	6	yes	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	—	2	12	no	—
148	C₂×C₄	—	2	12	no	—
149	C₂×C₄	—	118	48	no	—
150	C₂×C₄	—	236	8	no	—
151	C₂×C₄	—	236	24	no	—
152	C₂×C₄	C₂×C₄×Dic₅₉	1	12	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₅₉	1	12	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₅₉	1	12	yes	—
155	C₂×C₄	C₂×C₄⋊Dic₅₉	1	12	yes	—
156	C₂×C₄	#14273686	1	12	yes	—
157	C₂×C₄	C₄×D₂₃₆	1	24	yes	—
158	C₂×C₄	C₄×Dic₁₁₈	1	24	yes	—
159	C₂×C₄	#14273724	1	4	yes	—
160	C₂×C₄	#14273727	1	12	yes	—
161	C₂²	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
162	C₂²	C₂²×C₄⋊Dic₅₉	1	3	yes	—
163	C₂²	C₂×#14273686	1	3	yes	—
164	C₂²	C₂×C₄×D₂₃₆	1	6	yes	—
165	C₂²	C₂×C₄×Dic₁₁₈	1	6	yes	—
166	C₂²	C₂×#14273724	1	1	yes	characteristic
167	C₂²	C₂×#14273727	1	3	yes	—
168	C₂²	#16837952	1	6	yes	—
169	C₂²	#16837959	1	6	yes	—
170	C₄	—	2	12	no	—
171	C₄	—	236	16	no	—
172	C₄	C₄×C₄⋊Dic₅₉	1	24	yes	complemented
173	C₄	#16837272	1	8	yes	direct product factor
174	C₂	C₂×C₄×C₄⋊Dic₅₉	1	3	yes	—
175	C₂	C₂×#16837272	1	1	yes	characteristic
176	C₂	#21426503	1	3	yes	—
177	C₂	C₄×#14273724	1	2	yes	—
178	C₂	C₄×#14273727	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₄²
Frattini	C₂⁴	C₂³×D₅₉
Fitting	C₂×C₄³×C₅₉	C₂
socle, prime radical, cosocle	C₂⁴×C₅₉	C₂⁴
hypercenter	C₂×C₄³	D₅₉
hypocenter	C₅₉	C₄×#384
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. #16837272	C₄	direct (N × H)
13. C₄³×C₅₉	C₄	semidirect
47. C₄²×C₅₉	C₄²	semidirect
48. C₄²×C₅₉	C₄⋊C₄	semidirect
82. C₄×C₅₉	C₄×C₄⋊C₄	semidirect
83. C₄×C₅₉	#384	semidirect
102. C₄³	Dic₅₉	semidirect
103. C₅₉	C₄×#384	semidirect
139. C₄²	C₄×Dic₅₉	semidirect
140. C₄²	C₄⋊Dic₅₉	semidirect
172. C₄	C₄×C₄⋊Dic₅₉	semidirect
173. C₄	#16837272	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×#16837272	2
C₂×C₄×C₄⋊Dic₅₉	2
#21422326	2
C₂×C₄³×C₅₉	2
C₄×#384	59

Subgroups — C4×#16837272 #28509915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837272 #28509915