Subgroups — C₂³×#14273718 #29495356

Order 15104.

8414 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273718	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₉	C₂	1	2	yes	maximal
3	C₂³×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₅₉⋊C₄	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₅₉	C₂²	1	2	yes	—
10	C₂×#14273718	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊Dic₅₉	C₂²	1	56	yes	—
12	C₂²×C₄⋊Dic₅₉	C₂²	1	28	yes	—
13	C₂²×Dic₅₉⋊C₄	C₂²	1	56	yes	—
14	C₂²×C₄×Dic₅₉	C₂²	1	28	yes	—
15	C₂⁴×C₄×C₅₉	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₉	C₂²	1	28	yes	—
18	C₂³×Dic₅₉	—	2	8	no	—
19	C₂³×Dic₅₉	—	2	8	no	—
20	C₂³×Dic₅₉	—	2	16	no	—
21	C₂³×Dic₅₉	C₂³	1	14	yes	—
22	C₂³×Dic₅₉	C₂³	1	14	yes	—
23	C₂³×Dic₅₉	C₂³	1	28	yes	—
24	#14273718	C₂³	1	512	yes	direct product factor
25	C₂×C₄⋊Dic₅₉	C₂³	1	224	yes	—
26	C₂×C₄⋊Dic₅₉	C₂³	1	112	yes	—
27	C₂×Dic₅₉⋊C₄	C₂³	1	224	yes	—
28	C₂×C₄×Dic₅₉	C₂³	1	112	yes	—
29	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₅₉	—	2	16	no	—
31	C₂³×C₄×C₅₉	Q₈	1	16	yes	—
32	C₂³×C₄×C₅₉	C₂³	1	28	yes	—
33	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
34	C₂×C₄⋊C₄×C₅₉	C₂³	1	112	yes	—
35	C₂⁴×C₅₉	C₂×Q₈	1	8	yes	—
36	C₂⁴×C₅₉	#41	1	8	yes	—
37	C₂⁴×C₅₉	#41	1	8	yes	—
38	C₂⁴×C₅₉	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₅₉	—	2	112	no	—
40	C₂²×C₄×C₅₉	C₂×Q₈	1	112	yes	—
41	C₂²×C₄×C₅₉	C₂⁴	1	56	yes	—
42	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
43	C₂²×Dic₅₉	—	2	56	no	—
44	C₂²×Dic₅₉	—	2	56	no	—
45	C₂²×Dic₅₉	—	2	112	no	—
46	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
47	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
48	C₂²×Dic₅₉	C₂⁴	1	56	yes	—
49	C₄⋊C₄×C₅₉	C₂⁴	1	64	yes	—
50	C₄⋊Dic₅₉	C₂⁴	1	128	yes	—
51	C₄⋊Dic₅₉	C₂⁴	1	64	yes	—
52	Dic₅₉⋊C₄	C₂⁴	1	128	yes	—
53	C₄×Dic₅₉	C₂⁴	1	64	yes	—
54	C₂³×C₅₉	#125	1	64	yes	complemented
55	C₂³×C₅₉	C₂²×Q₈	1	28	yes	—
56	C₂³×C₅₉	C₂×#41	1	28	yes	—
57	C₂³×C₅₉	C₂×#41	1	28	yes	—
58	C₂³×C₅₉	C₂⁵	1	7	yes	—
59	C₂×C₄×C₅₉	—	2	112	no	—
60	C₂×C₄×C₅₉	C₂²×Q₈	1	112	yes	—
61	C₂×C₄×C₅₉	C₂⁵	1	16	yes	—
62	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
63	C₂×Dic₅₉	—	2	56	no	—
64	C₂×Dic₅₉	—	2	56	no	—
65	C₂×Dic₅₉	—	2	112	no	—
66	C₂×Dic₅₉	C₂⁵	1	8	yes	—
67	C₂×Dic₅₉	C₂⁵	1	8	yes	—
68	C₂×Dic₅₉	C₂⁵	1	16	yes	—
69	C₂³×#125	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	C₂×#125	1	112	yes	complemented
71	C₂²×C₅₉	C₂³×Q₈	1	14	yes	—
72	C₂²×C₅₉	C₂²×#41	1	14	yes	—
73	C₂²×C₅₉	C₂²×#41	1	14	yes	—
74	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
75	C₄×C₅₉	—	2	16	no	—
76	C₄×C₅₉	C₂³×Q₈	1	16	yes	—
77	Dic₅₉	—	2	8	no	—
78	Dic₅₉	—	2	8	no	—
79	Dic₅₉	—	2	16	no	—
80	C₂²×#125	—	59	56	no	—
81	C₂³×C₄⋊C₄	—	59	2	no	—
82	C₂³×C₄⋊C₄	—	59	2	no	—
83	C₂³×C₄⋊C₄	—	59	1	no	—
84	C₂³×C₄⋊C₄	D₅₉	1	1	yes	characteristic
85	C₂³×C₄²	—	59	1	no	—
86	C₂×C₅₉	C₂²×#125	1	28	yes	complemented
87	C₂×C₅₉	C₂⁴×Q₈	1	1	yes	characteristic
88	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
89	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
90	C₂⁴×C₄	—	59	1	no	—
91	C₂⁴×C₄	—	59	1	no	—
92	C₂⁴×C₄	—	59	2	no	—
93	C₂⁴×C₄	C₂×D₅₉	1	2	yes	—
94	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
95	C₂×#125	—	59	448	no	—
96	C₂²×C₄⋊C₄	—	59	56	no	—
97	C₂²×C₄⋊C₄	—	59	28	no	—
98	C₂²×C₄⋊C₄	—	59	56	no	—
99	C₂²×C₄⋊C₄	C₂×D₅₉	1	28	yes	—
100	C₂²×C₄²	—	59	28	no	—
101	C₅₉	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
102	C₂⁵	C₂²×D₅₉	1	1	yes	characteristic
103	C₂³×C₄	—	2	16	no	—
104	C₂³×C₄	—	59	28	no	—
105	C₂³×C₄	—	59	14	no	—
106	C₂³×C₄	—	59	14	no	—
107	C₂³×C₄	—	118	8	no	—
108	C₂³×C₄	—	118	8	no	—
109	C₂³×C₄	—	118	16	no	—
110	C₂³×C₄	Dic₁₁₈	1	16	yes	—
111	C₂³×C₄	C₂²×D₅₉	1	28	yes	—
112	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
113	#125	—	59	512	no	—
114	C₂×C₄⋊C₄	—	59	224	no	—
115	C₂×C₄⋊C₄	—	59	224	no	—
116	C₂×C₄⋊C₄	—	59	112	no	—
117	C₂×C₄⋊C₄	C₂²×D₅₉	1	112	yes	—
118	C₂×C₄²	—	59	112	no	—
119	C₂⁴	C₂×Dic₁₁₈	1	8	yes	—
120	C₂⁴	#11745687	1	8	yes	—
121	C₂⁴	#11745689	1	8	yes	—
122	C₂⁴	C₂³×D₅₉	1	7	yes	—
123	C₂²×C₄	—	2	112	no	—
124	C₂²×C₄	—	59	28	no	—
125	C₂²×C₄	—	59	28	no	—
126	C₂²×C₄	—	59	56	no	—
127	C₂²×C₄	—	118	56	no	—
128	C₂²×C₄	—	118	112	no	—
129	C₂²×C₄	—	118	56	no	—
130	C₂²×C₄	C₂×Dic₁₁₈	1	112	yes	—
131	C₂²×C₄	C₂³×D₅₉	1	56	yes	—
132	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
133	C₄⋊C₄	—	59	128	no	—
134	C₄⋊C₄	—	59	128	no	—
135	C₄⋊C₄	—	59	64	no	—
136	C₄⋊C₄	C₂³×D₅₉	1	64	yes	—
137	C₄²	—	59	64	no	—
138	C₂³	#14273718	1	64	yes	direct product factor
139	C₂³	C₂²×Dic₁₁₈	1	28	yes	—
140	C₂³	C₂×#11745687	1	28	yes	—
141	C₂³	C₂×#11745689	1	28	yes	—
142	C₂³	C₂⁴×D₅₉	1	7	yes	—
143	C₂×C₄	—	2	112	no	—
144	C₂×C₄	—	59	8	no	—
145	C₂×C₄	—	59	8	no	—
146	C₂×C₄	—	59	16	no	—
147	C₂×C₄	—	118	56	no	—
148	C₂×C₄	—	118	56	no	—
149	C₂×C₄	—	118	112	no	—
150	C₂×C₄	C₂²×Dic₁₁₈	1	112	yes	—
151	C₂×C₄	C₂⁴×D₅₉	1	16	yes	—
152	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
153	C₂²	C₂×#14273718	1	112	yes	direct product factor
154	C₂²	C₂³×Dic₁₁₈	1	14	yes	—
155	C₂²	C₂²×#11745687	1	14	yes	—
156	C₂²	C₂²×#11745689	1	14	yes	—
157	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
158	C₄	—	2	16	no	—
159	C₄	—	118	8	no	—
160	C₄	—	118	8	no	—
161	C₄	—	118	16	no	—
162	C₄	C₂³×Dic₁₁₈	1	16	yes	—
163	C₂	C₂²×#14273718	1	28	yes	direct product factor
164	C₂	C₂⁴×Dic₁₁₈	1	1	yes	characteristic
165	C₂	C₂³×#11745687	1	1	yes	characteristic
166	C₂	C₂³×#11745689	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₉
commutator, cosocle	C₂²×C₅₉	C₂⁶
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂³×C₄⋊C₄×C₅₉	C₂
socle, prime radical	C₂⁵×C₅₉	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₉
hypocenter	C₅₉	C₂³×#125
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273718	C₂	direct (N × H)
10. C₂×#14273718	C₂²	direct (N × H)
24. #14273718	C₂³	direct (N × H)
54. C₂³×C₅₉	#125	semidirect
70. C₂²×C₅₉	C₂×#125	semidirect
86. C₂×C₅₉	C₂²×#125	semidirect
101. C₅₉	C₂³×#125	semidirect
138. C₂³	#14273718	direct (N × H)
153. C₂²	C₂×#14273718	direct (N × H)
163. C₂	C₂²×#14273718	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273718	2
C₂³×C₄⋊Dic₅₉	2
C₂³×Dic₅₉⋊C₄	2
C₂³×C₄×Dic₅₉	2
C₂³×C₄⋊C₄×C₅₉	2
C₂³×#125	59

Subgroups — C23×#14273718 #29495356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14273718 #29495356