Subgroups — C₂³×#14273660 #29452333

Order 15104.

3614 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14273660	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₉	—	2	1	no	—
6	C₂³×C₅₉⋊C₈	C₄	1	16	yes	complemented
7	C₂³×C₅₉⋊C₈	C₂²	1	14	yes	—
8	C₂²×C₄⋊Dic₅₉	—	2	7	no	—
9	C₂²×C₄⋊Dic₅₉	C₂²	1	14	yes	—
10	C₂×#14273660	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄×C₅₉	—	2	1	no	—
12	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₅₉	—	2	7	no	—
14	C₂²×C₄⋊C₄×C₅₉	C₂²	1	14	yes	—
15	C₂³×Dic₅₉	—	2	14	no	—
16	C₂³×Dic₅₉	—	4	8	no	—
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	112	yes	complemented
18	C₂²×C₅₉⋊C₈	C₂³	1	28	yes	—
19	C₂×C₄⋊Dic₅₉	—	2	42	no	—
20	C₂×C₄⋊Dic₅₉	C₂³	1	28	yes	—
21	#14273660	C₂³	1	64	yes	direct product factor
22	C₂⁵×C₅₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₉	—	2	14	no	—
24	C₂³×C₄×C₅₉	—	4	8	no	—
25	C₂³×C₄×C₅₉	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₉	D₄	1	7	yes	—
27	C₂³×C₄×C₅₉	Q₈	1	8	yes	—
28	C₂³×C₄×C₅₉	C₂³	1	7	yes	—
29	C₂×C₄⋊C₄×C₅₉	—	2	42	no	—
30	C₂×C₄⋊C₄×C₅₉	C₂³	1	28	yes	—
31	C₂⁴×C₅₉	C₄⋊C₄	1	8	yes	—
32	C₂⁴×C₅₉	D₈	1	8	yes	—
33	C₂⁴×C₅₉	Q₁₆	1	8	yes	—
34	C₂⁴×C₅₉	C₂×D₄	1	7	yes	—
35	C₂²×C₄×C₅₉	—	2	28	no	—
36	C₂²×C₄×C₅₉	—	4	56	no	—
37	C₂²×C₄×C₅₉	C₄⋊C₄	1	56	yes	—
38	C₂²×C₄×C₅₉	C₂²×C₄	1	28	yes	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	21	yes	—
40	C₂²×C₄×C₅₉	C₂×Q₈	1	28	yes	—
41	C₂²×C₄×C₅₉	C₂⁴	1	7	yes	—
42	C₂²×Dic₅₉	—	2	28	no	—
43	C₂²×Dic₅₉	—	4	56	no	—
44	C₄⋊C₄×C₅₉	—	2	28	no	—
45	C₄⋊C₄×C₅₉	C₂⁴	1	8	yes	—
46	C₄⋊Dic₅₉	—	2	28	no	—
47	C₄⋊Dic₅₉	C₂⁴	1	8	yes	—
48	C₂×C₅₉⋊C₈	C₂²×C₄	1	112	yes	complemented
49	C₂×C₅₉⋊C₈	C₂⁴	1	8	yes	—
50	C₂³×C₅₉	#107	1	64	yes	complemented
51	C₂³×C₅₉	C₂×C₄⋊C₄	1	28	yes	—
52	C₂³×C₅₉	C₂×D₈	1	28	yes	—
53	C₂³×C₅₉	C₂×Q₁₆	1	28	yes	—
54	C₂³×C₅₉	C₂²×D₄	1	7	yes	—
55	C₂×C₄×C₅₉	—	2	8	no	—
56	C₂×C₄×C₅₉	—	4	56	no	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	84	yes	—
58	C₂×C₄×C₅₉	C₂³×C₄	1	14	yes	—
59	C₂×C₄×C₅₉	C₂²×D₄	1	7	yes	—
60	C₂×C₄×C₅₉	C₂²×Q₈	1	14	yes	—
61	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
62	C₂×Dic₅₉	—	2	8	no	—
63	C₂×Dic₅₉	—	4	56	no	—
64	C₅₉⋊C₈	C₂³×C₄	1	16	yes	complemented
65	C₂³×#107	—	59	1	no	maximal, Sylow
66	C₂²×C₅₉	C₂×#107	1	112	yes	complemented
67	C₂²×C₅₉	C₂²×C₄⋊C₄	1	14	yes	—
68	C₂²×C₅₉	C₂²×D₈	1	14	yes	—
69	C₂²×C₅₉	C₂²×Q₁₆	1	14	yes	—
70	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
71	C₄×C₅₉	—	4	8	no	—
72	C₄×C₅₉	C₂²×C₄⋊C₄	1	14	yes	—
73	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
74	C₄×C₅₉	C₂³×Q₈	1	1	yes	characteristic
75	Dic₅₉	—	4	8	no	—
76	C₂⁴×C₈	—	59	1	no	—
77	C₂³×C₄⋊C₄	—	59	1	no	—
78	C₂³×C₄⋊C₄	D₅₉	1	1	yes	characteristic
79	C₂²×#107	—	59	28	no	—
80	C₂×C₅₉	C₂²×#107	1	28	yes	complemented
81	C₂×C₅₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
83	C₂×C₅₉	C₂³×Q₁₆	1	1	yes	characteristic
84	C₂⁴×C₄	—	2	1	no	—
85	C₂⁴×C₄	—	118	1	no	—
86	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
87	C₂³×C₈	—	59	16	no	—
88	C₂³×C₈	—	59	14	no	—
89	C₂²×C₄⋊C₄	—	2	7	no	—
90	C₂²×C₄⋊C₄	—	59	14	no	—
91	C₂²×C₄⋊C₄	—	118	7	no	—
92	C₂²×C₄⋊C₄	C₂×D₅₉	1	14	yes	—
93	C₂×#107	—	59	112	no	—
94	C₅₉	C₂³×#107	1	1	yes	characteristic, complemented, Sylow
95	C₂⁵	C₂²⋊D₅₉	1	1	yes	characteristic
96	C₂³×C₄	—	2	14	no	—
97	C₂³×C₄	—	4	8	no	—
98	C₂³×C₄	—	118	14	no	—
99	C₂³×C₄	—	236	8	no	—
100	C₂³×C₄	Dic₁₁₈	1	8	yes	—
101	C₂³×C₄	C₄×D₅₉	1	8	yes	—
102	C₂³×C₄	C₂²⋊D₅₉	1	7	yes	—
103	C₂³×C₄	C₂²×D₅₉	1	7	yes	—
104	C₂²×C₈	—	59	112	no	—
105	C₂²×C₈	—	59	28	no	—
106	C₂×C₄⋊C₄	—	2	42	no	—
107	C₂×C₄⋊C₄	—	59	28	no	—
108	C₂×C₄⋊C₄	—	118	42	no	—
109	C₂×C₄⋊C₄	C₂²×D₅₉	1	28	yes	—
110	#107	—	59	64	no	—
111	C₂⁴	Dic₅₉⋊C₄	1	8	yes	—
112	C₂⁴	#11745669	1	8	yes	—
113	C₂⁴	#11745672	1	8	yes	—
114	C₂⁴	C₂×C₂²⋊D₅₉	1	7	yes	—
115	C₂²×C₄	—	2	28	no	—
116	C₂²×C₄	—	4	56	no	—
117	C₂²×C₄	—	118	28	no	—
118	C₂²×C₄	—	236	56	no	—
119	C₂²×C₄	Dic₅₉⋊C₄	1	56	yes	—
120	C₂²×C₄	C₂×Dic₁₁₈	1	28	yes	—
121	C₂²×C₄	C₂×C₄×D₅₉	1	28	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₉	1	21	yes	—
123	C₂²×C₄	C₂³×D₅₉	1	7	yes	—
124	C₂×C₈	—	59	112	no	—
125	C₂×C₈	—	59	8	no	—
126	C₄⋊C₄	—	2	28	no	—
127	C₄⋊C₄	—	59	8	no	—
128	C₄⋊C₄	—	118	28	no	—
129	C₄⋊C₄	C₂³×D₅₉	1	8	yes	—
130	C₂³	#14273660	1	64	yes	direct product factor
131	C₂³	C₂×Dic₅₉⋊C₄	1	28	yes	—
132	C₂³	C₂×#11745669	1	28	yes	—
133	C₂³	C₂×#11745672	1	28	yes	—
134	C₂³	C₂²×C₂²⋊D₅₉	1	7	yes	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	4	56	no	—
137	C₂×C₄	—	118	8	no	—
138	C₂×C₄	—	236	56	no	—
139	C₂×C₄	C₂×Dic₅₉⋊C₄	1	84	yes	—
140	C₂×C₄	C₂²×C₄×D₅₉	1	14	yes	—
141	C₂×C₄	C₂²×C₂²⋊D₅₉	1	7	yes	—
142	C₂×C₄	C₂²×Dic₁₁₈	1	14	yes	—
143	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
144	C₈	—	59	16	no	—
145	C₂²	C₂×#14273660	1	112	yes	direct product factor
146	C₂²	C₂²×Dic₅₉⋊C₄	1	14	yes	—
147	C₂²	C₂²×#11745669	1	14	yes	—
148	C₂²	C₂²×#11745672	1	14	yes	—
149	C₂²	C₂³×C₂²⋊D₅₉	1	1	yes	characteristic
150	C₄	—	4	8	no	—
151	C₄	—	236	8	no	—
152	C₄	C₂²×Dic₅₉⋊C₄	1	14	yes	—
153	C₄	C₂³×C₄×D₅₉	1	1	yes	characteristic
154	C₄	C₂³×Dic₁₁₈	1	1	yes	characteristic
155	C₂	C₂²×#14273660	1	28	yes	direct product factor
156	C₂	C₂³×Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₂	C₂³×#11745669	1	1	yes	characteristic
158	C₂	C₂³×#11745672	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₂³×C₄⋊C₄×C₅₉	C₂
socle, prime radical	C₂⁵×C₅₉	D₄
cosocle	C₂×C₄×C₅₉	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅₉
hypocenter	C₅₉	C₂³×#107
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#14273660	C₂	direct (N × H)
6. C₂³×C₅₉⋊C₈	C₄	semidirect
10. C₂×#14273660	C₂²	direct (N × H)
17. C₂²×C₅₉⋊C₈	C₂×C₄	semidirect
21. #14273660	C₂³	direct (N × H)
48. C₂×C₅₉⋊C₈	C₂²×C₄	semidirect
50. C₂³×C₅₉	#107	semidirect
64. C₅₉⋊C₈	C₂³×C₄	semidirect
66. C₂²×C₅₉	C₂×#107	semidirect
80. C₂×C₅₉	C₂²×#107	semidirect
94. C₅₉	C₂³×#107	semidirect
130. C₂³	#14273660	direct (N × H)
145. C₂²	C₂×#14273660	direct (N × H)
155. C₂	C₂²×#14273660	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×C₅₉⋊C₈	2
C₂³×C₄⋊Dic₅₉	2
C₂²×#14273660	2
C₂³×C₄⋊C₄×C₅₉	2
C₂³×#107	59

Subgroups — C23×#14273660 #29452333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14273660 #29452333