Subgroups — #28479878

Order 15104.

438 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	#21423949	C₂	1	2	yes	maximal
2	C₂×#16837153	C₂	1	2	yes	maximal
3	#21421931	C₂	1	2	yes	maximal
4	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
6	C₂×#14273638	—	2	2	no	—
7	#16837157	—	2	2	no	—
8	#16837153	C₄	1	8	yes	complemented
9	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄³×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₅₉⋊C₈	—	2	4	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
16	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
17	#14273638	—	4	4	no	—
18	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
25	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₅₉	—	2	4	no	—
27	C₄×C₈×C₅₉	—	2	4	no	—
28	C₄×C₈×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	C₄²	1	2	yes	—
39	C₂×C₈×C₅₉	C₄²	1	2	yes	—
40	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
41	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
42	C₄²×C₅₉	—	2	4	no	—
43	C₄²×C₅₉	—	2	2	no	—
44	C₄²×C₅₉	—	2	2	no	—
45	C₄²×C₅₉	—	2	2	no	—
46	C₄²×C₅₉	SD₁₆	1	4	yes	—
47	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
48	C₄²×C₅₉	#41	1	2	yes	—
49	C₂×C₅₉⋊C₈	—	2	8	no	—
50	C₂×C₅₉⋊C₈	—	4	4	no	—
51	C₂×C₅₉⋊C₈	—	4	4	no	—
52	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	#106	1	4	yes	—
59	C₂×C₄×C₅₉	#108	1	4	yes	—
60	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₅₉	#135	1	2	yes	—
68	C₈×C₅₉	—	2	2	no	—
69	C₈×C₅₉	—	2	4	no	—
70	C₈×C₅₉	#97	1	4	yes	complemented
71	C₅₉⋊C₈	—	4	8	no	—
72	#8273	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₅₉	#423	1	2	yes	—
75	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
76	C₂²×C₅₉	#427	1	1	yes	characteristic
77	C₂²×C₅₉	C₂×#108	1	2	yes	—
78	C₄×C₅₉	—	2	4	no	—
79	C₄×C₅₉	—	2	4	no	—
80	C₄×C₅₉	—	2	2	no	—
81	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
82	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
83	C₄×C₅₉	#432	1	2	yes	—
84	C₄×C₅₉	C₄×SD₁₆	1	4	yes	—
85	#1767	—	59	2	no	—
86	C₄²×C₈	D₅₉	1	1	yes	characteristic
87	C₂×#334	—	59	2	no	—
88	#1277	—	59	2	no	—
89	C₂×C₅₉	#1566	1	2	yes	—
90	C₂×C₅₉	#1568	1	2	yes	—
91	C₂×C₅₉	#1767	1	1	yes	characteristic
92	C₂×C₅₉	C₄×#106	1	1	yes	characteristic
93	C₂×C₅₉	C₄×#108	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	59	4	no	—
95	C₂×#97	—	118	2	no	—
96	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
97	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
100	C₄³	C₂×D₅₉	1	1	yes	characteristic
101	#336	—	118	2	no	—
102	#334	—	59	8	no	—
103	C₅₉	#8273	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	—	118	4	no	—
105	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
106	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
108	C₂×C₄²	Dic₁₁₈	1	2	yes	—
109	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
110	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
112	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
113	C₄⋊C₈	—	59	8	no	—
114	C₄⋊C₈	—	118	4	no	—
115	#97	—	236	4	no	—
116	C₄×C₈	—	2	4	no	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	C₄×D₅₉	1	4	yes	—
120	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
121	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
122	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
124	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	118	8	no	—
129	C₂×C₈	—	236	4	no	—
130	C₂×C₈	—	236	4	no	—
131	C₂×C₈	#11745664	1	4	yes	—
132	C₂×C₈	#11745664	1	4	yes	—
133	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
134	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	#11745660	1	4	yes	—
140	C₄²	C₂×C₄×D₅₉	1	2	yes	—
141	C₄²	#11745685	1	2	yes	—
142	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#14273659	1	4	yes	—
149	C₂×C₄	#14273663	1	4	yes	—
150	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
151	C₂×C₄	#14273686	1	1	yes	characteristic
152	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
153	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
154	C₂×C₄	C₂×#11745664	1	2	yes	—
155	C₂×C₄	C₂×#11745664	1	2	yes	—
156	C₂×C₄	C₂×#11745660	1	2	yes	—
157	C₂×C₄	#14273749	1	2	yes	—
158	C₈	—	2	4	no	—
159	C₈	—	2	2	no	—
160	C₈	—	236	8	no	—
161	C₈	#14273638	1	4	yes	complemented
162	C₂²	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
163	C₂²	#16837426	1	2	yes	—
164	C₂²	C₂×#14273659	1	1	yes	characteristic
165	C₂²	#16837440	1	1	yes	characteristic
166	C₂²	C₂×#14273663	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	C₂×#14273638	1	1	yes	characteristic
171	C₄	C₄×#11745664	1	1	yes	characteristic
172	C₄	#16837459	1	2	yes	—
173	C₄	C₄×#11745660	1	4	yes	—
174	C₂	#21422852	1	2	yes	—
175	C₂	#21422859	1	2	yes	—
176	C₂	#21423949	1	1	yes	characteristic
177	C₂	C₄×#14273659	1	1	yes	characteristic
178	C₂	C₄×#14273663	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#8273
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #16837153	C₄	semidirect
70. C₈×C₅₉	#97	semidirect
103. C₅₉	#8273	semidirect
161. C₈	#14273638	semidirect

Maximal subgroups

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

subgroup	index
#21423949	2
C₂×#16837153	2
#21421931	2
C₄²×C₈×C₅₉	2
#8273	59