Subgroups — #28589112

Order 15104.

1086 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837440	C₂	1	4	yes	maximal
2	#21426503	C₂	1	2	yes	maximal
3	#21424264	C₂	1	8	yes	maximal, complemented
4	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273686	C₂²	1	4	yes	—
6	#16837440	—	2	16	no	—
7	C₂×#14273661	C₂²	1	8	yes	—
8	C₂×#14273659	C₂²	1	8	yes	—
9	#16837272	C₂²	1	4	yes	—
10	#16837268	C₂²	1	4	yes	—
11	#16837262	—	2	4	no	—
12	C₄×C₄⋊Dic₅₉	—	2	4	no	—
13	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#97×C₅₉	C₄	1	8	yes	complemented
16	C₂×#97×C₅₉	C₂²	1	4	yes	—
17	#14273686	—	2	16	no	—
18	C₂×C₄⋊Dic₅₉	—	2	8	no	—
19	C₂×C₄⋊Dic₅₉	C₂³	1	8	yes	—
20	C₂×C₂²⋊Dic₅₉	—	2	4	no	—
21	C₂×C₄×Dic₅₉	—	2	4	no	—
22	#14273661	—	2	16	no	—
23	#14273659	—	2	16	no	—
24	C₅₉⋊#95	—	2	8	no	—
25	C₂³×C₄×C₅₉	D₄	1	2	yes	—
26	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₅₉	D₄	1	8	yes	—
29	C₂²×C₈×C₅₉	Q₈	1	8	yes	—
30	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
31	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅₉	D₄	1	2	yes	—
33	C₂×C₄²×C₅₉	Q₈	1	4	yes	—
34	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
35	#97×C₅₉	C₂×C₄	1	16	yes	—
36	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	—	4	4	no	—
38	C₂²×C₄×C₅₉	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
46	C₂²×Dic₅₉	—	2	8	no	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₅₉	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₅₉	C₂×Q₈	1	8	yes	—
51	C₄²×C₅₉	C₄⋊C₄	1	8	yes	—
52	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
53	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
54	C₂²⋊Dic₅₉	—	4	8	no	—
55	C₄⋊Dic₅₉	—	2	16	no	—
56	C₄⋊Dic₅₉	—	4	16	no	—
57	C₄×Dic₅₉	—	4	8	no	—
58	C₂³×C₅₉	—	2	4	no	—
59	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₅₉	#127	1	2	yes	—
61	C₂³×C₅₉	C₄⋊Q₈	1	2	yes	—
62	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	8	no	—
64	C₂×C₄×C₅₉	—	4	8	no	—
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
67	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅₉	#127	1	2	yes	—
69	C₂×C₄×C₅₉	C₄⋊Q₈	1	8	yes	—
70	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
73	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
74	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
75	C₂×Dic₅₉	—	4	8	no	—
76	C₂×Dic₅₉	—	4	16	no	—
77	C₈×C₅₉	C₂×C₄⋊C₄	1	16	yes	—
78	#20799	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	—	2	4	no	—
80	C₂²×C₅₉	—	2	4	no	—
81	C₂²×C₅₉	—	2	4	no	—
82	C₂²×C₅₉	#384	1	4	yes	—
83	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
85	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₅₉	#575	1	4	yes	—
87	C₄×C₅₉	—	2	8	no	—
88	C₄×C₅₉	#384	1	4	yes	—
89	C₄×C₅₉	C₂²×C₄⋊C₄	1	2	yes	—
90	C₄×C₅₉	C₂×C₄⋊Q₈	1	2	yes	—
91	Dic₅₉	—	8	8	no	—
92	C₂×#427	—	59	4	no	—
93	C₂²×#97	D₅₉	1	1	yes	characteristic
94	#2285	—	59	2	no	—
95	#1847	—	59	8	no	—
96	C₂×C₅₉	—	2	4	no	—
97	C₂×C₅₉	C₂×#384	1	1	yes	characteristic
98	C₂×C₅₉	#2911	1	2	yes	—
99	C₂×C₅₉	#3150	1	2	yes	—
100	C₂×C₅₉	#3162	1	2	yes	—
101	C₂³×C₈	C₂×D₅₉	1	2	yes	—
102	C₂×#117	—	59	4	no	—
103	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
104	#427	—	118	16	no	—
105	C₂×#107	—	59	8	no	—
106	C₂×#106	—	59	8	no	—
107	C₂×#97	Dic₅₉	1	8	yes	complemented
108	C₂×#97	C₂×D₅₉	1	4	yes	—
109	#384	—	59	4	no	—
110	#382	—	59	4	no	—
111	#380	—	118	4	no	—
112	C₄×C₄⋊C₄	—	118	4	no	—
113	C₅₉	#20799	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	D₂₃₆	1	2	yes	—
115	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
116	C₂²×C₈	Dic₁₁₈	1	8	yes	—
117	C₂²×C₈	D₂₃₆	1	8	yes	—
118	C₂²×C₈	C₂×Dic₅₉	1	8	yes	—
119	C₂²×C₈	C₂²×D₅₉	1	4	yes	—
120	#117	—	118	16	no	—
121	C₂×C₄⋊C₄	—	59	8	no	—
122	C₂×C₄⋊C₄	—	118	8	no	—
123	C₂×C₂²⋊C₄	—	118	4	no	—
124	C₂×C₄²	—	118	4	no	—
125	C₂×C₄²	Dic₁₁₈	1	4	yes	—
126	C₂×C₄²	D₂₃₆	1	2	yes	—
127	C₂×C₄²	C₂×Dic₅₉	1	4	yes	—
128	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
129	#107	—	118	16	no	—
130	#106	—	118	16	no	—
131	#97	C₂×Dic₅₉	1	16	yes	—
132	#95	—	118	8	no	—
133	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
134	C₂²×C₄	—	4	4	no	—
135	C₂²×C₄	—	118	8	no	—
136	C₂²×C₄	C₄⋊Dic₅₉	1	8	yes	—
137	C₂²×C₄	C₂×Dic₁₁₈	1	4	yes	—
138	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
139	C₂²×C₄	C₂×D₂₃₆	1	4	yes	—
140	C₂²×C₄	C₂×D₂₃₆	1	4	yes	—
141	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
142	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
143	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₅₉	1	32	yes	complemented
145	C₂×C₈	C₂×Dic₁₁₈	1	8	yes	—
146	C₂×C₈	C₂×D₂₃₆	1	8	yes	—
147	C₂×C₈	C₂²×Dic₅₉	1	8	yes	—
148	C₄⋊C₄	—	118	16	no	—
149	C₄⋊C₄	—	236	16	no	—
150	C₂²⋊C₄	—	236	8	no	—
151	C₄²	—	236	8	no	—
152	C₄²	C₄⋊Dic₅₉	1	8	yes	—
153	C₄²	C₂×Dic₁₁₈	1	4	yes	—
154	C₄²	C₂²×Dic₅₉	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
157	C₂³	#14273724	1	2	yes	—
158	C₂³	#14273727	1	2	yes	—
159	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	236	8	no	—
163	C₂×C₄	—	236	16	no	—
164	C₂×C₄	C₂×C₄⋊Dic₅₉	1	8	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₉	1	8	yes	—
167	C₂×C₄	#14273724	1	2	yes	—
168	C₂×C₄	#14273727	1	8	yes	—
169	C₂×C₄	#14273727	1	2	yes	—
170	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₂₃₆	1	2	yes	—
172	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
173	C₂×C₄	C₂²×Dic₁₁₈	1	2	yes	—
174	C₈	C₂×C₄⋊Dic₅₉	1	16	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	#16837272	1	4	yes	—
179	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
180	C₂²	C₂×#14273724	1	1	yes	characteristic
181	C₂²	C₂×#14273727	1	1	yes	characteristic
182	C₂²	#16838202	1	4	yes	—
183	C₄	—	2	8	no	—
184	C₄	—	472	8	no	—
185	C₄	#16837272	1	4	yes	—
186	C₄	C₂²×C₄⋊Dic₅₉	1	2	yes	—
187	C₄	C₂×#14273727	1	2	yes	—
188	C₂	—	2	4	no	—
189	C₂	C₂×#16837272	1	1	yes	characteristic
190	C₂	#21432686	1	2	yes	—
191	C₂	#21434542	1	2	yes	—
192	C₂	#21434620	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×D₄
prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×#97	D₅₉
hypocenter	C₅₉	#20799
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #21424264	C₂	semidirect
15. C₂×#97×C₅₉	C₄	semidirect
47. C₂×C₈×C₅₉	C₄⋊C₄	semidirect
107. C₂×#97	Dic₅₉	semidirect
113. C₅₉	#20799	semidirect
144. C₂×C₈	C₄⋊Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837440	2
#21426503	2
#21424264	2
C₂²×#97×C₅₉	2
#20799	59