Subgroups — #28479986

Order 15104.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21422385	C₂	1	2	yes	maximal
4	#21421933	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273663	C₄	1	2	yes	—
8	C₄×#11745664	C₂²	1	2	yes	—
9	#16837167	C₄	1	2	yes	—
10	#16837155	C₄	1	2	yes	—
11	#16837153	C₄	1	2	yes	—
12	#16837124	—	2	2	no	—
13	#16837123	—	2	2	no	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₈²×C₅₉	C₄	1	2	yes	—
18	C₈²×C₅₉	C₄	1	2	yes	—
19	C₂×#11745664	C₂×C₄	1	4	yes	—
20	#14273663	—	2	4	no	—
21	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
23	#14273638	—	2	2	no	—
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₅₉	—	2	1	no	—
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₉	—	2	1	no	—
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×C₈×C₅₉	—	2	4	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	2	4	no	—
47	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
54	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₉	#41	1	1	yes	characteristic
58	#11745664	—	2	4	no	—
59	#11745664	—	2	4	no	—
60	C₂×C₅₉⋊C₈	—	2	4	no	—
61	C₂×C₅₉⋊C₈	—	2	4	no	—
62	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₅₉	#131	1	2	yes	—
73	C₈×C₅₉	—	2	4	no	—
74	C₈×C₅₉	—	2	1	no	—
75	C₈×C₅₉	—	2	1	no	—
76	C₈×C₅₉	—	2	2	no	—
77	C₈×C₅₉	#97	1	2	yes	—
78	C₈×C₅₉	#97	1	2	yes	—
79	C₈×C₅₉	#108	1	4	yes	complemented
80	C₅₉⋊C₈	—	4	4	no	—
81	C₅₉⋊C₈	—	4	4	no	—
82	#8293	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₉	#446	1	2	yes	—
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
90	C₄×C₅₉	#405	1	1	yes	characteristic
91	C₄×C₅₉	C₂×#108	1	2	yes	—
92	C₄×C₅₉	#436	1	2	yes	—
93	C₄×C₅₉	#443	1	2	yes	—
94	C₄×#108	—	59	1	no	—
95	#1564	—	59	1	no	—
96	#1451	—	59	2	no	—
97	C₂×C₈²	D₅₉	1	1	yes	characteristic
98	#1278	—	59	2	no	—
99	C₂×C₅₉	—	2	2	no	—
100	C₂×C₅₉	#1770	1	1	yes	characteristic
101	C₂×C₅₉	#1772	1	1	yes	characteristic
102	C₂×C₅₉	C₄×#108	1	1	yes	characteristic
103	#431	—	59	2	no	—
104	C₂×#108	—	59	2	no	—
105	C₄×M₁₆	—	59	2	no	—
106	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#341	—	59	2	no	—
110	#335	—	59	2	no	—
111	#334	—	59	2	no	—
112	#322	—	118	2	no	—
113	#322	—	118	2	no	—
114	C₈²	Dic₅₉	1	2	yes	—
115	C₈²	Dic₅₉	1	2	yes	—
116	C₅₉	#8293	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	59	4	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	Dic₁₁₈	1	2	yes	—
120	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	#108	—	118	4	no	—
124	C₄⋊C₈	—	59	4	no	—
125	C₂²⋊C₈	—	118	2	no	—
126	#97	—	118	2	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	118	2	no	—
129	C₄×C₈	—	118	2	no	—
130	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
131	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
132	C₄×C₈	D₂₃₆	1	1	yes	characteristic
133	C₄×C₈	D₂₃₆	1	1	yes	characteristic
134	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
135	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
138	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂²×C₄	#11745685	1	1	yes	characteristic
141	M₁₆	—	118	4	no	—
142	M₁₆	—	118	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	118	4	no	—
148	C₂×C₈	—	118	4	no	—
149	C₂×C₈	#11745659	1	4	yes	—
150	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
151	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
154	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
155	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
156	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
157	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
158	C₄²	#11745685	1	1	yes	characteristic
159	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
160	C₂³	C₄×Dic₁₁₈	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
165	C₂×C₄	#14273686	1	1	yes	characteristic
166	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
167	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
169	C₂×C₄	C₂×#11745659	1	2	yes	—
170	C₂×C₄	#14273733	1	2	yes	—
171	C₈	—	2	4	no	—
172	C₈	—	2	2	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	236	4	no	—
176	C₈	—	236	4	no	—
177	C₈	#14273637	1	2	yes	—
178	C₈	#14273637	1	2	yes	—
179	C₈	#14273663	1	4	yes	complemented
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
183	C₂²	#16837532	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×#14273637	1	1	yes	characteristic
187	C₄	#16837357	1	1	yes	characteristic
188	C₄	C₂×#14273663	1	2	yes	—
189	C₄	#16837481	1	2	yes	—
190	C₄	#16837520	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#21423963	1	1	yes	characteristic
193	C₂	#21423975	1	1	yes	characteristic
194	C₂	C₄×#14273663	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₄⋊C₄
prime radical	C₂³×C₅₉	C₄×Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#8293
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
79. C₈×C₅₉	#108	semidirect
116. C₅₉	#8293	semidirect
179. C₈	#14273663	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273663	2
#21422844	2
#21422385	2
#21421933	2
C₂×C₈²×C₅₉	2
#8293	59