Subgroups — #28500119

Order 15104.

330 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425895	C₂	1	1	yes	maximal, characteristic
2	#21422855	C₂	1	1	yes	maximal, characteristic
3	#21422151	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic
5	#16837426	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	C₄×C₅₉⋊C₁₆	—	2	1	no	—
9	#16837155	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
12	C₅₉×#500	—	2	1	no	—
13	C₅₉×#500	—	2	1	no	—
14	C₅₉×#500	C₄	1	2	yes	—
15	C₅₉×#500	C₂²	1	2	yes	—
16	C₂×#107×C₅₉	—	2	1	no	—
17	C₂×#107×C₅₉	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#11745664	—	4	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
23	C₂²×Q₈×C₅₉	—	4	1	no	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊Q₈×C₅₉	—	2	2	no	—
27	C₄⋊Q₈×C₅₉	—	2	2	no	—
28	C₄⋊Q₈×C₅₉	—	4	1	no	—
29	C₄⋊Q₈×C₅₉	—	4	1	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
32	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
33	#107×C₅₉	—	2	2	no	—
34	#107×C₅₉	—	2	2	no	—
35	#107×C₅₉	—	2	2	no	—
36	#107×C₅₉	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₅₉	—	4	2	no	—
42	C₂×Q₈×C₅₉	—	4	2	no	—
43	C₂×Q₈×C₅₉	—	8	2	no	—
44	C₂×Q₈×C₅₉	—	8	2	no	—
45	C₂²×C₄×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₅₉	—	2	4	no	—
56	C₄⋊C₄×C₅₉	—	4	2	no	—
57	C₄⋊C₄×C₅₉	—	4	2	no	—
58	C₄⋊C₄×C₅₉	—	4	2	no	—
59	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₉	D₈	1	1	yes	characteristic
61	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
63	#11745664	—	8	2	no	—
64	C₂×C₅₉⋊C₈	—	4	2	no	—
65	C₅₉⋊C₁₆	—	4	2	no	—
66	C₂³×C₅₉	#102	1	1	yes	characteristic
67	Q₈×C₅₉	—	8	4	no	—
68	Q₈×C₅₉	—	8	2	no	—
69	Q₈×C₅₉	—	8	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	2	1	no	—
73	C₂×C₄×C₅₉	—	4	4	no	—
74	C₂×C₄×C₅₉	—	8	4	no	—
75	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
79	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
80	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₅₉	—	2	1	no	—
84	C₈×C₅₉	—	2	1	no	—
85	C₈×C₅₉	#101	1	2	yes	—
86	C₈×C₅₉	#137	1	2	yes	—
87	C₅₉⋊C₈	—	8	2	no	—
88	#11971	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	—	2	1	no	—
90	C₂²×C₅₉	—	2	1	no	—
91	C₂²×C₅₉	#357	1	2	yes	—
92	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	—	2	1	no	—
95	C₄×C₅₉	—	8	4	no	—
96	C₄×C₅₉	#357	1	2	yes	—
97	C₄×C₅₉	#362	1	2	yes	—
98	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
99	C₄×C₅₉	#419	1	1	yes	characteristic
100	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
101	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
102	C₂×#500	D₅₉	1	1	yes	characteristic
103	#2162	—	59	1	no	—
104	#1567	—	59	1	no	—
105	#1364	—	59	4	no	—
106	C₂×C₅₉	—	2	2	no	—
107	C₂×C₅₉	#1546	1	1	yes	characteristic
108	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
109	C₂×C₅₉	C₂×#362	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	Dic₅₉	1	2	yes	—
114	#500	C₂×D₅₉	1	2	yes	—
115	C₂×#107	—	2	1	no	—
116	C₂×#107	Dic₅₉	1	2	yes	—
117	#423	—	118	1	no	—
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	C₄⋊C₁₆	—	59	2	no	—
120	C₂²⋊C₁₆	—	118	1	no	—
121	C₄×C₁₆	—	118	1	no	—
122	#335	—	59	2	no	—
123	#335	—	118	1	no	—
124	C₅₉	#11971	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	4	1	no	—
126	C₂×M₁₆	—	236	1	no	—
127	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	1	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
136	C₂×C₁₆	—	118	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	2	no	—
139	#107	—	2	2	no	—
140	#107	C₂×Dic₅₉	1	4	yes	—
141	C₄⋊C₈	—	118	2	no	—
142	C₄⋊C₈	—	236	1	no	—
143	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	2	no	—
150	C₂×Q₈	—	8	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	#11745669	1	1	yes	characteristic
153	C₂²×C₄	#11745670	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
155	M₁₆	—	472	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	236	2	no	—
159	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	#11745669	1	1	yes	characteristic
168	C₄²	#11745670	1	1	yes	characteristic
169	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
171	C₁₆	—	236	2	no	—
172	C₂³	#14273648	1	1	yes	characteristic
173	Q₈	—	8	4	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	#14273648	1	1	yes	characteristic
182	C₂×C₄	#14273648	1	1	yes	characteristic
183	C₂×C₄	#14273648	1	1	yes	characteristic
184	C₂×C₄	#14273668	1	2	yes	—
185	C₂×C₄	#14273671	1	2	yes	—
186	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
188	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	472	2	no	—
192	C₈	#14273646	1	2	yes	—
193	C₈	#14273758	1	2	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#16837203	1	2	yes	—
197	C₂²	C₂×#14273648	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#16837203	1	2	yes	—
202	C₄	#16837218	1	2	yes	—
203	C₄	C₂×#14273646	1	1	yes	characteristic
204	C₄	#16837411	1	1	yes	characteristic
205	C₄	C₂×#14273668	1	1	yes	characteristic
206	C₄	C₂×#14273671	1	1	yes	characteristic
207	C₂	—	2	2	no	—
208	C₂	#21422768	1	1	yes	characteristic
209	C₂	C₂×#16837203	1	1	yes	characteristic
210	C₂	C₂×#16837218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#500	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	#11971
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #21422151	C₂	semidirect
124. C₅₉	#11971	semidirect

Maximal subgroups

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

subgroup	index
#21425895	2
#21422855	2
#21422151	2
C₂×C₅₉×#500	2
#11971	59