Subgroups — #28507303

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	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	#21422191	C₂	1	4	yes	maximal
4	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	#16837426	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	—	2	1	no	—
10	C₄×C₅₉⋊C₁₆	—	2	1	no	—
11	C₅₉×#533	—	2	1	no	—
12	C₅₉×#500	C₄	1	2	yes	—
13	C₅₉×#499	—	2	1	no	—
14	C₅₉×#499	C₄	1	2	yes	—
15	C₅₉×#498	—	2	1	no	—
16	C₅₉×#427	—	2	1	no	—
17	C₅₉×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#11745664	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
23	#14273663	—	2	4	no	—
24	#14273663	—	4	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₉	—	2	2	no	—
29	#125×C₅₉	—	2	2	no	—
30	C₂²⋊Q₈×C₅₉	—	4	2	no	—
31	C₄×Q₈×C₅₉	—	4	2	no	—
32	#117×C₅₉	—	2	2	no	—
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	#107×C₅₉	—	2	2	no	—
35	#107×C₅₉	C₂×C₄	1	4	yes	—
36	#106×C₅₉	—	2	2	no	—
37	#106×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₉	—	4	2	no	—
43	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
48	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
49	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₉	#41	1	2	yes	—
53	C₄⋊C₄×C₅₉	—	2	4	no	—
54	C₄⋊C₄×C₅₉	—	4	4	no	—
55	C₄⋊C₄×C₅₉	—	4	2	no	—
56	C₂²⋊C₄×C₅₉	—	4	2	no	—
57	C₄²×C₅₉	—	4	2	no	—
58	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₉	D₈	1	2	yes	—
61	#11745664	—	4	2	no	—
62	#11745664	—	4	2	no	—
63	C₂×C₅₉⋊C₈	—	4	2	no	—
64	C₅₉⋊C₁₆	—	2	4	no	—
65	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈×C₅₉	—	8	4	no	—
67	C₂×C₄×C₅₉	—	2	4	no	—
68	C₂×C₄×C₅₉	—	4	4	no	—
69	C₂×C₄×C₅₉	#102	1	2	yes	—
70	C₂×C₄×C₅₉	#102	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₅₉	#103	1	2	yes	—
78	C₈×C₅₉	#103	1	2	yes	—
79	C₈×C₅₉	C₄×Q₈	1	2	yes	—
80	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
81	C₅₉⋊C₈	—	8	2	no	—
82	#13285	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	2	no	—
84	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
85	C₂²×C₅₉	#487	1	1	yes	characteristic
86	C₂²×C₅₉	#498	1	1	yes	characteristic
87	C₄×C₅₉	—	2	2	no	—
88	C₄×C₅₉	—	8	4	no	—
89	C₄×C₅₉	#359	1	4	yes	—
90	C₄×C₅₉	#391	1	1	yes	characteristic
91	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
92	C₄×C₅₉	#486	1	1	yes	characteristic
93	C₄×C₅₉	#500	1	1	yes	characteristic
94	#3160	D₅₉	1	1	yes	characteristic
95	C₂×C₄×C₁₆	—	59	1	no	—
96	#1848	—	59	1	no	—
97	#1382	—	59	4	no	—
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#1990	1	1	yes	characteristic
101	C₂×C₅₉	C₂×#359	1	2	yes	—
102	#533	—	2	1	no	—
103	C₂²×C₁₆	—	59	2	no	—
104	#500	Dic₅₉	1	2	yes	—
105	#499	—	2	1	no	—
106	#499	Dic₅₉	1	2	yes	—
107	#498	—	2	1	no	—
108	C₂×#108	—	59	2	no	—
109	C₂×#108	—	118	1	no	—
110	#427	—	2	1	no	—
111	#427	C₂×D₅₉	1	2	yes	—
112	#423	—	118	1	no	—
113	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
114	C₄×C₁₆	—	118	1	no	—
115	C₄×C₁₆	—	118	1	no	—
116	C₅₉	#13285	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	118	2	no	—
118	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₄×Q₈	—	4	2	no	—
124	#117	—	2	2	no	—
125	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
126	C₂×C₁₆	—	59	4	no	—
127	C₂×C₁₆	—	118	2	no	—
128	C₂×C₁₆	—	118	2	no	—
129	#108	—	118	4	no	—
130	#108	—	236	2	no	—
131	#107	—	2	2	no	—
132	#107	C₂×Dic₅₉	1	4	yes	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	236	2	no	—
136	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
140	C₂×Q₈	—	4	2	no	—
141	C₂²×C₄	#11745670	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
143	M₁₆	—	236	2	no	—
144	M₁₆	—	236	2	no	—
145	C₂×C₈	—	236	2	no	—
146	C₂×C₈	#11745671	1	2	yes	—
147	C₂×C₈	#11745672	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₅₉	1	2	yes	—
151	C₂×C₈	#11745689	1	2	yes	—
152	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	#11745669	1	2	yes	—
160	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
161	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
162	C₁₆	—	118	4	no	—
163	C₂³	C₂×#11745670	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#14273648	1	2	yes	—
168	C₂×C₄	#14273648	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
170	C₂×C₄	#14273716	1	1	yes	characteristic
171	C₂×C₄	#14273725	1	1	yes	characteristic
172	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
173	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
174	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
175	C₈	—	472	2	no	—
176	C₈	#14273651	1	2	yes	—
177	C₈	#14273651	1	2	yes	—
178	C₈	Q₈×Dic₅₉	1	2	yes	—
179	C₈	#14273703	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14273648	1	1	yes	characteristic
182	C₂²	#16837754	1	1	yes	characteristic
183	C₂²	#16837798	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#16837209	1	4	yes	—
187	C₄	#16837309	1	1	yes	characteristic
188	C₄	C₂×#14273651	1	1	yes	characteristic
189	C₄	#16837750	1	1	yes	characteristic
190	C₄	#16837805	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#21425058	1	1	yes	characteristic
194	C₂	C₂×#16837209	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#3160	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#3160	D₅₉
hypocenter	C₅₉	#13285
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
22. C₂×C₅₉⋊C₁₆	Q₈	semidirect
116. C₅₉	#13285	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
#21424268	2
#21422191	2
C₅₉×#3160	2
#13285	59