Subgroups — #28505132

Order 15104.

350 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×#16837229	C₂	1	2	yes	maximal
2	C₂×#16837227	C₂	1	2	yes	maximal
3	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
4	#21424260	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14273661	C₂²	1	1	yes	characteristic
8	C₂×#14273661	C₂²	1	1	yes	characteristic
9	C₂×#14273659	—	2	1	no	—
10	#16837268	—	2	1	no	—
11	#16837229	—	2	4	no	—
12	#16837227	—	2	4	no	—
13	#16837178	C₄	1	4	yes	complemented
14	C₂×#107×C₅₉	—	2	1	no	—
15	C₂×#107×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#384	—	2	1	no	—
18	C₂×C₄⋊Dic₅₉	—	2	1	no	—
19	C₂×C₄⋊Dic₅₉	—	2	1	no	—
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
21	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	—	2	2	no	—
25	#14273659	—	4	2	no	—
26	C₅₉⋊#95	—	4	2	no	—
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
31	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
32	#107×C₅₉	—	2	2	no	—
33	#107×C₅₉	—	2	2	no	—
34	#107×C₅₉	—	4	2	no	—
35	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₉	—	4	2	no	—
38	C₂²×C₄×C₅₉	D₈	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	4	1	no	—
41	C₂²×Dic₅₉	—	4	1	no	—
42	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	D₈	1	2	yes	—
44	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
45	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₅₉	—	4	2	no	—
50	C₄⋊C₄×C₅₉	—	4	2	no	—
51	C₄⋊C₄×C₅₉	—	8	4	no	—
52	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₄²×C₅₉	Q₁₆	1	2	yes	—
54	C₄⋊Dic₅₉	—	4	1	no	—
55	C₄⋊Dic₅₉	—	4	1	no	—
56	C₄⋊Dic₅₉	—	4	1	no	—
57	C₄⋊Dic₅₉	—	4	1	no	—
58	C₅₉⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	—	8	2	no	—
63	C₂×C₄×C₅₉	—	8	4	no	—
64	C₂×C₄×C₅₉	#107	1	4	yes	—
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₅₉	—	8	1	no	—
72	C₂×Dic₅₉	—	8	1	no	—
73	C₂×Dic₅₉	—	8	2	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	C₈×C₅₉	#107	1	4	yes	—
76	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
78	#12908	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₉	#493	1	1	yes	characteristic
81	C₂²×C₅₉	#500	1	1	yes	characteristic
82	C₂²×C₅₉	#509	1	2	yes	—
83	C₂²×C₅₉	#510	1	2	yes	—
84	C₄×C₅₉	—	2	4	no	—
85	C₄×C₅₉	—	16	2	no	—
86	C₄×C₅₉	#384	1	1	yes	characteristic
87	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
88	C₄×C₅₉	#494	1	1	yes	characteristic
89	C₄×C₅₉	#500	1	1	yes	characteristic
90	Dic₅₉	—	16	1	no	—
91	Dic₅₉	—	16	1	no	—
92	C₂×#367	—	59	2	no	—
93	C₂×#366	—	59	2	no	—
94	C₂×#346	—	59	1	no	—
95	#1846	—	59	1	no	—
96	#1845	D₅₉	1	1	yes	characteristic
97	C₂×C₅₉	#1845	1	1	yes	characteristic
98	C₂×C₅₉	#2159	1	2	yes	—
99	C₂×C₅₉	#2250	1	1	yes	characteristic
100	C₂×C₅₉	#2251	1	1	yes	characteristic
101	C₂×C₅₉	#2255	1	2	yes	—
102	C₂²×C₁₆	—	59	2	no	—
103	C₂×#107	—	2	1	no	—
104	C₂×#107	—	59	1	no	—
105	C₂×#107	—	59	1	no	—
106	C₂×#107	C₂×D₅₉	1	2	yes	—
107	C₂×#106	—	118	1	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#384	—	2	1	no	—
110	#382	—	118	1	no	—
111	#367	—	118	4	no	—
112	#366	—	118	4	no	—
113	#346	—	59	4	no	—
114	C₅₉	#12908	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄⋊C₄	—	118	1	no	—
120	C₂×C₄⋊C₄	—	118	1	no	—
121	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
122	C₂×C₁₆	—	59	4	no	—
123	C₂×C₁₆	—	59	4	no	—
124	C₂×C₁₆	—	59	4	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	—	4	2	no	—
128	#107	—	118	2	no	—
129	#107	—	118	2	no	—
130	#106	—	236	2	no	—
131	C₄×C₈	Dic₁₁₈	1	2	yes	—
132	C₄×C₈	C₄×D₅₉	1	2	yes	—
133	#95	—	236	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	236	1	no	—
136	C₂²×C₄	—	236	1	no	—
137	C₂²×C₄	#11745669	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
140	C₂×C₈	#11745669	1	2	yes	—
141	C₂×C₈	#11745672	1	2	yes	—
142	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
144	C₂×C₈	D₄×D₅₉	1	2	yes	—
145	C₂×C₈	Q₈×D₅₉	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	8	4	no	—
149	C₄⋊C₄	—	236	1	no	—
150	C₄⋊C₄	—	236	1	no	—
151	C₄⋊C₄	—	236	1	no	—
152	C₄⋊C₄	—	236	1	no	—
153	C₄²	Dic₅₉⋊C₄	1	2	yes	—
154	C₄²	#11745672	1	2	yes	—
155	C₁₆	—	59	8	no	—
156	C₂³	C₂×#11745669	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	472	1	no	—
162	C₂×C₄	—	472	1	no	—
163	C₂×C₄	—	472	2	no	—
164	C₂×C₄	—	472	2	no	—
165	C₂×C₄	#14273660	1	4	yes	—
166	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#14273723	1	1	yes	characteristic
168	C₂×C₄	#14273725	1	1	yes	characteristic
169	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
170	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
171	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
172	C₈	#14273660	1	4	yes	—
173	C₈	C₄⋊C₄×D₅₉	1	2	yes	—
174	C₈	#14273726	1	2	yes	—
175	C₂²	C₂×#14273660	1	1	yes	characteristic
176	C₂²	#16837778	1	1	yes	characteristic
177	C₂²	#16837805	1	1	yes	characteristic
178	C₂²	#16837838	1	2	yes	—
179	C₂²	#16837843	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	16	2	no	—
182	C₄	—	944	1	no	—
183	C₄	—	944	1	no	—
184	C₄	#16837271	1	1	yes	characteristic
185	C₄	C₂×#14273660	1	1	yes	characteristic
186	C₄	#16837781	1	1	yes	characteristic
187	C₄	#16837805	1	1	yes	characteristic
188	C₂	#21424255	1	1	yes	characteristic
189	C₂	#21425882	1	2	yes	—
190	C₂	#21426360	1	1	yes	characteristic
191	C₂	#21426365	1	1	yes	characteristic
192	C₂	#21426379	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1845	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1845	D₅₉
hypocenter	C₅₉	#12908
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
13. #16837178	C₄	semidirect
58. C₅₉⋊C₁₆	C₄⋊C₄	semidirect
114. C₅₉	#12908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837229	2
C₂×#16837227	2
C₂×#16837178	2
#21424260	2
C₅₉×#1845	2
#12908	59