Subgroups — #28505134

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×#16837230	C₂	1	2	yes	maximal
2	C₂×#16837228	C₂	1	2	yes	maximal
3	#21424260	C₂	1	1	yes	maximal, characteristic
4	#21424256	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×#14273661	C₂²	1	1	yes	characteristic
9	C₂×#14273661	C₂²	1	1	yes	characteristic
10	C₂×#14273659	—	2	1	no	—
11	#16837272	—	2	1	no	—
12	#16837268	—	2	1	no	—
13	#16837230	—	2	4	no	—
14	#16837228	—	2	4	no	—
15	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#346	C₄	1	4	yes	complemented
18	C₂×C₄⋊Dic₅₉	—	2	1	no	—
19	C₂×C₄⋊Dic₅₉	—	2	1	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	4	2	no	—
22	#14273661	—	2	2	no	—
23	#14273661	—	2	2	no	—
24	#14273661	—	2	2	no	—
25	#14273661	—	2	2	no	—
26	#14273661	—	4	2	no	—
27	#14273659	—	4	2	no	—
28	C₅₉⋊#95	—	4	2	no	—
29	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
34	C₂×C₁₆×C₅₉	Q₈	1	4	yes	—
35	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₉	D₈	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₉	—	4	1	no	—
40	C₂²×Dic₅₉	—	4	1	no	—
41	C₂²×Dic₅₉	—	4	2	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	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
50	C₄²×C₅₉	Q₁₆	1	2	yes	—
51	C₄⋊Dic₅₉	—	4	1	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	4	1	no	—
54	C₄⋊Dic₅₉	—	4	1	no	—
55	C₄⋊Dic₅₉	—	4	1	no	—
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄⋊Dic₅₉	—	8	4	no	—
58	C₁₆×C₅₉	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	#107	1	4	yes	—
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₅₉	—	8	1	no	—
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	8	1	no	—
72	C₂×Dic₅₉	—	8	4	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	#384	1	1	yes	characteristic
86	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
87	C₄×C₅₉	#494	1	1	yes	characteristic
88	C₄×C₅₉	#500	1	1	yes	characteristic
89	Dic₅₉	—	16	2	no	—
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	D₅₉	1	1	yes	characteristic
95	#1846	—	59	1	no	—
96	#1845	—	59	1	no	—
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	C₂×D₅₉	1	2	yes	—
103	C₂×#107	—	59	2	no	—
104	C₂×#107	—	59	1	no	—
105	C₂×#107	—	59	1	no	—
106	C₂×#107	—	118	1	no	—
107	C₂×#106	—	118	1	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#384	—	118	1	no	—
110	#382	—	118	1	no	—
111	#367	—	118	4	no	—
112	#366	—	118	4	no	—
113	#346	Dic₅₉	1	4	yes	complemented
114	C₅₉	#12908	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	118	1	no	—
118	C₂×C₄⋊C₄	—	118	1	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄⋊C₄	—	236	2	no	—
121	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
122	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
123	C₂×C₁₆	D₂₃₆	1	4	yes	—
124	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
125	#107	—	118	2	no	—
126	#107	—	118	2	no	—
127	#107	—	118	2	no	—
128	#107	—	118	2	no	—
129	#107	—	236	2	no	—
130	#106	—	236	2	no	—
131	C₄×C₈	Dic₁₁₈	1	2	yes	—
132	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
133	#95	—	236	2	no	—
134	C₂²×C₄	—	236	1	no	—
135	C₂²×C₄	—	236	1	no	—
136	C₂²×C₄	—	236	2	no	—
137	C₂²×C₄	D₄₇₂	1	2	yes	—
138	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
139	C₂×C₈	D₄₇₂	1	2	yes	—
140	C₂×C₈	Dic₂₃₆	1	2	yes	—
141	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
142	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
144	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
145	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
146	C₄⋊C₄	—	472	4	no	—
147	C₄⋊C₄	—	236	1	no	—
148	C₄⋊C₄	—	236	2	no	—
149	C₄⋊C₄	—	236	1	no	—
150	C₄⋊C₄	—	236	1	no	—
151	C₄⋊C₄	—	236	1	no	—
152	C₄⋊C₄	—	236	2	no	—
153	C₄²	Dic₂₃₆	1	2	yes	—
154	C₄²	C₄⋊Dic₅₉	1	2	yes	—
155	C₁₆	C₄⋊Dic₅₉	1	8	yes	complemented
156	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	472	1	no	—
160	C₂×C₄	—	472	2	no	—
161	C₂×C₄	—	472	1	no	—
162	C₂×C₄	—	472	4	no	—
163	C₂×C₄	—	472	2	no	—
164	C₂×C₄	—	472	2	no	—
165	C₂×C₄	#14273661	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
167	C₂×C₄	#14273724	1	1	yes	characteristic
168	C₂×C₄	#14273727	1	1	yes	characteristic
169	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
172	C₈	#14273661	1	4	yes	—
173	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
174	C₈	#14273727	1	2	yes	—
175	C₂²	C₂×#14273661	1	1	yes	characteristic
176	C₂²	#16837779	1	1	yes	characteristic
177	C₂²	#16837807	1	1	yes	characteristic
178	C₂²	#16837840	1	2	yes	—
179	C₂²	#16837845	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	944	2	no	—
182	C₄	—	944	1	no	—
183	C₄	—	944	1	no	—
184	C₄	#16837272	1	1	yes	characteristic
185	C₄	C₂×#14273661	1	1	yes	characteristic
186	C₄	#16837782	1	1	yes	characteristic
187	C₄	#16837807	1	1	yes	characteristic
188	C₂	#21424256	1	1	yes	characteristic
189	C₂	#21425884	1	2	yes	—
190	C₂	#21426362	1	1	yes	characteristic
191	C₂	#21426367	1	1	yes	characteristic
192	C₂	#21426382	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#12908
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
17. C₅₉×#346	C₄	semidirect
58. C₁₆×C₅₉	C₄⋊C₄	semidirect
113. #346	Dic₅₉	semidirect
114. C₅₉	#12908	semidirect
155. C₁₆	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×#16837230	2
C₂×#16837228	2
#21424260	2
#21424256	2
C₂×C₅₉×#346	2
#12908	59