Subgroups — #28741337

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837819	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21425695	C₂	1	3	yes	maximal, complemented
4	#21425692	C₂	1	3	yes	maximal, complemented
5	M₃₂×Dic₅₉	C₂	1	3	yes	maximal, complemented
6	C₂×#16837177	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic
8	C₂×#14273734	C₄	1	1	yes	characteristic
9	#41×Dic₅₉	C₄	1	1	yes	characteristic
10	#16837819	C₄	1	8	yes	complemented
11	C₂×#14273667	C₂²	1	3	yes	—
12	C₂²×C₅₉⋊C₁₆	C₂²	1	3	yes	—
13	#16837360	C₄	1	3	yes	—
14	#16837357	C₂²	1	3	yes	—
15	M₁₆×Dic₅₉	C₄	1	3	yes	—
16	C₂×C₈×Dic₅₉	C₂²	1	3	yes	—
17	#16837177	C₂²	1	1	yes	characteristic
18	#16837177	C₂²	1	3	yes	—
19	#16837177	C₂²	1	3	yes	—
20	#16837177	C₂²	1	3	yes	—
21	C₁₆×Dic₅₉	C₂²	1	6	yes	—
22	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#504	C₄	1	8	yes	complemented
24	C₂×M₃₂×C₅₉	C₂²	1	3	yes	—
25	C₂²×C₁₆×C₅₉	C₂²	1	3	yes	—
26	#14273734	C₈	1	8	yes	—
27	C₂×#11745664	C₂×C₄	1	3	yes	—
28	C₂²×C₅₉⋊C₈	C₂×C₄	1	3	yes	—
29	Q₈×Dic₅₉	C₈	1	2	yes	—
30	D₄×Dic₅₉	C₈	1	6	yes	—
31	#14273686	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₅₉	C₂×C₄	1	3	yes	—
33	#14273667	C₂×C₄	1	12	yes	complemented
34	C₂×C₅₉⋊C₁₆	C₂×C₄	1	12	yes	complemented
35	C₂×C₅₉⋊C₁₆	C₂³	1	3	yes	—
36	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
37	#14273637	C₂×C₄	1	6	yes	—
38	C₈×Dic₅₉	C₂×C₄	1	3	yes	—
39	C₈×Dic₅₉	C₂×C₄	1	3	yes	—
40	C₈×Dic₅₉	C₂³	1	3	yes	—
41	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
42	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
43	#131×C₅₉	C₂×C₄	1	4	yes	—
44	#131×C₅₉	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₅₉	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
47	M₃₂×C₅₉	C₂×C₄	1	12	yes	complemented
48	C₂×C₁₆×C₅₉	C₂×C₄	1	12	yes	complemented
49	C₂×C₁₆×C₅₉	C₂³	1	3	yes	—
50	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
51	#41×C₅₉	C₄²	1	4	yes	—
52	#41×C₅₉	C₂×C₈	1	4	yes	—
53	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
54	C₂×D₄×C₅₉	C₂×C₈	1	3	yes	—
55	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
56	C₂²×Dic₅₉	C₂×C₈	1	6	yes	—
57	M₁₆×C₅₉	C₄²	1	12	yes	—
58	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
60	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
61	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₂²⋊Dic₅₉	C₂×C₈	1	6	yes	—
63	C₄⋊Dic₅₉	C₂×C₈	1	6	yes	—
64	C₄×Dic₅₉	C₂×C₈	1	6	yes	—
65	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₅₉	C₂²×C₄	1	3	yes	—
67	#11745664	C₂×C₈	1	12	yes	—
68	C₂×C₅₉⋊C₈	C₂×C₈	1	12	yes	—
69	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅₉⋊C₈	C₂²×C₄	1	3	yes	—
71	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
72	C₁₆×C₅₉	C₂²×C₄	1	6	yes	—
73	C₅₉⋊C₁₆	C₂²×C₄	1	2	yes	—
74	C₅₉⋊C₁₆	C₂²×C₄	1	6	yes	—
75	C₂³×C₅₉	C₂²×C₈	1	3	yes	—
76	Q₈×C₅₉	C₄×C₈	1	4	yes	—
77	D₄×C₅₉	C₄×C₈	1	12	yes	—
78	C₂×C₄×C₅₉	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₅₉	C₂²×C₈	1	6	yes	—
80	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
82	C₂×Dic₅₉	—	4	6	no	—
83	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
84	C₂×Dic₅₉	C₂²×C₈	1	6	yes	—
85	C₈×C₅₉	C₂×C₄²	1	6	yes	—
86	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
88	C₅₉⋊C₈	C₂²×C₈	1	2	yes	—
89	C₅₉⋊C₈	C₂²×C₈	1	6	yes	—
90	#33326	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	2	6	no	—
92	C₂²×C₅₉	C₂×C₄×C₈	1	6	yes	—
93	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₅₉	C₂×C₄×C₈	1	6	yes	—
95	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
96	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
97	Dic₅₉	—	2	2	no	—
98	Dic₅₉	—	2	6	no	—
99	C₂×#504	—	59	1	no	—
100	C₂×#504	D₅₉	1	1	yes	characteristic
101	C₄×#131	—	59	1	no	—
102	#2108	—	59	3	no	—
103	#2108	—	59	3	no	—
104	C₄×M₃₂	—	59	3	no	—
105	C₂×#346	—	59	3	no	—
106	C₂×C₅₉	—	2	6	no	—
107	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
108	C₂×C₅₉	#3407	1	2	yes	—
109	C₂×#131	—	59	1	no	—
110	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
111	C₄×#41	—	59	1	no	—
112	#504	—	59	8	no	—
113	#504	Dic₅₉	1	8	yes	complemented
114	C₂×M₃₂	—	59	3	no	—
115	C₂×M₃₂	C₂×D₅₉	1	3	yes	—
116	C₂²×C₁₆	—	59	3	no	—
117	C₂²×C₁₆	C₂×D₅₉	1	3	yes	—
118	#405	—	59	3	no	—
119	#405	—	59	3	no	—
120	C₄×M₁₆	—	59	3	no	—
121	C₂×C₄×C₈	—	59	3	no	—
122	#346	—	59	1	no	—
123	#346	—	59	3	no	—
124	#346	—	59	3	no	—
125	#346	—	59	3	no	—
126	C₄×C₁₆	—	59	6	no	—
127	C₅₉	#33326	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
129	#131	—	59	8	no	—
130	#131	C₄×D₅₉	1	4	yes	—
131	#131	C₂×Dic₅₉	1	4	yes	—
132	C₂×M₁₆	—	59	3	no	—
133	C₂×M₁₆	C₄×D₅₉	1	3	yes	—
134	C₂²×C₈	—	59	3	no	—
135	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
136	C₄×Q₈	—	59	2	no	—
137	C₄×D₄	—	59	6	no	—
138	#117	—	59	3	no	—
139	C₂×C₄²	—	59	3	no	—
140	M₃₂	—	59	12	no	—
141	M₃₂	C₂×Dic₅₉	1	12	yes	complemented
142	C₂×C₁₆	—	59	12	no	—
143	C₂×C₁₆	—	59	3	no	—
144	C₂×C₁₆	—	59	1	no	—
145	C₂×C₁₆	C₂×Dic₅₉	1	12	yes	complemented
146	C₂×C₁₆	C₂²×D₅₉	1	3	yes	—
147	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
148	#97	—	59	6	no	—
149	C₄×C₈	—	59	3	no	—
150	C₄×C₈	—	59	3	no	—
151	C₄×C₈	—	59	3	no	—
152	C₄×C₈	—	59	1	no	—
153	#41	C₈×D₅₉	1	4	yes	—
154	#41	C₄×Dic₅₉	1	4	yes	—
155	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
156	C₂×D₄	C₈×D₅₉	1	3	yes	—
157	C₂²×C₄	—	59	6	no	—
158	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
159	M₁₆	—	59	12	no	—
160	M₁₆	C₄×Dic₅₉	1	12	yes	—
161	C₂×C₈	—	59	12	no	—
162	C₂×C₈	—	59	1	no	—
163	C₂×C₈	—	59	3	no	—
164	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
165	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
166	C₂×C₈	C₂²×Dic₅₉	1	6	yes	—
167	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
168	C₄⋊C₄	—	59	6	no	—
169	C₂²⋊C₄	—	59	6	no	—
170	C₄²	—	59	6	no	—
171	C₄²	—	59	1	no	—
172	C₄²	—	59	3	no	—
173	C₁₆	—	59	2	no	—
174	C₁₆	—	59	6	no	—
175	C₁₆	C₂²×Dic₅₉	1	2	yes	—
176	C₁₆	C₂²×Dic₅₉	1	6	yes	—
177	C₂³	C₂×C₈×D₅₉	1	3	yes	—
178	Q₈	C₈×Dic₅₉	1	4	yes	—
179	D₄	C₈×Dic₅₉	1	12	yes	—
180	C₂×C₄	—	59	2	no	—
181	C₂×C₄	—	59	6	no	—
182	C₂×C₄	—	236	6	no	—
183	C₂×C₄	C₂×C₄×Dic₅₉	1	6	yes	—
184	C₂×C₄	C₂×C₈×D₅₉	1	6	yes	—
185	C₂×C₄	C₂×C₈×D₅₉	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
187	C₈	—	59	2	no	—
188	C₈	—	59	6	no	—
189	C₈	C₂×C₄×Dic₅₉	1	6	yes	—
190	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
191	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
192	C₂²	—	2	6	no	—
193	C₂²	C₂×C₈×Dic₅₉	1	6	yes	—
194	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
195	C₄	—	118	2	no	—
196	C₄	—	118	6	no	—
197	C₄	C₂×C₈×Dic₅₉	1	6	yes	—
198	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
199	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
200	C₂	—	2	6	no	—
201	C₂	C₂²×C₈×Dic₅₉	1	1	yes	characteristic
202	C₂	#21437197	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄×C₈
Fitting	C₂×C₅₉×#504	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	#33326
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
3. #21425695	C₂	semidirect
4. #21425692	C₂	semidirect
5. M₃₂×Dic₅₉	C₂	semidirect
6. C₂×#16837177	C₂	semidirect
10. #16837819	C₄	semidirect
23. C₅₉×#504	C₄	semidirect
33. #14273667	C₂×C₄	semidirect
34. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
47. M₃₂×C₅₉	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
113. #504	Dic₅₉	semidirect
127. C₅₉	#33326	semidirect
141. M₃₂	C₂×Dic₅₉	semidirect
145. C₂×C₁₆	C₂×Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837819	2
#131×Dic₅₉	2
#21425695	2
#21425692	2
M₃₂×Dic₅₉	2
C₂×#16837177	2
C₂×C₅₉×#504	2
#33326	59