Subgroups — #28503787

Order 15104.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#21425717	C₂	1	1	yes	maximal, characteristic
3	#21425693	C₂	1	2	yes	maximal, complemented
4	#21422182	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1757	C₂	1	1	yes	maximal, characteristic
6	#16837819	—	2	2	no	—
7	#16837819	—	2	2	no	—
8	C₂×#14273667	C₄	1	2	yes	complemented
9	C₂×#14273667	C₂²	1	2	yes	—
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₄	1	2	yes	complemented
12	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16837184	C₄	1	4	yes	complemented
14	#16837178	—	2	2	no	—
15	C₄×C₅₉⋊C₁₆	—	2	2	no	—
16	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#413	C₄	1	1	yes	characteristic
18	C₅₉×#410	C₄	1	1	yes	characteristic
19	C₅₉×#405	C₂²	1	2	yes	—
20	C₅₉×#344	C₄	1	2	yes	—
21	#14273667	—	2	2	no	—
22	#14273667	—	2	2	no	—
23	#14273667	—	2	4	no	—
24	#14273667	C₂×C₄	1	4	yes	—
25	C₂×C₅₉⋊C₁₆	—	2	2	no	—
26	C₂×C₅₉⋊C₁₆	—	2	2	no	—
27	C₂×C₅₉⋊C₁₆	—	2	2	no	—
28	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
32	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
33	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
34	#131×C₅₉	—	2	2	no	—
35	#131×C₅₉	—	2	2	no	—
36	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	#117×C₅₉	C₂×C₄	1	2	yes	—
42	#101×C₅₉	C₈	1	2	yes	—
43	#100×C₅₉	C₈	1	2	yes	—
44	#99×C₅₉	C₈	1	4	yes	—
45	#97×C₅₉	—	2	2	no	—
46	C₄×C₈×C₅₉	—	2	2	no	—
47	#41×C₅₉	—	2	2	no	—
48	#41×C₅₉	—	2	2	no	—
49	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
50	C₂×D₄×C₅₉	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₅₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₅₉	—	2	2	no	—
55	M₁₆×C₅₉	—	2	2	no	—
56	M₁₆×C₅₉	—	2	4	no	—
57	M₁₆×C₅₉	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	—	2	2	no	—
61	C₂×C₈×C₅₉	C₄²	1	2	yes	—
62	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₉	#41	1	2	yes	—
66	C₄⋊C₄×C₅₉	—	2	2	no	—
67	C₂²⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
68	C₄²×C₅₉	—	2	2	no	—
69	C₅₉⋊C₁₆	—	2	2	no	—
70	C₅₉⋊C₁₆	—	2	4	no	—
71	C₅₉⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₅₉⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
73	C₅₉⋊C₁₆	C₂²⋊C₄	1	2	yes	—
74	C₂³×C₅₉	C₄×C₈	1	2	yes	—
75	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
76	Q₈×C₅₉	—	2	2	no	—
77	D₄×C₅₉	—	2	4	no	—
78	D₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	2	no	—
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	4	no	—
82	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#131	1	2	yes	—
87	C₈×C₅₉	—	2	2	no	—
88	C₈×C₅₉	—	2	4	no	—
89	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₈×C₅₉	#117	1	1	yes	characteristic
91	C₈×C₅₉	C₄×D₄	1	2	yes	—
92	#12613	—	59	1	no	maximal, Sylow
93	C₂²×C₅₉	—	2	2	no	—
94	C₂²×C₅₉	—	4	1	no	—
95	C₂²×C₅₉	—	4	2	no	—
96	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂²×C₅₉	C₈×D₄	1	2	yes	—
98	C₄×C₅₉	—	2	2	no	—
99	C₄×C₅₉	—	4	4	no	—
100	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₅₉	#405	1	1	yes	characteristic
102	C₄×C₅₉	C₈×D₄	1	2	yes	—
103	C₂×#504	—	59	1	no	—
104	#2115	—	59	1	no	—
105	#2108	—	59	2	no	—
106	#1757	D₅₉	1	1	yes	characteristic
107	#1378	—	59	2	no	—
108	C₂×C₅₉	—	2	2	no	—
109	C₂×C₅₉	—	2	1	no	—
110	C₂×C₅₉	—	4	2	no	—
111	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
113	#504	—	118	2	no	—
114	#504	—	118	2	no	—
115	C₂×M₃₂	—	59	2	no	—
116	C₂×M₃₂	—	59	1	no	—
117	C₂×M₃₂	—	59	2	no	—
118	C₂²×C₁₆	—	59	1	no	—
119	C₂²×C₁₆	—	59	2	no	—
120	#413	Dic₅₉	1	1	yes	characteristic
121	#410	Dic₅₉	1	1	yes	characteristic
122	#405	C₂×D₅₉	1	2	yes	—
123	#349	—	59	4	no	—
124	#346	—	118	2	no	—
125	C₄×C₁₆	—	118	2	no	—
126	#344	Dic₅₉	1	2	yes	—
127	C₅₉	#12613	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₂×Dic₅₉	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
132	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
133	C₂×M₁₆	C₂×Dic₅₉	1	1	yes	characteristic
134	C₂²×C₈	C₄×D₅₉	1	2	yes	—
135	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	#117	C₂×Dic₅₉	1	2	yes	—
137	M₃₂	—	59	4	no	—
138	M₃₂	—	118	4	no	—
139	M₃₂	—	118	2	no	—
140	M₃₂	—	118	2	no	—
141	C₂×C₁₆	—	59	1	no	—
142	C₂×C₁₆	—	59	1	no	—
143	C₂×C₁₆	—	59	2	no	—
144	C₂×C₁₆	—	59	2	no	—
145	C₂×C₁₆	—	59	2	no	—
146	C₂×C₁₆	—	118	2	no	—
147	C₂×C₁₆	—	118	2	no	—
148	C₂×C₁₆	—	118	2	no	—
149	#101	C₅₉⋊C₈	1	2	yes	—
150	#100	C₅₉⋊C₈	1	2	yes	—
151	#99	C₅₉⋊C₈	1	4	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
157	C₂×D₄	C₂×C₅₉⋊C₈	1	2	yes	—
158	C₂×D₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
160	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	4	no	—
164	M₁₆	C₂×C₅₉⋊C₈	1	4	yes	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
171	C₂×C₈	D₄×D₅₉	1	2	yes	—
172	C₂×C₈	#11745687	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
175	C₄²	—	2	2	no	—
176	C₁₆	—	59	1	no	—
177	C₁₆	—	59	1	no	—
178	C₁₆	—	59	2	no	—
179	C₁₆	—	118	2	no	—
180	C₁₆	—	118	4	no	—
181	C₂³	C₄×C₅₉⋊C₈	1	2	yes	—
182	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
191	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
192	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
193	C₂×C₄	#14273734	1	2	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	2	no	—
196	C₈	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
197	C₈	#14273683	1	1	yes	characteristic
198	C₈	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
203	C₂²	D₄×C₅₉⋊C₈	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	4	4	no	—
206	C₄	C₂²⋊C₄×Dic₅₉	1	1	yes	characteristic
207	C₄	#16837358	1	1	yes	characteristic
208	C₄	D₄×C₅₉⋊C₈	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₂²⋊C₄×C₅₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊C₄×D₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1757	C₂
socle	C₂×C₅₉	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1757	D₅₉
hypocenter	C₅₉	#12613
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

Complements

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

N	H ≅ G/N	type
3. #21425693	C₂	semidirect
4. #21422182	C₂	semidirect
8. C₂×#14273667	C₄	semidirect
11. C₂²×C₅₉⋊C₁₆	C₄	semidirect
13. #16837184	C₄	semidirect
127. C₅₉	#12613	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
#21425717	2
#21425693	2
#21422182	2
C₅₉×#1757	2
#12613	59