Subgroups — #28602580

Order 15104.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432814	C₂	1	2	yes	maximal, complemented
2	#21424929	C₂	1	2	yes	maximal, complemented
3	#21424425	C₂	1	8	yes	maximal, complemented
4	#21424234	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423847	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273734	C₂²	1	4	yes	—
8	#16837474	—	2	8	no	—
9	#16837426	C₂²	1	2	yes	—
10	C₂×#14273654	C₂²	1	8	yes	—
11	#16837369	C₂²	1	4	yes	—
12	C₂×#14273638	C₂²	1	1	yes	characteristic
13	#16837248	C₂²	1	1	yes	characteristic
14	#16837167	C₂²	1	4	yes	—
15	#16837156	C₂²	1	4	yes	—
16	C₂×#135×C₅₉	C₂²	1	4	yes	—
17	C₅₉×#532	C₂²	1	2	yes	—
18	C₅₉×#495	C₄	1	4	yes	complemented
19	C₅₉×#494	C₄	1	1	yes	characteristic, complemented
20	C₅₉×#493	C₄	1	1	yes	characteristic, complemented
21	C₅₉×#492	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	#14273734	—	4	8	no	—
24	C₂×#11745664	—	2	4	no	—
25	C₂×#11745664	C₂³	1	4	yes	—
26	C₂²×C₅₉⋊C₈	—	2	4	no	—
27	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
28	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
29	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
30	#14273654	—	4	8	no	—
31	C₂²⋊(C₅₉⋊C₈)	—	2	8	no	—
32	#14273638	D₄	1	4	yes	—
33	C₅₉⋊#95	—	2	2	no	—
34	C₂×#41×C₅₉	C₂³	1	4	yes	—
35	#135×C₅₉	—	4	8	no	—
36	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
37	C₂×SD₁₆×C₅₉	C₂×C₄	1	8	yes	—
38	C₂×D₈×C₅₉	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
41	#127×C₅₉	C₂×C₄	1	2	yes	—
42	#124×C₅₉	C₂×C₄	1	4	yes	—
43	#121×C₅₉	—	2	4	no	—
44	C₄×D₄×C₅₉	—	2	4	no	—
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
49	#41×C₅₉	—	2	8	no	—
50	#41×C₅₉	—	2	8	no	—
51	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₅₉	—	2	4	no	—
53	C₂×D₄×C₅₉	—	2	4	no	—
54	C₂×D₄×C₅₉	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₉	—	2	4	no	—
56	C₂²×C₄×C₅₉	#41	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₉	—	2	2	no	—
59	Q₁₆×C₅₉	—	4	4	no	—
60	SD₁₆×C₅₉	—	4	8	no	—
61	D₈×C₅₉	—	4	4	no	—
62	C₂×C₈×C₅₉	—	2	4	no	—
63	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
65	C₄⋊C₄×C₅₉	—	2	4	no	—
66	C₂²⋊C₄×C₅₉	—	2	8	no	—
67	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₉	C₂×D₄	1	2	yes	—
70	C₄×Dic₅₉	—	2	4	no	—
71	#11745664	—	2	8	no	—
72	#11745664	—	4	8	no	—
73	C₂×C₅₉⋊C₈	—	2	4	no	—
74	C₂×C₅₉⋊C₈	—	2	4	no	—
75	C₂×C₅₉⋊C₈	—	4	8	no	—
76	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
77	C₂³×C₅₉	—	2	4	no	—
78	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
79	Q₈×C₅₉	—	2	8	no	—
80	D₄×C₅₉	—	2	8	no	—
81	D₄×C₅₉	—	4	8	no	—
82	D₄×C₅₉	—	8	4	no	—
83	C₂×C₄×C₅₉	—	2	4	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	—	4	4	no	—
86	C₂×C₄×C₅₉	—	4	4	no	—
87	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	H(5,𝔽₂)	1	4	yes	—
92	C₂×Dic₅₉	—	4	4	no	—
93	C₂×Dic₅₉	—	4	2	no	—
94	C₈×C₅₉	—	2	2	no	—
95	C₈×C₅₉	—	2	2	no	—
96	C₅₉⋊C₈	—	2	4	no	—
97	C₅₉⋊C₈	—	4	4	no	—
98	#22282	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	—	2	2	no	—
100	C₂²×C₅₉	—	4	4	no	—
101	C₂²×C₅₉	—	4	8	no	—
102	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂²×C₅₉	#540	1	2	yes	—
104	C₄×C₅₉	—	2	2	no	—
105	C₄×C₅₉	—	2	2	no	—
106	C₄×C₅₉	—	4	4	no	—
107	C₄×C₅₉	#518	1	2	yes	—
108	C₄×C₅₉	#535	1	2	yes	—
109	Dic₅₉	—	8	2	no	—
110	#3147	D₅₉	1	1	yes	characteristic
111	#2932	—	59	2	no	—
112	#1963	—	59	2	no	—
113	#1876	—	59	8	no	—
114	#1837	—	59	1	no	—
115	#1746	—	59	1	no	—
116	C₂×C₅₉	—	2	2	no	—
117	C₂×C₅₉	—	8	4	no	—
118	C₂×C₅₉	#2328	1	1	yes	characteristic
119	C₂×C₅₉	#2957	1	2	yes	—
120	C₂×#135	C₂×D₅₉	1	4	yes	—
121	C₂×#131	—	59	4	no	—
122	#532	C₂×D₅₉	1	2	yes	—
123	#495	Dic₅₉	1	4	yes	complemented
124	#494	Dic₅₉	1	1	yes	characteristic, complemented
125	#493	Dic₅₉	1	1	yes	characteristic, complemented
126	#492	Dic₅₉	1	2	yes	complemented
127	#435	—	118	8	no	—
128	#423	—	59	2	no	—
129	C₂×#104	—	59	8	no	—
130	#408	—	59	4	no	—
131	C₂×#97	—	59	1	no	—
132	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
133	#376	—	59	1	no	—
134	#341	—	59	4	no	—
135	#336	—	59	4	no	—
136	C₅₉	#22282	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	C₂²×D₅₉	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
140	C₂×SD₁₆	C₂×Dic₅₉	1	8	yes	—
141	C₂×D₈	C₂×Dic₅₉	1	4	yes	—
142	#131	—	236	8	no	—
143	C₂×M₁₆	—	59	4	no	—
144	C₂×M₁₆	—	118	4	no	—
145	C₂²×C₈	—	59	2	no	—
146	C₂²×C₈	—	118	4	no	—
147	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
149	#127	C₂×Dic₅₉	1	2	yes	—
150	#124	C₂×Dic₅₉	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	—	59	2	no	—
154	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
155	C₄⋊C₈	—	118	4	no	—
156	#104	—	236	8	no	—
157	C₂²⋊C₈	—	118	8	no	—
158	#97	—	59	4	no	—
159	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
162	#95	—	118	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×Dic₅₉	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	118	2	no	—
171	C₂²×C₄	#11745687	1	2	yes	—
172	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	118	8	no	—
177	M₁₆	—	236	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	59	4	no	—
180	C₂×C₈	—	118	4	no	—
181	C₂×C₈	—	118	4	no	—
182	C₂×C₈	—	236	8	no	—
183	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
184	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	118	4	no	—
188	C₄²	D₄×D₅₉	1	2	yes	—
189	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
190	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#11745687	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	236	4	no	—
202	C₂×C₄	—	236	2	no	—
203	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
204	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
205	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
207	C₂×C₄	#14273772	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	118	4	no	—
211	C₈	—	236	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
216	C₂²	#16838006	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	472	2	no	—
221	C₄	#16837877	1	2	yes	—
222	C₄	#16837965	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#21426822	1	1	yes	characteristic
226	C₂	#21433013	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting, prime radical	C₅₉×#3147	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#3147	D₅₉
hypocenter	C₅₉	#22282
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #21432814	C₂	semidirect
2. #21424929	C₂	semidirect
3. #21424425	C₂	semidirect
4. #21424234	C₂	semidirect
5. #21423847	C₂	semidirect
18. C₅₉×#495	C₄	semidirect
19. C₅₉×#494	C₄	semidirect
20. C₅₉×#493	C₄	semidirect
21. C₅₉×#492	C₄	semidirect
123. #495	Dic₅₉	semidirect
124. #494	Dic₅₉	semidirect
125. #493	Dic₅₉	semidirect
126. #492	Dic₅₉	semidirect
136. C₅₉	#22282	semidirect

Maximal subgroups

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

subgroup	index
#21432814	2
#21424929	2
#21424425	2
#21424234	2
#21423847	2
C₅₉×#3147	2
#22282	59