Subgroups — C₂³×#14273722 #29495361

Order 15104.

9486 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273722	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×Dic₅₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×(C₂×D₅₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂³×(C₂×D₅₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×(C₂×D₅₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₉	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×#14273722	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₅₉	C₂²	1	28	yes	complemented
15	C₂²×Dic₅₉⋊C₄	C₂²	1	28	yes	complemented
16	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	28	yes	—
17	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	28	yes	—
18	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	28	yes	—
19	C₂²×C₄×Dic₅₉	C₂²	1	28	yes	complemented
20	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₉	C₂²	1	28	yes	complemented
24	C₂⁴×D₅₉	—	2	8	no	—
25	C₂⁴×D₅₉	—	2	8	no	—
26	C₂⁴×D₅₉	—	2	8	no	—
27	C₂⁴×D₅₉	C₂³	1	14	yes	—
28	C₂³×Dic₅₉	—	2	8	no	—
29	C₂³×Dic₅₉	—	2	8	no	—
30	C₂³×Dic₅₉	—	2	8	no	—
31	C₂³×Dic₅₉	C₂³	1	14	yes	—
32	C₂³×Dic₅₉	C₂³	1	14	yes	—
33	C₂³×Dic₅₉	C₂³	1	14	yes	—
34	#14273722	C₂³	1	512	yes	direct product factor
35	C₂×C₄⋊Dic₅₉	C₂³	1	112	yes	complemented
36	C₂×Dic₅₉⋊C₄	C₂³	1	112	yes	complemented
37	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	112	yes	—
38	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	112	yes	—
39	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	112	yes	—
40	C₂×C₄×Dic₅₉	C₂³	1	112	yes	complemented
41	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₉	—	2	8	no	—
43	C₂³×C₄×C₅₉	—	2	8	no	—
44	C₂³×C₄×C₅₉	—	2	8	no	—
45	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
46	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
47	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
48	C₂×C₄⋊C₄×C₅₉	C₂³	1	112	yes	complemented
49	C₂⁴×C₅₉	#41	1	8	yes	—
50	C₂⁴×C₅₉	#41	1	8	yes	—
51	C₂⁴×C₅₉	#41	1	8	yes	—
52	C₂⁴×C₅₉	C₂⁴	1	7	yes	—
53	C₂³×D₅₉	—	2	56	no	—
54	C₂³×D₅₉	—	2	56	no	—
55	C₂³×D₅₉	—	2	56	no	—
56	C₂³×D₅₉	—	4	64	no	—
57	C₂³×D₅₉	C₂⁴	1	28	yes	—
58	C₂²×C₄×C₅₉	—	2	56	no	—
59	C₂²×C₄×C₅₉	—	2	56	no	—
60	C₂²×C₄×C₅₉	—	2	56	no	—
61	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
62	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
63	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
64	C₂²×Dic₅₉	—	2	56	no	—
65	C₂²×Dic₅₉	—	2	56	no	—
66	C₂²×Dic₅₉	—	2	56	no	—
67	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
68	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
69	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
70	C₄⋊C₄×C₅₉	C₂⁴	1	64	yes	complemented
71	(C₂×D₅₉)⋊C₄	C₂⁴	1	64	yes	—
72	(C₂×D₅₉)⋊C₄	C₂⁴	1	64	yes	—
73	(C₂×D₅₉)⋊C₄	C₂⁴	1	64	yes	—
74	C₄⋊Dic₅₉	C₂⁴	1	64	yes	complemented
75	Dic₅₉⋊C₄	C₂⁴	1	64	yes	complemented
76	C₄×Dic₅₉	C₂⁴	1	64	yes	complemented
77	C₂³×C₅₉	#126	1	64	yes	complemented
78	C₂³×C₅₉	C₂×#41	1	28	yes	—
79	C₂³×C₅₉	C₂×#41	1	28	yes	—
80	C₂³×C₅₉	C₂×#41	1	28	yes	—
81	C₂³×C₅₉	C₂⁵	1	7	yes	—
82	C₂²×D₅₉	—	2	56	no	—
83	C₂²×D₅₉	—	2	56	no	—
84	C₂²×D₅₉	—	2	56	no	—
85	C₂²×D₅₉	—	4	224	no	—
86	C₂²×D₅₉	C₂⁵	1	8	yes	—
87	C₂×C₄×C₅₉	—	2	56	no	—
88	C₂×C₄×C₅₉	—	2	56	no	—
89	C₂×C₄×C₅₉	—	2	56	no	—
90	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
91	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
92	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
93	C₂×Dic₅₉	—	2	56	no	—
94	C₂×Dic₅₉	—	2	56	no	—
95	C₂×Dic₅₉	—	2	56	no	—
96	C₂×Dic₅₉	C₂⁵	1	8	yes	—
97	C₂×Dic₅₉	C₂⁵	1	8	yes	—
98	C₂×Dic₅₉	C₂⁵	1	8	yes	—
99	C₂³×#126	—	59	1	no	maximal, Sylow
100	C₂²×C₅₉	C₂×#126	1	112	yes	complemented
101	C₂²×C₅₉	C₂²×#41	1	14	yes	—
102	C₂²×C₅₉	C₂²×#41	1	14	yes	—
103	C₂²×C₅₉	C₂²×#41	1	14	yes	—
104	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
105	C₂×D₅₉	—	2	8	no	—
106	C₂×D₅₉	—	2	8	no	—
107	C₂×D₅₉	—	2	8	no	—
108	C₂×D₅₉	—	4	112	no	—
109	C₄×C₅₉	—	2	8	no	—
110	C₄×C₅₉	—	2	8	no	—
111	C₄×C₅₉	—	2	8	no	—
112	Dic₅₉	—	2	8	no	—
113	Dic₅₉	—	2	8	no	—
114	Dic₅₉	—	2	8	no	—
115	C₂²×#126	—	59	56	no	—
116	C₂³×C₄⋊C₄	—	59	1	no	—
117	C₂³×C₄⋊C₄	—	59	1	no	—
118	C₂³×C₄⋊C₄	D₅₉	1	1	yes	characteristic, complemented
119	C₂³×C₂²⋊C₄	—	59	1	no	—
120	C₂³×C₂²⋊C₄	—	59	1	no	—
121	C₂³×C₂²⋊C₄	—	59	1	no	—
122	C₂³×C₄²	—	59	1	no	—
123	C₂×C₅₉	C₂²×#126	1	28	yes	complemented
124	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
125	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
126	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
127	D₅₉	—	4	8	no	—
128	C₂⁶	—	59	1	no	—
129	C₂⁴×C₄	—	59	1	no	—
130	C₂⁴×C₄	—	59	1	no	—
131	C₂⁴×C₄	—	59	1	no	—
132	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
133	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
135	C₂×#126	—	59	448	no	—
136	C₂²×C₄⋊C₄	—	59	28	no	—
137	C₂²×C₄⋊C₄	—	59	28	no	—
138	C₂²×C₄⋊C₄	C₂×D₅₉	1	28	yes	complemented
139	C₂²×C₂²⋊C₄	—	59	28	no	—
140	C₂²×C₂²⋊C₄	—	59	28	no	—
141	C₂²×C₂²⋊C₄	—	59	28	no	—
142	C₂²×C₄²	—	59	28	no	—
143	C₅₉	C₂³×#126	1	1	yes	characteristic, complemented, Sylow
144	C₂⁵	—	59	14	no	—
145	C₂⁵	—	118	8	no	—
146	C₂⁵	—	118	8	no	—
147	C₂⁵	—	118	8	no	—
148	C₂⁵	C₂²×D₅₉	1	1	yes	characteristic
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	2	8	no	—
152	C₂³×C₄	—	59	14	no	—
153	C₂³×C₄	—	59	14	no	—
154	C₂³×C₄	—	59	14	no	—
155	C₂³×C₄	—	118	8	no	—
156	C₂³×C₄	—	118	8	no	—
157	C₂³×C₄	—	118	8	no	—
158	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
159	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
160	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
161	#126	—	59	512	no	—
162	C₂×C₄⋊C₄	—	59	112	no	—
163	C₂×C₄⋊C₄	—	59	112	no	—
164	C₂×C₄⋊C₄	C₂²×D₅₉	1	112	yes	complemented
165	C₂×C₂²⋊C₄	—	59	112	no	—
166	C₂×C₂²⋊C₄	—	59	112	no	—
167	C₂×C₂²⋊C₄	—	59	112	no	—
168	C₂×C₄²	—	59	112	no	—
169	C₂⁴	—	59	28	no	—
170	C₂⁴	—	118	56	no	—
171	C₂⁴	—	118	56	no	—
172	C₂⁴	—	118	56	no	—
173	C₂⁴	—	236	64	no	—
174	C₂⁴	#11745685	1	8	yes	—
175	C₂⁴	#11745687	1	8	yes	—
176	C₂⁴	#11745689	1	8	yes	—
177	C₂⁴	C₂³×D₅₉	1	7	yes	—
178	C₂²×C₄	—	2	56	no	—
179	C₂²×C₄	—	2	56	no	—
180	C₂²×C₄	—	2	56	no	—
181	C₂²×C₄	—	59	28	no	—
182	C₂²×C₄	—	59	28	no	—
183	C₂²×C₄	—	59	28	no	—
184	C₂²×C₄	—	118	56	no	—
185	C₂²×C₄	—	118	56	no	—
186	C₂²×C₄	—	118	56	no	—
187	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
188	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
189	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
190	C₄⋊C₄	—	59	64	no	—
191	C₄⋊C₄	—	59	64	no	—
192	C₄⋊C₄	C₂³×D₅₉	1	64	yes	complemented
193	C₂²⋊C₄	—	59	64	no	—
194	C₂²⋊C₄	—	59	64	no	—
195	C₂²⋊C₄	—	59	64	no	—
196	C₄²	—	59	64	no	—
197	C₂³	—	59	8	no	—
198	C₂³	—	118	56	no	—
199	C₂³	—	118	56	no	—
200	C₂³	—	118	56	no	—
201	C₂³	—	236	224	no	—
202	C₂³	#14273722	1	64	yes	direct product factor
203	C₂³	C₂×#11745687	1	28	yes	—
204	C₂³	C₂×#11745685	1	28	yes	—
205	C₂³	C₂×#11745689	1	28	yes	—
206	C₂³	C₂⁴×D₅₉	1	7	yes	—
207	C₂×C₄	—	2	56	no	—
208	C₂×C₄	—	2	56	no	—
209	C₂×C₄	—	2	56	no	—
210	C₂×C₄	—	59	8	no	—
211	C₂×C₄	—	59	8	no	—
212	C₂×C₄	—	59	8	no	—
213	C₂×C₄	—	118	56	no	—
214	C₂×C₄	—	118	56	no	—
215	C₂×C₄	—	118	56	no	—
216	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
217	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
218	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
219	C₂²	—	118	8	no	—
220	C₂²	—	118	8	no	—
221	C₂²	—	118	8	no	—
222	C₂²	—	236	112	no	—
223	C₂²	C₂×#14273722	1	112	yes	direct product factor
224	C₂²	C₂²×#11745687	1	14	yes	—
225	C₂²	C₂²×#11745685	1	14	yes	—
226	C₂²	C₂²×#11745689	1	14	yes	—
227	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	2	8	no	—
230	C₄	—	2	8	no	—
231	C₄	—	118	8	no	—
232	C₄	—	118	8	no	—
233	C₄	—	118	8	no	—
234	C₂	—	236	8	no	—
235	C₂	C₂²×#14273722	1	28	yes	direct product factor
236	C₂	C₂³×#11745687	1	1	yes	characteristic
237	C₂	C₂³×#11745685	1	1	yes	characteristic
238	C₂	C₂³×#11745689	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₉
commutator, cosocle	C₂²×C₅₉	C₂⁶
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂³×C₄⋊C₄×C₅₉	C₂
socle	C₂⁵×C₅₉	C₂³
prime radical	C₂⁵×D₅₉	C₂²
hypercenter	C₂³×C₄⋊C₄	D₅₉
hypocenter	C₅₉	C₂³×#126
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273722	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₅₉	C₂	semidirect
3. C₂³×Dic₅₉⋊C₄	C₂	semidirect
7. C₂³×C₄×Dic₅₉	C₂	semidirect
8. C₂³×C₄⋊C₄×C₅₉	C₂	semidirect
13. C₂×#14273722	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₅₉	C₂²	semidirect
15. C₂²×Dic₅₉⋊C₄	C₂²	semidirect
19. C₂²×C₄×Dic₅₉	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₅₉	C₂²	semidirect
34. #14273722	C₂³	direct (N × H)
35. C₂×C₄⋊Dic₅₉	C₂³	semidirect
36. C₂×Dic₅₉⋊C₄	C₂³	semidirect
40. C₂×C₄×Dic₅₉	C₂³	semidirect
48. C₂×C₄⋊C₄×C₅₉	C₂³	semidirect
70. C₄⋊C₄×C₅₉	C₂⁴	semidirect
74. C₄⋊Dic₅₉	C₂⁴	semidirect
75. Dic₅₉⋊C₄	C₂⁴	semidirect
76. C₄×Dic₅₉	C₂⁴	semidirect
77. C₂³×C₅₉	#126	semidirect
100. C₂²×C₅₉	C₂×#126	semidirect
118. C₂³×C₄⋊C₄	D₅₉	semidirect
123. C₂×C₅₉	C₂²×#126	semidirect
138. C₂²×C₄⋊C₄	C₂×D₅₉	semidirect
143. C₅₉	C₂³×#126	semidirect
164. C₂×C₄⋊C₄	C₂²×D₅₉	semidirect
192. C₄⋊C₄	C₂³×D₅₉	semidirect
202. C₂³	#14273722	direct (N × H)
223. C₂²	C₂×#14273722	direct (N × H)
235. C₂	C₂²×#14273722	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273722	2
C₂³×C₄⋊Dic₅₉	2
C₂³×Dic₅₉⋊C₄	2
C₂³×(C₂×D₅₉)⋊C₄	2
C₂³×C₄×Dic₅₉	2
C₂³×C₄⋊C₄×C₅₉	2
C₂³×#126	59

Subgroups — C23×#14273722 #29495361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14273722 #29495361