Subgroups — C₂×#97×Dic₅₉ #28586011

Order 15104.

1790 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273638	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14273637	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₅₉	C₂	1	2	yes	maximal
4	C₂×C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic
5	#97×Dic₅₉	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₅₉⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₅₉	C₄	1	4	yes	—
9	C₂²×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×#14273638	C₄	1	8	yes	complemented
12	C₂×#14273638	C₂²	1	4	yes	—
13	C₂×#14273637	C₄	1	16	yes	complemented
14	C₂×#14273637	C₂²	1	8	yes	—
15	C₂×C₈×Dic₅₉	C₄	1	16	yes	complemented
16	C₂×C₈×Dic₅₉	C₂²	1	8	yes	—
17	C₄²×Dic₅₉	C₄	1	4	yes	—
18	C₄²×Dic₅₉	C₂²	1	4	yes	—
19	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
20	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₉	C₄	1	8	yes	complemented
22	C₂×#97×C₅₉	C₂²	1	4	yes	—
23	C₂³×Dic₅₉	C₂×C₄	1	2	yes	—
24	C₂³×Dic₅₉	C₂×C₄	1	2	yes	—
25	C₂²×C₅₉⋊C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₅₉⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₅₉⋊C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₅₉⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄×Dic₅₉	—	2	16	no	—
30	C₂×C₄×Dic₅₉	C₂×C₄	1	16	yes	—
31	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₅₉	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₅₉	C₂×C₄	1	8	yes	—
35	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
36	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
37	#14273638	C₂×C₄	1	16	yes	complemented
38	#14273637	C₂×C₄	1	32	yes	complemented
39	C₈×Dic₅₉	C₂×C₄	1	32	yes	complemented
40	C₂³×C₄×C₅₉	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
46	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
50	#97×C₅₉	C₂×C₄	1	16	yes	complemented
51	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	—	2	8	no	—
53	C₂²×C₄×C₅₉	C₄²	1	8	yes	—
54	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₅₉	—	2	8	no	—
62	C₂²×Dic₅₉	C₄²	1	8	yes	—
63	C₂²×Dic₅₉	C₄²	1	8	yes	—
64	C₂²×Dic₅₉	M₁₆	1	16	yes	—
65	C₂²×Dic₅₉	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₅₉	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₅₉	C₄²	1	32	yes	complemented
68	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
70	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
71	C₄²×C₅₉	C₄²	1	8	yes	—
72	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
73	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
74	C₄×Dic₅₉	—	2	32	no	—
75	C₄×Dic₅₉	C₄²	1	8	yes	—
76	C₄×Dic₅₉	C₄²	1	8	yes	—
77	C₄×Dic₅₉	C₂²×C₄	1	8	yes	—
78	C₄×Dic₅₉	C₂²×C₄	1	8	yes	—
79	C₂×C₅₉⋊C₈	C₄²	1	32	yes	complemented
80	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
81	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
82	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
83	C₂³×C₅₉	C₂×C₄²	1	2	yes	—
84	C₂³×C₅₉	C₂×C₄²	1	2	yes	—
85	C₂³×C₅₉	C₂×C₄²	1	2	yes	—
86	C₂³×C₅₉	C₂×M₁₆	1	8	yes	—
87	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	—	2	8	no	—
89	C₂×C₄×C₅₉	—	2	16	no	—
90	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₅₉	C₂×C₄²	1	8	yes	—
93	C₂×C₄×C₅₉	C₂×C₄²	1	8	yes	—
94	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₅₉	C₂×C₄²	1	8	yes	—
96	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
98	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₅₉	—	2	8	no	—
101	C₂×Dic₅₉	—	2	16	no	—
102	C₂×Dic₅₉	#97	1	32	yes	direct product factor
103	C₂×Dic₅₉	C₂×C₄²	1	8	yes	—
104	C₂×Dic₅₉	C₂×C₄²	1	8	yes	—
105	C₂×Dic₅₉	C₂×M₁₆	1	16	yes	—
106	C₈×C₅₉	C₂×C₄²	1	16	yes	complemented
107	C₅₉⋊C₈	C₂×C₄²	1	16	yes	complemented
108	C₂×C₄×#97	—	59	1	no	maximal, Sylow
109	C₂²×C₅₉	C₄³	1	4	yes	—
110	C₂²×C₅₉	C₂×#97	1	8	yes	—
111	C₂²×C₅₉	C₄×M₁₆	1	8	yes	—
112	C₂²×C₅₉	C₄×M₁₆	1	8	yes	—
113	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
115	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
116	C₂²×C₅₉	C₂²×M₁₆	1	4	yes	—
117	C₄×C₅₉	—	2	8	no	—
118	C₄×C₅₉	C₄³	1	4	yes	—
119	C₄×C₅₉	C₂²×C₄²	1	2	yes	—
120	C₄×C₅₉	C₂²×C₄²	1	2	yes	—
121	Dic₅₉	—	2	8	no	—
122	Dic₅₉	C₂×#97	1	16	yes	direct product factor
123	C₂²×#97	—	59	1	no	—
124	C₂²×#97	—	59	2	no	—
125	C₂²×#97	D₅₉	1	1	yes	characteristic
126	C₂²×C₄×C₈	—	59	2	no	—
127	C₂×C₄³	—	59	1	no	—
128	C₄×#97	—	59	8	no	—
129	C₂×C₅₉	C₄×#97	1	8	yes	complemented
130	C₂×C₅₉	C₂×C₄³	1	1	yes	characteristic
131	C₂×C₅₉	C₂²×#97	1	2	yes	—
132	C₂×C₅₉	C₂×C₄×M₁₆	1	2	yes	—
133	C₂×C₅₉	C₂×C₄×M₁₆	1	2	yes	—
134	C₂³×C₈	—	59	2	no	—
135	C₂³×C₈	C₂×D₅₉	1	2	yes	—
136	C₂²×C₄²	—	59	1	no	—
137	C₂²×C₄²	—	59	1	no	—
138	C₂²×C₄²	—	59	4	no	—
139	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
140	C₂×#97	—	59	16	no	—
141	C₂×#97	—	59	8	no	—
142	C₂×#97	—	59	8	no	—
143	C₂×#97	—	59	4	no	—
144	C₂×#97	Dic₅₉	1	8	yes	direct product factor
145	C₂×#97	C₂×D₅₉	1	4	yes	—
146	C₂×C₄×C₈	—	59	16	no	—
147	C₂×C₄×C₈	—	59	8	no	—
148	C₄³	—	59	4	no	—
149	C₄³	—	59	4	no	—
150	C₅₉	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	59	2	no	—
152	C₂³×C₄	—	59	2	no	—
153	C₂³×C₄	C₄×D₅₉	1	2	yes	—
154	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
155	C₂²×C₈	—	59	8	no	—
156	C₂²×C₈	—	59	8	no	—
157	C₂²×C₈	—	59	8	no	—
158	C₂²×C₈	—	59	4	no	—
159	C₂²×C₈	C₄×D₅₉	1	8	yes	—
160	C₂²×C₈	C₄×D₅₉	1	8	yes	—
161	C₂²×C₈	C₂×Dic₅₉	1	8	yes	—
162	C₂²×C₈	C₂²×D₅₉	1	4	yes	—
163	C₂×C₄²	—	59	2	no	—
164	C₂×C₄²	—	59	2	no	—
165	C₂×C₄²	—	59	8	no	—
166	C₂×C₄²	—	59	8	no	—
167	C₂×C₄²	—	59	2	no	—
168	C₂×C₄²	—	59	2	no	—
169	C₂×C₄²	—	59	16	no	—
170	C₂×C₄²	—	118	16	no	—
171	C₂×C₄²	C₄×D₅₉	1	2	yes	—
172	C₂×C₄²	C₄×D₅₉	1	4	yes	—
173	C₂×C₄²	C₂×Dic₅₉	1	4	yes	—
174	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
175	#97	—	59	32	no	—
176	#97	—	59	16	no	—
177	#97	C₂×Dic₅₉	1	16	yes	direct product factor
178	C₄×C₈	—	59	32	no	—
179	C₂⁴	C₂×C₄×D₅₉	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	59	8	no	—
182	C₂²×C₄	—	59	16	no	—
183	C₂²×C₄	—	59	4	no	—
184	C₂²×C₄	—	59	8	no	—
185	C₂²×C₄	—	59	4	no	—
186	C₂²×C₄	—	118	8	no	—
187	C₂²×C₄	C₄×Dic₅₉	1	8	yes	—
188	C₂²×C₄	C₂×C₄×D₅₉	1	4	yes	—
189	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₅₉	1	4	yes	—
192	C₂²×C₄	C₂×C₄×D₅₉	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
194	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
195	C₂×C₈	—	59	32	no	—
196	C₂×C₈	—	59	8	no	—
197	C₂×C₈	—	59	8	no	—
198	C₂×C₈	—	59	8	no	—
199	C₂×C₈	C₄×Dic₅₉	1	32	yes	complemented
200	C₂×C₈	C₂×C₄×D₅₉	1	8	yes	—
201	C₂×C₈	C₂×C₄×D₅₉	1	8	yes	—
202	C₂×C₈	C₂²×Dic₅₉	1	8	yes	—
203	C₄²	—	59	8	no	—
204	C₄²	—	59	8	no	—
205	C₄²	—	59	8	no	—
206	C₄²	—	59	8	no	—
207	C₄²	—	118	32	no	—
208	C₄²	C₄×Dic₅₉	1	8	yes	—
209	C₄²	C₂×C₄×D₅₉	1	4	yes	—
210	C₄²	C₂²×Dic₅₉	1	4	yes	—
211	C₂³	C₄²×D₅₉	1	2	yes	—
212	C₂³	C₄²×D₅₉	1	2	yes	—
213	C₂³	C₂×C₄×Dic₅₉	1	2	yes	—
214	C₂³	M₁₆×D₅₉	1	8	yes	—
215	C₂³	C₂²×C₄×D₅₉	1	1	yes	characteristic
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	16	no	—
218	C₂×C₄	—	59	16	no	—
219	C₂×C₄	—	59	8	no	—
220	C₂×C₄	—	59	32	no	—
221	C₂×C₄	—	59	8	no	—
222	C₂×C₄	—	118	8	no	—
223	C₂×C₄	—	118	16	no	—
224	C₂×C₄	C₄²×D₅₉	1	2	yes	—
225	C₂×C₄	C₄²×D₅₉	1	2	yes	—
226	C₂×C₄	C₄²×D₅₉	1	8	yes	—
227	C₂×C₄	C₂×C₄×Dic₅₉	1	8	yes	—
228	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₅₉	1	8	yes	—
230	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₄×D₅₉	1	2	yes	—
232	C₂×C₄	C₂²×C₄×D₅₉	1	2	yes	—
233	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
234	C₈	—	59	16	no	—
235	C₈	C₂×C₄×Dic₅₉	1	16	yes	complemented
236	C₂²	C₄²×Dic₅₉	1	4	yes	—
237	C₂²	#97×D₅₉	1	8	yes	—
238	C₂²	M₁₆×Dic₅₉	1	8	yes	—
239	C₂²	#16837354	1	8	yes	—
240	C₂²	C₂×C₄²×D₅₉	1	1	yes	characteristic
241	C₂²	C₂×C₄²×D₅₉	1	1	yes	characteristic
242	C₂²	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
243	C₂²	C₂×M₁₆×D₅₉	1	4	yes	—
244	C₄	—	2	8	no	—
245	C₄	—	59	16	no	—
246	C₄	—	118	8	no	—
247	C₄	C₄²×Dic₅₉	1	4	yes	—
248	C₄	C₂×C₄²×D₅₉	1	2	yes	—
249	C₄	C₂²×C₄×Dic₅₉	1	2	yes	—
250	C₂	#97×Dic₅₉	1	8	yes	direct product factor
251	C₂	C₂×C₄²×Dic₅₉	1	1	yes	characteristic
252	C₂	C₂×#97×D₅₉	1	2	yes	—
253	C₂	C₂×M₁₆×Dic₅₉	1	2	yes	—
254	C₂	C₂×#16837354	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₉
commutator	C₂×C₅₉	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×#97×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂²×C₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×#97	D₅₉
hypocenter	C₅₉	C₂×C₄×#97
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
5. #97×Dic₅₉	C₂	direct (N × H)
11. C₂×#14273638	C₄	semidirect
13. C₂×#14273637	C₄	semidirect
15. C₂×C₈×Dic₅₉	C₄	semidirect
21. C₂×#97×C₅₉	C₄	semidirect
37. #14273638	C₂×C₄	semidirect
38. #14273637	C₂×C₄	semidirect
39. C₈×Dic₅₉	C₂×C₄	semidirect
50. #97×C₅₉	C₂×C₄	semidirect
67. C₂×C₈×C₅₉	C₄²	semidirect
79. C₂×C₅₉⋊C₈	C₄²	semidirect
102. C₂×Dic₅₉	#97	direct (N × H)
106. C₈×C₅₉	C₂×C₄²	semidirect
107. C₅₉⋊C₈	C₂×C₄²	semidirect
122. Dic₅₉	C₂×#97	direct (N × H)
129. C₂×C₅₉	C₄×#97	semidirect
144. C₂×#97	Dic₅₉	direct (N × H)
150. C₅₉	C₂×C₄×#97	semidirect
177. #97	C₂×Dic₅₉	direct (N × H)
199. C₂×C₈	C₄×Dic₅₉	semidirect
235. C₈	C₂×C₄×Dic₅₉	semidirect
250. C₂	#97×Dic₅₉	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×#14273638	2
C₂²×#14273637	2
C₂²×C₈×Dic₅₉	2
C₂×C₄²×Dic₅₉	2
#97×Dic₅₉	2
C₂²×#97×C₅₉	2
C₂×C₄×#97	59

Subgroups — C2×#97×Dic59 #28586011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#97×Dic₅₉ #28586011