Subgroups — C₂×#21421941 #28478317

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837455	C₂	1	2	yes	maximal
2	#21421941	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837455	—	2	2	no	—
5	#16837455	—	2	2	no	—
6	#16837455	C₄	1	4	yes	—
7	#16837455	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₅₉⋊C₈)	C₄	1	4	yes	—
10	C₂×C₄×C₅₉⋊C₈	—	2	2	no	—
11	C₂³×C₈×C₅₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#97×C₅₉	—	2	2	no	—
14	C₂×#97×C₅₉	—	2	2	no	—
15	C₂×#97×C₅₉	C₄	1	2	yes	—
16	C₂×#97×C₅₉	C₂²	1	2	yes	—
17	C₂²×C₅₉⋊C₈	—	2	4	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
22	C₄×C₅₉⋊C₈	—	2	4	no	—
23	C₄×C₅₉⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₅₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	—	2	2	no	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	—	2	4	no	—
29	C₂²×C₈×C₅₉	C₈	1	8	yes	complemented
30	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	D₄	1	2	yes	—
36	C₂×C₄²×C₅₉	D₄	1	2	yes	—
37	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	#97×C₅₉	—	2	4	no	—
41	#97×C₅₉	—	2	4	no	—
42	C₂⁴×C₅₉	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	—	4	4	no	—
44	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
45	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₉	M₁₆	1	4	yes	—
56	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	—	2	8	no	—
59	C₂×C₈×C₅₉	—	2	8	no	—
60	C₂×C₈×C₅₉	—	2	4	no	—
61	C₂×C₈×C₅₉	—	2	4	no	—
62	C₂×C₈×C₅₉	C₂×C₈	1	8	yes	complemented
63	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₉	C₂×D₄	1	2	yes	—
72	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₅₉⋊C₈	—	2	8	no	—
74	C₂×C₅₉⋊C₈	—	4	8	no	—
75	C₂³×C₅₉	—	2	4	no	—
76	C₂³×C₅₉	#95	1	1	yes	characteristic
77	C₂³×C₅₉	#95	1	1	yes	characteristic
78	C₂³×C₅₉	C₄×C₈	1	2	yes	—
79	C₂³×C₅₉	#97	1	2	yes	—
80	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅₉	—	2	4	no	—
82	C₂×C₄×C₅₉	—	2	4	no	—
83	C₂×C₄×C₅₉	—	4	2	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	—	4	2	no	—
86	C₂×C₄×C₅₉	—	4	2	no	—
87	C₂×C₄×C₅₉	#95	1	2	yes	—
88	C₂×C₄×C₅₉	#95	1	2	yes	—
89	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₅₉	#97	1	2	yes	—
93	C₂×C₄×C₅₉	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₅₉	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
96	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
103	C₈×C₅₉	—	2	8	no	—
104	C₅₉⋊C₈	—	4	8	no	—
105	C₂×#1281	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	—	2	4	no	—
107	C₂²×C₅₉	—	2	4	no	—
108	C₂²×C₅₉	—	2	2	no	—
109	C₂²×C₅₉	—	2	2	no	—
110	C₂²×C₅₉	#336	1	2	yes	—
111	C₂²×C₅₉	#336	1	2	yes	—
112	C₂²×C₅₉	#344	1	4	yes	—
113	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
114	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
116	C₄×C₅₉	—	2	2	no	—
117	C₄×C₅₉	—	2	2	no	—
118	C₄×C₅₉	—	2	2	no	—
119	C₄×C₅₉	—	2	2	no	—
120	C₄×C₅₉	#336	1	2	yes	—
121	C₄×C₅₉	#336	1	2	yes	—
122	C₄×C₅₉	C₂×C₂²⋊C₈	1	2	yes	—
123	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
124	C₂×#431	—	59	2	no	—
125	C₂²×#97	D₅₉	1	1	yes	characteristic
126	#1281	—	59	4	no	—
127	C₂×C₅₉	—	2	2	no	—
128	C₂×C₅₉	—	2	1	no	—
129	C₂×C₅₉	—	2	1	no	—
130	C₂×C₅₉	#1281	1	4	yes	complemented
131	C₂×C₅₉	C₂×#336	1	1	yes	characteristic
132	C₂×C₅₉	C₂×#344	1	2	yes	—
133	C₂³×C₈	Dic₅₉	1	2	yes	—
134	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
135	#431	—	59	4	no	—
136	#431	—	59	4	no	—
137	#431	—	118	2	no	—
138	#431	—	118	2	no	—
139	C₂×C₄⋊C₈	—	118	2	no	—
140	C₂×C₂²⋊C₈	—	59	4	no	—
141	C₂×#97	—	2	2	no	—
142	C₂×#97	—	2	2	no	—
143	C₂×#97	Dic₅₉	1	2	yes	—
144	C₂×#97	C₂×D₅₉	1	2	yes	—
145	C₂×C₄×C₈	—	118	2	no	—
146	C₅₉	C₂×#1281	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	C₄×D₅₉	1	2	yes	—
148	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	118	4	no	—
153	C₂²×C₈	C₅₉⋊C₈	1	8	yes	complemented
154	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
155	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
156	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
157	C₂×C₄²	C₄×D₅₉	1	2	yes	—
158	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
159	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
161	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
162	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
163	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
164	C₄⋊C₈	—	118	4	no	—
165	C₄⋊C₈	—	118	4	no	—
166	C₂²⋊C₈	—	59	8	no	—
167	C₂²⋊C₈	—	118	4	no	—
168	#97	—	2	4	no	—
169	#97	—	2	4	no	—
170	C₄×C₈	—	118	4	no	—
171	C₄×C₈	—	118	4	no	—
172	C₂⁴	C₄×Dic₅₉	1	1	yes	characteristic
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
175	C₂²×C₄	#11745664	1	4	yes	—
176	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
177	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
178	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
179	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
180	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
182	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
183	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
184	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	2	8	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	118	8	no	—
193	C₂×C₈	—	236	8	no	—
194	C₂×C₈	C₂×C₅₉⋊C₈	1	8	yes	complemented
195	C₄²	Dic₅₉⋊C₄	1	2	yes	—
196	C₄²	Dic₅₉⋊C₄	1	2	yes	—
197	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
198	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
199	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
200	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
201	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
202	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
203	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
205	C₂³	—	2	4	no	—
206	C₂³	C₅₉⋊#95	1	1	yes	characteristic
207	C₂³	C₅₉⋊#95	1	1	yes	characteristic
208	C₂³	C₄×C₅₉⋊C₈	1	2	yes	—
209	C₂³	#14273638	1	2	yes	—
210	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₅₉⋊#95	1	2	yes	—
218	C₂×C₄	C₅₉⋊#95	1	2	yes	—
219	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
220	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
221	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
222	C₂×C₄	#14273638	1	2	yes	—
223	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	4	yes	—
224	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	4	yes	—
225	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
226	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
227	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
229	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
232	C₂×C₄	C₂×#11745664	1	2	yes	—
233	C₈	—	2	8	no	—
234	C₈	—	236	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	#16837157	1	2	yes	—
240	C₂²	#16837157	1	2	yes	—
241	C₂²	#16837173	1	4	yes	—
242	C₂²	C₂×C₅₉⋊#95	1	1	yes	characteristic
243	C₂²	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
244	C₂²	C₂×#14273638	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	#16837157	1	2	yes	—
250	C₄	#16837157	1	2	yes	—
251	C₄	C₂×C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
252	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	#21421941	1	4	yes	direct product factor
257	C₂	C₂×#16837157	1	1	yes	characteristic
258	C₂	C₂×#16837173	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂²×#97×C₅₉	C₂
socle	C₂³×C₅₉	#95
prime radical	C₂⁴×C₅₉	C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂²×#97	D₅₉
hypocenter	C₅₉	C₂×#1281
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #21421941	C₂	direct (N × H)
29. C₂²×C₈×C₅₉	C₈	semidirect
62. C₂×C₈×C₅₉	C₂×C₈	semidirect
130. C₂×C₅₉	#1281	semidirect
146. C₅₉	C₂×#1281	semidirect
153. C₂²×C₈	C₅₉⋊C₈	semidirect
194. C₂×C₈	C₂×C₅₉⋊C₈	semidirect
256. C₂	#21421941	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837455	2
#21421941	2
C₂²×#97×C₅₉	2
C₂×#1281	59

Subgroups — C2×#21421941 #28478317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21421941 #28478317