Subgroups — C₈×#14273719 #28596992

Order 15104.

742 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273719	C₂	1	1	yes	maximal, characteristic
2	#21424814	C₂	1	2	yes	maximal
3	#21424653	C₂	1	4	yes	maximal
4	#21424213	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₅₉⋊C₄	C₂	1	4	yes	maximal
6	C₈×C₄⋊Dic₅₉	C₂	1	2	yes	maximal
7	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14273719	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×Dic₅₉⋊C₈	C₂²	1	4	yes	—
11	C₂×C₈×Dic₅₉	C₂²	1	4	yes	—
12	#16837300	C₄	1	2	yes	—
13	#16837286	C₄	1	4	yes	—
14	#16837268	C₄	1	1	yes	characteristic
15	C₄×Dic₅₉⋊C₄	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
17	#16837157	C₂²	1	4	yes	—
18	#16837156	C₂²	1	4	yes	—
19	#16837156	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄³×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₅₉⋊C₈	C₂³	1	4	yes	—
26	#14273719	C₈	1	8	yes	direct product factor
27	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₅₉	C₂³	1	4	yes	—
32	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
33	Dic₅₉⋊C₈	—	2	8	no	—
34	C₈×Dic₅₉	—	2	8	no	—
35	C₅₉⋊#95	C₂×C₄	1	4	yes	—
36	C₅₉⋊#95	C₂×C₄	1	4	yes	—
37	C₅₉⋊#95	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₉	—	2	4	no	—
47	C₄×C₈×C₅₉	—	2	4	no	—
48	C₄×C₈×C₅₉	—	2	2	no	—
49	C₄×C₈×C₅₉	Q₈	1	4	yes	—
50	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₅₉	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₅₉	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₅₉	—	2	2	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₅₉	#41	1	4	yes	—
63	C₂×C₈×C₅₉	#41	1	4	yes	—
64	C₄²×C₅₉	—	2	4	no	—
65	C₄²×C₅₉	—	2	2	no	—
66	C₄²×C₅₉	—	2	2	no	—
67	C₄²×C₅₉	—	2	2	no	—
68	C₄²×C₅₉	C₂×C₈	1	4	yes	—
69	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
70	C₄²×C₅₉	#41	1	2	yes	—
71	C₄⋊Dic₅₉	—	2	4	no	—
72	C₄⋊Dic₅₉	C₂×C₈	1	8	yes	—
73	Dic₅₉⋊C₄	—	2	8	no	—
74	Dic₅₉⋊C₄	C₂×C₈	1	16	yes	—
75	C₄×Dic₅₉	—	2	4	no	—
76	C₄×Dic₅₉	—	2	4	no	—
77	C₂×C₅₉⋊C₈	—	2	4	no	—
78	C₂×C₅₉⋊C₈	—	2	4	no	—
79	C₂×C₅₉⋊C₈	—	2	4	no	—
80	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	2	2	no	—
86	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₉	#131	1	4	yes	—
91	C₂×C₄×C₅₉	#131	1	2	yes	—
92	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
94	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
96	C₂×Dic₅₉	—	2	4	no	—
97	C₂×Dic₅₉	—	2	4	no	—
98	C₂×Dic₅₉	—	2	4	no	—
99	C₂×Dic₅₉	—	2	4	no	—
100	C₂×Dic₅₉	—	2	4	no	—
101	C₂×Dic₅₉	C₂²×C₈	1	8	yes	—
102	C₈×C₅₉	—	2	4	no	—
103	C₈×C₅₉	—	2	2	no	—
104	C₈×C₅₉	#125	1	4	yes	complemented
105	C₅₉⋊C₈	—	4	4	no	—
106	C₈×#125	—	59	1	no	maximal, Sylow
107	C₂²×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₂²×C₅₉	C₄×#41	1	2	yes	—
109	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
110	C₂²×C₅₉	C₂×#131	1	1	yes	characteristic
111	C₂²×C₅₉	C₂×#131	1	2	yes	—
112	C₄×C₅₉	—	2	2	no	—
113	C₄×C₅₉	—	2	4	no	—
114	C₄×C₅₉	—	2	4	no	—
115	C₄×C₅₉	C₈×Q₈	1	4	yes	—
116	C₄×C₅₉	C₂×#125	1	1	yes	characteristic
117	C₄×C₅₉	#529	1	2	yes	—
118	C₄×C₅₉	#533	1	1	yes	characteristic
119	Dic₅₉	—	2	8	no	—
120	Dic₅₉	—	4	4	no	—
121	C₄×#125	—	59	1	no	—
122	#1939	—	59	2	no	—
123	#1916	—	59	4	no	—
124	#1831	—	59	1	no	—
125	C₈×C₄⋊C₄	—	59	2	no	—
126	C₈×C₄⋊C₄	—	59	4	no	—
127	C₄²×C₈	D₅₉	1	1	yes	characteristic
128	C₂×C₅₉	C₄×#125	1	1	yes	characteristic
129	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
130	C₂×C₅₉	#2960	1	1	yes	characteristic
131	C₂×C₅₉	C₈×#41	1	2	yes	—
132	C₂×C₅₉	#2966	1	2	yes	—
133	C₂×#125	—	59	1	no	—
134	C₂×C₄⋊C₈	—	59	4	no	—
135	C₂×C₄⋊C₈	—	59	2	no	—
136	C₂×C₄×C₈	—	59	4	no	—
137	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
140	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
141	#389	—	59	2	no	—
142	#387	—	59	4	no	—
143	#382	—	59	1	no	—
144	C₄×C₄⋊C₄	—	59	4	no	—
145	C₄×C₄⋊C₄	—	59	2	no	—
146	C₄³	C₂×D₅₉	1	1	yes	characteristic
147	#336	—	59	4	no	—
148	#336	—	59	4	no	—
149	#336	—	59	4	no	—
150	C₅₉	C₈×#125	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₈	—	59	4	no	—
152	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
154	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
155	#125	—	59	8	no	—
156	C₂×C₄⋊C₄	—	59	4	no	—
157	C₂×C₄⋊C₄	—	59	2	no	—
158	C₂×C₄⋊C₄	—	59	2	no	—
159	C₂×C₄⋊C₄	—	59	4	no	—
160	C₂×C₄²	—	59	4	no	—
161	C₂×C₄²	C₄×D₅₉	1	2	yes	—
162	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
165	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
166	C₄⋊C₈	—	118	8	no	—
167	C₄⋊C₈	—	118	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	118	8	no	—
172	C₄×C₈	Dic₁₁₈	1	4	yes	—
173	#95	—	59	4	no	—
174	#95	—	59	4	no	—
175	#95	—	59	4	no	—
176	C₂²×C₄	—	59	4	no	—
177	C₂²×C₄	—	59	4	no	—
178	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
181	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	118	4	no	—
187	C₂×C₈	—	118	4	no	—
188	C₂×C₈	—	118	4	no	—
189	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
190	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
191	C₂×C₈	#11745685	1	4	yes	—
192	C₂×C₈	#11745685	1	4	yes	—
193	C₄⋊C₄	—	59	16	no	—
194	C₄⋊C₄	—	59	8	no	—
195	C₄⋊C₄	—	118	8	no	—
196	C₄⋊C₄	—	118	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	118	4	no	—
202	C₄²	—	118	4	no	—
203	C₄²	C₈×D₅₉	1	4	yes	—
204	C₄²	C₂×Dic₁₁₈	1	2	yes	—
205	C₄²	#11745685	1	2	yes	—
206	C₂³	C₂²×C₄×D₅₉	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	59	8	no	—
213	C₂×C₄	—	118	4	no	—
214	C₂×C₄	—	118	4	no	—
215	C₂×C₄	—	118	4	no	—
216	C₂×C₄	—	118	4	no	—
217	C₂×C₄	—	118	4	no	—
218	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
219	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
220	C₂×C₄	C₂×C₈×D₅₉	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
222	C₂×C₄	#14273733	1	4	yes	—
223	C₂×C₄	#14273733	1	2	yes	—
224	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
225	C₂×C₄	C₂×#11745685	1	2	yes	—
226	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
227	C₂×C₄	C₂×#11745685	1	2	yes	—
228	C₈	—	2	4	no	—
229	C₈	—	2	2	no	—
230	C₈	—	236	4	no	—
231	C₈	#14273719	1	4	yes	direct product factor
232	C₂²	C₂×C₄×Dic₁₁₈	1	1	yes	characteristic
233	C₂²	C₄×#11745685	1	2	yes	—
234	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
235	C₂²	C₂×#14273733	1	1	yes	characteristic
236	C₂²	C₂×#14273733	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	118	8	no	—
241	C₄	—	236	4	no	—
242	C₄	C₈×Dic₁₁₈	1	4	yes	—
243	C₄	C₂×#14273719	1	1	yes	characteristic
244	C₄	#16837936	1	2	yes	—
245	C₄	#16837959	1	1	yes	characteristic
246	C₂	C₄×#14273719	1	1	yes	characteristic
247	C₂	C₂×C₈×Dic₁₁₈	1	1	yes	characteristic
248	C₂	#21433030	1	1	yes	characteristic
249	C₂	C₈×#11745685	1	2	yes	—
250	C₂	#21433071	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂³×C₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	C₈×#125
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. #14273719	C₈	direct (N × H)
104. C₈×C₅₉	#125	semidirect
150. C₅₉	C₈×#125	semidirect
231. C₈	#14273719	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14273719	2
#21424814	2
#21424653	2
#21424213	2
C₈×Dic₅₉⋊C₄	2
C₈×C₄⋊Dic₅₉	2
C₄²×C₈×C₅₉	2
C₈×#125	59

Subgroups — C8×#14273719 #28596992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14273719 #28596992