Subgroups — C₈×#14273645 #28479253

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273645	C₂	1	1	yes	maximal, characteristic
3	#21422419	C₂	1	1	yes	maximal, characteristic
4	C₈×#11745664	C₂	1	1	yes	maximal, characteristic
5	#21421947	C₂	1	1	yes	maximal, characteristic
6	#21421945	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₁₈	—	2	1	no	—
10	C₈×Dic₁₁₈	—	2	1	no	—
11	C₈×Dic₁₁₈	—	2	1	no	—
12	C₈×Dic₁₁₈	—	2	1	no	—
13	#16837455	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₉⋊C₈	C₄	1	2	yes	—
16	C₂×#14273645	C₄	1	2	yes	—
17	C₄×#11745664	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
19	#16837170	C₄	1	2	yes	—
20	#16837127	C₄	1	4	yes	—
21	#16837124	—	2	1	no	—
22	C₈×C₅₉⋊C₈	—	2	1	no	—
23	C₅₉×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#322	—	2	1	no	—
27	C₈²×C₅₉	—	2	1	no	—
28	C₂²×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
29	C₂×#11745664	C₂×C₄	1	2	yes	—
30	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₁₈	—	2	1	no	—
32	C₄×Dic₁₁₈	—	2	1	no	—
33	C₄×Dic₁₁₈	—	2	1	no	—
34	C₄×Dic₁₁₈	—	2	1	no	—
35	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
38	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
39	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
40	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
41	Dic₅₉⋊C₈	—	2	2	no	—
42	Dic₅₉⋊C₈	—	2	2	no	—
43	#14273645	C₈	1	8	yes	direct product factor
44	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
45	#14273638	—	2	1	no	—
46	C₄×C₅₉⋊C₈	—	2	1	no	—
47	C₄×C₅₉⋊C₈	—	2	1	no	—
48	C₈×Dic₅₉	—	2	2	no	—
49	C₈×Dic₅₉	—	2	2	no	—
50	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₉	—	2	1	no	—
55	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
56	#97×C₅₉	—	2	1	no	—
57	C₄×C₈×C₅₉	—	2	1	no	—
58	C₄×C₈×C₅₉	—	2	1	no	—
59	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₉	C₄²	1	2	yes	—
67	C₂²×Dic₅₉	C₄²	1	2	yes	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	—	2	1	no	—
70	C₂×Dic₁₁₈	—	2	2	no	—
71	C₂×Dic₁₁₈	C₂×C₈	1	4	yes	—
72	M₁₆×C₅₉	—	2	2	no	—
73	M₁₆×C₅₉	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	—	4	1	no	—
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₉	#41	1	1	yes	characteristic
85	C₄²×C₅₉	#41	1	1	yes	characteristic
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	Dic₅₉⋊C₄	—	2	2	no	—
89	Dic₅₉⋊C₄	—	2	2	no	—
90	C₄×Dic₅₉	—	2	2	no	—
91	C₄×Dic₅₉	—	2	2	no	—
92	#11745664	—	2	2	no	—
93	#11745664	C₂×C₈	1	4	yes	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	—	2	2	no	—
97	C₂×C₅₉⋊C₈	—	4	2	no	—
98	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	4	1	no	—
102	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₉	#131	1	2	yes	—
110	C₂×Dic₅₉	—	2	2	no	—
111	C₂×Dic₅₉	—	2	2	no	—
112	C₂×Dic₅₉	—	2	2	no	—
113	C₂×Dic₅₉	—	4	2	no	—
114	C₂×Dic₅₉	C₄×C₈	1	4	yes	—
115	Dic₁₁₈	—	2	4	no	—
116	Dic₁₁₈	—	2	4	no	—
117	C₈×C₅₉	—	2	1	no	—
118	C₈×C₅₉	—	2	1	no	—
119	C₈×C₅₉	—	2	1	no	—
120	C₈×C₅₉	—	2	4	no	—
121	C₈×C₅₉	—	4	2	no	—
122	C₈×C₅₉	#101	1	2	yes	complemented
123	C₅₉⋊C₈	—	2	4	no	—
124	C₅₉⋊C₈	—	4	2	no	—
125	C₅₉⋊C₈	—	4	2	no	—
126	C₈×#101	—	59	1	no	maximal, Sylow
127	C₂²×C₅₉	—	2	1	no	—
128	C₂²×C₅₉	—	2	1	no	—
129	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₅₉	#405	1	1	yes	characteristic
132	C₄×C₅₉	—	2	1	no	—
133	C₄×C₅₉	—	2	1	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
136	C₄×C₅₉	#413	1	1	yes	characteristic
137	C₄×C₅₉	C₈×D₄	1	2	yes	—
138	C₄×C₅₉	C₈×D₄	1	2	yes	—
139	Dic₅₉	—	2	4	no	—
140	Dic₅₉	—	4	2	no	—
141	C₂×C₈×Q₈	—	59	1	no	—
142	C₄×#101	—	59	1	no	—
143	#1461	—	59	1	no	—
144	C₈×M₁₆	—	59	1	no	—
145	C₈×M₁₆	D₅₉	1	1	yes	characteristic
146	#1283	—	59	1	no	—
147	#1283	—	59	1	no	—
148	C₂×C₅₉	—	2	1	no	—
149	C₂×C₅₉	—	2	1	no	—
150	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₅₉	C₄×#101	1	1	yes	characteristic
152	C₂×C₅₉	#1757	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	59	1	no	—
154	C₈×Q₈	—	118	1	no	—
155	C₈×Q₈	—	118	1	no	—
156	C₈×Q₈	—	118	1	no	—
157	C₈×Q₈	—	118	1	no	—
158	#431	—	59	1	no	—
159	#431	C₂×D₅₉	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	59	2	no	—
161	C₂×C₄⋊C₈	—	59	1	no	—
162	C₂×#101	—	59	2	no	—
163	C₄×M₁₆	—	59	1	no	—
164	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
165	C₂×C₄×C₈	—	59	2	no	—
166	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
167	#343	—	59	2	no	—
168	#324	—	59	4	no	—
169	#322	—	2	1	no	—
170	#322	—	118	1	no	—
171	C₈²	—	2	1	no	—
172	C₈²	—	118	1	no	—
173	C₅₉	C₈×#101	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	59	1	no	—
175	C₂×M₁₆	—	59	2	no	—
176	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
177	C₂²×C₈	—	59	2	no	—
178	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
180	C₄×Q₈	—	118	1	no	—
181	C₄×Q₈	—	118	1	no	—
182	C₄×Q₈	—	118	1	no	—
183	C₄×Q₈	—	118	1	no	—
184	C₂×C₄⋊C₄	—	59	2	no	—
185	C₂×C₄⋊C₄	—	59	1	no	—
186	C₂×C₄²	—	59	2	no	—
187	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	118	2	no	—
190	C₄⋊C₈	—	118	2	no	—
191	C₄⋊C₈	—	118	1	no	—
192	C₄⋊C₈	—	118	1	no	—
193	C₄⋊C₈	—	118	1	no	—
194	#101	—	59	8	no	—
195	C₂²⋊C₈	—	59	2	no	—
196	C₂²⋊C₈	C₄×D₅₉	1	2	yes	—
197	#97	—	2	1	no	—
198	#97	—	118	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	118	2	no	—
202	C₄×C₈	—	118	2	no	—
203	C₄×C₈	—	118	1	no	—
204	C₄×C₈	—	118	1	no	—
205	C₄×C₈	D₂₃₆	1	1	yes	characteristic
206	C₄×C₈	D₂₃₆	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
209	C₂×Q₈	—	59	4	no	—
210	C₂×Q₈	—	118	1	no	—
211	C₂×Q₈	—	118	1	no	—
212	C₂×Q₈	—	118	2	no	—
213	C₂²×C₄	—	59	2	no	—
214	C₂²×C₄	—	59	2	no	—
215	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	59	4	no	—
220	M₁₆	—	118	2	no	—
221	M₁₆	C₈×D₅₉	1	4	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	118	2	no	—
227	C₂×C₈	—	118	2	no	—
228	C₂×C₈	—	118	2	no	—
229	C₂×C₈	—	236	2	no	—
230	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
231	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
232	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
233	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
234	C₄⋊C₄	—	118	2	no	—
235	C₄⋊C₄	—	118	1	no	—
236	C₄⋊C₄	—	118	2	no	—
237	C₄⋊C₄	—	118	1	no	—
238	C₄²	—	118	2	no	—
239	C₄²	—	118	2	no	—
240	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
241	C₄²	#11745685	1	1	yes	characteristic
242	C₄²	#11745685	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
244	C₂³	C₄²×D₅₉	1	1	yes	characteristic
245	Q₈	—	118	4	no	—
246	Q₈	—	118	4	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	59	4	no	—
251	C₂×C₄	—	118	2	no	—
252	C₂×C₄	—	118	2	no	—
253	C₂×C₄	—	118	2	no	—
254	C₂×C₄	—	236	2	no	—
255	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#14273684	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
262	C₂×C₄	#14273733	1	2	yes	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	118	4	no	—
269	C₈	—	236	2	no	—
270	C₈	—	236	2	no	—
271	C₈	#14273645	1	2	yes	direct product factor
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
275	C₂²	C₄×C₈×D₅₉	1	1	yes	characteristic
276	C₂²	#16837356	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	118	4	no	—
281	C₄	—	236	2	no	—
282	C₄	C₂×#14273645	1	1	yes	characteristic
283	C₄	#16837384	1	1	yes	characteristic
284	C₄	C₈×C₂²⋊D₅₉	1	2	yes	—
285	C₄	C₈×D₂₃₆	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₈×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
289	C₂	C₄×#14273645	1	1	yes	characteristic
290	C₂	#21423897	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₅₉)⋊C₄
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₈×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₈×M₁₆	D₅₉
hypocenter	C₅₉	C₈×#101
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
43. #14273645	C₈	direct (N × H)
122. C₈×C₅₉	#101	semidirect
173. C₅₉	C₈×#101	semidirect
271. C₈	#14273645	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×C₈×Dic₁₁₈	2
C₄×#14273645	2
#21422419	2
C₈×#11745664	2
#21421947	2
#21421945	2
C₈×M₁₆×C₅₉	2
C₈×#101	59

Subgroups — C8×#14273645 #28479253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14273645 #28479253