Subgroups — C₈×#14273733 #28554274

Order 15104.

922 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	C₈×#11745685	C₂	1	2	yes	maximal
2	C₄×#14273733	C₂	1	1	yes	maximal, characteristic
3	#21422379	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422378	C₂	1	2	yes	maximal, complemented
5	C₈×#11745664	C₂	1	2	yes	maximal, complemented
6	C₈×#11745659	C₂	1	4	yes	maximal, complemented
7	C₈²×D₅₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273733	C₄	1	2	yes	—
10	C₄×#11745685	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₁₈	C₄	1	4	yes	—
12	C₈×D₂₃₆	C₄	1	4	yes	—
13	C₈×C₂²⋊D₅₉	C₄	1	8	yes	—
14	#16837455	C₂²	1	2	yes	—
15	#16837453	C₂²	1	4	yes	—
16	#16837357	C₄	1	2	yes	—
17	#16837356	C₄	1	4	yes	—
18	C₄×#11745664	C₂²	1	1	yes	characteristic
19	C₄×#11745659	C₂²	1	2	yes	—
20	C₄×C₈×D₅₉	C₂²	1	4	yes	—
21	C₄×C₈×D₅₉	C₂²	1	2	yes	—
22	#16837124	C₂²	1	4	yes	—
23	#16837123	C₂²	1	2	yes	—
24	C₈×C₅₉⋊C₈	C₂²	1	4	yes	—
25	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₈²×C₅₉	C₂²	1	2	yes	—
29	C₈²×C₅₉	C₂²	1	1	yes	characteristic
30	C₈²×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
32	#14273733	C₈	1	16	yes	direct product factor
33	C₂×#11745664	C₂×C₄	1	2	yes	—
34	C₂×#11745659	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₅₉	C₂×C₄	1	8	yes	—
36	C₂×C₈×D₅₉	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
38	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
39	C₄×C₂²⋊D₅₉	C₂×C₄	1	4	yes	—
40	#14273686	C₂×C₄	1	1	yes	characteristic
41	#14273684	C₂×C₄	1	2	yes	—
42	C₄²×D₅₉	C₂³	1	2	yes	—
43	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₅₉⋊C₈	C₂×C₄	1	8	yes	—
45	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
46	(C₂×D₅₉)⋊C₈	C₂×C₄	1	8	yes	—
47	#14273638	C₂×C₄	1	2	yes	—
48	#14273637	C₂×C₄	1	4	yes	—
49	C₄×C₅₉⋊C₈	C₂³	1	4	yes	—
50	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
51	C₈×Dic₅₉	C₂×C₄	1	8	yes	—
52	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂³	1	2	yes	—
60	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₉	C₂³	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₉	C₄²	1	2	yes	—
65	#11745685	C₂×C₈	1	8	yes	—
66	C₂×D₂₃₆	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₅₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₁₈	C₄²	1	1	yes	characteristic
70	C₂×C₈×C₅₉	—	2	8	no	—
71	C₂×C₈×C₅₉	C₂×C₈	1	8	yes	complemented
72	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
76	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₂²⋊Dic₅₉	C₄²	1	2	yes	—
80	(C₂×D₅₉)⋊C₄	C₄²	1	4	yes	—
81	C₄⋊Dic₅₉	C₄²	1	2	yes	—
82	Dic₅₉⋊C₄	C₄²	1	4	yes	—
83	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
84	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
85	#11745664	C₂×C₈	1	8	yes	complemented
86	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
87	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
88	#11745659	C₂×C₈	1	16	yes	complemented
89	C₈×D₅₉	—	2	16	no	—
90	C₈×D₅₉	C₂×C₈	1	16	yes	complemented
91	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
92	C₂²×D₅₉	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₅₉	—	2	4	no	—
94	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
99	C₂²⋊D₅₉	C₄×C₈	1	8	yes	—
100	C₂×Dic₅₉	C₂×C₄²	1	4	yes	—
101	C₂×Dic₅₉	C₂×C₄²	1	2	yes	—
102	D₂₃₆	C₄×C₈	1	4	yes	—
103	C₄×D₅₉	—	2	8	no	—
104	C₄×D₅₉	C₂²×C₈	1	8	yes	—
105	Dic₁₁₈	C₄×C₈	1	4	yes	—
106	C₈×C₅₉	—	2	4	no	—
107	C₈×C₅₉	C₂²×C₈	1	4	yes	—
108	C₈×C₅₉	C₂²×C₈	1	4	yes	—
109	C₈×C₅₉	#131	1	8	yes	complemented
110	C₅₉⋊C₈	—	2	8	no	—
111	C₅₉⋊C₈	C₂²×C₈	1	8	yes	—
112	C₈×#131	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	—	2	2	no	—
114	C₂²×C₅₉	C₂×C₄×C₈	1	2	yes	—
115	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
116	C₂×D₅₉	—	2	4	no	—
117	C₂×D₅₉	C₂×C₄×C₈	1	4	yes	—
118	C₄×C₅₉	—	2	2	no	—
119	C₄×C₅₉	C₂×C₄×C₈	1	2	yes	—
120	C₄×C₅₉	C₂³×C₈	1	2	yes	—
121	C₄×C₅₉	C₂×#131	1	4	yes	—
122	Dic₅₉	—	2	4	no	—
123	Dic₅₉	C₂×C₄×C₈	1	4	yes	—
124	C₈×#41	—	59	2	no	—
125	C₄×#131	—	59	1	no	—
126	#1450	—	59	1	no	—
127	#1450	—	59	2	no	—
128	C₈×M₁₆	—	59	2	no	—
129	C₈×M₁₆	—	59	4	no	—
130	C₂×C₈²	—	59	2	no	—
131	C₂×C₈²	D₅₉	1	1	yes	characteristic, complemented
132	C₂×C₅₉	—	2	2	no	—
133	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
134	C₂×C₅₉	C₄×#131	1	2	yes	—
135	D₅₉	—	2	4	no	—
136	C₂×#131	—	59	2	no	—
137	C₄×#41	—	59	1	no	—
138	C₈×Q₈	—	59	4	no	—
139	C₈×D₄	—	59	8	no	—
140	C₈×D₄	—	59	4	no	—
141	#431	—	59	2	no	—
142	#431	—	59	4	no	—
143	#405	—	59	2	no	—
144	#405	—	59	4	no	—
145	C₄×M₁₆	—	59	1	no	—
146	C₄×M₁₆	—	59	2	no	—
147	C₂×C₄×C₈	—	59	2	no	—
148	C₂×C₄×C₈	—	59	4	no	—
149	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
151	#322	—	59	2	no	—
152	#322	—	59	4	no	—
153	C₈²	—	59	4	no	—
154	C₈²	—	59	2	no	—
155	C₈²	C₂×D₅₉	1	2	yes	—
156	C₈²	C₂×D₅₉	1	1	yes	characteristic
157	C₈²	C₂×D₅₉	1	1	yes	characteristic
158	C₅₉	C₈×#131	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	59	1	no	—
160	#131	—	59	16	no	—
161	C₂×M₁₆	—	59	2	no	—
162	C₂×M₁₆	—	59	4	no	—
163	C₂²×C₈	—	59	4	no	—
164	C₂²×C₈	—	59	8	no	—
165	C₂²×C₈	C₄×D₅₉	1	4	yes	—
166	C₂²×C₈	C₄×D₅₉	1	2	yes	—
167	C₄×Q₈	—	59	2	no	—
168	C₄×D₄	—	59	4	no	—
169	C₄×D₄	—	59	2	no	—
170	#117	—	59	1	no	—
171	#117	—	59	2	no	—
172	C₂×C₄²	—	59	2	no	—
173	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
174	C₄⋊C₈	—	59	8	no	—
175	C₄⋊C₈	—	59	4	no	—
176	C₂²⋊C₈	—	59	4	no	—
177	C₂²⋊C₈	—	59	8	no	—
178	#97	—	59	4	no	—
179	#97	—	59	2	no	—
180	C₄×C₈	—	59	4	no	—
181	C₄×C₈	—	59	8	no	—
182	C₄×C₈	—	59	2	no	—
183	C₄×C₈	—	59	4	no	—
184	C₄×C₈	C₄×D₅₉	1	2	yes	—
185	C₄×C₈	C₄×D₅₉	1	4	yes	—
186	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₅₉	1	2	yes	—
188	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₅₉	1	2	yes	—
190	#41	—	59	8	no	—
191	C₂×Q₈	—	59	1	no	—
192	C₂×D₄	—	59	2	no	—
193	C₂×D₄	—	59	1	no	—
194	C₂²×C₄	—	59	2	no	—
195	C₂²×C₄	—	59	4	no	—
196	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
198	M₁₆	—	59	8	no	—
199	M₁₆	—	59	16	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	59	4	no	—
202	C₂×C₈	—	59	16	no	—
203	C₂×C₈	—	59	8	no	—
204	C₂×C₈	—	118	16	no	—
205	C₂×C₈	C₈×D₅₉	1	8	yes	complemented
206	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
207	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
210	C₄⋊C₄	—	59	4	no	—
211	C₄⋊C₄	—	59	2	no	—
212	C₂²⋊C₄	—	59	2	no	—
213	C₂²⋊C₄	—	59	4	no	—
214	C₄²	—	59	2	no	—
215	C₄²	—	59	4	no	—
216	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
217	C₄²	C₂×C₄×D₅₉	1	2	yes	—
218	C₄²	C₂³×D₅₉	1	1	yes	characteristic
219	C₂³	—	59	2	no	—
220	C₂³	C₄²×D₅₉	1	1	yes	characteristic
221	Q₈	—	59	4	no	—
222	D₄	—	59	8	no	—
223	D₄	—	59	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	59	2	no	—
226	C₂×C₄	—	59	8	no	—
227	C₂×C₄	—	59	4	no	—
228	C₂×C₄	—	118	8	no	—
229	C₂×C₄	C₄²×D₅₉	1	2	yes	—
230	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₅₉	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₅₉	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	59	8	no	—
236	C₈	—	118	8	no	—
237	C₈	C₂×C₈×D₅₉	1	4	yes	—
238	C₈	C₂×C₈×D₅₉	1	4	yes	—
239	C₈	#14273733	1	8	yes	direct product factor
240	C₂²	—	2	2	no	—
241	C₂²	—	59	4	no	—
242	C₂²	—	118	4	no	—
243	C₂²	C₄×C₈×D₅₉	1	2	yes	—
244	C₂²	C₂×C₄²×D₅₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	59	4	no	—
247	C₄	—	118	4	no	—
248	C₄	C₄×C₈×D₅₉	1	2	yes	—
249	C₄	C₂²×C₈×D₅₉	1	2	yes	—
250	C₄	C₂×#14273733	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	118	4	no	—
253	C₂	C₂×C₄×C₈×D₅₉	1	1	yes	characteristic
254	C₂	C₄×#14273733	1	2	yes	—

	subgroup	G/H
center	C₈²	C₂×D₅₉
commutator	C₂×C₅₉	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×C₄²
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	C₈×#131
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #21422379	C₂	semidirect
4. #21422378	C₂	semidirect
5. C₈×#11745664	C₂	semidirect
6. C₈×#11745659	C₂	semidirect
7. C₈²×D₅₉	C₂	semidirect
8. C₂×C₈²×C₅₉	C₂	semidirect
32. #14273733	C₈	direct (N × H)
71. C₂×C₈×C₅₉	C₂×C₈	semidirect
85. #11745664	C₂×C₈	semidirect
88. #11745659	C₂×C₈	semidirect
90. C₈×D₅₉	C₂×C₈	semidirect
109. C₈×C₅₉	#131	semidirect
131. C₂×C₈²	D₅₉	semidirect
158. C₅₉	C₈×#131	semidirect
205. C₂×C₈	C₈×D₅₉	semidirect
239. C₈	#14273733	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#11745685	2
C₄×#14273733	2
#21422379	2
#21422378	2
C₈×#11745664	2
C₈×#11745659	2
C₈²×D₅₉	2
C₂×C₈²×C₅₉	2
C₈×#131	59

Subgroups — C8×#14273733 #28554274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14273733 #28554274