Subgroups — C₂×#21426146 #28742939

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273673	C₂	1	2	yes	maximal
2	C₂×#16837782	C₂	1	1	yes	maximal, characteristic
3	#21426146	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#16837205	C₂	1	2	yes	maximal
6	C₂×C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	—	2	1	no	—
8	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
9	C₂×#14273727	—	2	1	no	—
10	C₂×#14273673	—	2	8	no	—
11	C₂×#14273673	C₂²	1	8	yes	—
12	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
13	#16837782	—	2	2	no	—
14	#16837782	C₂²	1	4	yes	—
15	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
16	#16837205	C₂²	1	8	yes	—
17	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
19	C₄×Q₁₆×C₅₉	—	2	2	no	—
20	C₄×Q₁₆×C₅₉	C₂²	1	4	yes	—
21	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#103×C₅₉	—	2	1	no	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₁₈	—	2	2	no	—
25	C₂×Dic₂₃₆	—	2	4	no	—
26	C₂×Dic₂₃₆	—	2	4	no	—
27	C₂×Dic₂₃₆	—	2	4	no	—
28	C₂×Dic₂₃₆	—	2	2	no	—
29	C₂×Dic₂₃₆	—	4	4	no	—
30	C₂×Dic₂₃₆	C₂³	1	4	yes	—
31	#14273727	—	2	4	no	—
32	#14273727	—	4	2	no	—
33	C₂×C₄⋊Dic₅₉	—	4	1	no	—
34	C₂×C₄⋊Dic₅₉	—	4	1	no	—
35	#14273673	—	2	16	no	—
36	C₂×C₅₉⋊C₁₆	—	2	4	no	—
37	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
38	C₂²×Q₈×C₅₉	—	2	1	no	—
39	C₂×Q₁₆×C₅₉	—	2	1	no	—
40	C₂×Q₁₆×C₅₉	—	2	2	no	—
41	C₂×Q₁₆×C₅₉	D₄	1	4	yes	—
42	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₉	—	2	4	no	—
46	C₄×Q₈×C₅₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
49	C₂×C₄²×C₅₉	—	4	1	no	—
50	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
51	#107×C₅₉	—	2	1	no	—
52	#107×C₅₉	C₂³	1	2	yes	—
53	#103×C₅₉	—	2	4	no	—
54	C₄×C₈×C₅₉	D₄	1	2	yes	—
55	C₄×C₈×C₅₉	C₂³	1	2	yes	—
56	C₂×Q₈×C₅₉	—	2	4	no	—
57	C₂×Q₈×C₅₉	—	2	2	no	—
58	C₂×Q₈×C₅₉	—	4	1	no	—
59	C₂×Q₈×C₅₉	—	4	2	no	—
60	C₂²×C₄×C₅₉	—	4	1	no	—
61	C₂²×C₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	4	2	no	—
66	C₂×Dic₁₁₈	—	2	4	no	—
67	C₂×Dic₁₁₈	—	4	4	no	—
68	C₂×Dic₁₁₈	—	4	4	no	—
69	C₂×Dic₁₁₈	—	4	2	no	—
70	Q₁₆×C₅₉	—	2	4	no	—
71	Q₁₆×C₅₉	—	2	2	no	—
72	Q₁₆×C₅₉	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₉	#41	1	2	yes	—
78	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	2	no	—
80	C₄⋊C₄×C₅₉	—	4	1	no	—
81	C₄⋊C₄×C₅₉	—	4	4	no	—
82	C₄²×C₅₉	—	4	4	no	—
83	C₄²×C₅₉	D₈	1	2	yes	—
84	C₄²×C₅₉	C₂×D₄	1	2	yes	—
85	C₄⋊Dic₅₉	—	4	4	no	—
86	C₄⋊Dic₅₉	—	4	4	no	—
87	Dic₂₃₆	—	2	4	no	—
88	Dic₂₃₆	—	2	4	no	—
89	Dic₂₃₆	—	4	8	no	—
90	Dic₂₃₆	—	4	4	no	—
91	C₅₉⋊C₁₆	—	2	4	no	—
92	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₅₉	—	2	4	no	—
94	Q₈×C₅₉	—	4	2	no	—
95	Q₈×C₅₉	—	4	4	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	4	4	no	—
99	C₂×C₄×C₅₉	—	4	2	no	—
100	C₂×C₄×C₅₉	—	8	2	no	—
101	C₂×C₄×C₅₉	Q₃₂	1	4	yes	complemented
102	C₂×C₄×C₅₉	#121	1	2	yes	—
103	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₅₉	—	4	4	no	—
109	C₂×Dic₅₉	—	8	4	no	—
110	Dic₁₁₈	—	4	4	no	—
111	Dic₁₁₈	—	4	4	no	—
112	Dic₁₁₈	—	8	4	no	—
113	C₈×C₅₉	—	2	2	no	—
114	C₈×C₅₉	#121	1	2	yes	—
115	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
117	C₂×#2209	—	59	1	no	maximal, Sylow
118	C₂²×C₅₉	#459	1	2	yes	—
119	C₂²×C₅₉	C₂×Q₃₂	1	2	yes	—
120	C₂²×C₅₉	#510	1	2	yes	—
121	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₅₉	—	2	2	no	—
123	C₄×C₅₉	—	4	4	no	—
124	C₄×C₅₉	—	8	2	no	—
125	C₄×C₅₉	#459	1	2	yes	—
126	C₄×C₅₉	C₂×Q₃₂	1	4	yes	complemented
127	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
128	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
129	Dic₅₉	—	8	4	no	—
130	C₂²×Q₃₂	—	59	2	no	—
131	C₂×#494	—	59	1	no	—
132	C₂×C₄×Q₁₆	D₅₉	1	1	yes	characteristic
133	#2209	—	59	8	no	—
134	C₂×C₄⋊C₁₆	—	59	1	no	—
135	C₂×#358	—	59	2	no	—
136	C₂×C₅₉	#2209	1	4	yes	complemented
137	C₂×C₅₉	C₂×#459	1	1	yes	characteristic
138	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
139	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
140	C₂²×Q₁₆	—	59	2	no	—
141	C₂²×Q₁₆	—	118	1	no	—
142	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	—	118	1	no	—
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×Q₃₂	—	59	8	no	—
146	C₂×Q₃₂	—	118	8	no	—
147	C₂²×C₁₆	—	59	2	no	—
148	#494	—	59	4	no	—
149	#494	—	118	2	no	—
150	C₄×Q₁₆	—	2	2	no	—
151	C₄×Q₁₆	C₂×D₅₉	1	4	yes	—
152	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
153	C₂×#103	—	2	1	no	—
154	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
155	C₄⋊C₁₆	—	59	4	no	—
156	#358	—	59	8	no	—
157	C₅₉	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×Q₈	—	118	2	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	—	59	4	no	—
163	C₂×Q₁₆	—	118	4	no	—
164	C₂×Q₁₆	—	118	4	no	—
165	C₂×Q₁₆	—	118	2	no	—
166	C₂×Q₁₆	—	118	4	no	—
167	C₂×Q₁₆	—	236	4	no	—
168	C₂×Q₁₆	D₂₃₆	1	4	yes	—
169	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
170	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
172	C₄⋊Q₈	—	118	4	no	—
173	C₄⋊Q₈	—	236	2	no	—
174	C₄×Q₈	—	2	4	no	—
175	C₄×Q₈	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	236	1	no	—
179	C₂×C₄⋊C₄	—	236	1	no	—
180	C₂×C₄²	—	4	1	no	—
181	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
182	Q₃₂	—	118	16	no	—
183	C₂×C₁₆	—	59	4	no	—
184	C₂×C₁₆	—	118	4	no	—
185	#107	—	2	1	no	—
186	#107	C₂²×D₅₉	1	2	yes	—
187	#103	—	2	4	no	—
188	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
189	C₄×C₈	C₂²×D₅₉	1	2	yes	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	118	4	no	—
195	C₂×Q₈	—	236	4	no	—
196	C₂×Q₈	—	236	4	no	—
197	C₂×Q₈	—	236	2	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	236	2	no	—
201	C₂²×C₄	#11745669	1	1	yes	characteristic
202	C₂²×C₄	#11745669	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	2	no	—
206	Q₁₆	—	118	4	no	—
207	Q₁₆	—	118	4	no	—
208	Q₁₆	—	236	8	no	—
209	Q₁₆	—	236	4	no	—
210	Q₁₆	C₂×D₂₃₆	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
213	C₂×C₈	#11745685	1	2	yes	—
214	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
215	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	236	4	no	—
221	C₄⋊C₄	—	236	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	#11745669	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
225	C₁₆	—	118	4	no	—
226	C₂³	C₂×#11745669	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	472	4	no	—
231	Q₈	—	236	4	no	—
232	Q₈	—	236	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	472	4	no	—
239	C₂×C₄	—	236	4	no	—
240	C₂×C₄	#14273673	1	4	yes	complemented
241	C₂×C₄	#14273701	1	2	yes	—
242	C₂×C₄	C₂×#11745669	1	2	yes	—
243	C₂×C₄	C₂×#11745669	1	2	yes	—
244	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
245	C₂×C₄	#14273752	1	2	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#14273701	1	2	yes	—
249	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
250	C₈	C₂×#11745685	1	1	yes	characteristic
251	C₂²	#16837597	1	2	yes	—
252	C₂²	C₂×#14273673	1	2	yes	—
253	C₂²	#16837842	1	2	yes	—
254	C₂²	C₂²×#11745669	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₄	—	472	4	no	—
259	C₄	#16837597	1	2	yes	—
260	C₄	C₂×#14273673	1	4	yes	complemented
261	C₄	C₂×#14273701	1	1	yes	characteristic
262	C₄	C₂×#14273752	1	1	yes	characteristic
263	C₂	#21426146	1	4	yes	direct product factor
264	C₂	C₂×#16837597	1	1	yes	characteristic
265	C₂	C₂²×#14273673	1	1	yes	characteristic
266	C₂	C₂×#16837842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×Q₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂×C₄×Q₁₆	D₅₉
hypocenter	C₅₉	C₂×#2209
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #21426146	C₂	direct (N × H)
101. C₂×C₄×C₅₉	Q₃₂	semidirect
126. C₄×C₅₉	C₂×Q₃₂	semidirect
136. C₂×C₅₉	#2209	semidirect
157. C₅₉	C₂×#2209	semidirect
240. C₂×C₄	#14273673	semidirect
260. C₄	C₂×#14273673	semidirect
263. C₂	#21426146	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×#14273673	2
C₂×#16837782	2
#21426146	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837205	2
C₂×C₄×Q₁₆×C₅₉	2
C₂×#2209	59

Subgroups — C2×#21426146 #28742939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426146 #28742939