Subgroups — C₂×#10598295 #12014136

Order 1280.

726 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂³×#672	C₂	1	1	yes	maximal, characteristic
4	#10598295	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
7	C₂³×#168	C₄	1	1	yes	characteristic
8	C₂²×C₅×C₁₆	—	2	2	no	—
9	C₂²×C₅×C₁₆	—	2	2	no	—
10	C₂²×C₅×C₁₆	C₄	1	4	yes	complemented
11	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
12	C₂×C₈×Dic₅	—	2	2	no	—
13	C₂×C₈×Dic₅	—	2	2	no	—
14	C₂×C₈×Dic₅	C₄	1	2	yes	—
15	C₂×C₈×Dic₅	C₂²	1	2	yes	—
16	C₂²×#672	—	2	2	no	—
17	C₂²×#672	—	2	2	no	—
18	C₂²×#672	C₄	1	4	yes	complemented
19	C₂²×#672	C₂²	1	2	yes	—
20	C₂×#1374	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₅	C₈	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	2	yes	—
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	D₄	1	2	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₅	—	2	2	no	—
32	C₂×C₄×Dic₅	—	2	2	no	—
33	C₂×C₄×Dic₅	C₈	1	2	yes	—
34	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
35	C₂²×#168	—	2	2	no	—
36	C₂²×#168	—	2	2	no	—
37	C₂²×#168	C₈	1	4	yes	—
38	C₂²×#168	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₁₆	—	2	8	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
43	C₈×Dic₅	—	2	4	no	—
44	C₈×Dic₅	—	2	2	no	—
45	C₈×Dic₅	—	2	2	no	—
46	C₂×#672	—	2	8	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	C₂×C₄	1	4	yes	complemented
50	C₂³×C₁₆	—	5	1	no	—
51	C₂³×C₁₆	D₅	1	1	yes	characteristic
52	C₂²×C₄×C₈	—	5	1	no	—
53	#1374	—	5	4	no	—
54	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₄²	1	2	yes	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	M₁₆	1	2	yes	—
61	C₂²×C₄×C₅	M₁₆	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅	—	2	4	no	—
64	C₂²×Dic₅	—	2	4	no	—
65	C₂²×Dic₅	C₁₆	1	8	yes	complemented
66	C₂²×Dic₅	C₂×C₈	1	4	yes	—
67	C₂×C₅×C₈	C₄²	1	2	yes	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
71	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₄×Dic₅	—	2	4	no	—
81	C₄×Dic₅	—	2	2	no	—
82	C₄×Dic₅	—	2	2	no	—
83	C₂×#168	—	2	8	no	—
84	C₂×#168	—	2	2	no	—
85	C₂×#168	—	2	2	no	—
86	C₂×#168	C₂×C₈	1	4	yes	—
87	C₅×C₁₆	—	2	4	no	—
88	#672	—	2	4	no	—
89	C₂³×C₈	—	5	1	no	—
90	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₄²	—	5	1	no	—
92	C₂²×C₁₆	—	2	2	no	—
93	C₂²×C₁₆	—	2	2	no	—
94	C₂²×C₁₆	—	5	4	no	—
95	C₂²×C₁₆	—	5	2	no	—
96	C₂²×C₁₆	—	10	2	no	—
97	C₂²×C₁₆	—	10	2	no	—
98	C₂²×C₁₆	Dic₅	1	4	yes	complemented
99	C₂²×C₁₆	C₂×D₅	1	2	yes	—
100	C₂×C₄×C₈	—	5	2	no	—
101	C₂×C₄×C₈	—	5	2	no	—
102	C₂×C₄×C₈	—	10	2	no	—
103	C₂×C₄×C₈	—	10	2	no	—
104	C₂³×C₅	C₄×C₈	1	2	yes	—
105	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
106	C₂³×C₅	C₄⋊C₈	1	2	yes	—
107	C₂³×C₅	C₂×C₁₆	1	4	yes	—
108	C₂³×C₅	M₃₂	1	4	yes	—
109	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	#95	1	4	yes	—
111	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₅	#97	1	4	yes	—
113	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
115	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
116	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
117	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
123	C₂×Dic₅	—	2	16	no	—
124	C₂×Dic₅	—	2	4	no	—
125	C₂×Dic₅	—	2	4	no	—
126	C₂×Dic₅	C₂×C₁₆	1	8	yes	complemented
127	C₅×C₈	#95	1	4	yes	—
128	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
132	#168	—	2	4	no	—
133	C₂³×C₄	—	5	2	no	—
134	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
135	C₂²×C₈	—	5	4	no	—
136	C₂²×C₈	—	5	2	no	—
137	C₂²×C₈	—	10	2	no	—
138	C₂²×C₈	—	10	2	no	—
139	C₂²×C₈	Dic₁₀	1	2	yes	—
140	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₅	1	2	yes	—
142	C₂²×C₈	D₂₀	1	2	yes	—
143	C₂²×C₈	C₂×Dic₅	1	2	yes	—
144	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
145	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
146	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₄²	—	5	2	no	—
148	C₂×C₄²	—	5	2	no	—
149	C₂×C₄²	—	10	2	no	—
150	C₂×C₄²	—	10	2	no	—
151	C₂×C₁₆	—	2	8	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	5	4	no	—
155	C₂×C₁₆	—	10	8	no	—
156	C₂×C₁₆	—	10	2	no	—
157	C₂×C₁₆	—	10	2	no	—
158	C₂×C₁₆	C₂×Dic₅	1	4	yes	complemented
159	C₄×C₈	—	10	4	no	—
160	C₄×C₈	—	10	2	no	—
161	C₄×C₈	—	10	2	no	—
162	C₂²×C₅	#336	1	4	yes	—
163	C₂²×C₅	C₄×C₁₆	1	4	yes	—
164	C₂²×C₅	#346	1	4	yes	—
165	C₂²×C₅	C₂²⋊C₁₆	1	8	yes	—
166	C₂²×C₅	C₄⋊C₁₆	1	8	yes	—
167	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
171	C₂²×C₅	C₂×M₃₂	1	2	yes	—
172	C₄×C₅	#336	1	4	yes	—
173	C₄×C₅	C₂×#95	1	1	yes	characteristic
174	C₄×C₅	C₂×#97	1	1	yes	characteristic
175	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
177	Dic₅	—	2	8	no	—
178	C₂⁴	C₈×D₅	1	1	yes	characteristic
179	C₂²×C₄	—	5	8	no	—
180	C₂²×C₄	—	5	4	no	—
181	C₂²×C₄	—	10	4	no	—
182	C₂²×C₄	—	10	4	no	—
183	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
184	C₂²×C₄	C₈×D₅	1	2	yes	—
185	C₂²×C₄	#676	1	2	yes	—
186	C₂²×C₄	#676	1	2	yes	—
187	C₂²×C₄	C₄×Dic₅	1	2	yes	—
188	C₂²×C₄	#683	1	2	yes	—
189	C₂²×C₄	#685	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
191	C₂×C₈	—	5	4	no	—
192	C₂×C₈	—	10	8	no	—
193	C₂×C₈	—	10	2	no	—
194	C₂×C₈	—	10	2	no	—
195	C₂×C₈	C₄×Dic₅	1	2	yes	—
196	C₂×C₈	#683	1	2	yes	—
197	C₂×C₈	#683	1	4	yes	—
198	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
199	C₂×C₈	#685	1	2	yes	—
200	C₂×C₈	#685	1	4	yes	—
201	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
202	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
204	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
205	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₄²	—	10	4	no	—
209	C₄²	—	10	2	no	—
210	C₄²	—	10	2	no	—
211	C₁₆	—	2	4	no	—
212	C₁₆	—	10	4	no	—
213	C₂×C₅	#1374	1	8	yes	complemented
214	C₂×C₅	C₂×#336	1	1	yes	characteristic
215	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
216	C₂×C₅	C₂×#346	1	1	yes	characteristic
217	C₂×C₅	C₂×C₂²⋊C₁₆	1	2	yes	—
218	C₂×C₅	C₂×C₄⋊C₁₆	1	2	yes	—
219	C₂³	D₅×C₁₆	1	4	yes	—
220	C₂³	#3930	1	4	yes	—
221	C₂³	C₈×Dic₅	1	2	yes	—
222	C₂³	#3946	1	2	yes	—
223	C₂³	#3952	1	2	yes	—
224	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
225	C₂×C₄	—	5	8	no	—
226	C₂×C₄	—	10	4	no	—
227	C₂×C₄	—	10	4	no	—
228	C₂×C₄	—	10	16	no	—
229	C₂×C₄	C₈×Dic₅	1	2	yes	—
230	C₂×C₄	#3946	1	2	yes	—
231	C₂×C₄	#3946	1	4	yes	—
232	C₂×C₄	#3947	1	4	yes	—
233	C₂×C₄	#3952	1	2	yes	—
234	C₂×C₄	#3952	1	4	yes	—
235	C₂×C₄	#3963	1	4	yes	—
236	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
237	C₂×C₄	C₂×#676	1	1	yes	characteristic
238	C₂×C₄	C₂×#676	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
240	C₂×C₄	C₂×#683	1	1	yes	characteristic
241	C₂×C₄	C₂×#685	1	1	yes	characteristic
242	C₈	—	10	4	no	—
243	C₈	#3963	1	4	yes	—
244	C₈	C₂×#683	1	1	yes	characteristic
245	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
246	C₈	C₂×#685	1	1	yes	characteristic
247	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
248	C₅	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
249	C₂²	C₁₆×Dic₅	1	4	yes	—
250	C₂²	#64721	1	8	yes	—
251	C₂²	#64722	1	4	yes	—
252	C₂²	#64727	1	8	yes	—
253	C₂²	#64770	1	4	yes	—
254	C₂²	C₂×D₅×C₁₆	1	2	yes	—
255	C₂²	C₂×#3930	1	2	yes	—
256	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
257	C₂²	C₂×#3946	1	1	yes	characteristic
258	C₂²	C₂×#3952	1	1	yes	characteristic
259	C₄	—	10	8	no	—
260	C₄	#64770	1	4	yes	—
261	C₄	C₂×#3946	1	1	yes	characteristic
262	C₄	C₂×#3947	1	1	yes	characteristic
263	C₄	C₂×#3952	1	1	yes	characteristic
264	C₄	C₂×#3963	1	1	yes	characteristic
265	C₂	#10598295	1	8	yes	direct product factor
266	C₂	C₂×C₁₆×Dic₅	1	1	yes	characteristic
267	C₂	C₂×#64722	1	1	yes	characteristic
268	C₂	C₂×#64721	1	2	yes	—
269	C₂	C₂×#64727	1	2	yes	—
270	C₂	C₂×#64770	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₅
commutator	C₂×C₅	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂³×C₅×C₁₆	C₂
socle, prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	C₂×#1374
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10598295	C₂	direct (N × H)
10. C₂²×C₅×C₁₆	C₄	semidirect
18. C₂²×#672	C₄	semidirect
42. C₂×C₅×C₁₆	C₂×C₄	semidirect
49. C₂×#672	C₂×C₄	semidirect
65. C₂²×Dic₅	C₁₆	semidirect
98. C₂²×C₁₆	Dic₅	semidirect
126. C₂×Dic₅	C₂×C₁₆	semidirect
158. C₂×C₁₆	C₂×Dic₅	semidirect
213. C₂×C₅	#1374	semidirect
248. C₅	C₂×#1374	semidirect
265. C₂	#10598295	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂²×C₈×Dic₅	2
C₂³×#672	2
#10598295	2
C₂×#1374	5

Subgroups — C2×#10598295 #12014136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598295 #12014136