Subgroups — C₂×#10599907 #12252419

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#349	C₂	1	1	yes	maximal, characteristic
3	C₂×#64773	C₂	1	1	yes	maximal, characteristic
4	C₂×#64732	C₂	1	2	yes	maximal, complemented
5	C₂×#64731	C₂	1	2	yes	maximal, complemented
6	#10599907	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
9	C₂²×#681	C₄	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	M₁₆×Dic₅	C₄	1	4	yes	—
12	M₁₆×Dic₅	C₂²	1	4	yes	—
13	C₂×#3947	C₄	1	1	yes	characteristic
14	C₂×#3947	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×#3944	C₂²	1	2	yes	—
18	C₅×#349	C₂²	1	4	yes	—
19	#64773	C₂²	1	4	yes	—
20	#64732	C₂²	1	8	yes	complemented
21	#64731	C₂²	1	8	yes	complemented
22	C₂×#2165	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×Dic₅	—	2	1	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×M₁₆	C₂³	1	2	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₅	—	2	1	no	—
32	C₂×C₄×Dic₅	—	2	2	no	—
33	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
36	C₂×#681	—	2	2	no	—
37	C₂×#681	—	2	2	no	—
38	C₂×#681	C₂×C₄	1	4	yes	—
39	C₂²×#168	C₂×C₄	1	1	yes	characteristic
40	C₂²×#168	C₂×C₄	1	1	yes	characteristic
41	C₅×M₃₂	—	2	2	no	—
42	C₅×M₃₂	C₂³	1	4	yes	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	#3947	C₂×C₄	1	4	yes	—
45	#3947	C₂×C₄	1	2	yes	—
46	#3947	C₂³	1	2	yes	—
47	C₈×Dic₅	C₂×C₄	1	4	yes	—
48	C₈×Dic₅	C₂×C₄	1	2	yes	—
49	C₈×Dic₅	C₂³	1	2	yes	—
50	#3944	—	2	2	no	—
51	#3944	C₂³	1	4	yes	—
52	C₂×#672	—	2	2	no	—
53	C₂×C₄×M₁₆	—	5	1	no	—
54	#2165	—	5	8	no	—
55	C₂×#364	—	5	2	no	—
56	C₂×#349	—	5	1	no	—
57	C₂×#349	D₅	1	1	yes	characteristic
58	C₂×#347	—	5	2	no	—
59	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅	M₁₆	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₅	—	2	1	no	—
69	C₂²×Dic₅	—	2	4	no	—
70	C₂²×Dic₅	—	2	1	no	—
71	C₂²×Dic₅	—	4	4	no	—
72	C₅×M₁₆	—	2	4	no	—
73	C₅×M₁₆	—	2	4	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	#41	1	2	yes	—
79	C₂×C₅×C₈	#41	1	2	yes	—
80	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
81	C₄×Dic₅	—	2	1	no	—
82	C₄×Dic₅	—	2	4	no	—
83	C₄×Dic₅	—	2	1	no	—
84	C₄×Dic₅	C₂²×C₄	1	2	yes	—
85	C₄×Dic₅	C₂²×C₄	1	2	yes	—
86	#681	—	2	8	no	—
87	C₂×#168	—	2	1	no	—
88	C₂×#168	—	2	1	no	—
89	C₂×#168	M₁₆	1	2	yes	—
90	C₂×#168	M₁₆	1	2	yes	—
91	C₂×#168	C₂²×C₄	1	2	yes	—
92	C₂×#168	C₂²×C₄	1	2	yes	—
93	C₅×C₁₆	—	2	4	no	—
94	#672	—	2	4	no	—
95	C₂²×M₁₆	—	5	1	no	—
96	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
97	C₂²×C₄²	—	5	1	no	—
98	C₂×M₃₂	—	5	2	no	—
99	C₂×M₃₂	C₂×D₅	1	2	yes	—
100	C₄×M₁₆	—	5	4	no	—
101	C₄×M₁₆	—	5	4	no	—
102	C₂×#97	—	5	1	no	—
103	C₂×#97	—	5	1	no	—
104	C₂×C₄×C₈	—	5	1	no	—
105	C₂×C₄×C₈	—	5	1	no	—
106	#364	—	5	8	no	—
107	#349	—	5	4	no	—
108	#349	C₂×D₅	1	4	yes	—
109	#347	—	5	8	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	1	no	—
112	C₂³×C₅	—	4	2	no	—
113	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₂³×C₅	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	#117	1	1	yes	characteristic
122	C₂×C₄×C₅	#117	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
128	C₂×Dic₅	—	2	4	no	—
129	C₂×Dic₅	—	2	2	no	—
130	C₂×Dic₅	—	2	2	no	—
131	C₂×Dic₅	—	4	8	no	—
132	C₂×Dic₅	—	4	2	no	—
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	#117	1	2	yes	—
135	C₅×C₈	C₂×#41	1	2	yes	—
136	#168	—	2	2	no	—
137	#168	C₂×M₁₆	1	4	yes	—
138	C₂³×C₄	—	10	1	no	—
139	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	5	4	no	—
143	C₂×M₁₆	—	10	2	no	—
144	C₂×M₁₆	—	10	2	no	—
145	C₂×M₁₆	C₄×D₅	1	2	yes	—
146	C₂×M₁₆	C₂²×D₅	1	2	yes	—
147	C₂²×C₈	—	5	1	no	—
148	C₂²×C₈	—	5	1	no	—
149	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₄²	—	5	1	no	—
152	C₂×C₄²	—	5	1	no	—
153	C₂×C₄²	—	5	4	no	—
154	C₂×C₄²	—	10	1	no	—
155	C₂×C₄²	—	10	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	5	4	no	—
158	M₃₂	—	10	2	no	—
159	M₃₂	C₂²×D₅	1	4	yes	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	10	2	no	—
162	#97	—	5	4	no	—
163	#97	—	5	2	no	—
164	#97	—	5	2	no	—
165	C₄×C₈	—	5	2	no	—
166	C₄×C₈	—	5	4	no	—
167	C₄×C₈	—	5	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	—	4	2	no	—
173	C₂²×C₅	#432	1	1	yes	characteristic
174	C₂²×C₅	#432	1	1	yes	characteristic
175	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	#432	1	1	yes	characteristic
178	C₄×C₅	#432	1	1	yes	characteristic
179	C₄×C₅	C₂×#117	1	1	yes	characteristic
180	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
181	Dic₅	—	2	4	no	—
182	Dic₅	—	4	2	no	—
183	C₂⁴	#676	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	10	1	no	—
189	C₂²×C₄	—	10	4	no	—
190	C₂²×C₄	—	10	1	no	—
191	C₂²×C₄	—	20	4	no	—
192	C₂²×C₄	#676	1	1	yes	characteristic
193	C₂²×C₄	#676	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	10	8	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	10	1	no	—
206	C₂×C₈	—	10	1	no	—
207	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₈	#711	1	2	yes	—
210	C₂×C₈	#711	1	2	yes	—
211	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
212	C₄²	—	5	2	no	—
213	C₄²	—	5	2	no	—
214	C₄²	—	10	1	no	—
215	C₄²	—	10	4	no	—
216	C₄²	—	10	1	no	—
217	C₁₆	—	2	4	no	—
218	C₁₆	—	10	4	no	—
219	C₂×C₅	—	2	1	no	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	#2165	1	2	yes	complemented
223	C₂×C₅	C₂×#432	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×#676	1	1	yes	characteristic
228	C₂³	C₂×#676	1	2	yes	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	10	4	no	—
236	C₂×C₄	—	10	2	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	—	20	8	no	—
239	C₂×C₄	—	20	2	no	—
240	C₂×C₄	#4023	1	1	yes	characteristic
241	C₂×C₄	#4023	1	1	yes	characteristic
242	C₂×C₄	C₂×#676	1	2	yes	—
243	C₂×C₄	C₂×#676	1	1	yes	characteristic
244	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
245	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	5	4	no	—
249	C₈	—	10	2	no	—
250	C₈	#4023	1	2	yes	—
251	C₈	C₂×#711	1	2	yes	—
252	C₅	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#65008	1	1	yes	characteristic
259	C₂²	#65008	1	1	yes	characteristic
260	C₂²	C₂²×#676	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	10	4	no	—
263	C₄	—	20	2	no	—
264	C₄	#65008	1	1	yes	characteristic
265	C₄	#65008	1	1	yes	characteristic
266	C₄	C₂×#4023	1	1	yes	characteristic
267	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	#10599907	1	2	yes	direct product factor
272	C₂	C₂×#65008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4023
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#349	C₂
socle	C₂²×C₅	#432
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#349	D₅
hypocenter	C₅	C₂×#2165
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
4. C₂×#64732	C₂	semidirect
5. C₂×#64731	C₂	semidirect
6. #10599907	C₂	direct (N × H)
20. #64732	C₂²	semidirect
21. #64731	C₂²	semidirect
222. C₂×C₅	#2165	semidirect
252. C₅	C₂×#2165	semidirect
271. C₂	#10599907	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×M₁₆×Dic₅	2
C₂×C₅×#349	2
C₂×#64773	2
C₂×#64732	2
C₂×#64731	2
#10599907	2
C₂×#2165	5

Subgroups — C2×#10599907 #12252419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10599907 #12252419