Subgroups — C₂×#10601716 #12252528

Order 1280.

662 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	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×#65429	C₂	1	2	yes	maximal, complemented
3	C₂×#64781	C₂	1	2	yes	maximal, complemented
4	#10601716	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64683	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#135	C₄	1	1	yes	characteristic
8	C₂×C₅×#131	C₂²	1	2	yes	—
9	C₅×#443	C₄	1	4	yes	—
10	C₅×#443	C₂²	1	4	yes	—
11	C₂×C₅×#108	C₄	1	1	yes	characteristic
12	C₂×C₅×#104	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65429	—	2	4	no	—
15	#65429	C₂²	1	8	yes	complemented
16	C₂×#3944	—	2	2	no	—
17	C₂×#3944	C₂²	1	2	yes	—
18	C₂²×#672	—	2	2	no	—
19	C₂²×#672	C₂²	1	1	yes	characteristic
20	C₂²×#672	C₂²	1	1	yes	characteristic
21	#64781	C₂²	1	8	yes	complemented
22	#64683	C₂²	1	4	yes	complemented
23	C₄×#672	C₂²	1	4	yes	complemented
24	C₂×#2170	—	5	1	no	maximal, Sylow
25	C₂×C₅×#41	C₂×C₄	1	2	yes	—
26	C₅×#135	C₈	1	4	yes	—
27	C₅×#135	C₂×C₄	1	4	yes	—
28	C₂×C₅×Q₁₆	C₈	1	1	yes	characteristic
29	C₂×C₅×SD₁₆	C₈	1	2	yes	—
30	C₂×C₅×D₈	C₈	1	1	yes	characteristic
31	C₅×#131	—	2	2	no	—
32	C₅×#131	—	2	2	no	—
33	C₅×#131	C₂×C₄	1	4	yes	—
34	C₅×#131	C₂³	1	4	yes	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₅×#108	C₂×C₄	1	4	yes	—
42	C₅×#104	C₂×C₄	1	8	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂³	1	2	yes	—
45	#3944	—	2	2	no	—
46	#3944	—	2	8	no	—
47	#3944	C₂³	1	4	yes	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	—	2	1	no	—
50	C₂×#672	—	2	1	no	—
51	C₂×#672	—	2	8	no	—
52	C₂×#672	D₄	1	1	yes	characteristic
53	C₂×#672	D₄	1	1	yes	characteristic
54	C₂×#672	D₄	1	2	yes	—
55	C₂×#672	C₂³	1	2	yes	—
56	C₂×#672	C₂³	1	2	yes	—
57	C₂×#504	—	5	2	no	—
58	C₂×#443	D₅	1	1	yes	characteristic
59	#2170	—	5	8	no	—
60	C₂×#364	—	5	1	no	—
61	C₂×#350	—	5	2	no	—
62	C₂×C₄×C₁₆	—	5	1	no	—
63	C₅×#41	—	2	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₅×#41	C₂×C₈	1	4	yes	—
66	C₅×#41	C₂²×C₄	1	4	yes	—
67	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₅×Q₁₆	C₂×C₈	1	4	yes	—
74	C₅×SD₁₆	C₂×C₈	1	8	yes	—
75	C₅×D₈	C₂×C₈	1	4	yes	—
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	C₂²×C₄	1	4	yes	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	#41	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	C₂²×C₄	1	2	yes	—
95	#672	—	2	4	no	—
96	#672	—	2	2	no	—
97	#672	C₂×D₄	1	2	yes	—
98	#672	C₂×D₄	1	2	yes	—
99	C₂×#135	Dic₅	1	1	yes	characteristic
100	C₂×#131	C₂×D₅	1	2	yes	—
101	#504	—	5	8	no	—
102	#504	—	10	4	no	—
103	C₂×M₃₂	—	5	2	no	—
104	C₂×M₃₂	—	10	2	no	—
105	C₂²×C₁₆	—	5	1	no	—
106	C₂²×C₁₆	—	5	1	no	—
107	C₂²×C₁₆	—	10	2	no	—
108	#443	Dic₅	1	4	yes	—
109	#443	C₂×D₅	1	4	yes	—
110	C₂×#108	Dic₅	1	1	yes	characteristic
111	C₂×#104	Dic₅	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	#364	—	5	4	no	—
114	#350	—	5	8	no	—
115	C₄×C₁₆	—	5	4	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	#131	1	1	yes	characteristic
118	C₅×Q₈	—	2	2	no	—
119	C₅×Q₈	C₂²×C₈	1	4	yes	—
120	C₅×D₄	—	2	4	no	—
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	2	2	no	—
123	C₅×D₄	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅	#131	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₅×C₈	—	2	4	no	—
136	C₅×C₈	C₄×D₄	1	1	yes	characteristic
137	C₅×C₈	C₄×D₄	1	1	yes	characteristic
138	C₅×C₈	C₂²×C₈	1	2	yes	—
139	C₅×C₈	C₂²×C₈	1	2	yes	—
140	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
141	C₅×C₈	C₂×#41	1	1	yes	characteristic
142	C₂×#41	C₂×Dic₅	1	2	yes	—
143	#135	#168	1	4	yes	—
144	#135	C₂×Dic₅	1	4	yes	—
145	C₂×Q₁₆	#168	1	1	yes	characteristic
146	C₂×SD₁₆	#168	1	2	yes	—
147	C₂×D₈	#168	1	1	yes	characteristic
148	#131	—	2	2	no	—
149	#131	—	2	2	no	—
150	#131	C₂×Dic₅	1	4	yes	—
151	#131	C₂²×D₅	1	4	yes	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
158	M₃₂	—	5	4	no	—
159	M₃₂	—	10	8	no	—
160	M₃₂	—	10	2	no	—
161	C₂×C₁₆	—	5	2	no	—
162	C₂×C₁₆	—	5	1	no	—
163	C₂×C₁₆	—	5	1	no	—
164	C₂×C₁₆	—	5	2	no	—
165	C₂×C₁₆	—	5	2	no	—
166	C₂×C₁₆	—	10	8	no	—
167	C₂×C₁₆	—	10	2	no	—
168	C₂×C₁₆	—	10	1	no	—
169	C₂×C₁₆	—	10	1	no	—
170	#108	C₂×Dic₅	1	4	yes	—
171	#104	C₂×Dic₅	1	8	yes	—
172	C₄×C₈	C₂×Dic₅	1	2	yes	—
173	C₄×C₈	C₂²×D₅	1	2	yes	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	4	no	—
176	C₂²×C₅	—	4	4	no	—
177	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
178	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
179	C₂²×C₅	C₂×#131	1	1	yes	characteristic
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	C₈×D₄	1	1	yes	characteristic
183	C₄×C₅	C₈×D₄	1	1	yes	characteristic
184	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	C₂×#168	1	4	yes	—
189	#41	C₂²×Dic₅	1	4	yes	—
190	C₂×Q₈	C₂×#168	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	C₂×#168	1	2	yes	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	Q₁₆	C₂×#168	1	4	yes	—
197	SD₁₆	C₂×#168	1	8	yes	—
198	D₈	C₂×#168	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂²×Dic₅	1	4	yes	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂×#168	1	1	yes	characteristic
207	C₂×C₈	C₂×#168	1	1	yes	characteristic
208	C₂×C₈	C₂×#168	1	2	yes	—
209	C₂×C₈	D₄×D₅	1	1	yes	characteristic
210	C₂×C₈	D₄×D₅	1	1	yes	characteristic
211	C₂×C₈	#711	1	1	yes	characteristic
212	C₂×C₈	#711	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₅	1	2	yes	—
214	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
215	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
216	C₄²	—	2	1	no	—
217	C₄²	C₂²×Dic₅	1	2	yes	—
218	C₁₆	—	5	2	no	—
219	C₁₆	—	5	2	no	—
220	C₁₆	—	10	4	no	—
221	C₁₆	—	10	2	no	—
222	C₂×C₅	—	2	1	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	—	4	4	no	—
225	C₂×C₅	#2170	1	2	yes	complemented
226	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	#4094	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	C₂²×#168	1	4	yes	—
231	D₄	—	2	4	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	C₂²×#168	1	4	yes	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂²×#168	1	1	yes	characteristic
241	C₂×C₄	C₂²×#168	1	1	yes	characteristic
242	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
244	C₂×C₄	#4094	1	1	yes	characteristic
245	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	C₂²×#168	1	2	yes	—
248	C₈	C₂²×#168	1	2	yes	—
249	C₈	D₄×Dic₅	1	1	yes	characteristic
250	C₈	D₄×Dic₅	1	1	yes	characteristic
251	C₈	C₂×D₄×D₅	1	1	yes	characteristic
252	C₈	C₂×#711	1	1	yes	characteristic
253	C₅	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	D₄×#168	1	1	yes	characteristic
258	C₂²	D₄×#168	1	1	yes	characteristic
259	C₂²	C₂×#4094	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	D₄×#168	1	1	yes	characteristic
263	C₄	D₄×#168	1	1	yes	characteristic
264	C₄	C₂³×#168	1	1	yes	characteristic
265	C₄	C₂×D₄×Dic₅	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	4	no	—
269	C₂	#10601716	1	2	yes	direct product factor
270	C₂	C₂×D₄×#168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#443	C₂
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×C₅×#135	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#443	D₅
hypocenter	C₅	C₂×#2170
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. C₂×#65429	C₂	semidirect
3. C₂×#64781	C₂	semidirect
4. #10601716	C₂	direct (N × H)
5. C₂×#64683	C₂	semidirect
6. C₂×C₄×#672	C₂	semidirect
15. #65429	C₂²	semidirect
21. #64781	C₂²	semidirect
22. #64683	C₂²	semidirect
23. C₄×#672	C₂²	semidirect
225. C₂×C₅	#2170	semidirect
253. C₅	C₂×#2170	semidirect
269. C₂	#10601716	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×C₅×#443	2
C₂×#65429	2
C₂×#64781	2
#10601716	2
C₂×#64683	2
C₂×C₄×#672	2
C₂×#2170	5

Subgroups — C2×#10601716 #12252528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601716 #12252528