Subgroups — C₂×#10598252 #12009015

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598252	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
7	C₅×#436	—	2	1	no	—
8	C₅×#436	—	2	1	no	—
9	C₅×#436	C₄	1	2	yes	—
10	C₅×#436	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²×#672	—	2	1	no	—
15	C₂²×#672	—	2	1	no	—
16	C₄⋊#672	—	2	1	no	—
17	C₄⋊#672	C₂²	1	2	yes	complemented
18	#64681	—	2	1	no	—
19	#64681	C₂²	1	2	yes	complemented
20	C₂×#1330	—	5	1	no	maximal, Sylow
21	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	—	2	1	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	4	2	no	—
26	C₂×C₅×M₁₆	—	4	2	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×D₄	—	2	1	no	—
31	C₄×C₅×D₄	—	2	1	no	—
32	C₄×C₅×D₄	C₈	1	2	yes	—
33	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×C₄⋊C₈	—	2	1	no	—
38	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₅×C₂²⋊C₈	—	2	2	no	—
40	C₅×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×C₈	D₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂×#672	—	2	4	no	—
46	C₂×#672	—	2	2	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	4	2	no	—
49	C₂×#436	D₅	1	1	yes	characteristic
50	C₂×C₄⋊C₁₆	—	5	1	no	—
51	C₂×#346	—	5	1	no	—
52	#1330	—	5	4	no	—
53	C₂⁴×C₅	—	2	1	no	—
54	C₂×C₅×D₄	—	2	1	no	—
55	C₂×C₅×D₄	—	4	2	no	—
56	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
65	C₅×M₁₆	—	4	4	no	—
66	C₅×M₁₆	—	4	2	no	—
67	C₅×M₁₆	—	4	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	D₈	1	2	yes	—
74	C₂×C₅×C₈	SD₁₆	1	2	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₅×C₂²⋊C₄	—	2	2	no	—
80	C₅×C₂²⋊C₄	—	2	2	no	—
81	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₅	M₁₆	1	1	yes	characteristic
84	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
85	#672	—	2	4	no	—
86	#672	—	4	2	no	—
87	C₂²×M₁₆	—	2	1	no	—
88	C₂×C₄×D₄	Dic₅	1	1	yes	characteristic
89	C₂²×C₁₆	—	10	1	no	—
90	C₂²×C₁₆	—	10	1	no	—
91	#436	—	2	1	no	—
92	#436	—	2	1	no	—
93	#436	Dic₅	1	2	yes	—
94	#436	C₂×D₅	1	2	yes	—
95	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
96	C₂×C₂²⋊C₈	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
98	C₄⋊C₁₆	—	5	2	no	—
99	C₄⋊C₁₆	—	10	1	no	—
100	#346	—	5	2	no	—
101	#346	—	10	1	no	—
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	4	2	no	—
104	C₂³×C₅	—	4	2	no	—
105	C₂³×C₅	—	4	2	no	—
106	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₅×D₄	—	4	2	no	—
108	C₅×D₄	—	4	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	4	4	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#102	1	2	yes	—
120	C₂×C₄×C₅	#104	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	4	2	no	—
127	C₅×C₈	#102	1	2	yes	—
128	C₅×C₈	C₂×D₈	1	1	yes	characteristic
129	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
130	C₂²×D₄	#168	1	1	yes	characteristic
131	C₂³×C₄	—	2	1	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	4	2	no	—
135	C₂×M₁₆	—	4	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	#168	1	2	yes	—
142	C₄×D₄	C₂×Dic₅	1	2	yes	—
143	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	2	1	no	—
145	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
146	C₂×C₁₆	—	10	4	no	—
147	C₂×C₁₆	—	10	2	no	—
148	C₂×C₁₆	—	10	2	no	—
149	C₂×C₁₆	—	20	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	—	4	2	no	—
161	C₂²×C₅	—	8	4	no	—
162	C₂²×C₅	D₄⋊C₈	1	2	yes	—
163	C₂²×C₅	#349	1	2	yes	—
164	C₂²×C₅	#350	1	2	yes	—
165	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	4	2	no	—
171	C₄×C₅	D₄⋊C₈	1	2	yes	—
172	C₄×C₅	C₂×#102	1	1	yes	characteristic
173	C₄×C₅	C₂×#104	1	1	yes	characteristic
174	C₂⁴	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	C₂×#168	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×#168	1	1	yes	characteristic
184	C₂²×C₄	#681	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
186	M₁₆	—	4	4	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	#686	1	2	yes	—
193	C₂×C₈	#687	1	2	yes	—
194	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂×#168	1	2	yes	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂×#168	1	1	yes	characteristic
203	C₄²	#681	1	1	yes	characteristic
204	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
205	C₄²	C₂²×Dic₅	1	1	yes	characteristic
206	C₁₆	—	10	4	no	—
207	C₁₆	—	20	2	no	—
208	C₂×C₅	—	8	2	no	—
209	C₂×C₅	#1330	1	4	yes	complemented
210	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
211	C₂×C₅	C₂×#349	1	1	yes	characteristic
212	C₂×C₅	C₂×#350	1	1	yes	characteristic
213	C₂³	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	C₂²⋊#168	1	1	yes	characteristic
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
230	C₂×C₄	#3964	1	2	yes	—
231	C₂×C₄	#3969	1	2	yes	—
232	C₂×C₄	C₂²×#168	1	1	yes	characteristic
233	C₂×C₄	C₂×#681	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	#3964	1	2	yes	—
239	C₈	C₂×#686	1	1	yes	characteristic
240	C₈	C₂×#687	1	1	yes	characteristic
241	C₅	C₂×#1330	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	8	4	no	—
246	C₂²	D₄⋊#168	1	2	yes	—
247	C₂²	#64773	1	2	yes	—
248	C₂²	#64781	1	2	yes	—
249	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	D₄⋊#168	1	2	yes	—
256	C₄	C₂×#3964	1	1	yes	characteristic
257	C₄	C₂×#3969	1	1	yes	characteristic
258	C₂	—	8	2	no	—
259	C₂	#10598252	1	4	yes	direct product factor
260	C₂	C₂×D₄⋊#168	1	1	yes	characteristic
261	C₂	C₂×#64773	1	1	yes	characteristic
262	C₂	C₂×#64781	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×#436	C₂
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×#436	D₅
hypocenter	C₅	C₂×#1330
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#672	C₂	semidirect
3. C₂×#64681	C₂	semidirect
4. #10598252	C₂	direct (N × H)
17. C₄⋊#672	C₂²	semidirect
19. #64681	C₂²	semidirect
209. C₂×C₅	#1330	semidirect
241. C₅	C₂×#1330	semidirect
259. C₂	#10598252	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×#436	2
C₂×C₄⋊#672	2
C₂×#64681	2
#10598252	2
C₂×#1330	5

Subgroups — C2×#10598252 #12009015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598252 #12009015