Subgroups — C₂×#10601671 #12068699

Order 1280.

894 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×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×#168	C₂	1	2	yes	maximal, complemented
3	C₂×D₄⋊#168	C₂	1	2	yes	maximal, complemented
4	#10601671	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64677	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
9	C₂³×#168	—	2	2	no	—
10	C₄×C₅×D₈	C₄	1	4	yes	—
11	C₄×C₅×D₈	C₂²	1	4	yes	—
12	C₂×C₅×#107	C₄	1	1	yes	characteristic
13	C₂×C₅×#102	C₄	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊#168	—	2	2	no	—
16	D₄×#168	—	2	4	no	—
17	D₄×#168	C₂²	1	8	yes	complemented
18	C₂×C₄⋊#168	C₂²	1	2	yes	—
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	D₄⋊#168	C₂²	1	8	yes	complemented
22	#64677	C₂²	1	4	yes	complemented
23	#64675	C₂²	1	4	yes	complemented
24	C₂×#1580	—	5	1	no	maximal, Sylow
25	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	—	2	2	no	—
27	C₂×C₅×D₈	C₈	1	8	yes	complemented
28	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	2	2	no	—
33	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
34	C₄×C₅×D₄	C₂³	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₂²×#168	—	2	2	no	—
39	C₂²×#168	—	2	1	no	—
40	C₂²×#168	—	2	8	no	—
41	C₂²×#168	—	4	8	no	—
42	C₂²×#168	D₄	1	2	yes	—
43	C₅×#107	C₂×C₄	1	4	yes	—
44	C₅×#102	C₂×C₄	1	8	yes	—
45	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂³	1	2	yes	—
47	C₂²⋊#168	—	2	8	no	—
48	C₄⋊#168	—	2	2	no	—
49	C₄⋊#168	C₂³	1	4	yes	—
50	C₄×#168	—	2	1	no	—
51	C₄×#168	D₄	1	2	yes	—
52	C₄×#168	C₂³	1	2	yes	—
53	C₄×#168	C₂³	1	2	yes	—
54	C₂×C₄×D₈	D₅	1	1	yes	characteristic
55	C₂×C₈×D₄	—	5	2	no	—
56	#1580	—	5	8	no	—
57	C₂×#334	—	5	1	no	—
58	C₂×D₄⋊C₈	—	5	2	no	—
59	C₂×#322	—	5	1	no	—
60	C₂⁴×C₅	—	2	2	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	2	4	no	—
63	C₂×C₅×D₄	C₂×C₈	1	8	yes	complemented
64	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	4	4	no	—
69	C₂²×C₄×C₅	—	4	4	no	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₅×D₈	C₂×C₈	1	16	yes	complemented
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	C₂×C₈	1	4	yes	complemented
76	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
80	C₅×C₂²⋊C₄	—	2	4	no	—
81	C₅×C₂²⋊C₄	—	2	4	no	—
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅	#41	1	1	yes	characteristic
85	C₄²×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×#168	—	2	4	no	—
87	C₂×#168	—	2	4	no	—
88	C₂×#168	—	2	2	no	—
89	C₂×#168	—	4	16	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	4	4	no	—
92	C₂×#168	C₂×D₄	1	4	yes	—
93	C₂²×D₈	Dic₅	1	1	yes	characteristic
94	C₂³×C₈	—	10	2	no	—
95	C₂×C₄×D₄	C₂×D₅	1	2	yes	—
96	C₄×D₈	Dic₅	1	4	yes	—
97	C₄×D₈	C₂×D₅	1	4	yes	—
98	C₈×D₄	—	5	8	no	—
99	C₈×D₄	—	10	4	no	—
100	C₂×#107	Dic₅	1	1	yes	characteristic
101	C₂×C₄⋊C₈	—	5	2	no	—
102	C₂×#102	Dic₅	1	2	yes	—
103	C₂×C₂²⋊C₈	—	10	2	no	—
104	C₂×C₄×C₈	—	5	1	no	—
105	C₂×C₄×C₈	—	5	1	no	—
106	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	#334	—	5	4	no	—
108	D₄⋊C₈	—	5	8	no	—
109	#322	—	5	4	no	—
110	C₂³×C₅	—	2	4	no	—
111	C₂³×C₅	—	2	8	no	—
112	C₂³×C₅	—	4	4	no	—
113	C₂³×C₅	—	4	4	no	—
114	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
115	C₅×D₄	—	2	4	no	—
116	C₅×D₄	—	2	8	no	—
117	C₅×D₄	C₂²×C₈	1	8	yes	complemented
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	8	no	—
124	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅	#131	1	2	yes	—
129	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	C₂²×C₈	1	4	yes	complemented
135	#168	—	2	4	no	—
136	#168	—	4	2	no	—
137	#168	—	4	4	no	—
138	C₂²×D₄	C₂×Dic₅	1	2	yes	—
139	C₂³×C₄	—	2	2	no	—
140	C₂×D₈	#168	1	8	yes	complemented
141	C₂×D₈	C₂×Dic₅	1	4	yes	—
142	C₂²×C₈	—	5	2	no	—
143	C₂²×C₈	—	10	8	no	—
144	C₂²×C₈	—	10	2	no	—
145	C₂²×C₈	—	10	1	no	—
146	C₂²×C₈	—	20	8	no	—
147	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	C₂×Dic₅	1	4	yes	—
152	C₄×D₄	C₂²×D₅	1	4	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
154	C₂×C₂²⋊C₄	—	2	2	no	—
155	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
156	#107	C₂×Dic₅	1	4	yes	—
157	C₄⋊C₈	—	5	4	no	—
158	C₄⋊C₈	—	10	2	no	—
159	#102	C₂×Dic₅	1	8	yes	—
160	C₂²⋊C₈	—	10	8	no	—
161	C₄×C₈	—	5	2	no	—
162	C₄×C₈	—	5	2	no	—
163	C₄×C₈	—	5	2	no	—
164	C₄×C₈	—	10	1	no	—
165	C₄×C₈	C₂×Dic₅	1	2	yes	—
166	C₄×C₈	C₂²×D₅	1	2	yes	—
167	C₂²×C₅	—	2	8	no	—
168	C₂²×C₅	—	4	16	no	—
169	C₂²×C₅	—	4	4	no	—
170	C₂²×C₅	—	4	4	no	—
171	C₂²×C₅	C₈×D₄	1	2	yes	—
172	C₂²×C₅	#442	1	2	yes	—
173	C₂²×C₅	#444	1	2	yes	—
174	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	4	4	no	—
178	C₄×C₅	C₈×D₄	1	2	yes	—
179	C₄×C₅	#442	1	2	yes	—
180	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
181	C₄×C₅	C₂×#131	1	1	yes	characteristic
182	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
183	C₂⁴	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	C₂×#168	1	8	yes	complemented
187	C₂×D₄	C₂²×Dic₅	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
194	C₂²×C₄	#711	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	D₈	C₂×#168	1	16	yes	complemented
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	5	4	no	—
199	C₂×C₈	—	10	4	no	—
200	C₂×C₈	—	10	4	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	20	16	no	—
203	C₂×C₈	—	20	2	no	—
204	C₂×C₈	—	20	4	no	—
205	C₂×C₈	C₂×#168	1	4	yes	complemented
206	C₂×C₈	C₂²×Dic₅	1	2	yes	—
207	C₂×C₈	C₂²×Dic₅	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	D₄×D₅	1	1	yes	characteristic
213	C₄²	#711	1	1	yes	characteristic
214	C₄²	C₂²×Dic₅	1	1	yes	characteristic
215	C₄²	C₂³×D₅	1	1	yes	characteristic
216	C₂×C₅	—	4	8	no	—
217	C₂×C₅	#1580	1	4	yes	complemented
218	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
219	C₂×C₅	C₂×#442	1	1	yes	characteristic
220	C₂×C₅	C₂×#444	1	1	yes	characteristic
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	D₄×Dic₅	1	1	yes	characteristic
226	D₄	—	2	8	no	—
227	D₄	—	2	4	no	—
228	D₄	C₂²×#168	1	8	yes	complemented
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	#4057	1	4	yes	—
236	C₂×C₄	C₂²×#168	1	2	yes	—
237	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
238	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
239	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
240	C₂×C₄	#4094	1	2	yes	—
241	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
242	C₂×C₄	C₂×#711	1	1	yes	characteristic
243	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	10	4	no	—
246	C₈	—	20	2	no	—
247	C₈	—	20	4	no	—
248	C₈	C₂²×#168	1	4	yes	complemented
249	C₅	C₂×#1580	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	16	no	—
254	C₂²	D₄×#168	1	2	yes	—
255	C₂²	#65441	1	2	yes	—
256	C₂²	#65486	1	2	yes	—
257	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	D₄×#168	1	2	yes	—
262	C₄	#65441	1	2	yes	—
263	C₄	C₂×#4057	1	2	yes	—
264	C₄	C₂³×#168	1	1	yes	characteristic
265	C₄	C₂×#4094	1	1	yes	characteristic
266	C₂	—	4	8	no	—
267	C₂	#10601671	1	4	yes	direct product factor
268	C₂	C₂×D₄×#168	1	1	yes	characteristic
269	C₂	C₂×#65441	1	1	yes	characteristic
270	C₂	C₂×#65486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₄×C₅×D₈	C₂
socle	C₂³×C₅	C₄×D₄
prime radical	C₂²×C₅×D₈	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₄×D₈	D₅
hypocenter	C₅	C₂×#1580
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×D₄×#168	C₂	semidirect
3. C₂×D₄⋊#168	C₂	semidirect
4. #10601671	C₂	direct (N × H)
5. C₂×#64677	C₂	semidirect
6. C₂×#64675	C₂	semidirect
17. D₄×#168	C₂²	semidirect
21. D₄⋊#168	C₂²	semidirect
22. #64677	C₂²	semidirect
23. #64675	C₂²	semidirect
27. C₂×C₅×D₈	C₈	semidirect
63. C₂×C₅×D₄	C₂×C₈	semidirect
73. C₅×D₈	C₂×C₈	semidirect
75. C₂×C₅×C₈	C₂×C₈	semidirect
117. C₅×D₄	C₂²×C₈	semidirect
134. C₅×C₈	C₂²×C₈	semidirect
140. C₂×D₈	#168	semidirect
186. C₂×D₄	C₂×#168	semidirect
196. D₈	C₂×#168	semidirect
205. C₂×C₈	C₂×#168	semidirect
217. C₂×C₅	#1580	semidirect
228. D₄	C₂²×#168	semidirect
248. C₈	C₂²×#168	semidirect
249. C₅	C₂×#1580	semidirect
267. C₂	#10601671	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×C₄×C₅×D₈	2
C₂×D₄×#168	2
C₂×D₄⋊#168	2
#10601671	2
C₂×#64677	2
C₂×#64675	2
C₂×#1580	5

Subgroups — C2×#10601671 #12068699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601671 #12068699