Subgroups — C₄×#65214 #12097425

Order 1280.

1318 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#65214	C₂	1	2	yes	maximal
3	C₄×C₂²⋊#168	C₂	1	4	yes	maximal
4	C₄×C₄⋊#168	C₂	1	4	yes	maximal, complemented
5	#10600181	C₂	1	2	yes	maximal, complemented
6	C₄²×#168	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄²×C₅	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₄	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₄³×C₅	C₄	1	2	yes	—
12	C₄³×C₅	C₄	1	2	yes	—
13	C₄³×C₅	C₂²	1	1	yes	characteristic
14	C₄³×C₅	C₂²	1	2	yes	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊#168	C₂²	1	4	yes	—
17	#65214	C₄	1	16	yes	direct product factor
18	C₂×C₄⋊#168	C₂²	1	4	yes	—
19	C₂×C₄×#168	C₂²	1	8	yes	—
20	C₂×C₄×#168	C₂²	1	4	yes	—
21	#64744	C₂²	1	8	yes	—
22	C₄×#431	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅	—	2	8	no	—
28	C₂×C₄²×C₅	—	2	8	no	—
29	C₂×C₄²×C₅	C₈	1	16	yes	complemented
30	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	2	yes	—
44	C₂²×#168	C₂³	1	8	yes	—
45	C₂²⋊#168	C₂×C₄	1	16	yes	—
46	C₄⋊#168	C₂×C₄	1	16	yes	complemented
47	C₄×#168	—	2	16	no	—
48	C₄×#168	C₂×C₄	1	16	yes	complemented
49	C₂×#431	—	5	2	no	—
50	C₂×C₄³	D₅	1	1	yes	characteristic
51	C₄×C₄⋊C₈	—	5	4	no	—
52	C₄×C₂²⋊C₈	—	5	4	no	—
53	#1744	—	5	2	no	—
54	C₄²×C₈	—	5	2	no	—
55	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	8	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	2	8	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	C₄²	1	4	yes	—
61	C₂²×C₄×C₅	C₄²	1	4	yes	—
62	C₂²×C₄×C₅	C₂×C₈	1	16	yes	—
63	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₄²×C₅	—	2	8	no	—
74	C₄²×C₅	—	2	4	no	—
75	C₄²×C₅	—	2	8	no	—
76	C₄²×C₅	—	2	4	no	—
77	C₄²×C₅	C₄²	1	4	yes	—
78	C₄²×C₅	C₄²	1	4	yes	—
79	C₄²×C₅	C₂×C₈	1	8	yes	—
80	C₄²×C₅	C₂×C₈	1	16	yes	—
81	C₄²×C₅	C₂×C₈	1	8	yes	—
82	C₄²×C₅	M₁₆	1	8	yes	—
83	C₄²×C₅	C₂²×C₄	1	4	yes	—
84	C₄²×C₅	C₂²×C₄	1	4	yes	—
85	C₄²×C₅	#41	1	8	yes	—
86	C₂×#168	—	2	16	no	—
87	C₂×#168	C₂²×C₄	1	16	yes	—
88	C₂²×C₄²	Dic₅	1	2	yes	—
89	C₂²×C₄²	Dic₅	1	2	yes	—
90	C₂²×C₄²	C₂×D₅	1	2	yes	—
91	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
92	#431	—	5	16	no	—
93	C₂×C₄⋊C₈	—	5	4	no	—
94	C₂×C₂²⋊C₈	—	5	4	no	—
95	C₂×C₄×C₈	—	5	4	no	—
96	C₂×C₄×C₈	—	5	8	no	—
97	C₄³	Dic₅	1	2	yes	—
98	C₄³	Dic₅	1	2	yes	—
99	C₄³	C₂×D₅	1	1	yes	characteristic
100	C₄³	C₂×D₅	1	2	yes	—
101	C₄³	C₂×D₅	1	1	yes	characteristic
102	#336	—	5	8	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	C₂×C₄²	1	2	yes	—
106	C₂³×C₅	C₂²×C₈	1	4	yes	—
107	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	2	8	no	—
114	C₂×C₄×C₅	—	2	32	no	—
115	C₂×C₄×C₅	C₄×C₈	1	16	yes	complemented
116	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₅	#117	1	8	yes	—
122	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
123	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
125	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
126	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
128	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×#41	1	4	yes	—
130	#168	—	2	16	no	—
131	C₂³×C₄	C₂×Dic₅	1	4	yes	—
132	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
134	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₂²×C₈	—	5	8	no	—
136	C₂×C₄²	—	2	8	no	—
137	C₂×C₄²	—	2	8	no	—
138	C₂×C₄²	#168	1	16	yes	complemented
139	C₂×C₄²	C₄×D₅	1	8	yes	—
140	C₂×C₄²	C₂×Dic₅	1	4	yes	—
141	C₂×C₄²	C₂×Dic₅	1	2	yes	—
142	C₂×C₄²	C₂×Dic₅	1	4	yes	—
143	C₂×C₄²	C₂×Dic₅	1	2	yes	—
144	C₂×C₄²	C₂×Dic₅	1	4	yes	—
145	C₂×C₄²	C₂×Dic₅	1	2	yes	—
146	C₂×C₄²	C₂×Dic₅	1	2	yes	—
147	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
148	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
149	C₂×C₄²	C₂²×D₅	1	2	yes	—
150	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₅	1	2	yes	—
153	C₄⋊C₈	—	5	16	no	—
154	C₂²⋊C₈	—	5	16	no	—
155	C₄×C₈	—	5	16	no	—
156	C₄×C₈	—	10	16	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	8	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
161	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
162	C₂²×C₅	C₂×#117	1	2	yes	—
163	C₂²×C₅	C₂³×C₈	1	2	yes	—
164	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	8	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
169	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
170	C₄×C₅	C₄×M₁₆	1	4	yes	—
171	C₄×C₅	#431	1	16	yes	complemented
172	C₄×C₅	C₄×#41	1	4	yes	—
173	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	C₂×#168	1	16	yes	—
179	C₂²×C₄	C₂×#168	1	8	yes	—
180	C₂²×C₄	C₄×Dic₅	1	4	yes	—
181	C₂²×C₄	C₄×Dic₅	1	4	yes	—
182	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
183	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
184	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
189	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
190	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
191	C₂×C₈	—	5	16	no	—
192	C₂×C₈	—	10	16	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	8	no	—
196	C₄²	—	2	8	no	—
197	C₄²	C₂×#168	1	8	yes	—
198	C₄²	C₂×#168	1	16	yes	—
199	C₄²	C₂×#168	1	8	yes	—
200	C₄²	#681	1	8	yes	—
201	C₄²	C₄×Dic₅	1	4	yes	—
202	C₄²	C₄×Dic₅	1	4	yes	—
203	C₄²	C₂×C₄×D₅	1	4	yes	—
204	C₄²	C₂×C₄×D₅	1	4	yes	—
205	C₄²	#709	1	8	yes	—
206	C₂×C₅	—	2	4	no	—
207	C₂×C₅	C₄×#117	1	1	yes	characteristic
208	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
209	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
210	C₂×C₅	C₂×#431	1	4	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₂²×#168	1	4	yes	—
214	C₂³	C₂×C₄×Dic₅	1	2	yes	—
215	C₂³	C₂³×Dic₅	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	32	no	—
223	C₂×C₄	C₄×#168	1	16	yes	complemented
224	C₂×C₄	C₂²×#168	1	8	yes	—
225	C₂×C₄	C₂²×#168	1	4	yes	—
226	C₂×C₄	C₂²×#168	1	4	yes	—
227	C₂×C₄	C₂²×#168	1	8	yes	—
228	C₂×C₄	C₂×#681	1	4	yes	—
229	C₂×C₄	C₂×#681	1	8	yes	—
230	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
231	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
232	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
233	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
234	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
235	C₂×C₄	#4072	1	8	yes	—
236	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
237	C₂×C₄	C₂×#709	1	4	yes	—
238	C₈	—	10	16	no	—
239	C₅	C₄×#431	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₂×C₄×#168	1	4	yes	—
244	C₂²	C₂³×#168	1	2	yes	—
245	C₂²	C₂²×#681	1	2	yes	—
246	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
247	C₂²	C₂×#4072	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	2	no	—
251	C₄	C₂×C₄×#168	1	4	yes	—
252	C₄	C₂×C₄×#168	1	4	yes	—
253	C₄	C₄×#681	1	4	yes	—
254	C₄	#65214	1	16	yes	direct product factor
255	C₄	C₄×#709	1	4	yes	—
256	C₂	—	2	4	no	—
257	C₂	C₂²×C₄×#168	1	1	yes	characteristic
258	C₂	C₂×C₄×#681	1	1	yes	characteristic
259	C₂	C₂×#65214	1	4	yes	—
260	C₂	C₄×#4072	1	1	yes	characteristic

	subgroup	G/H
center	C₄³	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₄³×C₅	C₂
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₄³	D₅
hypocenter	C₅	C₄×#431
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄³ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
4. C₄×C₄⋊#168	C₂	semidirect
5. #10600181	C₂	semidirect
6. C₄²×#168	C₂	semidirect
17. #65214	C₄	direct (N × H)
29. C₂×C₄²×C₅	C₈	semidirect
46. C₄⋊#168	C₂×C₄	semidirect
48. C₄×#168	C₂×C₄	semidirect
115. C₂×C₄×C₅	C₄×C₈	semidirect
138. C₂×C₄²	#168	semidirect
171. C₄×C₅	#431	semidirect
223. C₂×C₄	C₄×#168	semidirect
239. C₅	C₄×#431	semidirect
254. C₄	#65214	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×C₄³×C₅	2
C₂×#65214	2
C₄×C₂²⋊#168	2
C₄×C₄⋊#168	2
#10600181	2
C₄²×#168	2
C₄×#431	5

Subgroups — C4×#65214 #12097425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#65214 #12097425