Subgroups — C₂×#10598102 #11988736

Order 1280.

974 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#10598102	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄²×C₅	—	2	2	no	—
5	C₂²×C₄²×C₅	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄³×C₅	—	2	2	no	—
8	C₄³×C₅	C₄	1	2	yes	—
9	C₄³×C₅	C₂²	1	2	yes	—
10	C₂×C₂²⋊#168	—	2	2	no	—
11	#65214	—	2	2	no	—
12	#65214	—	2	2	no	—
13	#65214	C₄	1	4	yes	complemented
14	#65214	C₂²	1	4	yes	—
15	C₂×C₄⋊#168	C₄	1	4	yes	complemented
16	C₂×C₄×#168	—	2	2	no	—
17	C₂×#1276	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	—	2	2	no	—
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	D₄	1	2	yes	—
21	C₂×C₄²×C₅	—	2	16	no	—
22	C₂×C₄²×C₅	—	2	2	no	—
23	C₂×C₄²×C₅	—	2	4	no	—
24	C₂×C₄²×C₅	—	2	4	no	—
25	C₂×C₄²×C₅	—	2	4	no	—
26	C₂×C₄²×C₅	C₈	1	8	yes	complemented
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂²×#168	—	2	4	no	—
38	C₂²⋊#168	—	2	4	no	—
39	C₂²⋊#168	—	2	4	no	—
40	C₄⋊#168	—	2	4	no	—
41	C₄⋊#168	C₂×C₄	1	8	yes	complemented
42	C₄×#168	—	2	4	no	—
43	C₄×#168	—	2	4	no	—
44	C₂×#431	—	5	2	no	—
45	C₂×C₄³	D₅	1	1	yes	characteristic
46	#1276	—	5	4	no	—
47	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	16	no	—
49	C₂²×C₄×C₅	—	2	8	no	—
50	C₂²×C₄×C₅	—	2	4	no	—
51	C₂²×C₄×C₅	—	2	8	no	—
52	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₅	M₁₆	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₄²×C₅	—	2	16	no	—
67	C₄²×C₅	—	2	16	no	—
68	C₄²×C₅	—	2	8	no	—
69	C₄²×C₅	—	2	4	no	—
70	C₄²×C₅	C₄²	1	2	yes	—
71	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₅	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
76	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×C₈	1	8	yes	complemented
78	C₄²×C₅	C₂²×C₄	1	2	yes	—
79	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
81	C₂×#168	—	2	8	no	—
82	C₂×#168	—	4	8	no	—
83	C₂²×C₄²	—	2	2	no	—
84	C₂²×C₄²	Dic₅	1	2	yes	—
85	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
86	#431	—	5	4	no	—
87	#431	—	5	4	no	—
88	#431	—	10	2	no	—
89	#431	—	10	2	no	—
90	C₂×C₄⋊C₈	—	5	4	no	—
91	C₂×C₂²⋊C₈	—	10	2	no	—
92	C₂×C₄×C₈	—	10	2	no	—
93	C₄³	—	2	2	no	—
94	C₄³	Dic₅	1	2	yes	—
95	C₄³	C₂×D₅	1	2	yes	—
96	C₂³×C₅	—	2	4	no	—
97	C₂³×C₅	#95	1	1	yes	characteristic
98	C₂³×C₅	#95	1	1	yes	characteristic
99	C₂³×C₅	C₂²⋊C₈	1	4	yes	—
100	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	—	2	16	no	—
102	C₂×C₄×C₅	—	2	32	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	#95	1	2	yes	—
110	C₂×C₄×C₅	#95	1	1	yes	characteristic
111	C₂×C₄×C₅	#95	1	1	yes	characteristic
112	C₂×C₄×C₅	#95	1	2	yes	—
113	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
115	C₂×C₄×C₅	#97	1	2	yes	—
116	C₂×C₄×C₅	#97	1	2	yes	—
117	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
118	C₂×C₄×C₅	#104	1	8	yes	—
119	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
120	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
121	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
127	#168	—	4	8	no	—
128	C₂³×C₄	—	2	2	no	—
129	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
130	C₂³×C₄	C₂²⋊D₅	1	2	yes	—
131	C₂²×C₈	—	10	4	no	—
132	C₂×C₄²	—	2	16	no	—
133	C₂×C₄²	—	2	4	no	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	—	2	4	no	—
136	C₂×C₄²	—	2	4	no	—
137	C₂×C₄²	#168	1	8	yes	complemented
138	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
139	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₅	1	2	yes	—
141	C₂×C₄²	C₄×D₅	1	2	yes	—
142	C₂×C₄²	D₂₀	1	1	yes	characteristic
143	C₂×C₄²	D₂₀	1	1	yes	characteristic
144	C₂×C₄²	C₂×Dic₅	1	4	yes	—
145	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
147	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
148	C₄⋊C₈	—	5	8	no	—
149	C₄⋊C₈	—	10	4	no	—
150	C₂²⋊C₈	—	10	4	no	—
151	C₂²⋊C₈	—	10	4	no	—
152	C₄×C₈	—	10	4	no	—
153	C₄×C₈	—	10	4	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	2	8	no	—
157	C₂²×C₅	#336	1	2	yes	—
158	C₂²×C₅	#336	1	2	yes	—
159	C₂²×C₅	#339	1	4	yes	—
160	C₂²×C₅	C₂×#95	1	1	yes	characteristic
161	C₂²×C₅	C₂×C₂²⋊C₈	1	2	yes	—
162	C₄×C₅	—	2	16	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	#336	1	2	yes	—
166	C₄×C₅	#336	1	2	yes	—
167	C₄×C₅	#339	1	4	yes	—
168	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
169	C₄×C₅	C₂×#97	1	1	yes	characteristic
170	C₄×C₅	C₂×#104	1	4	yes	—
171	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
172	C₂⁴	C₂²⋊Dic₅	1	1	yes	characteristic
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	C₂×#168	1	4	yes	—
178	C₂²×C₄	#681	1	4	yes	—
179	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
180	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
181	C₂²×C₄	#683	1	2	yes	—
182	C₂²×C₄	#683	1	2	yes	—
183	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
185	C₂²×C₄	#685	1	2	yes	—
186	C₂²×C₄	#685	1	2	yes	—
187	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
188	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
191	C₂×C₈	—	10	8	no	—
192	C₂×C₈	—	20	8	no	—
193	C₄²	—	2	16	no	—
194	C₄²	—	2	16	no	—
195	C₄²	—	2	8	no	—
196	C₄²	—	2	4	no	—
197	C₄²	C₂×#168	1	8	yes	complemented
198	C₄²	C₄×Dic₅	1	2	yes	—
199	C₄²	#683	1	2	yes	—
200	C₄²	#683	1	2	yes	—
201	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
202	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
203	C₄²	#685	1	2	yes	—
204	C₄²	#685	1	2	yes	—
205	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
206	C₄²	C₂×C₄×D₅	1	2	yes	—
207	C₄²	C₂×D₂₀	1	1	yes	characteristic
208	C₂×C₅	—	2	2	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	#1276	1	4	yes	complemented
211	C₂×C₅	C₂×#336	1	1	yes	characteristic
212	C₂×C₅	C₂×#339	1	2	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	C₂²⋊#168	1	4	yes	—
215	C₂³	#3963	1	1	yes	characteristic
216	C₂³	#3963	1	1	yes	characteristic
217	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
218	C₂×C₄	—	2	16	no	—
219	C₂×C₄	—	2	32	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	C₄×#168	1	2	yes	—
227	C₂×C₄	C₄×#168	1	2	yes	—
228	C₂×C₄	#3935	1	2	yes	—
229	C₂×C₄	#3935	1	2	yes	—
230	C₂×C₄	C₄⋊#168	1	4	yes	—
231	C₂×C₄	C₄⋊#168	1	4	yes	—
232	C₂×C₄	#3937	1	8	yes	—
233	C₂×C₄	C₂²⋊#168	1	4	yes	—
234	C₂×C₄	#3963	1	2	yes	—
235	C₂×C₄	#3963	1	2	yes	—
236	C₂×C₄	#3963	1	1	yes	characteristic
237	C₂×C₄	#3963	1	1	yes	characteristic
238	C₂×C₄	C₂²×#168	1	2	yes	—
239	C₂×C₄	C₂×#681	1	2	yes	—
240	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
241	C₂×C₄	C₂×#683	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
243	C₂×C₄	C₂×#685	1	2	yes	—
244	C₈	—	20	8	no	—
245	C₅	C₂×#1276	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	8	no	—
249	C₂²	#64743	1	4	yes	—
250	C₂²	#64744	1	2	yes	—
251	C₂²	#64744	1	2	yes	—
252	C₂²	C₂×C₂²⋊#168	1	2	yes	—
253	C₂²	C₂×#3963	1	1	yes	characteristic
254	C₄	—	2	16	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	#64743	1	4	yes	—
258	C₄	#64744	1	2	yes	—
259	C₄	#64744	1	2	yes	—
260	C₄	C₂×C₄×#168	1	1	yes	characteristic
261	C₄	C₂×#3935	1	1	yes	characteristic
262	C₄	C₂×C₄⋊#168	1	2	yes	—
263	C₄	C₂×#3937	1	4	yes	—
264	C₂	—	2	2	no	—
265	C₂	—	2	2	no	—
266	C₂	#10598102	1	4	yes	direct product factor
267	C₂	C₂×#64743	1	2	yes	—
268	C₂	C₂×#64744	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₂₀
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₄³×C₅	C₂
socle	C₂³×C₅	#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₄³	D₅
hypocenter	C₅	C₂×#1276
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #10598102	C₂	direct (N × H)
13. #65214	C₄	semidirect
15. C₂×C₄⋊#168	C₄	semidirect
26. C₂×C₄²×C₅	C₈	semidirect
41. C₄⋊#168	C₂×C₄	semidirect
77. C₄²×C₅	C₂×C₈	semidirect
137. C₂×C₄²	#168	semidirect
197. C₄²	C₂×#168	semidirect
210. C₂×C₅	#1276	semidirect
245. C₅	C₂×#1276	semidirect
266. C₂	#10598102	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
#10598102	2
C₂×#1276	5

Subgroups — C2×#10598102 #11988736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598102 #11988736