Subgroups — C₂×#10601311 #12099755

Order 1280.

1046 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3954	C₂	1	4	yes	maximal
3	C₂×#65213	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
5	#10601311	C₂	1	8	yes	maximal, direct product factor
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×#681	C₂²	1	2	yes	—
9	C₂³×#168	C₂²	1	2	yes	—
10	C₅×#423	—	2	2	no	—
11	C₅×#423	C₂²	1	4	yes	—
12	C₂×C₅×C₄⋊C₈	—	2	2	no	—
13	C₂×#3954	—	2	16	no	—
14	C₂×#3954	C₂²	1	16	yes	—
15	#65213	—	2	2	no	—
16	#65213	C₂²	1	4	yes	—
17	C₂×C₄⋊#168	—	2	2	no	—
18	C₂×C₄×#168	C₄	1	2	yes	—
19	C₂×C₄×#168	C₄	1	4	yes	—
20	C₂×C₄×#168	C₄	1	2	yes	—
21	C₂×C₄×#168	C₂²	1	4	yes	—
22	C₂×#1848	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	Q₈	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	8	no	—
27	C₂×C₅×M₁₆	C₂³	1	4	yes	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	Q₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	2	yes	—
36	C₂×#681	—	2	2	no	—
37	C₂×#681	—	2	8	no	—
38	C₂×#681	C₂³	1	4	yes	—
39	C₂²×#168	—	2	2	no	—
40	C₂²×#168	C₂×C₄	1	4	yes	—
41	C₂²×#168	C₂×C₄	1	4	yes	—
42	C₂²×#168	D₄	1	8	yes	—
43	C₂²×#168	Q₈	1	8	yes	—
44	C₂²×#168	C₂³	1	4	yes	—
45	C₅×C₄⋊C₈	—	2	8	no	—
46	#3954	—	2	32	no	—
47	C₄⋊#168	—	2	8	no	—
48	C₄×#168	C₂×C₄	1	8	yes	—
49	C₄×#168	C₂×C₄	1	8	yes	—
50	C₂²×#108	—	5	4	no	—
51	C₂×#423	—	5	1	no	—
52	C₂×#423	D₅	1	1	yes	characteristic
53	C₂²×C₄×C₈	—	5	1	no	—
54	#1848	—	5	8	no	—
55	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	8	no	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₅×M₁₆	—	2	8	no	—
70	C₅×M₁₆	—	4	4	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	—	4	4	no	—
73	C₄²×C₅	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅	C₄⋊C₄	1	4	yes	—
75	C₄²×C₅	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅	C₂²×C₄	1	2	yes	—
77	C₄²×C₅	C₂²×C₄	1	2	yes	—
78	C₄²×C₅	C₂×D₄	1	4	yes	—
79	#681	—	2	8	no	—
80	#681	—	4	4	no	—
81	C₂×#168	—	2	4	no	—
82	C₂×#168	—	4	4	no	—
83	C₂×#168	C₄⋊C₄	1	16	yes	—
84	C₂×#168	C₄⋊C₄	1	16	yes	—
85	C₂×#168	C₂²×C₄	1	8	yes	—
86	C₂×#168	C₂×D₄	1	8	yes	—
87	C₂×#168	C₂×Q₈	1	8	yes	—
88	C₂²×M₁₆	—	5	2	no	—
89	C₂²×M₁₆	C₂×D₅	1	2	yes	—
90	C₂³×C₈	—	5	2	no	—
91	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
92	C₂×#108	—	5	16	no	—
93	C₂×#108	—	10	16	no	—
94	#423	—	2	2	no	—
95	#423	—	5	4	no	—
96	#423	—	10	2	no	—
97	#423	C₂×D₅	1	4	yes	—
98	C₂×C₄⋊C₈	—	2	2	no	—
99	C₂×C₄⋊C₈	—	10	2	no	—
100	C₂×C₄×C₈	—	5	2	no	—
101	C₂×C₄×C₈	—	5	4	no	—
102	C₂×C₄×C₈	—	5	2	no	—
103	C₂×C₄×C₈	—	5	4	no	—
104	C₂³×C₅	—	2	4	no	—
105	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
108	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	2	8	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	#108	1	8	yes	complemented
113	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₅	#127	1	2	yes	—
120	C₂×C₄×C₅	#127	1	2	yes	—
121	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
128	C₅×C₈	—	4	4	no	—
129	#168	—	4	4	no	—
130	#168	C₂×C₄⋊C₄	1	16	yes	—
131	C₂³×C₄	Dic₁₀	1	2	yes	—
132	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	8	no	—
135	C₂×M₁₆	—	5	4	no	—
136	C₂×M₁₆	—	10	2	no	—
137	C₂×M₁₆	—	10	8	no	—
138	C₂×M₁₆	C₂²×D₅	1	4	yes	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	5	4	no	—
141	C₂²×C₈	—	5	4	no	—
142	C₂²×C₈	—	5	8	no	—
143	C₂²×C₈	—	5	8	no	—
144	C₂²×C₈	—	5	4	no	—
145	C₂²×C₈	—	10	2	no	—
146	C₂×C₄²	Dic₁₀	1	2	yes	—
147	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₅	1	2	yes	—
149	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
150	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
151	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
152	C₂×C₄²	C₂²×D₅	1	2	yes	—
153	#108	—	10	32	no	—
154	C₄⋊C₈	—	2	8	no	—
155	C₄⋊C₈	—	10	8	no	—
156	C₄×C₈	—	5	8	no	—
157	C₄×C₈	—	5	8	no	—
158	C₂²×C₅	—	2	8	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	#384	1	2	yes	—
162	C₂²×C₅	#384	1	2	yes	—
163	C₂²×C₅	C₂×#108	1	4	yes	—
164	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
166	C₄×C₅	—	2	4	no	—
167	C₄×C₅	#384	1	2	yes	—
168	C₄×C₅	#384	1	2	yes	—
169	C₄×C₅	C₂×#108	1	8	yes	complemented
170	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
171	C₄×C₅	C₂×#127	1	1	yes	characteristic
172	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	#683	1	2	yes	—
176	C₂²×C₄	#683	1	4	yes	—
177	C₂²×C₄	#683	1	2	yes	—
178	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
179	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₅	1	4	yes	—
183	C₂²×C₄	Q₈×D₅	1	4	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	10	8	no	—
190	M₁₆	—	20	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	5	8	no	—
194	C₂×C₈	—	5	8	no	—
195	C₂×C₈	—	5	8	no	—
196	C₂×C₈	—	5	16	no	—
197	C₂×C₈	—	5	16	no	—
198	C₂×C₈	—	10	4	no	—
199	C₂×C₈	—	20	4	no	—
200	C₄²	#683	1	2	yes	—
201	C₄²	#683	1	4	yes	—
202	C₄²	#683	1	2	yes	—
203	C₄²	C₂×C₄×D₅	1	2	yes	—
204	C₄²	C₂×C₄×D₅	1	2	yes	—
205	C₄²	C₂×C₂²⋊D₅	1	4	yes	—
206	C₂×C₅	—	2	2	no	—
207	C₂×C₅	—	2	2	no	—
208	C₂×C₅	#1848	1	4	yes	complemented
209	C₂×C₅	C₂×#384	1	1	yes	characteristic
210	C₂×C₅	C₂²×#108	1	2	yes	—
211	C₂³	—	2	4	no	—
212	C₂³	#4034	1	4	yes	—
213	C₂³	C₂×#683	1	1	yes	characteristic
214	C₂³	C₂×#683	1	1	yes	characteristic
215	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	#3954	1	8	yes	complemented
220	C₂×C₄	#4034	1	4	yes	—
221	C₂×C₄	D₅×C₄⋊C₄	1	4	yes	—
222	C₂×C₄	D₅×C₄⋊C₄	1	4	yes	—
223	C₂×C₄	C₂×#683	1	4	yes	—
224	C₂×C₄	C₂×#683	1	4	yes	—
225	C₂×C₄	C₂×#683	1	1	yes	characteristic
226	C₂×C₄	C₂×#683	1	1	yes	characteristic
227	C₂×C₄	#4086	1	2	yes	—
228	C₂×C₄	#4086	1	2	yes	—
229	C₂×C₄	#4090	1	2	yes	—
230	C₂×C₄	#4090	1	2	yes	—
231	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×D₅	1	2	yes	—
233	C₂×C₄	C₂×Q₈×D₅	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
235	C₈	—	4	4	no	—
236	C₈	—	5	16	no	—
237	C₈	—	20	4	no	—
238	C₅	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	#65265	1	2	yes	—
243	C₂²	#65265	1	2	yes	—
244	C₂²	C₂×#3954	1	4	yes	—
245	C₂²	C₂×#4034	1	2	yes	—
246	C₂²	C₂²×#683	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	#65265	1	2	yes	—
249	C₄	#65265	1	2	yes	—
250	C₄	C₂×#3954	1	8	yes	complemented
251	C₄	C₂×D₅×C₄⋊C₄	1	2	yes	—
252	C₄	C₂×#4086	1	1	yes	characteristic
253	C₄	C₂×#4090	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	#10601311	1	4	yes	direct product factor
257	C₂	C₂×#65265	1	1	yes	characteristic
258	C₂	C₂²×#3954	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#423	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#423	D₅
hypocenter	C₅	C₂×#1848
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
5. #10601311	C₂	direct (N × H)
112. C₂×C₄×C₅	#108	semidirect
169. C₄×C₅	C₂×#108	semidirect
208. C₂×C₅	#1848	semidirect
219. C₂×C₄	#3954	semidirect
238. C₅	C₂×#1848	semidirect
250. C₄	C₂×#3954	semidirect
256. C₂	#10601311	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×#423	2
C₂²×#3954	2
C₂×#65213	2
C₂²×C₄×#168	2
#10601311	2
C₂×#1848	5

Subgroups — C2×#10601311 #12099755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601311 #12099755