Subgroups — C₂×#10598196 #11989244

Order 1280.

690 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	#10598196	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×#681	—	2	2	no	—
7	C₂×C₅×C₄⋊C₈	—	2	2	no	—
8	C₂×C₅×C₄⋊C₈	—	2	2	no	—
9	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	#65213	—	2	2	no	—
12	#65213	—	2	2	no	—
13	#65213	C₄	1	4	yes	—
14	#65213	C₂²	1	4	yes	—
15	C₂×C₄⋊#168	—	2	2	no	—
16	C₂×C₄⋊#168	C₄	1	4	yes	—
17	C₂×#1292	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	4	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×#681	—	2	4	no	—
34	C₂×#681	—	2	4	no	—
35	C₂×#681	—	4	4	no	—
36	C₂×#681	—	4	4	no	—
37	C₂²×#168	—	2	4	no	—
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₄⋊#168	—	2	4	no	—
41	C₄⋊#168	—	2	4	no	—
42	C₄⋊#168	—	2	4	no	—
43	C₄⋊#168	C₂×C₄	1	8	yes	—
44	C₂×#423	—	5	2	no	—
45	C₂²×C₄⋊C₈	D₅	1	1	yes	characteristic
46	#1292	—	5	4	no	—
47	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	D₈	1	2	yes	—
63	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
64	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	4	4	no	—
70	C₂×C₅×C₈	—	4	4	no	—
71	C₂×C₅×C₈	—	4	4	no	—
72	C₄²×C₅	C₄²	1	2	yes	—
73	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅	C₂²×C₄	1	2	yes	—
79	C₄²×C₅	C₂×D₄	1	2	yes	—
80	#681	—	4	8	no	—
81	#681	—	4	4	no	—
82	#681	—	4	4	no	—
83	C₂×#168	—	2	8	no	—
84	C₂×#168	—	4	8	no	—
85	C₂²×M₁₆	—	10	2	no	—
86	C₂³×C₈	Dic₅	1	2	yes	—
87	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
88	#423	—	5	4	no	—
89	#423	—	5	4	no	—
90	#423	—	10	2	no	—
91	#423	—	10	2	no	—
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	—	2	2	no	—
94	C₂×C₄⋊C₈	—	5	4	no	—
95	C₂×C₄⋊C₈	—	10	2	no	—
96	C₂×C₄⋊C₈	Dic₅	1	2	yes	—
97	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
98	C₂³×C₅	—	2	2	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	#95	1	1	yes	characteristic
101	C₂³×C₅	#95	1	1	yes	characteristic
102	C₂³×C₅	#106	1	2	yes	—
103	C₂³×C₅	#107	1	2	yes	—
104	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	#95	1	2	yes	—
117	C₂×C₄×C₅	#95	1	1	yes	characteristic
118	C₂×C₄×C₅	#95	1	2	yes	—
119	C₂×C₄×C₅	#95	1	1	yes	characteristic
120	C₂×C₄×C₅	#100	1	2	yes	—
121	C₂×C₄×C₅	#101	1	2	yes	—
122	C₂×C₄×C₅	#102	1	4	yes	—
123	C₂×C₄×C₅	#102	1	4	yes	—
124	C₂×C₄×C₅	#103	1	4	yes	—
125	C₂×C₄×C₅	#103	1	4	yes	—
126	C₂×C₄×C₅	#106	1	2	yes	—
127	C₂×C₄×C₅	#107	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
135	C₅×C₈	—	4	4	no	—
136	#168	—	4	8	no	—
137	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
138	C₂³×C₄	D₂₀	1	1	yes	characteristic
139	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
140	C₂×M₁₆	—	10	4	no	—
141	C₂×M₁₆	—	10	4	no	—
142	C₂×M₁₆	—	20	4	no	—
143	C₂×M₁₆	—	20	4	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	4	4	no	—
147	C₂²×C₈	—	10	4	no	—
148	C₂²×C₈	C₂×Dic₅	1	4	yes	—
149	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
150	C₂×C₄²	C₄×D₅	1	2	yes	—
151	C₂×C₄²	C₄×D₅	1	2	yes	—
152	C₂×C₄²	D₂₀	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
156	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	5	8	no	—
160	C₄⋊C₈	—	10	4	no	—
161	C₄⋊C₈	—	10	4	no	—
162	C₄⋊C₈	—	10	4	no	—
163	C₂²×C₅	—	2	4	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	#338	1	2	yes	—
170	C₂²×C₅	#340	1	2	yes	—
171	C₂²×C₅	#340	1	2	yes	—
172	C₂²×C₅	#343	1	2	yes	—
173	C₂²×C₅	C₂×#95	1	1	yes	characteristic
174	C₂²×C₅	C₂×#106	1	1	yes	characteristic
175	C₂²×C₅	C₂×#107	1	1	yes	characteristic
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	#340	1	4	yes	—
181	C₄×C₅	#343	1	2	yes	—
182	C₄×C₅	C₂×#100	1	1	yes	characteristic
183	C₄×C₅	C₂×#101	1	1	yes	characteristic
184	C₄×C₅	C₂×#102	1	2	yes	—
185	C₄×C₅	C₂×#103	1	2	yes	—
186	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	#677	1	4	yes	—
191	C₂²×C₄	D₄₀	1	2	yes	—
192	C₂²×C₄	Dic₂₀	1	2	yes	—
193	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
194	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
195	C₂²×C₄	#683	1	2	yes	—
196	C₂²×C₄	#683	1	2	yes	—
197	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
200	C₂²×C₄	#685	1	2	yes	—
201	C₂²×C₄	#685	1	2	yes	—
202	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
207	M₁₆	—	20	8	no	—
208	M₁₆	—	20	4	no	—
209	M₁₆	—	20	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	10	8	no	—
215	C₂×C₈	—	20	8	no	—
216	C₄²	C₄×Dic₅	1	2	yes	—
217	C₄²	#683	1	2	yes	—
218	C₄²	#683	1	2	yes	—
219	C₄²	#685	1	2	yes	—
220	C₄²	#685	1	2	yes	—
221	C₄²	C₂²⋊Dic₅	1	2	yes	—
222	C₄²	C₂×C₄×D₅	1	2	yes	—
223	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	#1292	1	4	yes	complemented
228	C₂×C₅	C₂×#338	1	1	yes	characteristic
229	C₂×C₅	C₂×#340	1	1	yes	characteristic
230	C₂×C₅	C₂×#343	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	#3949	1	2	yes	—
234	C₂³	#3950	1	2	yes	—
235	C₂³	#3963	1	1	yes	characteristic
236	C₂³	#3963	1	1	yes	characteristic
237	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	#3948	1	4	yes	—
250	C₂×C₄	#3948	1	4	yes	—
251	C₂×C₄	#3949	1	2	yes	—
252	C₂×C₄	#3950	1	2	yes	—
253	C₂×C₄	#3953	1	4	yes	—
254	C₂×C₄	#3953	1	4	yes	—
255	C₂×C₄	#3963	1	2	yes	—
256	C₂×C₄	#3963	1	1	yes	characteristic
257	C₂×C₄	#3963	1	2	yes	—
258	C₂×C₄	#3963	1	1	yes	characteristic
259	C₂×C₄	#3965	1	2	yes	—
260	C₂×C₄	#3968	1	2	yes	—
261	C₂×C₄	C₂×#677	1	2	yes	—
262	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
263	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
265	C₂×C₄	C₂×#683	1	2	yes	—
266	C₂×C₄	C₂×#685	1	2	yes	—
267	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	—	20	8	no	—
270	C₅	C₂×#1292	1	1	yes	characteristic, complemented, Sylow
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	#64750	1	2	yes	—
278	C₂²	#64771	1	2	yes	—
279	C₂²	#64771	1	2	yes	—
280	C₂²	#64780	1	2	yes	—
281	C₂²	C₂×#3949	1	1	yes	characteristic
282	C₂²	C₂×#3950	1	1	yes	characteristic
283	C₂²	C₂×#3963	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	#64750	1	2	yes	—
289	C₄	#64771	1	4	yes	—
290	C₄	C₂×#3948	1	2	yes	—
291	C₄	C₂×#3953	1	2	yes	—
292	C₄	C₂×#3965	1	1	yes	characteristic
293	C₄	C₂×#3968	1	1	yes	characteristic
294	C₂	—	2	2	no	—
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	#10598196	1	4	yes	direct product factor
298	C₂	C₂×#64750	1	1	yes	characteristic
299	C₂	C₂×#64771	1	1	yes	characteristic
300	C₂	C₂×#64780	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3963
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂²×C₅×C₄⋊C₈	C₂
socle	C₂³×C₅	#95
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂²×C₄⋊C₈	D₅
hypocenter	C₅	C₂×#1292
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #10598196	C₂	direct (N × H)
227. C₂×C₅	#1292	semidirect
270. C₅	C₂×#1292	semidirect
297. C₂	#10598196	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×C₄⋊C₈	2
C₂×#65213	2
#10598196	2
C₂×#1292	5

Subgroups — C2×#10598196 #11989244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598196 #11989244