Subgroups — C₂×#10598213 #12014176

Order 1280.

694 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3950	C₂	1	1	yes	maximal, characteristic
3	C₂³×#672	C₂	1	1	yes	maximal, characteristic
4	#10598213	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂²×C₄⋊Dic₅	—	2	1	no	—
8	C₂×C₅×#107	—	2	2	no	—
9	C₂×C₅×#107	—	2	1	no	—
10	C₂×C₅×#107	C₄	1	4	yes	complemented
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×#3950	—	2	2	no	—
13	C₂×#3950	—	2	1	no	—
14	C₂×#3950	C₄	1	4	yes	complemented
15	C₂×#3950	C₂²	1	2	yes	—
16	C₂²×#672	—	2	2	no	—
17	C₂²×#672	—	2	2	no	—
18	C₂²×#672	C₄	1	4	yes	complemented
19	C₂²×#672	C₂²	1	2	yes	—
20	C₂×#1380	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	—	4	1	no	—
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×Dic₅	—	4	1	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₂²×C₅×C₈	Q₈	1	2	yes	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	4	1	no	—
35	C₂×C₅×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄⋊Dic₅	—	2	4	no	—
37	C₂×C₄⋊Dic₅	—	2	2	no	—
38	C₂×C₄⋊Dic₅	—	4	1	no	—
39	C₂×C₄⋊Dic₅	—	4	2	no	—
40	C₅×#107	—	2	4	no	—
41	C₅×#107	—	2	2	no	—
42	C₅×#107	C₂×C₄	1	4	yes	complemented
43	#3950	—	2	4	no	—
44	#3950	—	2	2	no	—
45	#3950	C₂×C₄	1	4	yes	complemented
46	C₂×#672	—	2	8	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	C₂×C₄	1	4	yes	complemented
50	C₂³×C₁₆	—	5	1	no	—
51	C₂²×#107	—	5	1	no	—
52	C₂²×#107	D₅	1	1	yes	characteristic
53	#1380	—	5	4	no	—
54	C₂⁴×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	4	4	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	—	8	2	no	—
58	C₂²×C₄×C₅	—	8	2	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₅	—	4	4	no	—
68	C₂²×Dic₅	—	4	2	no	—
69	C₂²×Dic₅	—	8	2	no	—
70	C₂²×Dic₅	—	8	2	no	—
71	C₂×C₅×C₈	C₄²	1	4	yes	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
77	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
78	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₅×C₄⋊C₄	—	4	4	no	—
86	C₅×C₄⋊C₄	—	4	2	no	—
87	C₄⋊Dic₅	—	2	4	no	—
88	C₄⋊Dic₅	—	4	4	no	—
89	C₄⋊Dic₅	—	4	2	no	—
90	#672	—	2	4	no	—
91	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₄⋊C₄	—	2	1	no	—
93	C₂²×C₄⋊C₄	—	10	1	no	—
94	C₂²×C₁₆	—	5	4	no	—
95	C₂²×C₁₆	—	5	2	no	—
96	C₂²×C₁₆	—	10	2	no	—
97	C₂²×C₁₆	—	10	2	no	—
98	C₂×#107	—	2	1	no	—
99	C₂×#107	—	2	2	no	—
100	C₂×#107	—	5	4	no	—
101	C₂×#107	—	5	2	no	—
102	C₂×#107	—	10	2	no	—
103	C₂×#107	—	10	1	no	—
104	C₂×#107	Dic₅	1	4	yes	complemented
105	C₂×#107	C₂×D₅	1	2	yes	—
106	C₂³×C₅	#102	1	2	yes	—
107	C₂³×C₅	#102	1	2	yes	—
108	C₂³×C₅	#107	1	2	yes	—
109	C₂³×C₅	D₁₆	1	2	yes	—
110	C₂³×C₅	SD₃₂	1	2	yes	—
111	C₂³×C₅	SD₃₂	1	2	yes	—
112	C₂³×C₅	Q₃₂	1	2	yes	—
113	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	8	8	no	—
116	C₂×C₄×C₅	—	8	2	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	#95	1	4	yes	—
119	C₂×C₄×C₅	#102	1	2	yes	—
120	C₂×C₄×C₅	#102	1	2	yes	—
121	C₂×C₄×C₅	#103	1	4	yes	—
122	C₂×C₄×C₅	#103	1	4	yes	—
123	C₂×C₄×C₅	#106	1	4	yes	—
124	C₂×C₄×C₅	#107	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×Dic₅	—	4	4	no	—
132	C₂×Dic₅	—	8	8	no	—
133	C₂×Dic₅	—	8	2	no	—
134	C₂×Dic₅	—	8	2	no	—
135	C₅×C₈	#95	1	4	yes	—
136	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
137	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂³×C₄	—	4	1	no	—
141	C₂³×C₄	—	20	1	no	—
142	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
143	C₂²×C₈	Dic₁₀	1	2	yes	—
144	C₂²×C₈	C₄×D₅	1	2	yes	—
145	C₂²×C₈	C₄×D₅	1	2	yes	—
146	C₂²×C₈	D₂₀	1	2	yes	—
147	C₂²×C₈	C₂×Dic₅	1	2	yes	—
148	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
150	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	4	1	no	—
154	C₂×C₄⋊C₄	—	4	2	no	—
155	C₂×C₄⋊C₄	—	10	4	no	—
156	C₂×C₄⋊C₄	—	10	2	no	—
157	C₂×C₄⋊C₄	—	20	1	no	—
158	C₂×C₄⋊C₄	—	20	2	no	—
159	C₂×C₁₆	—	5	4	no	—
160	C₂×C₁₆	—	10	8	no	—
161	C₂×C₁₆	—	10	2	no	—
162	C₂×C₁₆	—	10	2	no	—
163	#107	—	2	4	no	—
164	#107	—	2	2	no	—
165	#107	—	5	4	no	—
166	#107	—	10	4	no	—
167	#107	—	10	2	no	—
168	#107	C₂×Dic₅	1	4	yes	complemented
169	C₂²×C₅	#340	1	4	yes	—
170	C₂²×C₅	#357	1	4	yes	—
171	C₂²×C₅	#357	1	4	yes	—
172	C₂²×C₅	#358	1	4	yes	—
173	C₂²×C₅	#358	1	4	yes	—
174	C₂²×C₅	#366	1	4	yes	—
175	C₂²×C₅	#367	1	4	yes	—
176	C₂²×C₅	C₂×#102	1	1	yes	characteristic
177	C₂²×C₅	C₂×#102	1	1	yes	characteristic
178	C₂²×C₅	C₂×#107	1	1	yes	characteristic
179	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
180	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
181	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
182	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
183	C₄×C₅	—	8	4	no	—
184	C₄×C₅	#340	1	4	yes	—
185	C₄×C₅	C₂×#95	1	1	yes	characteristic
186	C₄×C₅	C₂×#103	1	1	yes	characteristic
187	C₄×C₅	C₂×#103	1	1	yes	characteristic
188	C₄×C₅	C₂×#106	1	1	yes	characteristic
189	Dic₅	—	8	4	no	—
190	C₂⁴	#686	1	1	yes	characteristic
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	8	2	no	—
194	C₂²×C₄	—	8	2	no	—
195	C₂²×C₄	—	20	4	no	—
196	C₂²×C₄	—	20	2	no	—
197	C₂²×C₄	—	40	2	no	—
198	C₂²×C₄	—	40	2	no	—
199	C₂²×C₄	#683	1	2	yes	—
200	C₂²×C₄	#685	1	2	yes	—
201	C₂²×C₄	#686	1	1	yes	characteristic
202	C₂²×C₄	#687	1	2	yes	—
203	C₂²×C₄	#688	1	2	yes	—
204	C₂²×C₄	#689	1	2	yes	—
205	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
207	C₂×C₈	C₄×Dic₅	1	4	yes	—
208	C₂×C₈	#683	1	2	yes	—
209	C₂×C₈	#683	1	4	yes	—
210	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
211	C₂×C₈	#685	1	2	yes	—
212	C₂×C₈	#685	1	4	yes	—
213	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
214	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
217	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
218	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	10	4	no	—
224	C₄⋊C₄	—	20	4	no	—
225	C₄⋊C₄	—	20	2	no	—
226	C₁₆	—	10	4	no	—
227	C₂×C₅	#1380	1	8	yes	complemented
228	C₂×C₅	C₂×#340	1	1	yes	characteristic
229	C₂×C₅	C₂×#357	1	1	yes	characteristic
230	C₂×C₅	C₂×#357	1	1	yes	characteristic
231	C₂×C₅	C₂×#358	1	1	yes	characteristic
232	C₂×C₅	C₂×#358	1	1	yes	characteristic
233	C₂×C₅	C₂×#366	1	1	yes	characteristic
234	C₂×C₅	C₂×#367	1	1	yes	characteristic
235	C₂³	#3939	1	2	yes	—
236	C₂³	#3941	1	2	yes	—
237	C₂³	#3958	1	2	yes	—
238	C₂³	#3959	1	2	yes	—
239	C₂³	#3960	1	2	yes	—
240	C₂³	#3961	1	2	yes	—
241	C₂³	#3964	1	2	yes	—
242	C₂³	C₂×#686	1	1	yes	characteristic
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	20	4	no	—
248	C₂×C₄	—	40	2	no	—
249	C₂×C₄	—	40	2	no	—
250	C₂×C₄	—	40	8	no	—
251	C₂×C₄	#3939	1	2	yes	—
252	C₂×C₄	#3940	1	4	yes	—
253	C₂×C₄	#3941	1	2	yes	—
254	C₂×C₄	#3942	1	4	yes	—
255	C₂×C₄	#3963	1	4	yes	—
256	C₂×C₄	#3964	1	2	yes	—
257	C₂×C₄	#3967	1	4	yes	—
258	C₂×C₄	C₂×#683	1	1	yes	characteristic
259	C₂×C₄	C₂×#685	1	1	yes	characteristic
260	C₂×C₄	C₂×#687	1	1	yes	characteristic
261	C₂×C₄	C₂×#688	1	1	yes	characteristic
262	C₂×C₄	C₂×#689	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
264	C₈	#3963	1	4	yes	—
265	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
266	C₈	C₂×#683	1	1	yes	characteristic
267	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
268	C₈	C₂×#685	1	1	yes	characteristic
269	C₅	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
270	C₂²	#64709	1	4	yes	—
271	C₂²	#64710	1	4	yes	—
272	C₂²	#64711	1	4	yes	—
273	C₂²	#64712	1	4	yes	—
274	C₂²	#64749	1	4	yes	—
275	C₂²	#64782	1	4	yes	—
276	C₂²	#64784	1	4	yes	—
277	C₂²	C₂×#3939	1	1	yes	characteristic
278	C₂²	C₂×#3941	1	1	yes	characteristic
279	C₂²	C₂×#3958	1	1	yes	characteristic
280	C₂²	C₂×#3959	1	1	yes	characteristic
281	C₂²	C₂×#3960	1	1	yes	characteristic
282	C₂²	C₂×#3961	1	1	yes	characteristic
283	C₂²	C₂×#3964	1	1	yes	characteristic
284	C₄	—	8	4	no	—
285	C₄	—	40	4	no	—
286	C₄	#64749	1	4	yes	—
287	C₄	C₂×#3940	1	1	yes	characteristic
288	C₄	C₂×#3942	1	1	yes	characteristic
289	C₄	C₂×#3963	1	1	yes	characteristic
290	C₄	C₂×#3967	1	1	yes	characteristic
291	C₂	#10598213	1	8	yes	direct product factor
292	C₂	C₂×#64709	1	1	yes	characteristic
293	C₂	C₂×#64710	1	1	yes	characteristic
294	C₂	C₂×#64711	1	1	yes	characteristic
295	C₂	C₂×#64712	1	1	yes	characteristic
296	C₂	C₂×#64749	1	1	yes	characteristic
297	C₂	C₂×#64782	1	1	yes	characteristic
298	C₂	C₂×#64784	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#686
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×#107	C₂
socle, prime radical	C₂⁴×C₅	D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×#107	D₅
hypocenter	C₅	C₂×#1380
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
4. #10598213	C₂	direct (N × H)
10. C₂×C₅×#107	C₄	semidirect
14. C₂×#3950	C₄	semidirect
18. C₂²×#672	C₄	semidirect
42. C₅×#107	C₂×C₄	semidirect
45. #3950	C₂×C₄	semidirect
49. C₂×#672	C₂×C₄	semidirect
104. C₂×#107	Dic₅	semidirect
168. #107	C₂×Dic₅	semidirect
227. C₂×C₅	#1380	semidirect
269. C₅	C₂×#1380	semidirect
291. C₂	#10598213	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×#107	2
C₂²×#3950	2
C₂³×#672	2
#10598213	2
C₂×#1380	5

Subgroups — C2×#10598213 #12014176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598213 #12014176