Subgroups — C₂×#10604107 #13043409

Order 1280.

694 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65721	C₂	1	2	yes	maximal
2	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#10604107	C₂	1	8	yes	maximal, direct product factor
4	C₂×#64850	C₂	1	2	yes	maximal, complemented
5	C₂×#64849	C₂	1	2	yes	maximal, complemented
6	C₂×#4130	—	2	2	no	—
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	#65721	—	2	4	no	—
9	#65721	C₂²	1	8	yes	—
10	C₂×#3994	C₂²	1	4	yes	—
11	C₂×#3993	—	2	2	no	—
12	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
13	C₂²×#672	C₄	1	1	yes	characteristic, complemented
14	C₂×D₅×C₁₆	C₄	1	4	yes	complemented
15	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×D₅×C₁₆	C₂²	1	4	yes	—
20	#64850	C₂²	1	8	yes	complemented
21	#64849	C₂²	1	8	yes	complemented
22	C₂×#2158	—	5	1	no	maximal, Sylow
23	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
24	C₂×C₂²⋊AGL(1,𝔽₅)	—	4	2	no	—
25	#4130	—	2	8	no	—
26	#4130	—	4	4	no	—
27	C₂×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
28	C₂×C₄×AGL(1,𝔽₅)	—	4	2	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅	D₄	1	1	yes	characteristic
34	C₂×C₈×D₅	D₄	1	2	yes	—
35	C₂×C₈×D₅	Q₈	1	2	yes	—
36	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₅	C₂³	1	2	yes	—
39	#3994	—	2	4	no	—
40	#3994	C₂³	1	8	yes	—
41	#3993	—	2	8	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
45	C₂×#672	C₂×C₄	1	1	yes	characteristic
46	C₂×#672	C₂×C₄	1	1	yes	characteristic
47	C₂×#672	C₂×C₄	1	4	yes	complemented
48	D₅×C₁₆	C₂×C₄	1	2	yes	—
49	D₅×C₁₆	C₂×C₄	1	2	yes	—
50	D₅×C₁₆	C₂×C₄	1	4	yes	complemented
51	D₅×C₁₆	C₂×C₄	1	4	yes	complemented
52	D₅×C₁₆	C₂³	1	2	yes	—
53	D₅×C₁₆	C₂³	1	2	yes	—
54	C₂³×C₁₆	—	5	1	no	—
55	C₂×#427	—	5	2	no	—
56	#2158	—	5	8	no	—
57	C₂×#367	—	5	2	no	—
58	C₂×#366	—	5	2	no	—
59	C₂³×D₅	D₈	1	1	yes	characteristic
60	C₂²×AGL(1,𝔽₅)	—	4	4	no	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₅	Q₁₆	1	1	yes	characteristic
63	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₅	D₈	1	1	yes	characteristic
66	C₂×C₄×D₅	D₈	1	2	yes	—
67	C₂×C₄×D₅	Q₁₆	1	2	yes	—
68	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
69	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
71	C₂²⋊AGL(1,𝔽₅)	—	4	8	no	—
72	C₄⋊AGL(1,𝔽₅)	—	2	8	no	—
73	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
74	C₄×AGL(1,𝔽₅)	—	4	8	no	—
75	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
80	C₂×#168	C₄⋊C₄	1	2	yes	—
81	C₂×#168	C₂×Q₈	1	1	yes	characteristic
82	C₂×#168	C₂×Q₈	1	2	yes	—
83	C₈×D₅	C₄⋊C₄	1	2	yes	—
84	C₈×D₅	C₄⋊C₄	1	2	yes	—
85	C₈×D₅	C₄⋊C₄	1	2	yes	—
86	C₈×D₅	C₂²×C₄	1	1	yes	characteristic
87	C₈×D₅	C₂²×C₄	1	2	yes	—
88	C₈×D₅	C₂×D₄	1	2	yes	—
89	C₈×D₅	C₂×D₄	1	1	yes	characteristic
90	C₈×D₅	C₂×Q₈	1	1	yes	characteristic
91	C₈×D₅	C₂×Q₈	1	2	yes	—
92	C₈×D₅	C₂⁴	1	1	yes	characteristic
93	C₅×C₁₆	C₂²×C₄	1	2	yes	—
94	C₅×C₁₆	C₂²×C₄	1	2	yes	—
95	#672	C₂²×C₄	1	2	yes	—
96	#672	C₂²×C₄	1	2	yes	—
97	C₂³×C₈	—	5	1	no	—
98	C₂×#117	—	10	2	no	—
99	C₂²×C₁₆	—	5	1	no	—
100	C₂²×C₁₆	—	5	1	no	—
101	C₂²×C₁₆	—	5	1	no	—
102	C₂²×C₁₆	—	5	1	no	—
103	C₂²×C₁₆	—	5	1	no	—
104	C₂²×C₁₆	—	5	4	no	—
105	C₂²×C₁₆	—	5	4	no	—
106	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
107	#427	—	5	8	no	—
108	#427	—	10	4	no	—
109	C₂×#107	—	5	4	no	—
110	C₂×#106	—	10	2	no	—
111	#367	—	5	8	no	—
112	#366	—	5	8	no	—
113	C₂³×C₅	#107	1	1	yes	characteristic
114	C₂²×D₅	—	2	4	no	—
115	C₂²×D₅	#107	1	1	yes	characteristic
116	C₂²×D₅	#107	1	2	yes	—
117	C₂²×D₅	C₂×D₈	1	2	yes	—
118	C₂²×D₅	C₂×D₈	1	1	yes	characteristic
119	C₂×AGL(1,𝔽₅)	—	4	8	no	—
120	C₂×AGL(1,𝔽₅)	—	8	8	no	—
121	C₂×C₄×C₅	#107	1	2	yes	—
122	C₂×C₄×C₅	#107	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×Dic₅	#107	1	2	yes	—
126	C₂×Dic₅	#107	1	1	yes	characteristic
127	C₂×Dic₅	C₂×Q₁₆	1	2	yes	—
128	C₂×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
129	C₄×D₅	#107	1	2	yes	—
130	C₄×D₅	#107	1	2	yes	—
131	C₄×D₅	#107	1	2	yes	—
132	C₄×D₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₄×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₄×D₅	C₂×D₈	1	2	yes	—
135	C₄×D₅	C₂×D₈	1	1	yes	characteristic
136	C₄×D₅	C₂×Q₁₆	1	2	yes	—
137	C₄×D₅	C₂×Q₁₆	1	1	yes	characteristic
138	C₄×D₅	C₂²×D₄	1	1	yes	characteristic
139	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
140	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
142	#168	C₂×C₄⋊C₄	1	1	yes	characteristic
143	#168	C₂×C₄⋊C₄	1	2	yes	—
144	#168	C₂²×Q₈	1	1	yes	characteristic
145	C₂³×C₄	—	5	1	no	—
146	C₂²×C₈	—	5	1	no	—
147	C₂²×C₈	—	5	1	no	—
148	C₂²×C₈	—	5	2	no	—
149	C₂²×C₈	—	5	1	no	—
150	C₂²×C₈	—	5	2	no	—
151	C₂²×C₈	—	5	1	no	—
152	C₂²×C₈	—	5	2	no	—
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	5	2	no	—
155	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	#117	—	10	8	no	—
157	#117	—	20	4	no	—
158	C₂×C₄⋊C₄	—	10	4	no	—
159	C₂×C₂²⋊C₄	—	20	2	no	—
160	C₂×C₄²	—	20	2	no	—
161	C₂×C₁₆	—	5	1	no	—
162	C₂×C₁₆	—	5	1	no	—
163	C₂×C₁₆	—	5	4	no	—
164	C₂×C₁₆	—	5	2	no	—
165	C₂×C₁₆	—	5	2	no	—
166	C₂×C₁₆	—	5	4	no	—
167	C₂×C₁₆	—	5	4	no	—
168	C₂×C₁₆	—	5	2	no	—
169	C₂×C₁₆	—	5	2	no	—
170	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
173	#107	—	5	8	no	—
174	#107	—	10	4	no	—
175	#106	—	10	8	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	C₂×#107	1	2	yes	—
178	C₂²×C₅	C₂×#107	1	1	yes	characteristic
179	C₂×D₅	—	2	2	no	—
180	C₂×D₅	—	2	4	no	—
181	C₂×D₅	—	2	4	no	—
182	C₂×D₅	C₂×#107	1	2	yes	—
183	C₂×D₅	C₂×#107	1	1	yes	characteristic
184	C₂×D₅	#508	1	4	yes	—
185	C₂×D₅	C₂²×D₈	1	1	yes	characteristic
186	AGL(1,𝔽₅)	—	8	8	no	—
187	C₄×C₅	C₂×#107	1	1	yes	characteristic
188	C₄×C₅	C₂×#107	1	2	yes	—
189	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
190	Dic₅	C₂×#107	1	2	yes	—
191	Dic₅	C₂×#107	1	1	yes	characteristic
192	Dic₅	C₂²×Q₁₆	1	1	yes	characteristic
193	C₂⁴	—	5	1	no	—
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂²×C₄	—	5	2	no	—
197	C₂²×C₄	—	5	2	no	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	2	no	—
200	C₂²×C₄	—	5	2	no	—
201	C₂²×C₄	—	5	1	no	—
202	C₂²×C₄	—	5	1	no	—
203	C₂²×C₄	—	20	4	no	—
204	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₂×C₈	—	5	1	no	—
206	C₂×C₈	—	5	2	no	—
207	C₂×C₈	—	5	1	no	—
208	C₂×C₈	—	5	2	no	—
209	C₂×C₈	—	5	2	no	—
210	C₂×C₈	—	5	1	no	—
211	C₂×C₈	—	5	2	no	—
212	C₂×C₈	—	5	2	no	—
213	C₂×C₈	—	5	2	no	—
214	C₂×C₈	—	5	1	no	—
215	C₂×C₈	—	5	2	no	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	—	5	1	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
221	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
223	C₄⋊C₄	—	10	8	no	—
224	C₄⋊C₄	—	20	4	no	—
225	C₂²⋊C₄	—	20	8	no	—
226	C₄²	—	20	8	no	—
227	C₁₆	—	5	2	no	—
228	C₁₆	—	5	2	no	—
229	C₁₆	C₂²×AGL(1,𝔽₅)	1	2	yes	—
230	C₁₆	C₂²×AGL(1,𝔽₅)	1	2	yes	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	#2158	1	2	yes	complemented
233	C₂×C₅	C₂²×#107	1	1	yes	characteristic
234	D₅	—	2	2	no	—
235	D₅	#2158	1	2	yes	complemented
236	D₅	C₂×#508	1	2	yes	—
237	C₂³	—	5	1	no	—
238	C₂³	—	5	2	no	—
239	C₂³	—	5	2	no	—
240	C₂³	—	5	1	no	—
241	C₂³	—	10	4	no	—
242	C₂³	#3994	1	1	yes	characteristic
243	C₂×C₄	—	5	2	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	2	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	2	no	—
250	C₂×C₄	—	5	2	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	5	1	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	2	no	—
257	C₂×C₄	—	20	8	no	—
258	C₂×C₄	—	40	8	no	—
259	C₂×C₄	#3994	1	2	yes	—
260	C₂×C₄	#3994	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
263	C₈	—	5	2	no	—
264	C₈	—	5	1	no	—
265	C₈	—	5	1	no	—
266	C₈	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
267	C₈	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
268	C₈	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₅	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	5	1	no	—
272	C₂²	—	5	1	no	—
273	C₂²	—	5	4	no	—
274	C₂²	—	5	2	no	—
275	C₂²	—	10	4	no	—
276	C₂²	—	10	4	no	—
277	C₂²	—	10	2	no	—
278	C₂²	C₂×#3994	1	2	yes	—
279	C₂²	C₂×#3994	1	1	yes	characteristic
280	C₄	—	5	1	no	—
281	C₄	—	5	2	no	—
282	C₄	—	5	1	no	—
283	C₄	—	40	8	no	—
284	C₄	C₂×#3994	1	1	yes	characteristic
285	C₄	C₂×#3994	1	2	yes	—
286	C₄	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	5	2	no	—
289	C₂	—	5	2	no	—
290	C₂	—	10	2	no	—
291	C₂	#10604107	1	2	yes	direct product factor
292	C₂	C₂²×#3994	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3994
commutator	C₅×C₈	C₂³×C₄
Frattini	C₈	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₁₆	C₄
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×D₅	D₈
cosocle	C₈×D₅	C₂⁴
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#2158
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #10604107	C₂	direct (N × H)
4. C₂×#64850	C₂	semidirect
5. C₂×#64849	C₂	semidirect
12. C₂²×C₅×C₁₆	C₄	semidirect
13. C₂²×#672	C₄	semidirect
14. C₂×D₅×C₁₆	C₄	semidirect
15. C₂×D₅×C₁₆	C₄	semidirect
16. C₂×D₅×C₁₆	C₄	semidirect
20. #64850	C₂²	semidirect
21. #64849	C₂²	semidirect
44. C₂×C₅×C₁₆	C₂×C₄	semidirect
47. C₂×#672	C₂×C₄	semidirect
50. D₅×C₁₆	C₂×C₄	semidirect
51. D₅×C₁₆	C₂×C₄	semidirect
106. C₂²×C₁₆	AGL(1,𝔽₅)	semidirect
172. C₂×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
232. C₂×C₅	#2158	semidirect
235. D₅	#2158	semidirect
269. C₅	C₂×#2158	semidirect
291. C₂	#10604107	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×#65721	2
C₂²×D₅×C₁₆	2
#10604107	2
C₂×#64850	2
C₂×#64849	2
C₂×#2158	5

Subgroups — C2×#10604107 #13043409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10604107 #13043409