Subgroups — C₂×D₄×#1042 #13107787

Order 1600.

879 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1042	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×D₄×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#1042	C₂	1	2	yes	maximal, complemented
4	D₄×#1042	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊#1042	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#1042	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₅	C₂²	1	2	yes	—
8	C₂²×#1042	—	2	8	no	—
9	C₂²×#1042	C₂²	1	1	yes	characteristic
10	C₂²×#1042	C₂²	1	2	yes	—
11	C₂²×#1042	C₂²	1	1	yes	characteristic
12	C₂²×#1042	C₂²	1	8	yes	complemented
13	C₂×C₅²×D₄	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×C₂²⋊D₅	C₄	1	2	yes	complemented
15	C₅×D₄×D₅	C₄	1	4	yes	complemented
16	C₅×D₄×D₅	C₂²	1	4	yes	—
17	C₂×C₅×D₂₀	C₄	1	1	yes	characteristic, complemented
18	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
19	C₂²⋊#1042	C₂²	1	8	yes	complemented
20	C₄⋊#1042	C₂²	1	4	yes	complemented
21	C₄×#1042	C₂²	1	4	yes	complemented
22	C₂×D₄×AGL(1,𝔽₅)	—	5	1	no	maximal
23	C₂×D₄×Dic₅	—	5	1	no	maximal
24	C₂³×C₅²	C₂×C₄	1	2	yes	complemented
25	C₂²×C₅×D₅	—	2	4	no	—
26	C₂²×C₅×D₅	—	2	4	no	—
27	C₂²×C₅×D₅	C₂×C₄	1	4	yes	complemented
28	C₂²×C₅×D₅	C₂×C₄	1	2	yes	complemented
29	C₂²×C₅×D₅	C₂³	1	4	yes	—
30	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
31	C₂×#1042	—	2	4	no	—
32	C₂×#1042	—	2	2	no	—
33	C₂×#1042	—	2	16	no	—
34	C₂×#1042	D₄	1	4	yes	direct product factor
35	C₂×#1042	C₂³	1	2	yes	—
36	C₂×#1042	C₂³	1	4	yes	—
37	C₂×#1042	C₂³	1	2	yes	—
38	C₅²×D₄	C₂×C₄	1	4	yes	complemented
39	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic, complemented
40	C₅×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
41	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
42	C₅×D₂₀	C₂×C₄	1	4	yes	complemented
43	C₄×C₅×D₅	C₂×C₄	1	2	yes	complemented
44	C₄×C₅×D₅	C₂³	1	2	yes	—
45	C₂³×AGL(1,𝔽₅)	—	5	2	no	—
46	C₂²×C₅×D₄	—	5	1	no	—
47	C₂³×Dic₅	—	5	2	no	—
48	C₂×D₄×D₅	D₅	1	1	yes	characteristic
49	C₂×C₂²⋊AGL(1,𝔽₅)	—	5	2	no	—
50	D₄×AGL(1,𝔽₅)	—	5	8	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	—	5	1	no	—
52	C₂×C₄×AGL(1,𝔽₅)	—	5	1	no	—
53	C₂×C₂²⋊Dic₅	—	5	2	no	—
54	D₄×Dic₅	—	5	8	no	—
55	C₂×C₄⋊Dic₅	—	5	1	no	—
56	C₂×C₄×Dic₅	—	5	1	no	—
57	C₂²×C₅²	—	2	4	no	—
58	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₅²	C₂²×C₄	1	4	yes	complemented
60	C₂×C₅×D₅	—	2	8	no	—
61	C₂×C₅×D₅	—	2	4	no	—
62	C₂×C₅×D₅	—	2	4	no	—
63	C₂×C₅×D₅	—	2	4	no	—
64	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₅×D₅	C₂²×C₄	1	4	yes	complemented
66	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
67	C₂×C₅×D₅	#41	1	2	yes	—
68	C₂×C₅×D₅	C₂⁴	1	1	yes	characteristic
69	#1042	—	2	4	no	—
70	#1042	—	2	2	no	—
71	#1042	C₂×D₄	1	4	yes	direct product factor
72	C₄×C₅²	C₂²×C₄	1	2	yes	complemented
73	C₅×Dic₅	C₂²×C₄	1	2	yes	complemented
74	C₂⁴×C₅	—	5	2	no	—
75	C₂³×D₅	C₂×D₅	1	2	yes	—
76	C₂²×AGL(1,𝔽₅)	—	5	1	no	—
77	C₂²×AGL(1,𝔽₅)	—	5	1	no	—
78	C₂²×AGL(1,𝔽₅)	—	5	2	no	—
79	C₂²×AGL(1,𝔽₅)	—	5	8	no	—
80	C₂²×AGL(1,𝔽₅)	—	10	8	no	—
81	C₂×C₅×D₄	—	4	1	no	—
82	C₂×C₅×D₄	—	5	2	no	—
83	C₂×C₅×D₄	—	5	4	no	—
84	C₂×C₅×D₄	—	5	4	no	—
85	C₂×C₅×D₄	—	5	1	no	—
86	C₂×C₅×D₄	Dic₅	1	1	yes	characteristic, complemented
87	C₂×C₅×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
88	C₂²×C₄×C₅	—	5	1	no	—
89	C₂×C₂²⋊D₅	Dic₅	1	2	yes	complemented
90	C₂²×Dic₅	—	5	1	no	—
91	C₂²×Dic₅	—	5	1	no	—
92	C₂²×Dic₅	—	5	2	no	—
93	C₂²×Dic₅	—	5	8	no	—
94	C₂²×Dic₅	—	10	8	no	—
95	D₄×D₅	Dic₅	1	4	yes	complemented
96	D₄×D₅	C₂×D₅	1	4	yes	—
97	C₂×D₂₀	Dic₅	1	1	yes	characteristic, complemented
98	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
99	C₂²⋊AGL(1,𝔽₅)	—	5	8	no	—
100	C₄⋊AGL(1,𝔽₅)	—	5	4	no	—
101	C₄×AGL(1,𝔽₅)	—	5	4	no	—
102	C₂²⋊Dic₅	—	5	8	no	—
103	C₄⋊Dic₅	—	5	4	no	—
104	C₄×Dic₅	—	5	4	no	—
105	C₂×C₄×D₄	—	25	1	no	Sylow
106	C₂×C₅²	—	2	4	no	—
107	C₂×C₅²	C₄×D₄	1	2	yes	complemented
108	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
109	C₅×D₅	—	2	4	no	—
110	C₅×D₅	C₄×D₄	1	2	yes	complemented
111	C₅×D₅	C₂²×D₄	1	1	yes	characteristic
112	C₅×D₅	C₂×#41	1	1	yes	characteristic
113	C₂³×C₅	—	4	2	no	—
114	C₂³×C₅	—	5	1	no	—
115	C₂³×C₅	—	5	2	no	—
116	C₂³×C₅	—	5	4	no	—
117	C₂³×C₅	—	5	4	no	—
118	C₂³×C₅	—	10	4	no	—
119	C₂³×C₅	—	10	4	no	—
120	C₂³×C₅	C₂×Dic₅	1	2	yes	complemented
121	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
122	C₂²×D₅	—	2	4	no	—
123	C₂²×D₅	—	2	4	no	—
124	C₂²×D₅	C₂×Dic₅	1	4	yes	complemented
125	C₂²×D₅	C₂×Dic₅	1	2	yes	complemented
126	C₂²×D₅	C₂²×D₅	1	4	yes	—
127	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
128	C₂×AGL(1,𝔽₅)	—	5	4	no	—
129	C₂×AGL(1,𝔽₅)	—	5	2	no	—
130	C₂×AGL(1,𝔽₅)	—	5	2	no	—
131	C₂×AGL(1,𝔽₅)	—	5	4	no	—
132	C₂×AGL(1,𝔽₅)	—	10	16	no	—
133	C₂×AGL(1,𝔽₅)	—	10	4	no	—
134	C₂×AGL(1,𝔽₅)	—	10	2	no	—
135	C₅×D₄	—	4	4	no	—
136	C₅×D₄	—	5	8	no	—
137	C₅×D₄	—	5	4	no	—
138	C₅×D₄	C₂×Dic₅	1	4	yes	complemented
139	C₅×D₄	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	5	1	no	—
142	C₂×C₄×C₅	—	5	2	no	—
143	C₂×C₄×C₅	—	5	2	no	—
144	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic, complemented
145	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
146	C₂²⋊D₅	C₂×Dic₅	1	8	yes	complemented
147	C₂×Dic₅	—	5	2	no	—
148	C₂×Dic₅	—	5	4	no	—
149	C₂×Dic₅	—	5	2	no	—
150	C₂×Dic₅	—	5	4	no	—
151	C₂×Dic₅	—	10	16	no	—
152	C₂×Dic₅	—	10	2	no	—
153	C₂×Dic₅	—	10	4	no	—
154	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic, complemented
155	D₂₀	C₂×Dic₅	1	4	yes	complemented
156	C₄×D₅	C₂×Dic₅	1	2	yes	complemented
157	C₄×D₅	C₂²×D₅	1	2	yes	—
158	C₂²×D₄	—	5	1	no	—
159	C₂³×C₄	—	25	2	no	—
160	C₄×D₄	—	25	8	no	—
161	C₂×C₄⋊C₄	—	25	1	no	—
162	C₂×C₂²⋊C₄	—	25	2	no	—
163	C₂×C₄²	—	25	1	no	—
164	C₅²	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	4	no	—
167	C₂²×C₅	—	4	1	no	—
168	C₂²×C₅	—	4	4	no	—
169	C₂²×C₅	—	5	2	no	—
170	C₂²×C₅	—	5	2	no	—
171	C₂²×C₅	—	5	1	no	—
172	C₂²×C₅	—	5	1	no	—
173	C₂²×C₅	—	5	4	no	—
174	C₂²×C₅	—	8	4	no	—
175	C₂²×C₅	—	10	4	no	—
176	C₂²×C₅	—	10	4	no	—
177	C₂²×C₅	—	10	8	no	—
178	C₂²×C₅	—	10	4	no	—
179	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
180	C₂²×C₅	C₂²×Dic₅	1	4	yes	complemented
181	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	4	yes	complemented
183	C₂×D₅	—	2	4	no	—
184	C₂×D₅	—	2	4	no	—
185	C₂×D₅	—	2	4	no	—
186	C₂×D₅	—	2	8	no	—
187	C₂×D₅	D₄×D₅	1	2	yes	—
188	C₂×D₅	#711	1	2	yes	—
189	C₂×D₅	C₂²×Dic₅	1	1	yes	characteristic
190	C₂×D₅	C₂²×Dic₅	1	4	yes	complemented
191	C₂×D₅	C₂³×D₅	1	1	yes	characteristic
192	AGL(1,𝔽₅)	—	5	4	no	—
193	AGL(1,𝔽₅)	—	10	4	no	—
194	AGL(1,𝔽₅)	—	10	2	no	—
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	—	5	2	no	—
197	C₄×C₅	C₂²×Dic₅	1	2	yes	complemented
198	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	complemented
199	Dic₅	—	5	4	no	—
200	Dic₅	—	10	2	no	—
201	Dic₅	—	10	4	no	—
202	Dic₅	C₂²×Dic₅	1	2	yes	complemented
203	C₂⁴	—	5	2	no	—
204	C₂×D₄	—	5	2	no	—
205	C₂×D₄	—	5	4	no	—
206	C₂×D₄	—	5	4	no	—
207	C₂×D₄	—	5	1	no	—
208	C₂×D₄	#1042	1	1	yes	characteristic, direct product factor
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	25	1	no	—
211	C₂²×C₄	—	25	1	no	—
212	C₂²×C₄	—	25	2	no	—
213	C₂²×C₄	—	25	8	no	—
214	C₂²×C₄	—	50	8	no	—
215	C₄⋊C₄	—	25	4	no	—
216	C₂²⋊C₄	—	25	8	no	—
217	C₄²	—	25	4	no	—
218	C₂×C₅	—	2	4	no	—
219	C₂×C₅	—	2	4	no	—
220	C₂×C₅	—	4	2	no	—
221	C₂×C₅	—	4	1	no	—
222	C₂×C₅	—	5	1	no	—
223	C₂×C₅	—	5	1	no	—
224	C₂×C₅	—	5	2	no	—
225	C₂×C₅	—	8	4	no	—
226	C₂×C₅	—	10	4	no	—
227	C₂×C₅	D₄×Dic₅	1	2	yes	complemented
228	C₂×C₅	D₄×AGL(1,𝔽₅)	1	2	yes	complemented
229	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
230	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
231	D₅	—	2	4	no	—
232	D₅	D₄×Dic₅	1	2	yes	complemented
233	D₅	C₂×D₄×D₅	1	1	yes	characteristic
234	D₅	C₂×#711	1	1	yes	characteristic
235	C₂³	—	5	1	no	—
236	C₂³	—	5	2	no	—
237	C₂³	—	5	4	no	—
238	C₂³	—	5	4	no	—
239	C₂³	—	10	4	no	—
240	C₂³	—	10	4	no	—
241	C₂³	C₂×#1042	1	2	yes	complemented
242	D₄	—	5	8	no	—
243	D₄	—	5	4	no	—
244	D₄	C₂×#1042	1	4	yes	direct product factor
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	25	2	no	—
249	C₂×C₄	—	25	4	no	—
250	C₂×C₄	—	25	2	no	—
251	C₂×C₄	—	25	4	no	—
252	C₂×C₄	—	50	16	no	—
253	C₂×C₄	—	50	4	no	—
254	C₂×C₄	—	50	2	no	—
255	C₂×C₄	C₂×#1042	1	1	yes	characteristic, complemented
256	C₅	—	4	1	no	—
257	C₅	C₂×D₄×Dic₅	1	1	yes	characteristic, complemented
258	C₅	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
259	C₂²	—	2	4	no	—
260	C₂²	—	5	2	no	—
261	C₂²	—	5	2	no	—
262	C₂²	—	5	1	no	—
263	C₂²	—	5	1	no	—
264	C₂²	—	5	4	no	—
265	C₂²	—	10	4	no	—
266	C₂²	—	10	4	no	—
267	C₂²	—	10	8	no	—
268	C₂²	—	10	4	no	—
269	C₂²	C₂²×#1042	1	1	yes	characteristic
270	C₂²	C₂²×#1042	1	4	yes	complemented
271	C₄	—	5	2	no	—
272	C₄	—	25	4	no	—
273	C₄	—	50	4	no	—
274	C₄	—	50	2	no	—
275	C₄	C₂²×#1042	1	2	yes	complemented
276	C₂	—	2	4	no	—
277	C₂	—	5	1	no	—
278	C₂	—	5	1	no	—
279	C₂	—	5	2	no	—
280	C₂	—	10	4	no	—
281	C₂	D₄×#1042	1	2	yes	direct product factor
282	C₂	C₂³×#1042	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1042
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂	C₂³×#1042
Fitting	C₂×C₅²×D₄	C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂×C₅×D₄×D₅	C₂
cosocle	C₂×C₅×D₅	C₂⁴
hypercenter	C₂×D₄	#1042
hypocenter	C₅²	C₂×C₄×D₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#1042	C₂	semidirect
3. C₂×C₂²⋊#1042	C₂	semidirect
4. D₄×#1042	C₂	direct (N × H)
5. C₂×C₄⋊#1042	C₂	semidirect
6. C₂×C₄×#1042	C₂	semidirect
12. C₂²×#1042	C₂²	semidirect
13. C₂×C₅²×D₄	C₄	semidirect
14. C₂×C₅×C₂²⋊D₅	C₄	semidirect
15. C₅×D₄×D₅	C₄	semidirect
17. C₂×C₅×D₂₀	C₄	semidirect
19. C₂²⋊#1042	C₂²	semidirect
20. C₄⋊#1042	C₂²	semidirect
21. C₄×#1042	C₂²	semidirect
24. C₂³×C₅²	C₂×C₄	semidirect
27. C₂²×C₅×D₅	C₂×C₄	semidirect
28. C₂²×C₅×D₅	C₂×C₄	semidirect
34. C₂×#1042	D₄	direct (N × H)
38. C₅²×D₄	C₂×C₄	semidirect
39. C₂×C₄×C₅²	C₂×C₄	semidirect
40. C₅×C₂²⋊D₅	C₂×C₄	semidirect
41. C₂×C₅×Dic₅	C₂×C₄	semidirect
42. C₅×D₂₀	C₂×C₄	semidirect
43. C₄×C₅×D₅	C₂×C₄	semidirect
59. C₂²×C₅²	C₂²×C₄	semidirect
65. C₂×C₅×D₅	C₂²×C₄	semidirect
71. #1042	C₂×D₄	direct (N × H)
72. C₄×C₅²	C₂²×C₄	semidirect
73. C₅×Dic₅	C₂²×C₄	semidirect
86. C₂×C₅×D₄	Dic₅	semidirect
87. C₂×C₅×D₄	AGL(1,𝔽₅)	semidirect
89. C₂×C₂²⋊D₅	Dic₅	semidirect
95. D₄×D₅	Dic₅	semidirect
97. C₂×D₂₀	Dic₅	semidirect
107. C₂×C₅²	C₄×D₄	semidirect
110. C₅×D₅	C₄×D₄	semidirect
120. C₂³×C₅	C₂×Dic₅	semidirect
121. C₂³×C₅	C₂×AGL(1,𝔽₅)	semidirect
124. C₂²×D₅	C₂×Dic₅	semidirect
125. C₂²×D₅	C₂×Dic₅	semidirect
138. C₅×D₄	C₂×Dic₅	semidirect
139. C₅×D₄	C₂×AGL(1,𝔽₅)	semidirect
144. C₂×C₄×C₅	C₂×Dic₅	semidirect
145. C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	semidirect
146. C₂²⋊D₅	C₂×Dic₅	semidirect
154. C₂×Dic₅	C₂×Dic₅	semidirect
155. D₂₀	C₂×Dic₅	semidirect
156. C₄×D₅	C₂×Dic₅	semidirect
164. C₅²	C₂×C₄×D₄	semidirect
180. C₂²×C₅	C₂²×Dic₅	semidirect
182. C₂²×C₅	C₂²×AGL(1,𝔽₅)	semidirect
190. C₂×D₅	C₂²×Dic₅	semidirect
197. C₄×C₅	C₂²×Dic₅	semidirect
198. C₄×C₅	C₂²×AGL(1,𝔽₅)	semidirect
202. Dic₅	C₂²×Dic₅	semidirect
208. C₂×D₄	#1042	direct (N × H)
227. C₂×C₅	D₄×Dic₅	semidirect
228. C₂×C₅	D₄×AGL(1,𝔽₅)	semidirect
232. D₅	D₄×Dic₅	semidirect
241. C₂³	C₂×#1042	semidirect
244. D₄	C₂×#1042	direct (N × H)
255. C₂×C₄	C₂×#1042	semidirect
257. C₅	C₂×D₄×Dic₅	semidirect
258. C₅	C₂×D₄×AGL(1,𝔽₅)	semidirect
270. C₂²	C₂²×#1042	semidirect
275. C₄	C₂²×#1042	semidirect
281. C₂	D₄×#1042	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1042	2
C₂×C₅×D₄×D₅	2
C₂×C₂²⋊#1042	2
D₄×#1042	2
C₂×C₄⋊#1042	2
C₂×C₄×#1042	2
C₂×D₄×AGL(1,𝔽₅)	5
C₂×D₄×Dic₅	5

Subgroups — C2×D4×#1042 #13107787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#1042 #13107787