Subgroups — C₂×#11717480 #13107826

Order 1600.

1744 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717480	C₂	1	16	yes	maximal, direct product factor
5	C₂×#87873	C₂	1	3	yes	maximal, complemented
6	C₂×#87871	C₂	1	3	yes	maximal, complemented
7	C₂×#87870	C₂	1	3	yes	maximal, complemented
8	C₂×D₅×Dic₁₀	C₂	1	3	yes	maximal, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	Q₈×C₅⋊D₅	C₂²	1	8	yes	complemented
11	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	—
12	C₂×#6045	C₂²	1	3	yes	—
13	C₅×#713	C₂²	1	8	yes	complemented
14	C₅×Q₈×D₅	C₂²	1	8	yes	complemented
15	C₂×C₅×D₂₀	C₂²	1	3	yes	—
16	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
17	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
18	C₂×C₅×Dic₁₀	C₂²	1	3	yes	—
19	C₂×#6039	C₂²	1	3	yes	—
20	C₂×#6038	C₂²	1	3	yes	—
21	C₂×#6038	C₂²	1	1	yes	characteristic
22	C₂×#6037	C₂²	1	3	yes	—
23	C₂×D₅×Dic₅	C₂²	1	6	yes	—
24	#87873	C₂²	1	24	yes	complemented
25	#87871	C₂²	1	24	yes	complemented
26	#87870	C₂²	1	24	yes	complemented
27	D₅×Dic₁₀	C₂²	1	24	yes	complemented
28	C₂×#4150	—	5	1	no	maximal
29	C₂×#4147	—	5	1	no	maximal
30	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×D₅	C₂³	1	3	yes	—
32	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
33	C₅²×Q₈	C₂³	1	4	yes	—
34	C₂×C₄×C₅²	C₂³	1	3	yes	—
35	C₂×#1039	C₂³	1	3	yes	—
36	C₄×C₅⋊D₅	C₂³	1	12	yes	—
37	#6045	C₂³	1	12	yes	—
38	C₂×C₅×Dic₅	C₂³	1	3	yes	—
39	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
40	C₅×D₂₀	C₂³	1	12	yes	—
41	C₄×C₅×D₅	C₂³	1	12	yes	—
42	C₄×C₅×D₅	C₂³	1	12	yes	—
43	C₅×Dic₁₀	C₂³	1	12	yes	—
44	#6039	C₂³	1	12	yes	—
45	#6038	C₂³	1	12	yes	—
46	#6038	C₂³	1	4	yes	—
47	#6037	C₂³	1	12	yes	—
48	D₅×Dic₅	C₂³	1	24	yes	—
49	C₂×C₅×#41	—	5	1	no	—
50	C₂²×C₅×Q₈	—	5	1	no	—
51	#4150	—	5	16	no	—
52	#4147	—	5	16	no	—
53	C₂×#713	—	5	3	no	—
54	C₂×#713	D₅	1	1	yes	characteristic, complemented
55	C₂×Q₈×D₅	—	5	1	no	—
56	C₂×Q₈×D₅	—	5	1	no	—
57	C₂×Q₈×D₅	—	5	3	no	—
58	C₂×Q₈×D₅	—	10	2	no	—
59	C₂×Q₈×D₅	D₅	1	1	yes	characteristic, complemented
60	C₂×#711	—	5	3	no	—
61	C₂×#711	—	5	3	no	—
62	C₂×#709	—	5	3	no	—
63	C₂×#709	—	5	3	no	—
64	C₂×#709	—	5	3	no	—
65	C₂²×Dic₁₀	—	5	3	no	—
66	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
67	C₂×C₅⋊D₅	—	2	2	no	—
68	C₂×C₅⋊D₅	C₂⁴	1	2	yes	—
69	C₂×C₅×D₅	—	2	2	no	—
70	C₂×C₅×D₅	—	2	6	no	—
71	C₂×C₅×D₅	C₂⁴	1	6	yes	—
72	C₂×C₅×D₅	C₂⁴	1	2	yes	—
73	C₄×C₅²	C₂⁴	1	6	yes	—
74	#1039	C₂⁴	1	6	yes	—
75	C₅×Dic₅	C₂⁴	1	6	yes	—
76	C₅×Dic₅	C₂⁴	1	2	yes	—
77	C₅×#41	—	5	8	no	—
78	C₂×C₅×Q₈	—	2	2	no	—
79	C₂×C₅×Q₈	—	5	8	no	—
80	C₂×C₅×Q₈	—	5	3	no	—
81	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
82	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₅×D₄	—	5	3	no	—
84	C₂²×C₄×C₅	—	5	3	no	—
85	C₂²×C₄×C₅	—	5	3	no	—
86	C₂×C₂²⋊D₅	—	5	3	no	—
87	C₂×C₂²⋊D₅	—	5	3	no	—
88	C₂×C₂²⋊D₅	—	5	1	no	—
89	C₂×C₂²⋊D₅	—	5	3	no	—
90	C₂²×Dic₅	—	5	6	no	—
91	#713	—	5	24	no	—
92	#713	C₂×D₅	1	8	yes	complemented
93	Q₈×D₅	—	5	8	no	—
94	Q₈×D₅	—	5	8	no	—
95	Q₈×D₅	—	5	24	no	—
96	Q₈×D₅	—	10	16	no	—
97	Q₈×D₅	C₂×D₅	1	8	yes	complemented
98	#711	—	5	24	no	—
99	#711	—	5	24	no	—
100	#709	—	5	24	no	—
101	#709	—	5	24	no	—
102	#709	—	5	24	no	—
103	C₂×D₂₀	—	5	1	no	—
104	C₂×D₂₀	—	5	3	no	—
105	C₂×D₂₀	C₂×D₅	1	3	yes	—
106	C₂×C₄×D₅	—	5	6	no	—
107	C₂×C₄×D₅	—	5	3	no	—
108	C₂×C₄×D₅	—	5	3	no	—
109	C₂×C₄×D₅	—	10	6	no	—
110	C₂×C₄×D₅	C₂×D₅	1	3	yes	—
111	C₂×C₄×D₅	C₂×D₅	1	3	yes	—
112	C₂×Dic₁₀	—	5	3	no	—
113	C₂×Dic₁₀	—	5	3	no	—
114	C₂×Dic₁₀	—	5	3	no	—
115	C₂×Dic₁₀	—	5	3	no	—
116	C₂×Dic₁₀	—	5	24	no	—
117	C₂×Dic₁₀	—	10	6	no	—
118	C₂×Dic₁₀	C₂×D₅	1	3	yes	—
119	C₂×2₋¹⁺⁴	—	25	1	no	Sylow
120	C₂×C₅²	2₋¹⁺⁴	1	2	yes	complemented
121	C₂×C₅²	C₂⁵	1	1	yes	characteristic
122	C₅⋊D₅	—	2	2	no	—
123	C₅×D₅	—	2	2	no	—
124	C₅×D₅	—	2	6	no	—
125	C₂³×C₅	—	5	3	no	—
126	C₂³×C₅	—	5	1	no	—
127	C₂²×D₅	—	5	1	no	—
128	C₂²×D₅	—	5	1	no	—
129	C₂²×D₅	—	10	2	no	—
130	C₂²×D₅	C₂²×D₅	1	3	yes	—
131	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
132	C₅×Q₈	—	2	8	no	—
133	C₅×Q₈	—	5	12	no	—
134	C₅×Q₈	C₂²×D₅	1	4	yes	—
135	C₅×Q₈	C₂²×D₅	1	4	yes	—
136	C₅×D₄	—	5	12	no	—
137	C₂×C₄×C₅	—	2	6	no	—
138	C₂×C₄×C₅	—	5	1	no	—
139	C₂×C₄×C₅	—	5	3	no	—
140	C₂×C₄×C₅	—	5	12	no	—
141	C₂×C₄×C₅	—	5	12	no	—
142	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
143	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
144	C₂²⋊D₅	—	5	12	no	—
145	C₂²⋊D₅	—	5	12	no	—
146	C₂²⋊D₅	—	5	4	no	—
147	C₂²⋊D₅	—	5	12	no	—
148	C₂×Dic₅	—	5	3	no	—
149	C₂×Dic₅	—	5	3	no	—
150	C₂×Dic₅	—	5	24	no	—
151	C₂×Dic₅	—	10	6	no	—
152	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
153	C₂×Dic₅	C₂²×D₅	1	3	yes	—
154	D₂₀	—	5	4	no	—
155	D₂₀	—	5	12	no	—
156	D₂₀	C₂²×D₅	1	12	yes	—
157	C₄×D₅	—	5	24	no	—
158	C₄×D₅	—	5	12	no	—
159	C₄×D₅	—	5	12	no	—
160	C₄×D₅	—	10	24	no	—
161	C₄×D₅	C₂²×D₅	1	12	yes	—
162	C₄×D₅	C₂²×D₅	1	12	yes	—
163	Dic₁₀	—	5	12	no	—
164	Dic₁₀	—	5	12	no	—
165	Dic₁₀	—	5	12	no	—
166	Dic₁₀	—	5	12	no	—
167	Dic₁₀	—	10	24	no	—
168	Dic₁₀	C₂²×D₅	1	12	yes	—
169	2₋¹⁺⁴	—	25	16	no	—
170	C₂×#41	—	5	1	no	—
171	C₂×#41	—	25	3	no	—
172	C₂×#41	—	25	3	no	—
173	C₂×#41	—	25	3	no	—
174	C₂²×Q₈	—	5	1	no	—
175	C₂²×Q₈	—	25	3	no	—
176	C₂²×Q₈	—	25	1	no	—
177	C₅²	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	5	6	no	—
180	C₂²×C₅	—	5	2	no	—
181	C₂²×C₅	—	10	6	no	—
182	C₂²×C₅	—	10	2	no	—
183	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
184	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
185	C₂×D₅	—	2	2	no	—
186	C₂×D₅	—	2	6	no	—
187	C₂×D₅	—	5	2	no	—
188	C₂×D₅	—	5	2	no	—
189	C₂×D₅	—	10	4	no	—
190	C₂×D₅	—	10	8	no	—
191	C₂×D₅	—	10	2	no	—
192	C₂×D₅	—	10	2	no	—
193	C₂×D₅	C₂³×D₅	1	6	yes	—
194	C₂×D₅	C₂³×D₅	1	2	yes	—
195	C₄×C₅	—	2	12	no	—
196	C₄×C₅	—	5	2	no	—
197	C₄×C₅	—	5	6	no	—
198	C₄×C₅	C₂³×D₅	1	6	yes	—
199	C₄×C₅	C₂³×D₅	1	6	yes	—
200	Dic₅	—	5	6	no	—
201	Dic₅	—	5	6	no	—
202	Dic₅	—	10	12	no	—
203	Dic₅	C₂³×D₅	1	2	yes	—
204	Dic₅	C₂³×D₅	1	6	yes	—
205	#41	—	5	8	no	—
206	#41	—	25	24	no	—
207	#41	—	25	24	no	—
208	#41	—	25	24	no	—
209	C₂×Q₈	—	5	8	no	—
210	C₂×Q₈	—	5	3	no	—
211	C₂×Q₈	—	25	3	no	—
212	C₂×Q₈	—	25	24	no	—
213	C₂×Q₈	—	25	8	no	—
214	C₂×Q₈	—	25	3	no	—
215	C₂×Q₈	D₅²	1	1	yes	characteristic
216	C₂×D₄	—	5	3	no	—
217	C₂×D₄	—	25	1	no	—
218	C₂×D₄	—	25	3	no	—
219	C₂×D₄	—	25	3	no	—
220	C₂²×C₄	—	5	3	no	—
221	C₂²×C₄	—	5	3	no	—
222	C₂²×C₄	—	25	6	no	—
223	C₂²×C₄	—	25	3	no	—
224	C₂×C₅	—	2	4	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	10	6	no	—
227	C₂×C₅	—	10	2	no	—
228	C₂×C₅	#4147	1	2	yes	complemented
229	C₂×C₅	#4150	1	2	yes	complemented
230	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
231	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
232	D₅	—	2	2	no	—
233	D₅	—	2	6	no	—
234	D₅	—	10	8	no	—
235	D₅	—	10	2	no	—
236	D₅	—	10	2	no	—
237	C₂³	—	5	1	no	—
238	C₂³	—	5	3	no	—
239	C₂³	—	25	1	no	—
240	Q₈	—	5	12	no	—
241	Q₈	—	25	12	no	—
242	Q₈	—	25	12	no	—
243	Q₈	C₂×D₅²	1	4	yes	—
244	D₄	—	5	12	no	—
245	D₄	—	25	4	no	—
246	D₄	—	25	12	no	—
247	D₄	—	25	12	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	3	no	—
250	C₂×C₄	—	5	12	no	—
251	C₂×C₄	—	5	12	no	—
252	C₂×C₄	—	25	24	no	—
253	C₂×C₄	—	25	3	no	—
254	C₂×C₄	—	25	12	no	—
255	C₂×C₄	C₂×D₅²	1	3	yes	—
256	C₅	—	2	2	no	—
257	C₅	C₂×#4147	1	1	yes	characteristic, complemented
258	C₅	C₂×#4150	1	1	yes	characteristic, complemented
259	C₂²	—	5	2	no	—
260	C₂²	—	5	6	no	—
261	C₂²	—	10	2	no	—
262	C₂²	—	10	6	no	—
263	C₂²	—	25	2	no	—
264	C₂²	—	50	2	no	—
265	C₂²	C₂²×D₅²	1	1	yes	characteristic
266	C₄	—	5	2	no	—
267	C₄	—	5	6	no	—
268	C₄	—	25	6	no	—
269	C₄	C₂²×D₅²	1	6	yes	—
270	C₂	—	10	2	no	—
271	C₂	—	10	6	no	—
272	C₂	—	50	2	no	—
273	C₂	#11717480	1	2	yes	direct product factor
274	C₂	C₂³×D₅²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅²
commutator, cosocle	C₂×C₅²	C₂⁵
Frattini	C₂	C₂³×D₅²
Fitting	C₂×C₅²×Q₈	C₂²
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₅⋊D₅	C₂	semidirect
2. C₂×C₅×#713	C₂	semidirect
3. C₂×C₅×Q₈×D₅	C₂	semidirect
4. #11717480	C₂	direct (N × H)
5. C₂×#87873	C₂	semidirect
6. C₂×#87871	C₂	semidirect
7. C₂×#87870	C₂	semidirect
8. C₂×D₅×Dic₁₀	C₂	semidirect
10. Q₈×C₅⋊D₅	C₂²	semidirect
13. C₅×#713	C₂²	semidirect
14. C₅×Q₈×D₅	C₂²	semidirect
24. #87873	C₂²	semidirect
25. #87871	C₂²	semidirect
26. #87870	C₂²	semidirect
27. D₅×Dic₁₀	C₂²	semidirect
54. C₂×#713	D₅	semidirect
59. C₂×Q₈×D₅	D₅	semidirect
92. #713	C₂×D₅	semidirect
97. Q₈×D₅	C₂×D₅	semidirect
120. C₂×C₅²	2₋¹⁺⁴	semidirect
177. C₅²	C₂×2₋¹⁺⁴	semidirect
228. C₂×C₅	#4147	semidirect
229. C₂×C₅	#4150	semidirect
257. C₅	C₂×#4147	semidirect
258. C₅	C₂×#4150	semidirect
273. C₂	#11717480	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×Q₈×C₅⋊D₅	2
C₂×C₅×#713	2
C₂×C₅×Q₈×D₅	2
#11717480	2
C₂×#87873	2
C₂×#87871	2
C₂×#87870	2
C₂×D₅×Dic₁₀	2
C₂×#4150	5
C₂×#4147	5

Subgroups — C2×#11717480 #13107826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11717480 #13107826